1.1.1 Waveform Modulation Descriptions

This section characterizes signal waveforms comprised of baseband information modulated on an arbitrary carrier frequency, denoted as f_c Hz. The baseband information is characterized as having a lowpass bandwidth of B Hz and, in typical applications, f_c >> B. In many communication system applications, the carrier frequency facilitates the transmission between the transmitter and receiver terminals and can be removed without effecting the information. When the carrier frequency is removed from the received signal the signal processing sampling requirements are dependent only on the bandwidth B.

The signal modulations described in Sections 1.1.1.1 through 1.1.1.4 are amplitude, phase, frequency, and suppressed carrier modulations. The amplitude, phase, and frequency modulations are often applied to the transmission of analog information; however, they are also used in various applications involving digital data transmission. For example, these modulations, to varying degrees, are the underlying waveforms used in the U.S. Air Force Satellite Control Network (AFSCN) involving satellite uplink and downlink control, status, and ranging.

In describing the demodulator processing of the received waveforms, the information, following removal of the carrier frequency, is associated with in‐phase and quadphase (I/Q) baseband channels or rails. Although these I/Q channels are described as containing quadrature real signals, they are characterized as complex signals with real and imaginary parts. This complex signal description is referred to as complex envelope or analytic signal representations and is discussed in Section 1.1.1.5. Suppressed carrier modulation and the analytic signal representation emphasize quadrature data modulation that leads to a discussion of the Hilbert transform in Section 1.1.1.6. Section 1.1.1.7 discusses conventional heterodyning of the received signal to baseband followed by data demodulation.

1.1.1.1 Amplitude Modulation

Conventional amplitude modulation (AM) is characterized as

(1.1)

where A is the peak carrier voltage, is the modulation index, m(t) is the information modulation function, ω_m is the modulation angular frequency, and ω_c is the AM carrier angular frequency. Upon multiplying (1.1) through by sin(ω_ct) and applying elementary trigonometric identities, the AM‐modulated signal is expressed as

(1.2)

Therefore, s(t) represents the conventional double sideband (DSB) AM waveform with the upper and lower sidebands at equally spaced about the carrier at ω_c. With the information modulation function m(t) normalized to unit power, the power in each sideband is m_IP_S/4 where P_S is the power in the carrier frequency f_c.

1.1.1.2 Phase Modulation

Conventional phase modulation (PM) is characterized as

(1.3)

where A is the peak carrier voltage, ω_c is the carrier angular frequency, and φ(t) is an arbitrary phase modulation function containing the information. The commonly used phase function is expressed as

(1.4)

where ϕ is the peak phase deviation. Substituting (1.4) into (1.3), the phase‐modulated signal is expressed as

(1.5)

and, upon applying elementary trigonometric identities, (1.5) yields

(1.6)

The trigonometric functions involving sinusoidal arguments can be expanded in terms of Bessel functions [2] and (1.6) simplifies to

(1.7)

Equation (1.7) is characterized by the carrier frequency with peak amplitude AJ₀(ϕ) and upper and lower sideband pairs at with peak amplitudes AJ_n(ϕ). For small arguments the Bessel functions reduce to the approximations with and (1.7) reduces to

(1.8)

Under these small argument approximations, the similarities between (1.8) and (1.2) are apparent.

1.1.1.3 Frequency Modulation

The frequency‐modulated (FM) waveform is described as

(1.9)

where A is the peak carrier voltage, ω_c is the carrier angular frequency, Δf is the peak frequency deviation of the modulation frequency f_m, and ω_m is the modulation angular frequency. The ratio Δf/f_m is the frequency modulation index. Noting the similarities between (1.9) and (1.5), the expression for the frequency‐modulated waveform is expressed, in terms of the Bessel functions, as

(1.10)

with the corresponding small argument approximation for the Bessel function expressed as

(1.11)

The similarities between (1.11), (1.8), and (1.2) are apparent.

1.1.1.4 Suppressed Carrier Modulation

A commonly used form of modulation is suppressed carrier modulation expressed as

(1.12)

In this case, when the carrier is mixed to baseband, information modulation function m(t) does not have a direct current (DC) spectral component involving δ(ω). So, upon multiplication by the carrier, there is no residual carrier component ω_c in the received baseband signal. Because the carrier is suppressed it is not available at the receiver/demodulator to provide a coherent reference, so special considerations must be given to the carrier recovery and subsequent data demodulation. Suppressed carrier‐modulated waveforms are efficient, in that, all of the transmitted power is devoted to the information. Suppressed carrier modulation and the various methods of carrier recovery are the central focus of the digital communication waveforms discussed in the following chapters.

1.1.1.5 Real and Analytic Signals

The earlier modulation waveforms are described mathematically as real waveforms that can be transmitted over real or physical channels. The general description of the suppressed carrier waveform, described in (1.12), can be expressed in terms of in‐phase and quadrature modulation functions m_c(t) and m_s(t) as

(1.13)

The quadrature modulation functions are expressed as

(1.14)

and

(1.15)

With PM the data {d_c, d_s} may be contained in a phase function φ_d(t), m(t) is a unit energy symbol shaping function that provides for spectral control relative to the commonly used rect(t/T) function, and A represents the peak carrier voltage on each rail. With quadrature modulations, unique symbol shaping functions, m_c(t) and m_s(t), may be applied to each rail; for example, unbalanced quadrature modulations involve different data rates on each quadrature rail. With quadrature amplitude modulation (QAM) the data is described in terms of the multilevel quadrature amplitudes {α_c, α_s} that are used in place of {d_c, d_s} in (1.14) and (1.15).

Equation (1.13) can also be expressed in terms of the real part of a complex function as

(1.16)

where

(1.17)

The function is referred to as the complex envelope or analytic representation of the baseband signal and plays a fundamental role in the data demodulation, in that, it contains all of the information necessary to optimally recover the transmitted information. Equation (1.17) applies to receivers that use linear frequency translation to baseband. Linear frequency translation is typical of heterodyne receivers using intermediate frequency (IF) stages. This is a significant result because the system performance can be evaluated using the analytic signal without regard to the carrier frequency [3]; this is particularly important in computer performance simulations.

Evaluation of the real part of the signal expressed in (1.16) is performed using the complex identity No. 4 in Section 1.14.6 with the result

(1.18)

A note of caution is in order, in that, the received signal power based on the analytic signal is twice that of the power in the carrier. This results because the analytic signal does not account for the factor of 1/2 when mixing or heterodyning with a locally generated carrier frequency and is directly related the factor of 1/2 in (1.18). The signal descriptions expressed in (1.12) through (1.18) are used to describe the narrowband signal characteristics used throughout much of this book.

1.1.1.6 Hilbert Transform and Analytic Signals

The Hilbert transform of the real s(t) is defined as

(1.19)

The second expression in (1.19) represents the convolution of s(t) with a filter with impulse response where h(t) represents the response to a Hilbert filter with frequency response H(ω) characterized as

(1.20)

The Hilbert transform of s(t) results in a spectrum that is zero for all negative frequencies with positive frequencies representing a complex spectrum associated with the real and imaginary parts of an analytic function. Applying (1.20) to the signal spectrum results in the spectrum of the Hilbert transformed signal

(1.21)

Applying (1.21) to the spectrum S(ω) of (1.12) or (1.13), the bandwidth B of m(t) must satisfy the condition B << f_c. In this case, the inverse Fourier transform of the spectrum yields the Hilbert filter output given by

(1.22)

where T_H[s(t)] represents the Hilbert transform of s(t).

The function expressed by (1.22) is orthogonal to s(t) and, if the carrier frequency were removed following the Hilbert transform, the result would be identical to the imaginary part of the analytic signal expressed by (1.17). The processing is depicted in Figure 1.1.

1.1.1.7 Conventional and Complex Heterodyning

Conventional heterodyning is depicted in Figure 1.2. The zonal filters are ideal low‐pass filters with frequency response given by

(1.23)

These filters remove the 2ω_c term that results from the mixing operation and, for s(t) as expressed by (1.13), the quadrature outputs are given by

(1.24)

and

(1.25)

With ideal phase tracking the phase term ϕ(t) is zero resulting in the quadrature modulation functions m_c(t) and m_s(t) in the respective low‐pass channels.

1.2.1 The Transform Pair rect(t/T) ⇔ Tsinc(fT)

The transform relationship rect(t/T) ⇔ Tsinc(fT) occurs so often that it deserves special consideration. For example, consider the following function:

(1.35)

where ω_c, τ, and ϕ represent arbitrary angular frequency, delay, and phase parameters. The signal s(t) is depicted in Figure 1.3.

The Fourier transform of s(t) is evaluated as

(1.36)

Expressing the cosine function in terms of complex exponential functions and performing some simplifications results in the expression

(1.37)

Evaluation of the integrals in (1.37) appears so often that it is useful to generalize the solutions as follows:

Consider the integral

(1.38)

The general solution involves multiplying the last equality in (1.38) by the factors and , having a product of one, where is the average of the integration limits. Distributing the second factor over the numerator of (1.38) and then simplifying yields the result

(1.39)

Applying (1.39) to (1.37) and simplifying gives the desired result

(1.40)

When , the positive and negative frequency spectrums do not influence one another and, in this case, the positive frequency spectrum is defined as

(1.41)

On the other hand, when the carrier frequency and phase are zero, (1.40) simplifies to the baseband spectrum, evaluated as

(1.42)

Using (1.42), the baseband Fourier transform pair, corresponding to of (1.35) with f_c = 0, is established as

(1.43)

and, with τ = 0,

(1.44)

1.2.2 The sinc(x) Function

The sinc(x) function is defined as

(1.45)

and is depicted in Figure 1.4. When x is expressed as the normalized frequency variable x = fT then (1.45), when scaled by T, is the frequency spectrum of the unit amplitude pulse rect(t/T) of duration T seconds such that t ≤ |T/2|. This function is symmetrical in x and the maximum value of the first sidelobe occurs at x = 1.431 with a level of 10log(sinc²(x)) = −13.26 dB; the peak sidelobe levels decrease in proportion to 1/|x|. The noise bandwidth of a filter function H(f) is defined as

(1.46)

where f_o is the filter frequency corresponding to the maximum response. When a receiver filter is described as H(f) = sinc(fT) the receiver low‐pass noise bandwidth is evaluated as B_n = 1/T where T is the duration of the filter impulse response.

It is sometimes useful to evaluate the area of the sinc(x) function and, while there is no closed form solutions, the solution can be evaluated in terms of the sine‐integral S_i(x)⁴ as

(1.47)

where the sine‐integral is defined as the integral of sin(λ)/λ. Equation (1.47) is shown in Figure 1.5. The limit of S_i(πz) as |z| → ∞ is⁵ π sign(1,z)/2 so the corresponding limit of (1.47) is 0.5sgn(z).

A useful parameter, often used as a benchmark for comparing spectral efficiencies, is the area under sinc²(x) as a function of x. The area is evaluated in terms of the sine‐integral as

(1.48)

Equation (1.48) is plotted in Figure 1.6 as a percent of the total area and it is seen that the spectral containment of 99% is in excess of 18 spectral sidelobes, that is, x = fT = 18. In the following chapters, spectral efficient waveforms are examined with 99% containment within 2 or 3 sidelobes, so the sinc(x) function does not represent a spectrally efficient waveform modulation.

1.2.3 The Fourier Transform Pair

The evaluation of this Fourier transform pair is fundamental to Nyquist sampling theory and is demonstrated in Section 2.3 in the evaluation of discrete‐time sampling. In this case, the function f(t) is an infinite repetition of equally spaced delta functions δ(t) with intervals T seconds as expressed by

(1.49)

The challenge is to show that the Fourier transform of (1.49) is equal to an infinite repetition of equally spaced and weighted frequency domain delta functions expressed as

(1.50)

with weighting ω_o and frequency intervals . Direct application of the Fourier transform to (1.49) leads to the spectrum but this does not demonstrate the equality in (1.50). Similarly, evaluation of the inverse Fourier transform of (1.50) results in the time‐domain expression

(1.51)

So, by showing that , the transform pair between (1.49) and (1.50) will be established. Consider g_N(t) to be a finite summation of terms in (1.51) given by

(1.52)

The second equality in (1.52) can be shown using the finite series identity No. 12, Section 1.14.1. Equation (1.52) is referred to by Papoulis [7] as the Fourier‐series kernel and appears in a number of applications involving the Fourier transform.

The function g_N(t) is plotted in Figure 1.7 for N = 8. The abscissa is time normalized by the pulse repetition interval such that, , and there are a total of peaks of which three are shown in the figure. Furthermore, there are eight time sidelobes between t/T = 0 and 0.5 with the first nulls from the peak value at t/T = 0 occurring at ; the peak values are in this example.

