11.2.1.1 Case Study: Bandpass Sampled AGC Performance Evaluation

This case study examines the performance of a UHF modem AGC with the gain control derived from the demodulator input IF of 455 kHz sampled at a rate of f_s = 6144 kHz. The noise bandwidth of the antialiasing filter is 80 kHz. The maximum receiver gain is 135 dB with a minimum detectable input signal level of −135 dBm. Referring to Figure 11.4, a 10‐bit DAC is used and the AGC reference input is V_ref = V_rms where V_rms is the root‐mean‐square voltage of the sampled carrier. The reference voltage is 12 dB below V_m leaving two magnitude bits for additive noise and peak signal fluctuations above V_ref. The following simulated performance of the AGC is based on a noise‐free CW received signal with the demodulator input signal at 455 kHz under two conditions of the receiver input level: −5 and −60 dBm. In both cases, the receiver gain is set to the maximum gain of 135 dB and, prior to the received signal, the receiver input is zero, that is, receiver noise is not included.

The low‐pass AGC filter is a cascade of four synchronously tuned single‐pole filters with an overall bandwidth of 200 Hz. In the following description, the LPF output is denoted as s_o. The operation of the AGC control is similar to all adaptive feedback control systems, in that, the error signal s_o forms a discriminator S‐curve providing positive and negative gain adjustments resulting in zero average filter output, <s_o> = 0, under steady‐state conditions. Under the steady‐state conditions ideal integrator output corresponds to the optimum receiver gain setting of V_rms = V_ref.

To provide more control over the AGC performance, than by simply letting the ideal integrator output control the receiver gain, the simulated gain control function provides discrete gain adjustment based on the filter output s_o and various thresholds as outlined in Figure 11.7. To this end, the ideal integrator output is replaced by the gain control logic using three fixed gain increments Δ_i that are applied in succession as the filter output falls to zero. The last two gain increments are weighted by the filter output and are proportionally decreased as s_o approaches zero. The thresholds T₀ and N₀ establish the conditions for the declaration of initial AGC acquisition and T₁ and N₁ establish the conditions for declaring the loss of AGC acquisition. The parameter I₀ is the number samples corresponding to one‐third of the LPF time constant and invokes the final gain control increment Δ₃ and the declaration of AGC acquisition. Taken together, these parameters establish the AGC attack and decay time. The selection of the AGC control parameters offers considerable design flexibility in the AGC performance and the logic is easily expanded to include additional capabilities. For example, using logic to track signal fade rates will allow for longer and deeper signal fading conditions or temporary loss of signal power before declaring lost AGC. The gain control logic can also be used to suspend demodulator symbol time tracking during fading and loss‐of‐signal conditions. Although not shown in Figure 11.7, the control logic also distributes the gain among the various gain‐controlled amplifies so as to preserve the receiver noise figure.⁷

The simulated performance of the AGC with a noise‐free CW received signal is shown in Figure 11.8 for received signal power levels of −5 and −60 dBm. The curves represent the receiver gain and the declaration of AGC acquisition or detection via the parameter L = 2. The first 10 ms of the 50 ms AGC simulated response is shown. The AGC reference voltage is set at V_ref = V_rms where V_rms is the voltage of the ADC input signal and is 12 dB below saturation of the 10‐bit DAC. The receiver gain is initially set to the maximum gain of 135 dB and, with the noise‐free assumption, the receiver input is zero, so under these conditions, application of the CW input signal at t = 0 results in an acquisition time 1.9 ms for both the −5 and −60 dBm received signal levels. The acquisition time is essentially determined by the bandwidth of the LPF.

Figure 11.9 shows the AGC response over the full 50 ms of the simulation using a rate 1/2 convolutional coded 19.2 kbps QPSK‐modulated waveform. The simulated signal‐to‐noise performance of this coded waveform, using a constraint length seven Viterbi decoder with infinite quantization, corresponds to an E_b/N_o of 4.25 dB at P_be = 10⁻⁵. Therefore, because the ADC quantization noise is negligible compared to the receiver noise, the signal‐to‐noise ratio in the 80 kHz noise bandwidth of the antialiasing filter is −1.95 dB. Under these conditions the AGC acquisition time is 2.03 ms.⁸ The samples s_o in Figure 11.9 are obtained from the AGC LPF and occur at a rate of 2 kHz.