The maximum values of , occurring at , are determined by applying L’Hospital’s rule to (1.52), which is rewritten as

(1.53)

The approximation in (1.53) is obtained by noting that as N increases the rate of the sinusoidal variations in the numerator term increases with a frequency of Hz while the rate of sinusoidal variation in the denominator remains unchanged. Therefore, in the vicinity of , and (1.53) reduces to a sin(x)/x function with and a peak amplitude (2N + 1). The proof of the transform pair is completed by showing that f(t) = g(t). Referring to (1.51) g(t) is expressed as

(1.54)

From (1.53) as N approaches infinity the sin(x)/x sidelobe nulls converge to t/T = n, the peak values become infinite, and the corresponding area over the interval |t/T| = n ± 1/2 approaches unity. Therefore, g(t) resembles a periodic series of delta functions resulting in the equality

(1.55)

thus completing the proof that (1.49) and (1.50) correspond to a Fourier transform pair. Papoulis (Reference 7, pp. 50–52) provides a more eloquent proof that the limiting form of g_N(t) is indeed an infinite sequence of delta functions.

1.2.4 The Discrete Fourier Transform

The DFT pair relating the discrete‐time function f(mΔt) ≡ f(m) and discrete‐frequency function F(nΔf) ≡ F(n) is denoted as where

(1.56)

With the DFT the number of time and frequency samples can be chosen independently. This is advantageous when preparing presentation material or examining fine spectral or temporal details, as might be useful when debugging simulation programs, by the independent selection of the integers m and n.

1.2.5 The Fast Fourier Transform

As discussed in the preceding section, the DFT pair, relating the discrete‐time function f(mΔt) ≡ f(m) and the discrete‐frequency function F(nΔf) ≡ F(n), is denoted as where f(m) and F(n) are characterized by the expressions for the DFT. The FFT [11–17], is a special case corresponding to m and n being equal to N as described in the remainder of this section. In these relationships N is the number of time samples and is defined as the power of a fixed radix‐r FFT or as the powers of a mixed radix‐r_j FFT.⁶ The fixed radix‐2 FFT, with r = 2 and N = 2ⁱ, results in the most processing efficient implementation.

Defining the time window of the FFT as T_w results in an implicit periodicity of f(t) such that f(t) = f(t ± kT_w) and Δt = T_w/N. The sampling frequency is defined as and, based on Shannon’s sampling theorem, the periodicity does not pose a practical problem as long as the signal bandwidth is completely contained in the interval |B| ≤ f_s/2 = N/(2T_w). Since the FFT results in an equal number of time and frequency domain samples, that is, Δf = f_s/N and Δt = T_w/N, it follows that ΔfΔt = f_s T_w/N² = 1/N. Normalizing the expression of the time function, f(m), in (1.56), that is, multiplying the inverse DFT (IDFT) by Δt requires dividing the expression for F(n) by Δt. Upon substituting these results into (1.56), the FFT transform pairs become

(1.57)

The time and frequency domain sampling characteristics of the FFT are shown in Figure 1.8. This depiction focuses on a communication system example, in that, the time samples over the FFT window interval T_w are subdivided into N_sym symbol intervals of duration T seconds with N_s samples/symbol.

Typically the bandwidth of the modulated waveform is taken to be the reciprocal of the symbol duration, that is, 1/T Hz; however, the receiver bandwidth required for low symbol distortion is typically several times greater than 1/T depending upon the type of modulation. Referring to Figure 1.8 the sampling frequency is f_s = 1/Δt, the sampling interval is Δt = T/N_s, the size of the FFT is N_fft = N_sN_sym, and the frequency sampling increment is Δf = f_s/N_fft. Upon using these relationships, the frequency resolution, or frequency samples per symbol bandwidth B = 1/T, is found to be

(1.58)

and the number of spectral sidelobes⁷ or symbol bandwidths over the sampling frequency range is

(1.59)

Therefore, to increase the resolution of the sampled signal spectrum, the number of symbols must be increased and this is comparable to increasing T_w. On the other hand, to increase the number of signal sidelobes contained in the frequency spectrum the number of samples per symbol must be increased and this is comparable to decreasing Δt. Both of these conditions require increasing the size (N) of the FFT. However, for a given size, the FFT does not allow independent selection of the frequency and time resolution as determined, respectively, by (1.58) and (1.59). This can be accomplished by using the DFT as discussed in Section 1.2.4. Since the spectrum samples in the range 0 ≤ f < f_s/2 represent the positive frequency signal spectrum and those over the range f_s/2 ≤ f < f_s represent the negative frequency signal spectrum, the range of signal sidelobes of interest is ±f_s/(2B) = ±N_s/2. As a practical matter, if the signal carrier frequency is not zero then the sampling frequency must be increased to maintain the signal sidelobes aliasing criterion. The sampling frequency selection is discussed in Chapter 11 in the context of signal acquisition when the received signal frequency is estimated based on locally known conditions.

The following implementation of the FFT is based on the Cooley and Tukey [18] decimation‐in‐time algorithm as described by Brigham and Morrow [19] and Brigham [20]. Although (1.57) characterizes the FFT transform pairs, the real innovation leading to the fast transformation is realized by the efficient algorithms used to execute the transformation. Considering the radix‐2 FFT with N = 2ⁿ, this involves defining the constant

(1.60)

and recognizing that

(1.61)

Equation (1.61) can be expressed in matrix form, using N = 4 for simplicity, as

(1.62)

Recognizing that W⁰ = 1 and the exponent nm is modulo(N), upon factoring the matrix in (1.62) into the product of two submatrices (in general the product of log₂N submatrices) leads to the implementation involving the minimum number of computations expressed as

(1.63)

The simplifications result in the outputs F(2) and F(1) being scrambled and the unscrambling to the natural‐number ordering simply involves reversing the binary number equivalents, that is, with F′(1) = F(2) and F′(2) = F (1); therefore, the unscrambling is accomplished as F(1) = F (01) = F′(2) = F′(10) and F (2) = F (10) = F′(1) = F′(01). The radix‐2 with N = 4 FFT, described by (1.63), is implemented as shown in the diagram of Figure 1.9.

The inverse FFT (IFFT) is implemented by changing the sign of the exponent of W in (1.60), interchanging the roles of F(n) and f(m), as described earlier, and replacing Δt by Δf. Recognizing that ΔtΔf = 1/N, it is a common practice not to weight the FFT but to weight the IFFT by 1/N as indicated in (1.57). The number of complex multiplication is determined from (1.63) by recognizing that W² = −W⁰ and not counting previous products like W⁰f(2) from row 1 and W²f(2) = −W⁰f(0) from row 3 in the first matrix multiplication on the rhs of (1.63). For the commonly used radix‐2 FFT, the number of complex multiplications is (N/2)log₂(N) and the number of complex additions is Nlog₂(N). By comparison, the number of complex multiplications and additions in the direct Fourier transform are N² and N(N − 1), respectively. These computational advantages are enormous for even modest transform sizes.

1.2.5.1 The Pipeline FFT

The FFT algorithm discussed in the preceding section involves decimation‐in‐time processing and requires collecting an entire block of time‐sampled data prior to performing the Fourier transform. In contrast, the pipeline FFT [21] processes the sampled data sequentially and outputs a complete Fourier transform of the stored data at each sample. The implementation of a radix‐2, N = 8‐point pipeline FFT is shown in Figure 1.10. The pipeline FFT inherently scrambles the outputs F′(n) and the unscrambled outputs are not shown in the figure; the unscrambling is accomplished by simply reversing the order of the binary representation of the output locations, n, as described in the preceding section.

In general, the number of complex multiplications for a complete transform is (N/2)(N − 1). In Chapter 11 the pipeline FFT is applied in the acquisition of a waveform where a complete N‐point FFT output is not required at every sample. For example, if the complete N‐point FFT is only required at sample intervals of N_sT_s, the number of complex multiplications can be significantly reduced (see Problem 10). The pipeline FFT can be used to interpolate between the fundamental frequency cells by appending zeros to the data samples and appropriately increasing the size of the FFT; it can also be used with data samples requiring mixed radix processing. The pipeline FFT is applicable to radar and sonar signal detection processing [21] using a variety of spectral shaping windows; however, the intrinsic rect(t/T) FFT window is nearly matched for the detection of orthogonally spaced M‐ary FSK modulated frequency tones.

1.2.6 The FFT as a Detection Filter

The pipeline Fourier transform is made up of a cascade of transversal filter building blocks shown in Figure 1.10. The transfer function of this building block is

(1.64)

The overall transfer function from the input to a particular output is evaluated as

(1.65)

where k = log₂(N) and k_i = 2i − 1, i = 1, …, k. The complex weights are given by

(1.66)

where

(1.67)

Substitution of W_ℓ,i into (1.65) results in

(1.68)

where

(1.69)

This transfer function is expressed in terms of a magnitude and phase functions in ω by substituting s = jω with the result

(1.70)

where

(1.71)

Therefore, the FFT forms N filters, each having a maximum response that occurs at the frequencies . As N increases these transfer functions result in the response

(1.72)

The magnitude of (1.72) is the sinc(x) function associated with the uniformly weighted envelope modulation function and, therefore, the FFT filter functions as a matched detection filter for these forms of modulations. Examples of these modulated waveforms are binary phase shift keying (BPSK), quadrature phase shift keying (QPSK), offset quadrature phase shift keying (OQPSK), and M‐ary FSK.

The FFT detection filter loss relative to the ideal matched filter is examined as N increases. The input signal is expressed as

(1.73)

and the corresponding signal spectrum for positive frequencies with ω_c ≫ 2π/T is

(1.74)

The matched filter for the optimum detection of s(t) in additive white noise with spectral density N_o is defined as

(1.75)

where K is an arbitrary scale factor and T_o is an arbitrary delay influencing the causality of the filter. By letting , , T_o = (N − 1)T_s/2, and it is seen that the FFT approaches a matched filter as N increases.

The question of how closely the FFT approximates a matched filter detector is examined in terms of the loss in signal‐to‐noise ratio. The filter loss is expressed in dB as

(1.76)

where (SNR_o)_opt = 2E/N_o is the signal‐to‐noise ratio out of the matched filter and E is the signal energy. The signal‐to‐noise ratio out of the FFT filter is expressed in terms of the peak signal output of the detection filter and the output noise power as

(1.77)

where B_n is the detection filter noise bandwidth. For convenience the zero‐frequency FFT filter output is considered, that is, for , and letting the signal phase ϕ = 0, the response of interest is

(1.78)

and, from (1.74),

(1.79)

To evaluate SNR_o at the output of the FFT filter, g_o(t)_max and B_n are computed as

(1.80)

and

(1.81)

Substituting these results into (1.77) and using (1.76), the parameter ρ is evaluated as

(1.82)

Equation (1.82) is evaluated numerically for several values of N and the results are tabulated in Table 1.2. These results indicate, for example, that detecting an 8‐ary FSK‐modulated waveform with orthogonal tone spacing using an N = 8‐point FFT results in a performance loss of 0.116 dB relative to an ideal matched filter.

1.2.7 Interpolation Using the FFT

When an FFT is performed on a uniformly weighted set of N data samples a set of N sinc(fT_w) orthogonal filters is generated where T_w = NT_s is the sampled data window and T_s is the sampling interval. The N filters span the frequency range f_s = 1/T_s and provide N frequency estimates that are separated by Δf = f_s/N Hz. Frequency interpolation is achieved if the FFT window is padded by adding nN zero‐samples, thereby increasing the window by nNT_s seconds. In this case, a set of (n + 1)N sinc(fT_w) filters spanning the frequency f_s is generated that provides n‐point interpolation between each of the original N filters.

The FFT can also be used to interpolate between time samples. For example, consider a sampled time function characterized by N samples over the interval T_w = NT_s where T_s is the sampling interval. The corresponding N‐point FFT has N filters separated by Δf = f_s/N where f_s = 1/T_s. If nN zero‐frequency samples are inserted between frequency samples N/2 and N/2 + 1 and the IFFT is taken on the resulting (n + 1)N samples, the resulting time function contains n interpolation samples between each of the original N time samples. These interpolations methods increase the size of the FFT or IFFT and thereby the computational complexity.

1.2.8 Spectral Estimation Using the FFT

Many applications involve the characterization of the PSD of a finite sequence of random data. A random data sequence represents a stochastic process, for which, the PSD is defined as the Fourier transform of the autocorrelation function of the sequence. If the random process is such that the statistical averages formed among independent stochastic process are equal to the time averages of the sequences, then the Fourier transform will converge in some sense⁸ to the true PSD, S²(ω); however, this typically requires very long sequences that are seldom available. Furthermore, the classical approach, using the Fourier transform of the autocorrelation function, is processing intense and time consuming, requiring long data sequences to yield an accurate representation to the PSD. A much simpler approach, analyzed by Oppenheim and Schafer [22], is to recognize that the Fourier transform of a relatively short data sequence x(n) of N samples is

(1.83)

and, defining the Fourier transform of the autocorrelation function C_xx(m) of x(n) as the periodogram

(1.84)

However, the periodogram is not a consistent estimate⁹ of the true PSD, having a large variance about the true values resulting in wild fluctuations. Oppenheim and Schafer then show that Bartlett’s procedure [10, 23] of averaging periodograms of independent data sequences results in a consistent estimate and, if K periodograms are averaged, the resulting variance is decreased by K. In this case, the PSD estimate is evaluated as

(1.85)

Oppenheim and Schafer also discuss the application of windows to the periodograms and Welch [17] describes a procedure involving the averaging of modified periodograms.

1.2.9 Fourier Transform Properties

The following Fourier transform properties are based on the transform pairs and where x(t) and y(t) may be real or complex.

1.2.9.1 Linearity

(1.86)

1.2.9.2 Translation

(1.87)

and

(1.88)

1.2.9.3 Conjugation

(1.89)

and

(1.90)

1.2.9.4 Differentiation

With and then

(1.91)

and

(1.92)

1.2.9.5 Integration

Defining and then

(1.93)

and

(1.94)

1.2.10 Fourier Transform Relationships

The following Fourier transform relationships are based on the transform pairs and where x(t) and y(t) may be real or complex.

1.2.10.1 Convolution

Defining the Fourier transforms and then

(1.95)

and

(1.96)

1.2.10.2 Integral of Product (Parseval’s Theorem)

(1.97)

Letting y(t) = x(t) results in Parseval’s Theorem that equates the signal energy in the time and frequency domains as

(1.98)

1.2.11 Summary of Some Fourier Transform Pairs

Some often used transform relationships are listed in Table 1.3.

Waveform f(t)	Spectrum F(f)
1	δ(f)
f(t − τ)	F(f)exp(−j2πfτ)
δ(t)
δ(t − τ)	exp(−j2πfτ)
f(at)	(1/a)F(f/a)
cos(2πfot)
sin(2π fot)
	(j2πf)nF(f)
f(t) = x(t) y(t)	X(f)*Y(f) =
f(t) = x(t)*y(t)	F(f) = X(f) Y(f)
	U(f)exp(−j2πfτ)
a
	Tsinc(fT)b
sinc(2t/T)

^*Denotes convolution.

^aThe signum function sgn(x) is also denoted as signum(x).

^bWoodward [24].

1.3.1 Ideal Amplitude and Zero Phase Filter

In this example, the filter is characterized in the frequency domain as having a constant unit amplitude over the bandwidth f ≤ |B| with zero amplitude otherwise and a zero phase function. Using the previous notation the filter is characterized in the frequency and time domains as

(1.100)

The frequency characteristics of the signal and filter are shown in Figure 1.11.

The easiest way to evaluate the filter response to a pulse input signal is by convolving the functions as

(1.101)

The rect(•) function determines the integration limits with the upper and lower limits evaluated for τ when the argument equals ±½, respectively. This evaluation leads to the integration

(1.102)

Equation (1.102) is evaluated in terms of the sine integral [25]

(1.103)

resulting in the filter output g(t) expressed as

(1.104)

Defining the normalized variable y = t/T and the parameter ρ = BT, Equation (1.104) is expressed as

(1.105)

Equation (1.105) is plotted in Figure 1.12 for several values of the time‐bandwidth (BT) parameter. Range resolution is proportional to bandwidth and the increased rise time or smearing of the pulse edges with decreasing bandwidth is evident. The ISI that degrades the performance of a communication system results from the symbol energy that occurs in adjacent symbols due to the filtering.

This analysis considers only the pulse distortion caused by constant amplitude filter response and, as will be seen in the following section, filter amplitude ripple and nonlinear phase functions also result in additional signal distortion. If the filter were to exhibit a linear phase function ϕ(f) = −2πfT_o where T_o represents a constant time delay, then, referring to Table 1.3, the output is simply delayed by T_o without any additional distortion. If T_o is sufficiently large, the filter can be viewed as a causal filter, that is, no output is produced before the input signal is applied.

1.3.2 Nonideal Amplitude and Phase Filters: Paired Echo Analysis

In this section the pulse distortion caused by a filter with prescribed amplitude and phase functions is examined using the analysis technique of paired echoes [26]. A practical application of paired echo analysis occurred when a modem production line was stopped at considerable expense due to noncompliance of the bit‐error test involving a few tenths of a decibel. The required confidence level of the bit‐error performance under various IF filter conditions precluded the use of Monte Carlo simulations; however, much to the pleasure of management, the paired echo analysis was successfully applied to identify the cause of the subtle filter distortion losses.

Consider a filter with amplitude and phase functions expressed as

(1.106)

where the amplitude and phase fluctuations with frequency are expressed as

(1.107)

and

(1.108)

The parameters a and τ_a represent the amplitude and period of the amplitude ripple and b and τ_b represent the amplitude and period of the phase ripple. Using these functions in (1.106) and separating the constant delay term involving T_o, results in the filter function

(1.109)

Equation (1.109) is simplified by using the trigonometric identity

(1.110)

and the Bessel function identity [27]

(1.111)¹⁰

In arriving at the last expression in (1.111), the following identities were used

(1.112)

Upon substituting (1.110) and (1.111) into (1.109), and performing the multiplications to obtain additive terms representing unique delays results in the filter frequency response

(1.113)

Upon performing the inverse Fourier transform of each term in (1.113), the filter impulse response, h(t), becomes a summation of weighted and delayed sinc(x) functions of the form 2BKsinc(2B(t − T_d)) where K and T_d are the amplitude and delay associated with each of the terms. Performing the convolution indicated by the first equality in (1.101), that is, for an arbitrary signal s(t), the ideally filtered response g(t) is expressed as

(1.114)

When g(t) is passed through the filter H(f) with amplitude and phase described, respectively, by (1.107) and (1.108), the distorted output g_o(t) is evaluated as

(1.115)

If the input signal is described by the rect(t/T) function, then g(t) is the response expressed by (1.104) and depicted in Figure 1.12. The distortion terms appear as paired echoes of the filtered input signal and Figure 1.13 shows the relative delay and amplitude of each echo of the filtered output g(t). For b << 1 the approximations J₀(b) = 1.0 and J₁(b) = b/2 apply and when a = b = 0 the filter response is simply the delayed but undistorted replica of the input signal, that is, g_o(t) = g(t − T_o). More complex filter amplitude and phase distortion functions can be synthesized by applying Fourier series expansions that yield paired echoes that can be viewed as noisy interference terms that degrade the system performance; however, the analysis soon becomes unwieldy so computer simulation of the echo amplitudes and delays must be undertaken.

1.3.3 Example of Delay Distortion Loss Using Paired Echoes

The evaluation of the signal‐to‐interference ratio resulting from the delay distortion of a filter is examined using paired echo analysis. The objective is to examine the distortion resulting from a specification of the filters peak phase error and group delay within the filter bandwidth. The filter phase response is characterized as

(1.116)

where T_o is the filter delay resulting from the linear phase term, ϕ_o is the peak phase deviation from linearity over the filter bandwidth, and τ is the period of the sinusoidal phase distortion function. The linear phase term introduces the filter delay T_o that does not result in signal distortion; however, the sinusoidal phase term does cause signal distortion. In this example, the phase deviation over the filter bandwidth is specified parametrically as ϕ_o(deg) = 3 and 7°. The parameter τ is chosen to satisfy the peak delay distortion defined as

(1.117)

where ϕ_o is in radians. The peak delay, evaluated for fτ = 0, is specified as T_d = 34 and 100 ns and, using (1.117), the period of the sinusoidal phase function, τ = T_d/ϕ_o, is tabulated in Table 1.4 for the corresponding peak phase errors and peak delay specification. Practical maximum limits of the group delay normalized by the symbol rate, R_s, are also specified.

Considering an ideal unit gain filter with amplitude response of A(ω) = 1, the filter transfer function is expressed as

(1.118)

Upon taking the inverse Fourier transform of (1.118), the filter impulse response is evaluated as

(1.119)

The parameter τ determines the delay spread of all the interfering terms; however, for small arguments the interference is dominated by the J₁(ϕ_o) term and the signal‐to‐interference ratio is defined as

(1.120)

For ϕ_o(deg) = 3 and 7°, the respective signal‐to‐interference ratios are 32 and 24.3 dB and under these conditions, a 10 dB filter input signal‐to‐noise ratio results in the output signal‐to‐noise ratio degraded by 0.02 and 0.17 dB, respectively.

1.5.1 Probability and Cumulative Distribution and Probability Density Functions

The mathematical description [6, 8, 24, 28, 30–32] of the random variable X resulting from the mapping X(χ) given the random event χ ∈ S is based on the statistical properties of the random event characterized by the probability P({X ≤ x}) where {X ≤ x} denotes all of the events X(χ) in S. For continuous random variables P(X = x) = 0. The probability function P(X_i ∈ S_i) satisfies the following axioms:

1. A1. P(X(χ)∈S) ≥ 0
2. A2. P({X(χ)∈S}) = 1
3. A3. If P(S_i∩S_j) = Ø ∀i ≠ j then

Axiom A3 applies for infinite event spaces by letting J = ∞. Several corollaries resulting from these axioms are as follows:

1. C1. P(χ^c) = 1 − P(χ) where χ^c is the complement of χ such that χ^c∩χ = Ø
2. C2. P(χ) ≤ 1
3. C3. P(χ_i ∪ χ_j) = P(χ_i) + P(χ_j) − P(χ_i ∩ χ_j)
4. C4. If P(Ø) = 0

The cumulative distribution function (cdf) of the variable X is defined in terms of the value of x on the real line as

(1.125)

where F_X(x) has the following properties:

1. P1. 0 ≤ F_X(x) ≤ 1
2. P2. In the limit as x approaches ∞, F_X(x) = 1
3. P3. In the limit as x approaches −∞, F_X(x) = 0
4. P4. F_X(x) is a nondecreasing function of x
5. P5. In the limit as ε approaches 0, F_X(x_i) = F_X(x_i + ε)
6. P5. The probability in the interval x_i < x ≤ x_j is: P(x_i < x ≤ x_j) = F_X(x_j) – F_X(x_i)
7. P6. In the limit as ε approaches 0, the probability of the event x_i is P(x_i − ε < x ≤ x_i) = F_X(x_i) − F_X(x_i − ε).

Property P5 is referred to as being continuous from the right and is particularly important with discrete random variables, in that, F_X(x_i) includes a discrete random variable at x_i. Property P7, for a continuous random variable, states that P(x_i) = 0; however, for a discrete random variable, P(x_i) = p_X(x_i) where p_X(x_i) is the probability mass function (pmf) defined in Section 1.5.1.2.

The probability density function¹³ (pdf) of X is defined as

(1.126)

The pdf is frequency used to characterize a random variable because, compared to the cdf, it is easier to describe and visualize the characteristics of the random variable.

1.5.1.1 Continuous Random Variables

A random variable is continuous if the cdf is continuous so that F_X(x) can be expressed by the integral of the pdf. The mapping in Figure 1.14 results in the continuous real variable x. From (1.125) and (1.126) it follows that

(1.127)

A frequently encountered and simple example of a continuous random variable is characterized by the uniformly distributed pdf shown in Figure 1.15 with the corresponding cdf and probability function.

From property P7, the probability of X = x_i is evaluated as

(1.128)

However, for continuous random variables, the limit in (1.128) is equal to F_X(x_i) so P(X = x_i) = 0; this event is handled as described in Section 1.5.2.

1.5.1.2 Discrete Random Variables

The probability mass function [8, 28, 29] (pmf) of the discrete random variable X is defined in terms of the discrete probabilities on the real line as

(1.129)

The corresponding cdf is expressed as

(1.130)

where u(x − x_i) is the unit‐step function occurring at x = x_i and is defined as

(1.131)

Using (1.126), and recognizing that the derivative of u(x − x_i) is the delta function δ(x − x_i), the pdf of the discrete random variable is expressed as

(1.132)

The pmf p_X(x_i) results in a weighted delta function and, from (1.130), (1.131), and property P2, the summation must satisfy the condition .

The pdf, cdf, and the corresponding probability for the discrete random variable corresponding to binary data {0,1} with pmf functions p_X(0) = 1/3 and p_X(1) = 2/3 are shown in Figure 1.16. The importance of property P5 is evident in Figure 1.16, in that, the delta function at x = 1 is included in the cdf resulting in P(X ≤ 1) = 1. Regarding property P7, the limit in (1.128) approaches X = x_i from the left, corresponding to the base of the discontinuity, so that P(X = x_i) = p_X(x_i).

1.5.1.3 Mixed Random Variables

Mixed random variables are composed of continuous and discrete random variables and the following example is a combination of the continuous and discrete random variables in the examples of Sections 1.5.1.1 and 1.5.1.2. The major consideration in this case is the determination of the event pmf for the continuous (C) and discrete (D) random variables to satisfy property P2. Considering equal pmfs, such that, p_X(S = C) = p_X(S = D) = 1/2, the pdf, cdf, and probability are depicted in Figure 1.17.