11.2.2.1 Frequency Estimation Using the FFT

In this section, the determination of the received carrier frequency error using the CW preamble segment is accomplished by performing an N_fft‐point FFT as depicted in Figure 11.10. To account for the uncertainty of not knowing where the preamble starts, FFTs must be performed sequentially until detection is declared. Declaration of signal detection is based on a FFT frequency cell exceeding the CFAR threshold. To maximize the use of the CW preamble duration, overlapping FFTs are used and, to increase the correct frequency detection probability, two successive FFT detections are required with the second occurring within ±1 frequency cell of that declared during the previous FFT. Requiring two successive detections under these conditions also reduces the false‐detection probability. A CW segment duration of T_cw = 2T_fft will guarantee that two FFTs with one overlapping FFT can be performed; T_cw = 2.5T_fft guarantees three FFTs with one overlapping FFT can be performed; T_cw = 3T_fft guarantees four FFTs with two overlapping FFTs can be performed. In general, for 50% FFT window overlap with T_cw = ρT_fft where ρ = 1, 1.5, 2, 2.5, 3, 3.5, … guarantees that N FFTs can be performed with and . The fundamental frequency resolution of the FFT is defined as the reciprocal of the FFT window, that is,

(11.9)

For a given CW segment duration, increasing the number of FFTs requires a smaller FFT window resulting in less frequency resolution. Various trade‐offs between the CW segment duration, the detection performance, the frequency estimation accuracy, and the FFT parameters are discussed in the remainder of this section.

The received CW signal spectrum shown in Figure 11.11 is similar to that shown in Figure 11.3 with the modulated signal spectrum replaced by the discrete spectral line S(f) = δ(f − f_ε); in the figure the frequency error is shown as the maximum specified error⁹ f_εmax. In this case, the transition or guard frequency (Δf) is related exclusively to the transition band of the antialiasing filter, that is, there are no spectral sidelobes to contend with as shown in Figure 11.3.

The key performance parameter for the CW signal detection is the carrier power to noise power spectral density ratio (C/N_o) that is related to E_b/N_o as

(11.10)

For comparable performance, independent of the bit rate, the duration of the CW segment is often specified in terms of the number of baseband data bits (N_B) as

(11.11)

Referring to Figure 11.11 the sampling frequency is expressed as

(11.12)

The number of waveform samples in the CW segment is determined as N_s = T_cw/T_s = T_cwf_s and, upon expressing the bit rate in terms of the symbol rate as R_b = kr_cR_s and using these results and (11.11) with , the number of CW samples is evaluated as

(11.13)

Using (11.13) the size of the FFT is determined as

(11.14)

where ρ is the number of FFTs to be performed over the CW interval¹⁰ T_cw. Equation (11.14) generally requires using a mixed radix FFT; however, a radix‐2 FFT can be used by choosing N_fft = 2ⁿ: n ∈ positive integer such that . Unfortunately, the radix‐2 FFT often results in and corresponds to the inefficient use of the CW interval and less frequency resolution.¹¹ The CW window utilization efficiency is defined as

(11.15)

where T_fft = N_fft/f_s is the FFT window duration.

The sampling frequency can be increased to improve the utilization efficiency of the CW interval, thereby improving the frequency resolution, using a radix‐2 FFT. This is accomplished by choosing a larger number of samples per CW interval. For example, by choosing n′ such that , the number of samples per CW interval becomes and, using (11.13), the adjusted normalized sampling frequency becomes

(11.16)

The FFT provides for a frequency estimation accuracy of |f_acc| = f_res/2; however, this estimation accuracy can be improved by using interpolation between the FFT frequency cells that are separated by f_res. Improving the FFT resolution using zero padding is discussed in Section 1.2.7 and 2 : 1 zero padding results in a frequency estimation accuracy of . The frequency resolution and accuracy are depicted in Figure 11.12 for the rectangular weighted FFT with and without zero padding.