1.5.2 Definitions and Fundamental Relationships for Continuous Random Variables

For the continuous random variables X, such that the events X(χ_j) ∈ S_i, the joint cdf is determined by integrating the joint pdf expressed as

(1.133)

and, provided that is continuous and exists, it follows that

(1.134)

The probability function is then evaluated by integrating x_i over the appropriate regions x_i1 < r_i ≤ x_i2: i = 1, …, N with the result

(1.135)

1.5.2.1 Marginal pdf of Continuous Random Variables

The marginal pdf is determined by integrating over the entire region of all the random variables except for the desired marginal pdf. For example, the marginal pdf for x₁ is evaluated as (see Problem 17)

(1.136)

The random variables X_i are independent iff the joint cdf can be expressed as product of the each cdf, that is

(1.137)

In addition, if X_i ∀ i are jointly continuous, the random variables are independent if the joint pdf can be expressed as the product of each pdf as

(1.138)

Therefore, the joint pdf of independent random variables is the same as the product of each marginal pdf computed sequentially as in (1.136).

The joint cdf of two continuous random variables is defined as

(1.139)

with the following properties,

(1.140)

and the joint pdf is defined as

(1.141)

with the following properties,

(1.142)

1.5.2.2 Conditional pdf and cdf of Continuous Random Variables

The conditional pdf is expressed as

(1.143)

and the conditional cdf is evaluated as

(1.144)

A basic rule for removing random variables from the left and right side of the conditional symbol ( | ) is given by Papoulis [33]. To remove random variables from the left side simply integrate each variable x_j from −∞ to ∞: j ≤ i. To remove random variables from the right side, for example, x_j and x_k: i + 1 ≤ j,k ≤ n, multiply by the conditional pdfs of x_j and x_k with respect to the remaining variables and integrate x_j and x_k from −∞ to ∞. For example, referring to (1.143) and considering f_X1(x₁|x₂,x₃,x₄), eliminating the random variables x₃ and x₄ from the right side is evaluated as

(1.145)

The conditional probability of Y ∈ S₁ given X(χ) = x is expressed as

(1.146)

Since P(X = x) = 0 for the continuous random variable X, (1.146) is undefined; however, if X and Y are jointly continuous with continuous joint cdfs, as defined in (1.139), then the conditional cdf of Y, given X, is defined as

(1.147)

and differentiating (1.147) with respect to y results in

(1.148)

If f_X(x) ≠ 0, the conditional cdf of y, given X = x, is expressed as [34]

(1.149)

and the corresponding conditional pdf is evaluated by differentiating (1.149) with respect to y and is expressed as

(1.150)

If X and Y are independent random variables then and (1.147) and (1.150) become and .

Upon rearranging (1.150), the joint pdf of X and Y is expressed as

(1.151)

Considering the probability space S₁ = S_Y|X ∩ S_X, such that Ø, the probability P(Y ∈S_X) is determined by the total probability law defined as

(1.152)

In this case, the subspace S_X can be examined as if it were a total probability space obeying the axioms, corollaries, and properties stated earlier.

1.5.2.3 Expectations of Continuous Random Variables

In general, the k‐th moment of the random variable X is defined as the expectation

(1.153)

and the k‐th central moments are defined as the expectation

(1.154)

The mean value m_x of X is defined as the expectation

(1.155)

The second central moment of X is evaluated as

(1.156)

where Var[x] is the variance of x. An efficient approach in evaluating the k‐th moments of a random variable, without performing the integration in (1.153) or (1.155), is based on the moment theorem as expressed by the moment generation function (1.241) in Section 1.5.6.

The expectation of the function g(x) is evaluated as

(1.157)

and the expectation of the function g(X,Y) of two continuous random variables is

(1.158)

The expectation is distributive over summation so that

(1.159)

and

(1.160)

The following relationships between X and Y apply under the indicated conditions:

(1.161)

From (1.160) and (1.161) it is seen that if X and Y are uncorrelated random variables they are also orthogonal random variables if the mean of either X or Y is zero. The following example demonstrates that if two jointly Gaussian distributed random variables are orthogonal they are also independent.

The conditional expectation of X given Y is defined as

(1.162)

However, if Y is a random variable the function g₂(Y) = E(X|Y) is also a random variable and, using (1.157), the expectation (1.162) becomes

(1.163)

Papoulis [35] establishes the basic theorem for the conditional expectation of the function g(X,Y) conditioned on X = x, expressed as the random variable E[g(X,Y)|X = x]. The theorem is:

(1.164)

with the corollary relationship

(1.165)

Papoulis refers to (1.165) as a powerful formula.

The Bivariate Distribution—An Example of Conditional Distributions

Consider that x₁ and x₂ are Gaussian random variables with means m₁, m₂ and variances σ₁, σ₂, respectively, with the joint pdf is expressed as [36]

(1.166)

where ρ is the correlation coefficient, such that, |ρ| ≤ 1, expressed as

(1.167)

Using (1.150), the distribution of x₁ conditioned on x₂ is expressed as

(1.168)

If x₁ and x₂ are uncorrelated random variables then E[x₁x₂] = E[x₁]E[x₂] and, from (1.167), the correlation coefficient is zero and (1.168) reduces to the Gaussian distribution of x₁ with . Therefore, two jointly Gaussian distributed random variables are orthogonal and independent if they are uncorrelated.

Referring to (1.165), the first and second conditional moments of the second equality in (1.168) are evaluated using as E[g₁(X₁)g₂(X₂)] and , respectively, with and In the evaluation, the conditional mean of the Gaussian distribution is established from (1.168) by observation as

(1.169)

and the desired result is evaluated as

(1.170)

where and . The evaluation of is left as an exercise in Problem 12. The evaluation of (1.169) could have been performed using the integration in (1.155); however, it is significantly easier and less prone to error to simply associate the required parameters with the known form of the conditional Gaussian distribution as indicated in (1.168).

With zero‐mean random variables X₁ and X₂, that is, when m₁ = m₂ = 0, the second equality in (1.168) results in (see Papoulis [37])

(1.171)

and

(1.172)

The time correlated zero‐mean, equal‐variance Gaussian random variables denoted as x_i and x_i−1 taken at t_i = t_i−1 + Δt are characterized, using the last equality in (1.168), as

(1.173)

Equation (1.173) is used to model Gaussian fading channels with the fade duration dependent on Δt and ρ and the fade depth dependent on σ₁.

1.5.3 Definitions and Fundamental Relationships for Discrete Random Variables

In the following relationships, x_i, y_i, x, and y are considered to be discrete random variables corresponding to the event probabilities P_X(x_i), P_Y(y_i), P_X(x), and P_Y(y) with the corresponding pmfs p_Z(z) = P_Z(Z = z) : Z = {X,Y}, z = {x_i,y_i,z,y} corresponding to the amplitude of the discrete delta functions. In general, the characterization of discrete random variables is similar to that of continuous random variables with the integrations replaced by summations and the pdf replaced with the pmf.

1.5.3.1 Statistical Independence

If X(χ_i) = x with χ_i ∈ S and the events χ_i are independent ∀ i, then the joint probabilities are expressed as the product

(1.174)

or, in terms of the pmf, p_X(x_i) = P(X = x_i)

(1.175)

If S = S₁∩S₂ such that X(χ_i) = x_i with χ_i ∈ S₁, Y(χ_j) = y_j with χ_j∈ S₂, and the individual mdfs satisfy (1.175), then

(1.176)

Therefore, if the joint pmfs are independent, X and Y are also independent and, from the last equality in (1.176), S₁ and S₂ are also independent. Consequently, {X,Y} are independent iff the pmfs of X and Y can be expressed in the product form as in (1.175).

The expectation of x is evaluated as

(1.177)

For the discrete sampled function g(X,Y), the expectation value is evaluated as

(1.178)

where the pmf is expressed as .

1.5.3.2 Conditional Probability

The conditional probability of X given Y = y_j is expressed as

(1.179)

and, in terms of the conditional pmfs, (1.179) becomes

(1.180)

The pmf behaves like the pdf of continuous random variables, in that, if the event X(χ_i) = x_i with χ_i ∈ S₁, the probability of X ∈ S₁ given Y = y_j is evaluated as

(1.181)

If X and Y are independent (1.180) becomes

(1.182)

1.5.3.3 Bayes Rule

Bayes rule is expressed, in terms of the condition probability, as

(1.183)

and, in terms of probabilities and pmfs, Bayes rule is expressed as

(1.184)

The probability state transition diagram is shown in Figure 1.18 for N‐dimensional input and output states x_i and y_i, respectively. The outputs are completely defined by the conditional, or transition, probabilities P(y_j|x_i) and the input a priori probabilities P(x_i). Upon choosing the state y_j, that is, given y_j, the a posteriori probability P(x_i|y_j) is the conditional probability that the input state was x_i. Wozencraft and Jacobs (Reference 30, p. 34) point out that, “The effect of the transmission [decision] is to alter the probability of each possible input from its a priori to its a posteriori value.”

The conditional expectation of X given Y = y is

(1.185)

where the pmf p_X(x_i|y) = P(X = x_i|y).

1.5.4 Functions of Random Variables

Applications involving random variables that are functions of random variables, that is, z = g(x₁, …, x_M), require that the density function f_Z(z) be determined given : n = 1, …, M. In the following subsections, the transformation from to f_Z(z) is discussed for the relatively easy case involving functions of one random variables, that is, M = 1. More complicated cases are also discussed involving functions of two random variables and M random variables of the form . The following descriptions involve continuous random variables and cases involving discrete and mixed random variables are discussed in References 6, 8, 29.

1.5.4.1 Functions of One Random Variable

In the following description, the mapping of the random variable X = x is continuous and F_X(x) is differentiable at x as in (1.126), with finite values of f_X(x). The transformation from X to Z can be based on the functional relationships z = g(x) or x = h(z) with the requirements that corresponding to unit areas under each transformation. These transformations correspond, respectively, to

(1.186)

and

(1.187)

Equations (1.186) and (1.187) require the inverse relationship

(1.188)

The function z = h(x) typically has a finite number of solutions x_n, corresponding to the roots z = h(x₁), h(x₂),…, h(x_N) of the transformation and, under these conditions, the solution to f_Z(z) given f_X(x_n) is determined using the fundamental theorem [38, 39],

(1.189)

where h(z_n) corresponds to the transformation of x_n expressed in terms of z_n and .

As an example, consider a sinusoidal signal z, with constant amplitude a and random phase φ uniformly distributed between ±π, expressed as

(1.190)

Referring to Figure 1.19, and noting that , the problem is to determine the pdf f_Z(z) using the two roots of and . Using (1.190), is evaluated as

(1.191)

and

(1.192)

Therefore, evaluating (1.189) with f_Φ(φ) = 1/(2π) results in

(1.193)

1.5.4.2 Functions of Two or More Random Variables

The concepts involving a function of one random variable can also be applied when the random variable Z is a function of several random variables; for example, the dependence on two random variables, such that, z = g(x,y) is discussed at length by Papoulis (Reference 8, Chapters 6 and 7) where the subjects involving marginal distributions, joint density functions, probability masses, conditional distributions and densities, and independence are introduced. According to (1.126), the probability density function f_Z(z) is determined from the distribution function F_Z(z) as

(1.194)

and the joint pfd of X and Y is characterized for continuous distributions as

(1.195)

where the joint cdf is given by

(1.196)

Based on the conditions for the equality of the probabilities, that is,

the pdfs are equated as

(1.197)

Upon differentiating (1.197) with respect to z yields the desired result expressed as

(1.198)

As an example application consider the random variable Z = X + Y; Papoulis states that, “This is the most important example of a function involving two random variables.” Upon letting y = z – x and using (1.198) the density function of Z is evaluated as

(1.199)

and, when X and Y are independent, (1.199) is simply the convolution of f_X(x) with f_Y(y). Several examples involving the use of (1.199) are given in Section 1.5.6.1.

Using the joint probability density function of two continuous random variables x and y, as expressed in (1.195), the marginal pdfs f_X(x) and f_Y(y) are obtained by integrating over y and x, respectively, resulting in

(1.200)

and

(1.201)

These results can also be generalized to apply to the joint density function of any number of continuous random variables by integrating over each of the undesired variables.

1.5.5 Probability Density Functions

The following two subsections examine the probability density function [40] of the magnitude and phase of a sinusoidal signal with additive noise and the probability density function of the product of two zero‐mean equal‐variance Gaussian distributions. In these cases, the random variables of interest involve functions of two random variables. In Section 1.5.6, the characteristic function is defined and examined for several probability distribution functions demonstrating the central limit theorem with increasing summation of random variables. In Section 1.5.7, many of the probability distributions used in the following chapters are summarized and compared.