An important consideration in the frequency estimation processing is establishing a signal detection algorithm for declaring signal present and estimating the signal frequency error. For example, for an N_fft‐point FFT there are N_fft possible frequency locations, however, when the signal is present only one location, or possibly two contiguous locations, correspond to the frequency of the received signal. The signal detection algorithm must then provide for declaring that a signal is present for subsequent acquisition processing, when the signal is not present it must provide for continued searching within the CW signal segment. The importance of the signal detection algorithm cannot be overstated: it must provide for a low probability of false detection and a high probability of correct signal detection with an acceptable coarse frequency estimation.

The signal detection algorithm used in conjunction with the coarse frequency estimation processing is the CFAR algorithm that uses a detection threshold based on the magnitude of the signal plus noise in the frequency cells around a selected frequency cell, referred to as the cell under test. The cell under test is defined as the FFT frequency cell currently being examined under the hypothesis that it corresponds to the correct received signal frequency. The threshold is based on a two‐parameter censored CFAR with the two parameters computed as the mean and standard deviation of the cells excluding the cell under test and N_censor cells on either side of the cell under test. With this definition N_censor represents the number of one‐sided censored cells and the threshold is computed as

(11.17)

where κ is the threshold factor selected to meet the specified system detection and false‐alarm probabilities. The mean and standard deviation are based on a finite sample population size as outlined in Section 1.13.3. Denoting the complex spectrum sample in each of FFT frequency cell as c_n: n = 1, …, N_fft the mean and standard deviation are computed in consideration of the censoring, as

(11.18)

and

(11.19)

The primed summations signify that the summation excludes the cell under test and the 2N_censor censored cells such that . The censoring reduces the influence of the cell under test and the adjacent cells on the censored mean and standard deviation. The influence of the censored cells is related to the spectral sidelobes of the signal. For example, the censoring for a rectangular windowed FFT without interpolation is typically N_censor = 2 and with a nonuniform weighted FFT window and/or interpolation, censoring values of 4–6 are often used.

The frequency cell identified by the CFAR processing is used to compute the frequency estimate using early–late (E/L) gate interpolation. The frequency estimates from multiple FFTs separated by known intervals of ΔT seconds are used to estimate the received signal Doppler or frequency rate as

(11.20)

where f_est(1) and f_est(2) are successive FFT frequency estimates. These design concepts involving signal present detection and coarse frequency estimation are discussed in more detail using the example in the following case study.

11.2.2.2 Case Study: FFT Signal Detection and Frequency Estimation

In this case study, signal present detection and carrier frequency error estimation are examined using an example involving a 19.2 kHz BPSK‐modulated waveform without FEC coding. The maximum specified frequency error is f_εmax = 10 kHz and the FFT processing of the CW preamble segment uses ρ = 2.5 that guarantees that three FFT can be performed with one overlapping FFT. The signal detection is based on two (N_det = 2) consecutive FFT detections. The first FFT detection is declared if the maximum cell magnitude exceeds the CFAR threshold. The second FFT detection is declared if the threshold is exceeded in the same or an adjacent cell to that of the first detection; this corresponds to a frequency estimate within f_acc of that estimated in the first detection. The CFAR threshold is based on a two‐parameter CFAR with N_censor = 2 cell censoring. The frequency estimation uses parabolic E/L interpolation. In this evaluation, the signal detection and frequency estimation are examined for two FFT windows and interpolation conditions: the rectangular window with zero padding and the Hanning window with and without zero padding.

The sampling frequency is evaluated using (11.12) with a guard frequency Δf = 2R_s = 38.4 kHz yielding f_s = 96.8 kHz. For this uncoded BPSK example k = r_c = 1, R_s = R_b and, using N_B = 88 information bits per CW segment, the number of samples is found from (11.13) to be N_s = 443 and the FFT size is computed using (11.14) with the result N_fft = 177. Therefore, a 177‐point FFT can be used for the signal detection and frequency estimation; however, because 177 is only divisible by 3 and 59 a mixed radix FFT must be used. The more computationally efficient radix‐2 FFT with N_fft less than 177 uses N_fft = 128; however, from (11.15) the CW window utilization efficiency is only 72% and the frequency resolution is f_res = 756.25 Hz.

Because of the poor utilization efficiency and resolution frequency, the FFT size is increased to  = 256. With this modification, the total number of FFT samples becomes  = 640 and the adjusted sampling frequency is computed using (11.16) and found to be ; this is rounded up to yield  kHz. In this case, the FFT efficiency is 99.7% with f_res = 546.875 Hz. The greatest common divisor of the sampling frequency and symbol rate is gcd(140K, 19.2K) = 800 so these rates are derived from a high‐frequency clock of f_clk = 19.2K(140,000/800) = 140K(19,200/800) = 3.36 MHz. Table 11.2 summarizes the parameters used in this case study for the CW segment acquisition processing.

Parameter	Value	Comments
Specified parameters
Data modulation	—	BPSK
Bit rate	19.2	Rb (kbps)
Symbol rate	19.2	Rs (ksps)
Bits per symbol	1	k
FEC code rate	1	rc
Maximum frequency error	10	fεmax (kHz)
Guard frequency	2Rs	Δf (kHz)
Bits per CW segment	88	NB
FFTs per CW segment	2.5	ρ
Consecutive detections	2	Ndet
Cell censoring	2	Ncensor
E/L interpolation	—	Parabolic
FFT window	—	Rectangular and Hanning
Computed parameters
Sample rate	140	(kHz)
System clock	3.36	fclk (MHz)
Samples per CW segment	640
FFT sizea	256	Nfft without padding
Frequency resolution	546.875	fres (Hz)
Frequency accuracy	273.438	facc (Hz), without FFT padding
	136.719	facc (Hz), with FFT padding

^aFFT size is without zero padding; with 2 : 1 zero padding FFT size is doubled.

The CW preamble segment signal detection and frequency estimation performance, operating under the conditions listed in Table 11.2, are evaluated using computer simulations and the results are shown in the following figures. Figures 11.13 and 11.14 show the signal detection performance in terms of the detection and false‐alarm probabilities as a function of the CFAR threshold factor k using rectangular and Hanning FFT windows, respectively, with zero padding and equivalent E_b/N_o signal‐to‐noise ratios of 0 and −3 dB. Referring to Figure 11.12b, the peaks or maximum magnitudes of the FFT outputs occur at 273.4375i Hz: i = 0, …, N_fft − 1 and these conditions correspond to the best case performance, whereas the minimum magnitudes correspond to 273.4375i + 136.71875 Hz and represent the worst case performance. The false‐alarm probability is conditioned on two consecutive FFT detections and is obtained with a noise‐only input. All of the simulation results are based on Monte Carlo simulations of 5000 trials for each threshold so the false‐alarm results are projected below about P_fa = 10⁻⁴.

The simulated frequency estimation performance is characterized in terms of a histogram representing the cumulated distribution function (cdf) shown in the following figures. The normalized frequency error is expressed as where f_est is the estimate of the received signal frequency error based on the parabolic E/L estimation algorithm. The histogram consists of 400 bins over the positive frequency range of 0 to f_res corresponding to a bin resolution of δf = 1.3671875 Hz. The abscissa of the cdf is the normalized frequency ratio f_norm = i|δf|/f_res: i = 1, …, 400 with f_norm limited to f_norm(max) = 1/2 in the cdf plots.¹² The cdf is shown for the best and worst cases as defined earlier and the random frequency case where the frequency error is uniformly distributed between ±f_acc.