1.5.5.1 Distributions of Sinusoidal Signal Magnitude and Phase in Narrowband Additive White Gaussian Noise

This example involves the evaluation of the pdf of the magnitude and phase at the output of a narrowband filter when the input is a sinusoidal signal with uniformly distributed phase and zero‐mean additive white Gaussian noise [41] (AWGN). In this case, the output of the narrowband filter is a narrowband random process. The evaluation involves three random variables: the input signal phase φ and the two independent‐identically distributed (iid) zero‐mean quadrature noise random variables with variance . The signal plus noise out of the filter is expressed as

(1.202)

where the third equality in (1.202) emphasizes the in‐phase and quadrature functions of the signal and noise terms and, when sampled at t = iT_s, represent the random variables x_c, n_c, x_s, and n_s. The functional relationships are and with n_c and n_s representing zero‐mean quadrature Gaussian random variables. The signal phase, φ, is uniformly distributed between 0 and 2π. Under these conditions, the quadrature signal and noise components x_c and x_s are independent random variables¹⁴ and the pdfs of x_c and x_s are expressed as

(1.203)

and

(1.204)

The pdf of the phase is

(1.205)

Using (1.203), (1.204), and (1.205) the joint pdf is expressed as

(1.206)

The evaluation of the joint pdf of the magnitude and phase of the sampled sine‐wave plus noise involves the transformation of variables from (x_c,x_s) to (r,θ) as depicted in Figure 1.20. The magnitude is described as

(1.207)

and the in‐phase and quadrature components, x_c and x_s, are described in terms of the angle θ as

(1.208)

Expressing the phase angle in (1.208) as a function of x_c and x_s leads to the expressions

(1.209)

and

(1.210)

The Jacobian of the transformation is defined as [6, 8, 28, 29]

(1.211)

and, using the Jacobian, the transformation from (x_c,x_s) to (r,θ) is expressed as

(1.212)

To evaluate the Jacobian for this transformation, the functions g_ij(x_c,x_s) are defined in terms of (1.207), (1.209), and (1.210) as follows:

(1.213)

(1.214)

and

(1.215)

Upon evaluating the partial derivatives in (1.211), the Jacobian is found to be¹⁵

(1.216)

and, using (1.208), the functions h₁(r,θ) and h₂(r,θ) are expressed as

(1.217)

and

(1.218)

Substituting (1.216), (1.217), and (1.218) into (1.212) and applying the independence of x_c, x_s, and φ, as in (1.206), the pdf of the transformed variables r and θ is expressed as

(1.219)

where r ≥ 0, otherwise the pfd is zero, and θ and φ are uniformly distributed over the range 0 ≤ θ, φ ≤ 2π. The pdf for the magnitude r is determined by computing the marginal distribution M_R(r) by integrating over the ranges of θ and φ. Defining ψ = θ − φ, the marginal is evaluated as

(1.220)

Davenport and Root [42] point out that the integrand of the bracketed integral is periodic in the uniformly distributed phase ψ and can be integrated over the interval 0 to 2π. With this integration range, the bracketed integral is identified as the zero‐order modified Bessel function expressed as [43]

(1.221)

Therefore, upon using (1.221) and performing the integration over φ, the marginal distribution function M_R(r) simplifies, at least in notation, to

(1.222)

Equation (1.222) is the Rice distribution or, as referred to throughout this book, the Ricean distribution that, as developed in the forgoing analysis, characterizes the baseband magnitude distribution of a CW signal with narrowband additive white Gaussian noise. The Ricean distribution also characterizes the magnitude distribution of a received signal from a channel with multipath interference; this channel is referred to as a Ricean fading channel. The Ricean distribution becomes the Rayleigh distribution as A → 0 and the Gaussian distribution as A → ∞; the proof of these two limits is the subject of Problems 19 and 20. The Rayleigh distribution characterizes the amplitude distribution of narrowband noise or, in the case of multipath interference, the composite signal magnitude of many random scatter returns without a dominant specular return or signal component. The multipath interference is the subject of Chapter 18. Defining the signal‐to‐noise ratio as γ = A²/(2σ²), (1.222) is expressed as

(1.223)

The pdf of the phase function is evaluated by computing the marginal distribution M_ΘΦ(θ,φ) by integrating over the range of the magnitude r. By forming or completing the square of the exponent in the last equality in (1.219) the integration is performed as

(1.224)

Davenport and Root [44] provide an approximate solution to (1.224), under the condition Acos(θ – φ) >> σ. The approximation is expressed as

(1.225)

where γ is the signal‐to‐noise ratio defined earlier. An alternate solution, without the earlier restriction, is expressed by Hancock [45], with ψ = θ − φ, as

(1.226)

where P(z) is the probability integral defined in Section 3.5. Hancock’s phase function is used in Section 4.2.1 to characterize the performance of phase‐modulated waveforms.

As γ → 0 in (1.226) the function f_Ψ(ψ) → 1/2π resulting in the uniform phase pdf. However, for γ greater than about 3, the probability integral is approximated as [26]

(1.227)

Using (1.227), the phase pdf is approximated as

(1.228)

With |ψ| ≅ 0 such that sin²(ψ) ≅ ψ² and defining (1.228) is approximated as

(1.229)

Equation (1.229) describes a zero‐mean Gaussian phase pdf with the phase variance . Hancock’s phase function, expressed in (1.226), is plotted in Figure 4.3 for various signal‐to‐noise ratios.

1.5.5.2 Distribution of the Product of Two Independent Gaussian Random Variables

In this section the pdf of the product z = xy of two zero‐mean equal‐variance iid Gaussian random variables X and Y is determined. The solution involves defining an auxiliary random variable w = h(x) = x with z = g(x,y) = xy and evaluating f_Z,W(w,z) characterized as

(1.230)

where J_X,Y(x,y) is the Jacobian of the transformation evaluated as

(1.231)

Using (1.231) and the joint Gaussian pfd of X and Y, expressed by (1.230), with x = w and y = z/w, the marginal pdf of z is evaluated as

(1.232)

However, since X and Y are independent

(1.233)

and, upon substituting x = w and y = z/w into (1.233), (1.232) is expressed as

(1.234)

where the second equality recognizes that the first equality is symmetrical in w. Letting λ = w²/2σ² (1.234) is expressed as

(1.235)

The solution to the integral in (1.235) appears in the table of integrals by Gradshteyn and Ryzhik (Reference 46, p. 340, pair No. 12) as

(1.236)

where K_v(u) is the modified Bessel function of the second kind of order v. With v = 0 and u = z/σ², (1.235) is evaluated as

(1.237)

The magnitude of z in (1.237) is used because of the even symmetry of f_Z(z) with respect to z. The symmetry of f_Z(z) results in a zero‐mean value so the variance is evaluated as

(1.238)

The solution to the integral in (1.238) is found in Gradshteyn and Ryzhik (Reference 46, p. 684, Integral No. 16) and the variance f_Z(z) is evaluated as

(1.239)

where the second equality in (1.239) results from the value of the Gamma function . In Example 4 of Section 1.5.6.1, the pdf of the summation of N iid random variables with pdfs expressed by (1.237) is examined.

1.5.6 The Characteristic Function

The characteristic function of the random variable X is defined as the average value of e^jvx and is expressed as

(1.240)

With v = −ω and x = t (1.240) is similar to the Fourier transform of a time‐domain function. The characteristic function is also referred to as the moment‐generating function, in that, the nth moment of the random variable X, defined as the expected value E[xⁿ], is evaluated (see Problem 26) as

(1.241)

The Fourier transform relationship between time domain convolution and frequency domain multiplication also applies to the convolution of random variables and the multiplication of the corresponding characteristic functions. Therefore, based on the discussion in Section 1.5.6.1, the summation of N identically distributed (id) random variables corresponds to the product of their individual characteristic functions, that is,

(1.242)

This is a very useful result, in that, the distribution of the summation of N independent random variables is obtained as the inverse transform [47] of (1.242) expressed as

(1.243)

Campbell and Foster [47] provide an extensive listing of Fourier transform pairs defined as

(1.244)

and, by defining v = −2πf, the Fourier transform pairs apply to the transform pairs between f_X(x) and C_X(v) as expressed in (1.240).

1.5.6.1 Summation of Independently Distributed Random Variables

If two random variables X and Y are independent then the probability density f_Z(z) of their sum Z = X + Y is determined from the convolution of f_X(x) with f_Y(y) so that¹⁶

(1.245)

For multiple summations of a random variable, the convolution is repeated for each random variable in the summation.

Example 1

Consider the summation of N zero‐mean uniformly distributed random variables X_i expressed as

(1.246)

with

(1.247)

For N = 2 the convolution involves two ranges of the variable z as shown in Figure 1.21 and the integrations are evaluated as

(1.248)

and

(1.249)

Upon evaluation of (1.248) and (1.249) and recognizing the symmetry about z the density function is expressed as

(1.250)

Repeating the application of the convolution for N = 3 and 4 (see Problem 24) results in the probability density functions shown in Figure 1.22 with the corresponding cdf results shown in Figure 1.23. As the probability density and characteristic functions will approach those of the Gaussian distributed random variable (see Problem 23).

The moments of the random variable X are evaluated using the characteristic function

(1.251)

In regions where the characteristic function converges, the moments E[xⁿ] completely define the characteristic function and the pdf of the random variable X, so, upon expanding (1.251) as the power series

(1.252)

the moments are easily evaluated using (1.241). The moments for the random variable Z, formed as in (1.246), are determined using (1.242) and, with X_i : i = 1, …, N iid random variables, the characteristic function for Z is approximated as

(1.253)

The first and second moments for N = 1, …, 4 are listed in Table 1.6. These results are also obtained by evaluating f_Z(z) using (1.250) and then evaluating the moments (see Problem 25) as

(1.254)

N	E[z]	E[z2]
1	0	a2/3
2	0	2a2/3
3	0	a2
4	0	4a2/3

However, it is much easier to use the characteristic function.

Example 2

As another example, consider the summation of N random variables X_i characterized as the sinusoidal function

(1.255)

with constant amplitudes A_i and zero‐mean uniformly distributed phase, expressed as

(1.256)

The resulting pdf of the random variable X_i for ϕ = π, is evaluated in (1.193) as

(1.257)

and is plotted in Figure 1.24.

The pdf of the random variable Z, expressed as in (1.246),¹⁷ is evaluated by successive convolutions as in (1.245) and the results for N = 2, 3, and 4 are plotted in Figure 1.25 with the corresponding cdf functions shown in Figure 1.26. The results in Figures 1.25 and 1.26 for N > 1 are obtained by numerical evaluations of the convolutions using incremental values of Δz = 2.5 × 10⁻⁵; this is a reasonable compromise between simulation time and fidelity in dealing with the infinite value at |x| = 1.0.

In this case, the mean and variance of the random variable X are evaluated using the characteristic function of (1.257) found in (Reference 47, p. 123, Transform Pair 914.5); the result is

(1.258)

where I_o(−) is the modified Bessel function of order zero. Expanding (1.258) for Av < 1 as a power series, (Reference 46, p. 375, Ascending Series 9.6.10), results in¹⁸

(1.259)

and the moments are easily evaluated using (1.241). The first and second moments are listed as the theoretical values in Table 1.7. The moments for the random variable Z, formed as in (1.246) with X_i iid random variables for all i as expressed by the pdf in (1.255), are determined using the characteristic function expressed as

(1.260)

N	Theoretical		Numericala (A = 1)
	E[z]	E[z2]	E[z]	E[z2]
1	0	A2/2	0	0.4999
2	0	A2	0	1.0044
3	0	3A2/2	0	1.5055
4	0	2A2	0	2.0146

^aNumerical values are sampled with Δz = 2.5 × 10⁻⁵.

The corresponding first two moments of the random variable Z for N = 2, 3, and 4 are also listed in Table 1.7. The numerical results listed in Table 1.7 are based on computer evaluations of the various convolutions resulting in the pdfs shown in Figures 1.24 and 1.25.

A major observation in these two examples is that the probability distribution of the random variable Z approaches a Gaussian distribution as N increases (see Problem 27). This is evidence of the central limit theorem which states that (see Davenport and Root, Reference 6, p. 81) the sample mean of the sum of N arbitrarily distributed statistically independent samples becomes normally distributed as N increases. This is referred to the equal‐components case of the central limit theorem. However, as pointed out by Papoulis (Reference 8, p. 266), a consequence of the central limit theorem is that the distribution f_Z(z) of the sum of N statistically independent distributions having arbitrary pdf’s tends to a normal distribution as N increases. This is a stronger statement and suggests that the probability P(z) = f_Z(Z < z) can be considered a Gaussian distribution for all z as is frequency assumed to be the case in practice. Davenport and Root also point out that, even though N is seemingly large, the tails of the resulting distribution may result in a poor approximation to the Gaussian distribution.

Upon computing the mean and variance using the power series expansion of C_Z(v) expressed by (1.252) with av << 1, the approximate expression for the corresponding Gaussian distribution is easily obtained. After summing N uniformly distributed amplitudes the expression for the pdf is

(1.261)

Similarly, for the summation of N sinusoids with Av << 1, the pdf in Example 2 is expressed as

(1.262)

It is interesting to note that the second moments are Nλ² for all values of N including those for which the pdf does not have the slightest resemblance to the Gaussian pdf. In these cases, the important difference is that the corresponding probabilities P(x) = F_X(X < x) are entirely different from those of the Gaussian distribution with the possible exception of the median value. Finally, it is noted that the limiting behavior for λv << 1 and N → ∞ applies to the summation of independently distributed distributions that may, or may not, be identically distributed distributions.