Figure 11.15 shows the three performance conditions for signal‐to‐noise ratios equivalent to E_b/N_o = 0, 3, and 6 dB using a rectangular window with zero padding corresponding to 2 : 1 FFT interpolation. As an example application, the arrows in Figure 11.15 correspond to the worst case performance at a signal‐to‐noise ratio of 3 dB and, under this condition, the frequency estimation error is δf ≤ 0.19f_res = 103.9 Hz with a probability of 0.99. Referring to (11.3) and (11.4) and using B_LT = 0.1 for BPSK modulation, the lock‐in frequency and time for a second‐order PLL are 806 Hz and 0.28 ms, respectively, and f_est = 103.9 Hz is well within the lock‐in frequency range of the PLL.

These conditions are repeated in Figures 11.16 and 11.17 using a Hanning window with and without zero padding, respectively. The performance of the Hanning window with zero padding is considerably degraded from that of the rectangular window performance shown in Figure 11.15. The performance difference is attributed to the lower discriminator gain that is a consequence of the inherent wider spectral bandwidth of the Hanning window.

11.2.2.3 Frequency Estimation Using the Pipeline FFT

The pipeline FFT described in Section 1.2.5.1, although more signal processing intense than the block FFT described earlier, provides an efficient method of simultaneous signal detection and frequency estimation that allows for a shorter CW preamble segment. The benefits are a consequence of the sequential processing that results in a continuous push‐broom acquisition over the range of the frequency uncertainty. An example of the pipeline FFT is shown in Figure 11.18.

This pipeline FFT example uses a 32‐point FFT with 2 : 1 zero padding that corresponds to N_fft = 32 and an interpolation factor¹³ of N_I = 2. All of the operating parameters for the acquisition are based on the maximum frequency uncertainty of the received signal f_εmax, the frequency guard‐band Δf, and the sampling frequency f_s as described by (11.7). In this case, the determination of the sampling frequency is somewhat simpler than that discussed in Section 11.2.2.1 because overlapping block FFTs are not involved, also, because of the CW signal, the guard band depends solely on the transition band of the antialiasing filter. Therefore, upon determining the sampling frequency, the estimation interval is determined using the sampling interval δt = 1/f_s as¹⁴

(11.21)

and the frequency resolution is f_res = 1/T_e = f_s/N_fft. With zero padding the accuracy of the frequency measurement is f_acc = f_res/N_I. Using these relationships, the frequency axis in Figure 11.18 spans the frequency range 32 f_acc and the time axis spans the range 32δt or 2T_e.

In the noise‐free simulation of Figure 11.18, the CW frequency tone is placed in the center of the frequency cell f/f_acc = C₀ = 7 (cell 0 corresponds to the zero frequency) and Figure 11.19 shows magnitude response of cell C₀ and several neighboring cells. The time is normalized to the estimation interval T_e and the optimum cell output increases linearly, reaching the optimum value at t = T_e. The cells C₀ ± 1 are used for E/L gate frequency tracking and are included in the CFAR censoring during acquisition. When searching for an acquisition detection and performing detection verifications the CFAR detection algorithm, discussed in the preceding section, is executed at regular intervals of, for example, T_e/2.

The resolution bandwidth, f_res, of a uniformly weighted FFT with an underlying size N_fft/N_I ≥ 8 is essentially equal to the noise bandwidth, so increasing the underlying FFT size reduces the noise power in each FFT cell; however, the scalloping, leakage, and aliasing losses must also be dealt with in the acquisition processing. FFT interpolation reduces the scalloping loss and the worst case signal‐to‐noise loss due to interpolation.

11.2.2.4 Frequency Estimation Using Discriminator

In this section, the FD is examined that provides an estimate of a received signal carrier frequency error during acquisition. The basic implementation of the FD is then extended to further resolve the frequency estimate by using smaller delays that are related by powers of two and, in this regard, this implementation is similar to the butterfly element of the FFT. A fine frequency estimator is then described that essentially zooms in on the initial coarse estimate to provide considerably higher resolution.