Example 3

This example involves the summation of random chips {±1} in a direct‐sequence spread‐spectrum (DSSS) waveform. In this case, the chips occur with equal probabilities according to the pdf expressed as

(1.263)

Using (1.240), the characteristic function is evaluated as

(1.264)

The DSSS waveform uses N chips per bit and the demodulator correlation sums the N chips to form the correlation output with the corresponding characteristic function given by

(1.265)

To evaluate the first and second moments of y only the first two terms in the expansion of cos^N(v) are required and, upon using (1.241), these moments are evaluated as

(1.266)

and

(1.267)

The variance of y is defined as the second central moment and with zero‐mean the variance is .

Example 4

The pdf f_Z(z) of the product of two, zero‐mean, equal‐variance, iid Gaussian random variables, z = xy, is expressed in (1.237) as a function of the zero‐order modified Bessel function K_o(|z|/σ²) where the magnitude of z provides for the range: −∞ ≤ z ≤ ∞. In this example, the pdf is evaluated where : i = 1, …, N. The evaluation is based on the N‐th power of the characteristic function C_Z(v) and, from the work of Campbell and Foster (Reference 47, p. 60, pair No. 558), the characteristic function is evaluated as¹⁹

(1.268)

The characteristic function of is the N‐th power of (1.268) expressed as

(1.269)

and, using the transform pair of Campbell and Foster (Reference 47, p. 61, pair No. 569.0), the pdf of is evaluated as

(1.270)

As in the case for f_Z(z), the pdf applies for −∞ ≤ ≤ ∞ and is symmetrical with respect to resulting in a zero‐mean value with the variance expressed as

(1.271)

The solution to the integral in (1.271) is found in Gradshteyn and Ryzhik (Reference 46, p. 684, Integral No. 16) and the variance f_Z(z) is evaluated using

(1.272)

Substituting the solution to the integral in (1.272) into (1.271), with u = (N + 3)/2, v = (N − 1)/2, and a = 1/σ², the solution to variance simplifies to

(1.273)

In the earlier evaluation, the integer argument Gamma function is related to the factorial as ! and . This result could also be evaluated using the movement generating function of (1.241), however, using the integral solution as in (1.272) it is sometimes easier to evaluate the moments. With a sufficiently large value of N the pdf is approximated as the Gaussian pfd expressed as

(1.274)

The probability density functions discussed earlier and others encountered in the following chapters are summarized in Table 1.8 with the corresponding mean values, variances, and characteristic functions.

Name	fX(x)	E[x]	Var[x]	CX(v)	Conditions
Uniform
Bernoulli		p	p(1 − p)	(1 − p) + pv	Discrete binary variable x = ki : i = 1, 2
Binomial		np	np(1 – p)		Discrete variable
Poisson		α	α		Discrete variable
Exponential		1/α	1/α2
Gaussian (normal)		m	σ2
Chi‐square (N = 2) Exponential (α = 1/2)		2	4		x ≥ 0
Chi‐squared (N‐degrees)		N	2N		N‐degrees of freedom x ≥ 0
Rayleigh		as γ→∞	as γ→∞		x > 0
Ricean		a	a	a	x > 0
Gamma		α/β	α/β2		x > 0 β > 0; λ > 0
Lognormal				b	y is lognormal y = ex ≥ 0 x = N(m,σ)
Nakagami‐m		c	c	c	x ≥ 0

Notes: γ = A²/(2σ²) is the signal‐to‐noise ratio. γ→0 f_X(x) = Rayleigh with E[x] = , Var[x] = .

^aγ→∞ f_X(x) = Gaussian with E[x] = A, Var[x] = σ².

^bApproximated using a series expansion of e^jvy.

^cRefer to special cases in Section 1.5.7.2.

1.5.7 Relationships between Distributions

In the following two subsections, the relationship between various probability density functions is examined by straightforward parameter transformations, allowing parameters to approach limits, or simply altering various parameter values. The most notable relationship is based on the central limit theorem in which a distribution approaches the Gaussian distribution by increasingly summing the operative random variable.

1.5.7.1 Relationship between Chi‐Square, Gaussian, Rayleigh, and Ricean Distributions

A random variable has a chi‐square (χ²) distribution with N degrees of freedom if it has the same distribution as the sum of the squares of N‐independent, normally distributed random variables, each with zero‐mean and unit variance.²⁰

Consider the zero‐mean Gaussian or normal distributed random variable x with variance and pdf expressed as

(1.275)

The pdf of a new random variable y = x², obtained by simply squaring x, is determined by considering the positive and negative regions of as shown in Figure 1.27.

The pdf of y is determined using the incremental intervals dy = 2xdx at such

(1.276)

The characteristic function of (1.276) is evaluated as

(1.277)

Consider now the random variable z resulting from the summation of N independent random variables y_i such that

(1.278)

The characteristic function of z is simply the N‐th power of C_Y(v) so that

(1.279)

Equation (1.279) transforms to the pdf of z, resulting in

(1.280)

In conforming to the earlier definition, the chi‐square distribution is expressed by letting in (1.280) or, more formally, using the transformation ; therefore, the pdf of the chi‐square random variable χ with N degrees of freedom is

(1.281)

and the corresponding characteristic, or moment generating, function is

(1.282)

Equation (1.281) is occasionally referred to as the central χ² distribution because it is based on noise only, that is, the underlying zero‐mean Gaussian random variables x_i with distribution given by (1.275) do not contain a signal component.²¹

Special Case for N = 2

Under this special case (1.280) reduces to the exponential distribution

(1.283)

So the resulting chi‐square χ² distribution is obtained from (1.281) with N = 2. This is an important case because x₁ and x₂ can be thought of as orthogonal components in the complex description of a baseband data sample. Urkowitz [48] shows that the energy of a wide‐sense stationary narrowband white noise Gaussian random process with bandwidth –W to W Hz and measured over a finite interval of T seconds is approximated by N = 2WT terms or degrees of freedom. The frequency W is the noise bandwidth of the narrowband baseband filter and the approximation error in the energy measurement decreases with increasing 2WT. The factor of two can be thought of as the computation of complex orthogonal baseband functions so N = 2 degrees of freedom correspond to WT = 1. For example, the rect(t/T) function observed over the interval T seconds has a noise bandwidth of W = 1/T Hz corresponding to WT = 1 resulting in 2 degrees of freedom.

Upon letting , the random variable w is described in terms of the Rayleigh distribution

(1.284)

So the Rayleigh distribution is derived from the magnitude of the quadrature zero‐mean Gaussian distributed random variables, x = N(0,σ).²²

1.5.8 Order Statistics

Communication systems analysis and performance evaluations often involve a large number of random samples taken from an underlying continuous or discrete probability distribution function. The various parameters, used to characterize the system performance, result in limiting distributions with associated means, variances, and confidence levels as dictated, for example, by an underlying distribution. Order statistics [31, 50, 51], on the other hand, involves a distribution‐free or nonparametric analysis that requires only that the probability distribution functions be continuous and not necessary related to the underlying distribution from which the samples are taken. However, the randomly drawn samples are considered to be statistically independent.

Consider that the n random samples {X₁, X₂, …, X_n} are taken from the continuous pdf f_X(x) over the range . Now consider reordering the random variables X_i : i = 1, …, n to form the random variables {Y₁, Y₂, …, Y _n} arranged in ascending order of magnitude, such that, where is uniformly distributed over the interval b − a. The joint pdf of the ordered samples [52] is expressed as

(1.285)

for and n! is the number of mutually disjoint sets of x₁, x₂, …, x_n. For example, for n = 4 the set x₁, x₂, x₃, x₄ results in n! = 24 mutually disjoint sets determined as shown in Table 1.9. The first six mutually disjoint sets are determined by cyclically left shifting the indicated subsets of original set x₁, x₂, x₃, x₄; a cyclic left shift of a subset is obtained by shifting each element of the subset to the left and replacing the leftmost element in the former position of the rightmost element. Following the first six sets shown in the figure, the original set is cyclically shifted three more times each leading to six mutually disjoint sets by shifting subsets resulting in a total of 24 mutually disjoint sets.

The ordered sample Y_i is referred to as the i‐th order statistic of the sample set. The marginal pfd of the n‐th order statistic Y_n, that is, the maximum of {X₁, X₂, …, X_n}, is evaluated using (1.285) by performing the integrations in the ascending order i = 1, 2, …, n − 1 as follows²³:

(1.286)

The solution (see Problem 15) to (1.286) is

(1.287)

where Fⁿ⁻¹(y_n) is the cdf evaluated as

(1.288)

Using the marginal pdf of Y_n given by (1.287), the probability of selecting the maximum of value Y_n is determined as

(1.289)

These results are distribution free, in that, the pdf has not been defined; however, from a practical point of view (1.289) can be evaluated for any continuous pdf.

The distributions from which the x_i are taken need not be identical²⁴; for example, the samples x₁ through x_j can be taken from a distribution involving signal plus noise (or clutter) and those from x_j+1 through x_n corresponding to noise (or clutter) only. Using this example the distribution in (1.287) is expressed as

(1.290)

where F_sn(y) is the distribution corresponding to signal plus noise and F_n(y) is the noise‐only distribution.

Example distributions used to evaluate the performance of communication and radar systems are Gaussian, Ricean, lognormal, and Weibull distributions. Table 1.10 lists the false‐detection probabilities, for the indicated signal‐to‐noise ratios γ_dB, associated with the detection of j = 1 signal‐plus‐noise event and k = n − j = (1,2,4, and 8) noise‐only events.

1.5.9 Properties of Correlation Functions

Correlation processing is used in nearly every aspect of demodulator signal detection from energy detection, waveform acquisition, waveform tracking, parameter estimation, and information recovery processing. With this wide range of applications, the theoretical analyst, algorithm developer, software coder, and hardware developer must be thoroughly familiar with the properties and implementation of waveform correlators. An equally important processing function is that of convolution or linear filtering. The equivalence between matched filtering and correlation is established in Section 1.7.2 and involves a time delay in the correlation response; with this understanding, the properties of correlation can be applied to convolution or filtering. The correlation response can be exploited to determine the signal signature regarding the location of a signal in time and frequency, the duration and bandwidth of the signal, the shape of the modulated signal waveform, and the estimate of the information contained in the modulated waveform.

The correlation function²⁵ is evaluated for the complex functions as the integral

(1.291)

and

(1.292)

where the asterisk denotes complex conjugation.

Autocorrelation processing examines the correlation characteristics of a single random process with the maximum magnitude corresponding to the zero‐lag condition that is equal to the maximum energy over the correlation interval. The correlation response is indicative of the shape of and the duration, τ_d, of the principal correlation response is indicative of the correlation time. For deterministic signals, the correlation time (τ_o) is usually characterized in terms of the one‐sided width of the principal correlation lobe; however, for stochastic processes the correlation interval is defined when || decreases monotonically from to a defined level; for example, when the normalized correlation response first reaches the level . The normalized correlation response is referred to as the correlation coefficient as defined in (1.295) or (1.296). The parameters related to the correlation of the function have equivalent Fourier transform frequency‐domain definitions. In the case of stochastic processes, the Fourier transform of is defined as the PSD of the process.

Expanding (1.292) in terms of the real and imaginary with and results in

(1.293)

This evaluation requires four real multiplies and integrations for each lag, whereas, if and ỹ(t) were real functions only one multiplication and integration is required for each lag. With discrete‐time sampling, the integrations are replaced by summations over the finite sample values and ỹ_n where t = nT_s: n = 0, …, N − 1 and T_s is the sampling interval; in this case, the computational complexity is proportional to N². The computation complexity can be significantly reduced by performing the correlation in the frequency domain using FFT [53], in which case, for a radix‐2 FFT with N = 2^k, the computation complexity is proportional to Nlog₂(N). Brigham [54] provides detailed descriptions of the implementation and advantages of FFT correlation and convolution processing. The correlation results throughout the following chapters use the direct and FFT approaches without distinction.

Referring to (1.291) the zero‐lag correlation is expressed as

(1.294)

where E_x is the total energy in the received signal. Using (1.294), the normalized correlation is defined in terms of the normalized autocorrelation coefficient as

(1.295)

with |ρ_x(τ)| ≤ 1. From (1.292), the normalized cross‐correlation coefficient is defined as

(1.296)

with |ρ_xy(τ)| ≤ 1.

The correlation may also be defined in terms of the long‐term average over the interval T as

(1.297)

However, most practical waveforms are limited to a finite duration T_c = NT_s and, in these cases, is zero outside of the range T_c. Therefore, dividing the zero‐lag correlation by T_c results in the second‐order moment where is the DC or mean signal power. Removing the mean signal level prior to performing the correlation results in the autocovariance with . Table 1.11 summarized several properties of correlation functions.

Property	Comments
	Autocorrelation
	x(t) is real
	Autocovariance
	Cross‐covariance

Consider, for example, that is a received signal plus AWGN, the correlation is performed in the demodulator using the known reference signal . The dynamic range of the demodulator detection processing is minimized by the normalization in (1.296) and the optimum signal detection corresponds to ρ_xy(0). On the other hand, if the optimum timing is not known, near optimum detection can be achieved by choosing the maximum correlation output over the uncertainty range of the correlation lag about τ = 0. During initial signal acquisition, the constant false‐alarm rate (CFAR) threshold, described in Section 11.2.2.1, is an effective algorithm for signal presence detection and coarse synchronization.

1.6.1 Stochastic Processes

The subject of stochastic processes is discussed in considerable detail by Papoulis [55] and Davenport and Root [56] and the following definitions are often stated or implied in the applications discussed in throughout the following chapters. A stochastic process is defined as a random variable that is a function of time and the random events χ in S as depicted in Figure 1.14. In this context the random variable is characterized as x(t,χ). For a fixed value of t = t_i, x(t_i,χ) is a random variable and χ = χ_i, x(t,χ_i) denotes as the real random process x(t) such that x(t_i) is a random variable with pdf f_X(x:t_i); in general, the pdf of x(t) is defined as f_X(x:t).

1.6.1.1 Stationarity

There are several ways to define the stationarity of a stochastic process, for example, stationarity of finite order, asymptotic stationary, and periodic stationarity; however, the following two are the most frequently encountered.

Strict‐Sense Stationary Process

The stochastic process x(t) is strict‐sense stationary, or simply stationary, if the statistics are unaltered by a shift in the time axis. Furthermore, two random variables are jointly stationary if the joint statistics are unaltered by an equal time shift of each random variable, that is, the probability density function f(x ; t) is the same for all time shifts τ. This is characterized as

(1.298)

Wide‐Sense Stationary Process

The stochastic process x(t) is wide‐sense stationary (WSS) if its expected value is constant and autocorrelation function is a function of the time shift τ = t₂ − t₁ ∀ t₁ and t₂. WSS stationarity is characterized as

(1.299)

and

(1.300)

Because wide‐sense stationarity depends on only the first and second moments it is also referred to as weak stationarity. A function of two random processes is wide‐sense stationary if each process is wide‐sense stationary and their cross‐correlation function is dependent only the time shift, that is,

(1.301)

1.6.1.2 Ergodic Random Process

The random process x(t), defined earlier, is an ergodic random process if the statistics of x(t) are completely defined by the statistics of x(t,χ). Denoting the random process x(t_i,χ) as an ensemble of x(t,χ), then ergodicity ensures that the statistics x(t_i) are identical to those of the ensemble; in short, the time statistics are identical to the ensemble statistics.²⁶ Ergodicity of the mean, of the stochastic process x(t,χ), exists under the condition

(1.302)

where the time average is defined as

(1.303)

and the ensemble average is defined as

(1.304)

Since the mean value of a random process must be a constant, the ergodic of the mean theorem states that the equality condition in (1.302) is satisfied when where η is a constant. This is a nontrivial task to prove, however, following the discussion by Papoulis [57], the ergodic of the mean theorem states that

(1.305)

The iff condition in (1.305) is formally expressed in terms of the autocovariance function for which the limit T → ∞ is expressed as the variance . However, from (1.305), the expectation E[x(t)] = η resulting in . Therefore, the limit T → ∞ of the autocovariance function converges in probability with the conclusion that proving ergodicity of the mean.²⁷ Demonstration of ergodicity of the autocorrelation function is considerably more involved, requiring the fourth‐order moments.

1.6.2 Narrowband Gaussian Noise

Consider the noise described by the narrowband process [58] with bandwidth B << f_c expressed as

(1.306)

where N(t) and θ(t) represent, respectively, the envelop and phase of the noise and ω_c = 2πf_c is the angular carrier frequency. Upon expanding the trigonometric functions, (1.306) can also be expressed as

(1.307)

where

(1.308)

and

(1.309)

The noise terms n_c(t) and n_s(t) are uncorrelated with spectrum S(f) and bandwidth B, such that S(f) = 0 for |f − f_c| > B/2. This is the general characterization of a narrowband noise process; however, in the following analysis, n_c(t) and n_s(t) are also considered to be statistically independent, stationary zero‐mean white noise Gaussian processes with one‐sided spectral density N_o watts/Hz.

Because of the stationarity, the noise autocorrelation is dependent only on the correlation lag τ and is evaluated as

(1.310)

Upon evaluating the product in (1.310) and distributing the expectation, it is found that the conditions for stationarity require²⁸ R_ss(τ) = R_cc(τ) and R_cs(τ) = −R_sc(τ) so that (1.310) reduces to

(1.311)

The noise power is evaluated using (1.311) with τ = 0 with the result R_nn(0) = R_cc(0) = . This evaluation can be carried further using the Wiener–Khinchin theorem²⁹ which states that the power spectral density of a WSS random process is the Fourier transform of the autocorrelation function, that is,

(1.312)

From (1.312) the inverse Fourier transform is

(1.313)

and, substituting the condition that the single‐sided noise spectral density is defined as N_o watts/Hz, (1.313) becomes

(1.314)

In (1.314) the single‐sided noise density is divided by two because of the two‐sided integration, that is, the integration includes negative frequencies. In this case, the noise power, defined for τ = 0, is infinite, however, when the ideal band‐limited filter, with bandwidth B, is considered the noise power in the filter centered at f_c is computed as

(1.315)

In this case the one‐sided noise density N_o is used instead of N_o/2 because the one‐sided integration is over positive frequencies.

If a linear filter with impulse response h(t) is used, the frequency response is given by

(1.316)

The corresponding unit gain normalizing factor is |H(0)|. With the stationary noise process n(t) applied to the input of the filter, the output is determined using the convolution integral and the result is as follows:

(1.317)

Using (1.317) it can be shown (see Problem 33) that the normalized spectrum of the output noise is expressed, in terms of the input noise PSD S_n(f), as

(1.318)

where |H(0)| is the normalizing gain of the filter. Using (1.318), with S_n(f) = N_o/2 corresponding to white noise, the output noise power is evaluated as

(1.319)

where the second integral in (1.319) is recognized as the definition of the noise bandwidth of the bandpass filter with lowpass bandwidth B_n.

1.7.1 Example Application of Matched Filtering

In this example, a BPSK‐modulated received signal is considered with binary source data bits b_i = {0,1} expressed as the unipolar data d_i = (1 − 2b_i) = {1,−1} over the data intervals of the bit duration T. The received signal plus noise is expressed as

(1.330)

The signal is described as

(1.331)

The noise is zero‐mean additive white Gaussian noise with one‐sided spectral density N_o described as

(1.332)

The receiver‐matched filter impulse response and Fourier transform are given by

(1.333)

In (1.333) the signal spectrum defined as S(f) and the squared magnitude of the matched filter output at the optimum sampling point is

(1.334)

where the gain G = |H(0)| is normalized to one resulting in a unit gain‐matched filter response H(f).

Referring to the additive noise described by (1.332) and Section 1.6.2, the noise power at the output of the matched filter is expressed as

(1.335)

where B/2 is the baseband bandwidth of the matched filter.

The received signal, as expressed in (1.330), can be rewritten in terms of the optimally sampled matched filter output as

(1.336)

where n_i are iid zero‐mean, unit variance, white Gaussian noise samples. Upon dividing (1.336) by the sampled matched filter output is expressed as

(1.337)

The sampled values l(r(iT_o)) and l′(r_i) are referred to as sufficient statistics, in that, they contain all of the information in r(t), expressed in (1.330), to make a maximum‐likelihood estimate of the source‐bit d_i. The normalized form in (1.337) is used as the turbo decoder input discussed in Section 8.12. In Section 1.8 the sufficient statistic is seen to be a direct consequence of the log‐likelihood ratio.

1.7.2 Equivalence between Matched Filtering and Correlation

Consider the receiver input as the sum of the transmitted signal plus noise expressed as

(1.338)

The cross‐correlation of r(t) with a replica of the received signal is computed as

(1.339)

Defining the matched filter impulse response as h(t), the matched filter output response to the input r(t) is

(1.340)

However, referring to the preceding matched filter discussion, the matched filter response is equal to the delayed image of the signal, such that,

(1.341)

As mentioned previously, the delay T_o ensures that the filter response is causal and, therefore, realizable. To substitute (1.341) into (1.340) first let so that and substitute this result in (1.340) to get

(1.342)

So that the convolution response is equal to the cross‐correlation response delayed by T_o. If the input noise is zero, so that r(t) = s(t), the same conclusion can be drawn regarding the autocorrelation response.

1.8.1 Example of Likelihood and Log‐Likelihood Ratio Detection

Consider the two hypotheses H₁ and H₀ mentioned earlier with d_i = {1,−1} and the observation r_i in (1.345) with the additive noise n_i characterized as iid zero‐mean white Gaussian noise, denoted as N(0,σ_n). The transition probabilities are expressed in terms of the Gaussian noise pdf as

(1.349)

Upon forming the likelihood ratio and recognizing that = ±d_i, the likelihood ratio decision simplifies to

(1.350)

and the log‐likelihood ratio decision simplifies to

(1.351)

Recognizing that l(r_i) is a sufficient statistic, (1.351) is rewritten as

(1.352)

When C₁₀ = C₀₁ = 1, C₀₀ = C₁₁ = 0, and P₀ = P₁ the LLR simplifies to

(1.353)

Therefore, the data estimate is when l(r_i) > 0, otherwise, . Recall that observations r_i : t = iT_o are made at the output of the matched filter. These concepts involving the LR and LLR surface again in Section 3.2 and the notion of the natural logarithm of the transitions probabilities is discussed in the following section involving parameter estimation.

1.9.1 Example of MS and MAP Parameter Estimation

As an example application of the parameter estimation procedures discussed earlier, consider the Poisson distribution that is used to predict population growth, telephone call originations, gamma ray emissions from radioactive materials, and is central in the development of queueing theory [68]. For this example, the Poisson distribution is characterized as

(1.368)

In the application of (1.368) to queueing theory, a = λt is the average number of people entering a queueing line in the time interval 0 to t and λ is the arrival rate. The a posteriori distribution is the probability of a conditioned on exactly n arrivals occurring in the time interval. A fundamental relationship in the Poisson distribution is that the time interval between people entering the queueing line is exponentially distributed and is characterized by the a priori distribution

(1.369)

The a posteriori pdf in (1.368) is expressed in terms of the a priori and transition pdfs as

(1.370)

where the constant k is a normalizing constant that includes 1/p_n(n). Integrating of the second equality in (1.370) with respect to da over the range 0 to ∞ and setting the result equal to one, the value of k is found to be k = 2ⁿ⁺¹ and (1.370) becomes

(1.371)

Using (1.356) and (1.371) the MS estimate is evaluated as

(1.372)

Also, using (1.357) and (1.371) the MAP estimate is evaluated as

(1.373)

and solving the second equality in (1.373) for a results in . As is typical in many cases, the MS and MAP estimation procedures result in the same estimate. It is left as an exercise (see Problem 38) to determine the bias of the estimates, compute the Cramér–Rao bound, and using (1.366), determine if the estimates are efficient, that is, if the Cramér–Rao equality condition applies.

1.9.2 Constant‐Parameter Estimation in Gaussian Noise

To simplify the description of the estimation processing, the analysis in this section considers the single constant‐parameter case with zero‐mean narrowband additive Gaussian noise. Under these conditions, the estimation is based on the solution to the maximum‐likelihood equation with the joint pdf of the received signal and noise written as

(1.374)

where a is the constant parameter to be estimated and r_i = s_i + n_i represents the received signal samples. The sampling rate satisfies the Nyquist sampling frequency and the second equality in (1.374) recognizes that the noise samples are independent. The following analysis is based on the work of the Woodard [24] and Skolnik [61] and uses the maximum‐likelihood estimate of (1.359) with the Cramér–Rao bound expressed by (1.365).

Using (1.374) with zero‐mean AWGN, the minimum Cramér–Rao bound on the variance of the estimate is expressed as

(1.375)

In arriving at the third equality in (1.375) the factor k is independent of the parameter a and, it is recognized that, where B = 1/T_e is the bandwidth corresponding to the estimation time. The integral is formed by letting Δt → 0 as the number of samples N → ∞ over the estimation interval T_e. Upon taking the logarithm of the product kexp(−) and performing the partial derivatives on the integrand, (1.375) simplifies to

(1.376)

The last equality in (1.376) is the basis for determining the variance and is obtained by moving the expectation inside of the integral and recognizing that . The following example outlines the procedures for estimating the variance of the estimate using the ML procedures.