To clarify the description of the FD, a noise‐free received CW signal is expressed as

(11.22)

where ω_o represents the transmitted carrier angular frequency, ω_ε is an unknown angular frequency error involving the Doppler frequency and various oscillator frequency errors, ϕ_o is an arbitrary phase angle, and A is the peak carrier voltage. The complex envelope of s_i(t) is given by

(11.23)

and represents the respective baseband in‐phase and quadrature terms are s_ci(t) and s_si(t). In the simplified noise‐free environment, the FD output is computed as the autocorrelation of using a fixed lag‐delay τ and is expressed as

(11.24)

The complex implementation of the correlator is shown in Figure 11.20a, and Figure 11.20b shows an equivalent implementation involving real functions identified as the in‐phase and quadrature components of . The LPFs provide time averaging over the estimation interval T_e as indicated in (11.24). The integration or filtering is fundamental to the correlation processing and is intended to reduce the influence of the additive noise associated with the received signal; the output noise is complicated by the multiplication resulting in products involving S × N and N × N. The correlation lag delay τ is selected to provide the greatest unambiguous range in the frequency estimation as described later.

Referring to (11.24), the angle between the in‐phase and quadrature terms of is given by¹⁵

(11.25)

To avoid ambiguities with ± frequency uncertainties it is necessary that the maximum unknown frequency correspond to a correlator output phase <π radians and by solving (11.25) for f_ε with ϕ = π the condition is

(11.26)

Allowing for some guard range in a noisy environment, it is prudent to require that

(11.27)

where |ϕ_max| < π radians. These relationships are depicted by the FD phase diagrams in Figure 11.21 where the frequency error is given by

(11.28)

Defining ϕ_max in terms of the guard range as a fraction, η of π radians, such that,

(11.29)

then (11.27) is expressed as

(11.30)

Equation (11.28) is the estimate of the frequency error based on the discriminator phase error. Although evaluation of the phase estimate using the inverse tangent function is computationally complex, there are two major advantages: the estimate is linear with frequency and independent of signal amplitude A.

The imaginary part of can also be used to form the discriminator response expressed as

(11.31)

Approximating (11.31) for small arguments and solving for the frequency error results in

(11.32)

This form of the discriminator is similar to that used in the Costas implementation of the PLL; however, unlike (11.25), the response given by (11.31) is not linear over the entire unambiguous frequency range as seen by the two responses shown in Figure 11.22. Furthermore, the unambiguous range of (11.31) is limited to |fτ| < 0.25. The following case study uses the linear atan2 discriminator implementation and examines the frequency estimation error under several signal‐to‐noise conditions.

The FD can also be used to estimate various derivatives of the signal phase function by cascading additional fixed lag‐delay correlators. For example, the frequency estimate and frequency‐rate estimate are implemented as shown in Figure 11.23. In this case, the input phase function is expressed as

(11.33)

The details in demonstrating the frequency and frequency rate estimates in Figure 11.23 are left as an exercise (see Problem 9).

11.2.2.5 Case Study: Discriminator Frequency Estimation

The frequency estimation performance and the probability of correctly declaring the frequency using the discriminator shown in Figure 11.20 are examined in this case study. The evaluation is based on the normalized form of the key equation (11.26) obtained by dividing by the Nyquist band frequency yielding

(11.34)

Two practical observations are made concerning (11.34). First, to provide a frequency guard range against an unambiguous frequency estimate with additive noise, the normalized form of (11.30) is

(11.35)

The second observation is based on the sampled data processing requiring that the discriminator delay be integrally related to the sampling frequency, that is, τ = n/f_s where n is an integer. Recognizing that f_s = 2f_N, the denominator of the right‐hand side of (11.34) and (11.35) is simply f_sτ and the maximum normalized frequency range satisfying the integer requirement occurs when f_sτ = 1, that is, when the discriminator delay is one sample, and (11.35) becomes

(11.36)

The phase and frequency estimates are computed at the output of the LPFs shown in Figure 11.20 and the accuracy of these estimates with signal and noise is a function of the signal‐to‐noise ratio and the estimation interval T_e = 1/f_B where f_B is the bandwidth of the LPFs as discussed in Section 1.9.2.1.