1.9.2.1 Example of ML Estimate Variance Evaluation

Consider the signal s(t) expressed as

(1.377)

where A is the peak carrier voltage, ω_o is the IF angular frequency, is the angular frequency rate, and ϕ is a constant phase angle; the signal power is defined P_s = A²/2.

The variance of the frequency estimate is determined by squaring the partial derivative of s(t) respect to ω_o and integrating over the estimation interval T_e as indicated in (1.376). Under these conditions the analysis of the Cramér–Rao lower bound is performed as follows.

(1.378)

Upon neglecting the term involving 2ω_o and performing the integration, (1.378) becomes

(1.379)

where γ_e = P_sT_e/N_o is the signal‐to‐noise ratio in the estimation bandwidth of 1/T_e. In terms of the carrier frequency f_o in hertz, the standard deviation of the estimate is

(1.380)

In a similar manner, the standard deviation of the frequency rate and phase are evaluated as

(1.381)

and

(1.382)

The evaluation of the standard deviation of the signal amplitude (A) estimation error is left as an exercise (see Problem 39).

1.9.3 Received Signal Delay and Frequency Estimation Errors

Accurate estimation of the signal delay and frequency are essential in aiding the signal acquisition processing by minimizing the overall time and frequency search ranges. The delay estimation accuracy (σ_Td) is inversely related to the signal bandwidth (B) and the signal frequency estimation accuracy (σ_fd) is inversely related to the time duration (T) of the signal. Neglecting the signal‐to‐noise dependence of each measurement, these inverse dependencies are evident, in that, the product σ_Tdσ_fd ∝ 1/TB where TB is the time‐bandwidth product of the waveform. For typical modulated waveforms T and B are inversely related so that simultaneous accurate time and frequency estimates are not attainable. However, the use of spread‐spectrum (SS) waveform modulation provides for arbitrarily large BT products with simultaneous accurate estimates of T_d and f_d. The analysis of delay and frequency estimation errors in the following sections is based on the work of Skolnik [61] and can be applied to conventional or SS‐modulated waveforms. In Section 1.9.3.3 delay and frequency estimation is examined using a DSSS‐modulated waveform.

1.9.3.1 Delay Estimation Error Based on Effective Bandwidth

The signal delay measurement accuracy using the effective signal bandwidth was introduced by Gabor [69] and is discussed by Woodward [24] and defined as the standard deviation of the delay measurement expressed as³⁴

(1.383)

where γ_e = P_s/N_oW = E/N_o is the signal‐to‐noise ratio³⁵ measured in the two‐sided bandwidth W, N_o is the one‐sided noise density, P_s is the signal power, and β is the effective bandwidth of the signal. β² is the normalized second moment of the waveform spectrum |S(f)|², defined as

(1.384)

The denominator in (1.384) is the signal energy and the integration limits extend over the frequency range f ≤ |W/2| corresponding to the nonzero signal spectrum. The one‐way range error corresponding to (1.383) is meters where c is the free‐space velocity of light in meter/second.

For the rectangular symbol modulation function rect(t/T), band limited to W Hz with β² ≅ W/T and large time‐bandwidth products WT/2, (1.383) is evaluated as

(1.385)

1.9.3.2 Frequency Estimation Error Based on Effective Signal Duration

In a manner similar to the analysis of the delay estimation error in the preceding section, Manasse [70] shows that the, minimum root‐mean‐square (rms) error in the frequency estimate is given by³⁶

(1.386)

where γ_e = E/N_o is the signal‐to‐noise ratio measured in the two‐sided bandwidth W, N_o is the one‐sided noise density, P_s is the signal power, and α is the effective time duration of the received signal. The parameter α² is the normalized second moment of the waveform s(t) and is defined as

(1.387)

The Doppler frequency results from the velocity (v) and the carrier frequency (f_c) and is expressed as f_d = (v/c)f_c. Frequency errors resulting from hardware oscillators are usually treated separately and combined as the root‐sum‐square (RSS) of the respective standard deviations.

For the band‐limited rect(t/T) symbol modulation used in the preceding section, the normalized second moment is evaluated as α² ≅ (πT)²/3 and (1.386) is expressed as

(1.388)

Comparison of (1.385) and (1.388) demonstrates the inverse relationship between the estimation accuracy of the range‐delay and frequency errors for conventional (unspread) modulations. For example, for a given time bandwidth (WT) product and signal‐to‐noise ratio (γ_e), the delay estimate error decreases with decreasing symbol duration; however, the frequency estimate error increases. The issue resolves about the signal‐to‐noise ratio in the estimation bandwidth. For example, with conventional waveform modulations, WT = 2BT = 2 so BT = 1 and the bandwidth changes inversely with the symbol duration. Consequently, by decreasing symbol duration, the bandwidth increases resulting in a signal‐to‐noise γ_s, measured in the symbol bandwidth, degradation of B/B′ where . Therefore, in the previous example, to maintain a constant signal‐to‐noise ratio γ_e the estimation interval must be appropriately adjusted. As mentioned previously, the solution to simultaneously obtaining accurate estimates of range delay and frequency while maintaining a constant γ_s, involves the use of a SS‐modulated signals with an inherently large WT product as discussed in the following section.

1.9.3.3 Improved Frequency and Time Estimation Errors Using the DSSS Waveform

The DSSS waveform uses a pseudo‐noise (PN) sequence of chips with and instantaneous bandwidth (W) over the estimation interval (T) as shown in Figure 1.29.³⁷ The resulting large WT product signal provides for arbitrarily low time and frequency estimation errors. This is accomplished by the respective selection of a high bandwidth (short duration) chip interval (τ) and the low bandwidth (long duration) estimation interval T. The estimation interval can be increased to improve the frequency estimate; conversely, the chip interval can be decreased to improve the range‐delay estimate; however, to maintain the accuracy of the other, the number of chips per estimation interval (N) must be increased. These relationships are described in terms of the pulse compression ratio, defined as ρ = T/τ = N. In Figure 1.29 the chips are depicted as appropriately delayed Ad_nrect((t−n)/τ − 0.5): n = 0, …, N − 1 functions and, because of the equivalence between the correlator and matched filter, the peak correlator output is a triangular function with a peak value³⁸ of AN. When sampled at t = Nτ, the correlator output results in the maximum signal‐to‐noise ratio measured in the bandwidth of 1/T Hz.

Based on the fundamental principles for jointly achieving accurate time and frequency estimates as stated earlier, the triangular shape of the wide bandwidth correlator output is related to the accuracy of the time estimate and the low bandwidth sampled output at interval of T = Nτ determines the accuracy of the frequency estimate. Therefore, evaluation of the time and frequency estimation accuracies of the DSSS waveform involves evaluating, respectively, the effective bandwidth (β) of the triangular function and the effective time duration (α) of the rect(t/T − 0.5) function.

Delay Estimation Error of the DSSS Waveform

The delay estimation error is based on detecting the changes in the leading and trailing edge of wideband signals. This does not require that the signal has a short duration but that the bandwidth is sufficiently wide to preserve the rapid rise and fall times of the correlator response. On the other hand, received signals with additive noise must be detected and the parameters estimated under the optimum signal‐to‐noise conditions as provided by matched filtering or correlation. In this regard, the correlator output in Figure 1.29 is examined in the context of the signal delay estimate error.

The triangular function, corresponding to the correlator output, is an isosceles triangle with base and height equal to 2τ and AN, respectively, and is described as

(1.389)

where, for convenience, ξ = t − Nτ such that the time axis is shifted so that the isosceles triangle is symmetrical about ξ = 0. The effective bandwidth of R_s(ξ) is evaluated (see Problem 41) as

(1.390)

and the corresponding standard deviation of the delay estimate is

(1.391)

Frequency Estimation Error of the DSSS Waveform

The frequency estimation error is based on the interval T of the PN sequence under the conditions corresponding to the local PN reference being exactly synchronized with and multiplied by the received signal; in other words, with zero frequency and phase errors, the integrand of the correlation integral is constant over the interval T. However, with a frequency error of f_ε Hz the correlator response is computed as

(1.392)

The principal frequency error in the main lobe of the sinc(f_εT) function corresponds to| f_ε| ≤ 1/T which defines the fundamental resolution accuracy of the frequency estimate. However, the effective duration of the correlator of length T = Nτ is evaluated (see Problem 42) as

(1.393)

and the corresponding standard deviation of the frequency estimate is

(1.394)

Considering the SS pulse compression ratio, or processing gain, ρ = T/τ, the correlator output signal‐to‐noise ratio (γ_e) in (1.391) and (1.394) is measured in the bandwidth of 1/T. The product of the estimation accuracies of the SS waveform is

(1.395)

Therefore, the time and frequency estimates accuracies can be made arbitrarily low, even in low signal‐to‐noise ratio environments, by designing a SS waveform with a sufficiently high pulse compression ratio.

1.9.3.4 Effective Bandwidth of SRRC and SRC Waveforms

In view of the increasing demands on bandwidth, the spectral containment of the spectral raised‐cosine (SRC) waveform meets the corresponding need for spectrum conservation. Although the spectral root‐raised‐cosine (SRRC) waveform has a slightly wider bandwidth than the SRC waveform, it is preferred because of the improved matched filter detection³⁹ and, in the context of range delay estimation, provides for a somewhat better range delay estimate. The spectrum of the SRC waveform is characterized, in the context of a spectral windowing function, in Section 1.11.4.1 and the spectrum of the SRRC is characterized in Section 4.3.2 in the context of the optimum transmitted waveform for root‐raised‐cosine (RRC) waveform modulation. The following analysis compares the effective bandwidth of the SRC and SRRC waveforms with the understanding that the SRC delay estimate is based on the matched filter out samples taken symmetrically about the optimum matched filter sample at t = T_o.

The dependence of the effective bandwidth of the SRRC and SRC waveforms on the excess bandwidth parameter α is expressed as

(1.396)

and

(1.397)

Equations (1.396) and (1.397) are plotted in Figure 1.30 that demonstrates the advantages of the wider bandwidth SRRC waveform in providing short rise‐times symbols with the associated improvement in range‐delay detection. In this regard, the rect(t/T) modulated received symbol, as characterized by the BPSK‐modulated waveform, has zero rise‐time and results in perfect range‐delay detection in a noise‐free channel and receiver; however, infinite bandwidth is required to achieve this performance. The dashed curve in Figure 1.30 shows the normalized effective bandwidth of the rect(t/T) modulated symbol after passing through an ideal (1/B)rect(f/B) filter with one‐sided bandwidth B/2 Hz; this is referred to a filtered BPSK and is discussed in the following section.

The noise bandwidth of the SRRC frequency function is significant, in that, it corresponds to the demodulator‐matched filter response used in the detection of the SRRC‐modulated waveform. On the other hand, the interest in the noise bandwidth of the SRC is more academic in nature because of its application as a windowing function. In either event, the noise bandwidth of the SRRC and SRC frequency functions is examined in Problem 44.

1.9.3.5 Effective Bandwidth of the Ideally Filtered rect(t/T) Waveform

In this case the effective bandwidth of the ideal symbol modulation, characterized by rect(t/T), is evaluated after passing through an ideal filter with frequency response (1/B)rect(f/B) where B/2 is the one‐sided or low‐pass bandwidth of the filter. The filter response is examined in Section 1.3 and the normalized effective bandwidth of the filtered symbol is characterized by Skolnik [71] as

(1.398)

This result is plotted in Figure 1.31 as a function of BT where T is the symbol duration. From this plot it is evident that as BT approaches infinity the standard deviation of the range‐delay estimate approaches zero resulting in an exact estimate of the true range delay. A practical application is to define a finite bandwidth which is sufficiently wide so as not to degrade the symbol detection through intersymbol interference.

Defining the excess bandwidth factor for the ideal filter as , where R_s = 1/T is the rect(t/T) symbol rate, in terms of the excess bandwidth factor α of the raised‐cosine (RC) waveform, . The corresponding range of BT is 1/2 ≤ BT ≤ 1; this range of the filtered rect(t/T) effective bandwidth from Figure 1.31 is plotted as the dashed curve in Figure 1.30. The range of BT results in significant intersymbol interference and received symbol energy loss even under the ideal conditions of symbol time and frequency correction. However, under the same conditions, if the SRRC waveform and matched filter responses are sufficiently long, the intersymbol interference and symbol energy loss will be negligible. At the maximum SRRC normalized effective bandwidth of βT = 2.27, the filter time bandwidth product corresponds to BT = 2.64 or a 32% increase with the filter bandwidth spanning the main signal spectral lobe and 16% of the adjacent sidelobes. In other words, the one‐sided filter bandwidth spans 1.32 lobes and, referring to Appendix A, this results in a performance loss of about 1.25 dB for BPSK waveform modulation; for a loss of less than 0.3 dB the BT product should be greater than 5 with a resulting effective bandwidth of βT = 3.23 corresponding to a (3.23/2.27 − 1)100 = 42% improvement relative to the best SRRC range‐delay estimation error; however, the required bandwidth is 150% wider. The bandwidth and range‐delay estimation accuracy are design trade‐off in the waveform selection.