As an example of this analysis, consider a CW carrier sampled at f_s = 19.2 kHz with f_N = 9.6 kHz and a guard interval of η = 0.3 (30%). Based on a simulation of the signal processing in Figure 11.20, with a CW signal source and AWGN channel, the frequency estimation performance of the discriminator is shown in Figures 11.24 and 11.25 as a function for the signal‐to‐noise ratio measured in a sampling frequency bandwidth of 19.2 kHz. The ordinates are plotted in terms of the normalized frequency estimation performance and the example case using the 19.2 kHz sampling frequency. The performance in Figure 11.24 corresponds to a LPF¹⁶ bandwidth of f_B = 38 Hz with samples, whereas the performance in Figure 11.25 corresponds to f_B = 218 Hz and N = 88 samples. This case was selected to correspond to the 88 bit CW preamble length in the FFT case study of Section 11.2.2.2 with a bit rate of 19.2 kbps. In both cases the number of Monte Carlo acquisition trials at each signal‐to‐noise ratio is 10,000 and the maximum and minimum values correspond to the extremes recorded among the trials for each signal‐to‐noise ratio.

These results correspond to the worst case frequency error because the input signal frequency error corresponds to f_εmax = 6720 Hz as expressed in the normalized form by (11.36). The impact of the LPF bandwidth is evident in these two figures, for example, the performance using the lower bandwidth results in significantly improved estimations and there is no evidence that an unambiguous phase estimate occurred. To the contrary, the degraded performance for the 218 Hz bandwidth case, shown in Figure 11.25, clearly shows the result of the phase measurement reaching through the guard range and causing ambiguous estimates for signal‐to‐noise ratios less than 0 dB. The occurrences of a frequency estimate exceeding the unambiguous frequency range of the discriminator are obvious in the simulation program with a fixed frequency error at, for example, a positive value of f_εmax. That is, referring to the phase diagram in Figure 11.21, when ϕ = ϕ_max a frequency estimate resulting in a phase estimate π + φ is computed by the atan2(y,x) function as the phase −π + φ radians that corresponds to a negative frequency most likely in the range −f_N to −f_εmax. Therefore, referring to Figure 11.25, the abrupt negative frequency jump at 0 dB results from the positive frequency exceeding f_N. For the 218 Hz LPF bandwidth case, the probability of a phase over‐flow, as described earlier, for signal‐to‐noise ratios of −4.0 and −0.5 dB is 1.49e⁻² and 1e⁻⁴, respectively.

Although, as described earlier, the ambiguous frequency estimate cannot be discerned in the CW segment, signal verification performed during the synchronization preamble segment will most likely fail because the PLL pull‐in frequency is exceeded. However, the initial CW segment ambiguous frequency estimate can also be verified by correcting the frequency and repeating the CW segment estimation processing using a smaller estimation range f_ε by increasing the value of τ as in (11.27); this verification processing may have to be repeated to account for the sign ambiguity of the initial estimate. When τ is varied in multiples of two the processing is similar to that of a radix‐2 pipeline FFT thus improving the frequency estimation accuracy.

The frequency detection or CW acquisition probability is evaluated by examining the standard deviation of the phase estimates ϕ_εn computed by the atan2(y,x) function over the estimation interval of N samples as defined earlier. This criterion is chosen because during the CW preamble, the phase standard deviation is zero without noise and because it increases inversely with the signal‐to‐noise ratio. The functional processing for the CW acquisition or signal present detection is shown in Figure 11.26; also shown is the frequency correction of the input signal that is used in the symbol synchronization preamble segment. The angular frequency estimate is where the index i is available from the system sample clock generator such that t = t_i = iΔt = i/f_s.

The frequency detection analysis evaluates the performance for the case of f_B = 218 Hz (N = 88 samples) corresponding to Figure 11.25. Because the processing of the received signal samples by the FD is sequential, in that, the sampled data is continuously passing through the detection algorithm, much like the pipeline FFT, a detection hypotheses is made at regularly spaced intervals using 75% or of the most recently collected samples; this reduces the possibility of initial transients influencing the detection. With this understanding, the mean and standard deviation are computed over the most recent N′ = 66 phase samples ϕ_εn as expressed by (11.18) and (11.19) by substituting and using the unprimed summations, that is, no censoring of the N′ samples is used. Letting and using the normalizing frequency f_norm = 200 Hz the threshold, T_hr, is selected to meet the detection and false‐alarm requirements defined as

(11.37)

and

(11.38)

These probabilities are evaluated using a histogram, with 200 bins spanning or 2 kHz, that is used as a probability distribution function. For each signal‐to‐noise ratio, the detection probability is based to 10,000 frequency acquisition trials and the false‐alarm probability is based on 100,000 acquisitions trials with noise only. The performance with and without noise uses an ideal AGC; however, the atan2(y,x) function also provides immunity to the level of the received signal. Based on this description, the detection and false‐alarm performance is shown in Figure 11.27 as the solid and dashed curves, respectively, as a function of the threshold.

The frequency detection results are summarized in Table 11.3 for the indicated detection probabilities and the corresponding threshold and false‐alarm probability. The acceptable operating signal‐to‐noise ratio depends on the application and system specifications and these results illustrate the relationship between the detection and false‐alarm probabilities. Typically the detection probability is specified and the false‐alarm probability is chosen as low as possible to meet other system requirements like the message throughput delay and processor loading. Applications involving automatic repeat request (ARQ) and nonreal time message processing may tolerate higher false‐alarm probabilities. The detection performance characterized for the 88‐bit CW segment, corresponding to f_B = 38 Hz, results in reasonable detection probabilities for signal‐to‐noise ratios greater than 0 or 3 dB. Recall that these results represent the worst case conditions corresponding to an initial frequency error of and the performance with uniformly distributed frequency errors will be somewhat better. Furthermore, by decreasing the LPF bandwidth the operating signal‐to‐noise ratios can be extended into the negative region.

The frequency estimation can be improved by passing the frequency corrected input signal through the FD multiple times, each time improving the frequency of the previous estimate. The estimation improvement is accomplished using the same estimation time T_e by decreasing the sampling frequency after the initial estimate has been removed, thereby, reducing the signal bandwidth uncertainty as shown in Figure 11.26. In other words, instead of passing the initial frequency‐corrected signal to the synchronization segment as shown, the corrected signal samples are passed through the frequency decimator a second time using , corresponding to and . This requires that the signal samples over the sliding window of T_e seconds be stored in memory and that the rate of the signal processor is commensurate with real‐time processing; the sliding window refers to the estimation interval in consideration of the sequential processing. This refinement of the frequency estimate can be repeated until the estimate falls within the PLL bandwidth as given by (11.3). For a given PLL B_LT product, the frequency estimation limit is also dependent on the symbol rate of the underlying received signal modulation and the critical signal‐to‐noise ratio as discussed in Section 10.6.11.

The frequency estimate resulting from a second pass through the discriminator is shown in Figure 11.28 corresponding to the first pass estimation results shown in Figure 11.25. The parameters for the pass correspond to the example un‐normalized conditions: f_εmax = 0.7f_N, f_B = 218 Hz, and N = 88 samples with f_s = 19.2 kHz, f_N = 9.6 kHz, and η = 0.3 (30%). Recalling that the first pass was evaluated for a constant, worse case, frequency offset of f_ε = f_εmax = 6,720 Hz with 10,000 Monte Carlo trials for each signal‐to‐noise ratio, so, for each signal‐to‐noise ratio there are 10,000 independent randomly distributed frequency estimates. Each of these frequency estimates is applied to the discriminator on the second pass using k = 2 with  = f_s/2, /2, and η′ = η resulting in the performance in Figure 11.28. The means and standard deviations corresponding, respectively, to the first and second passes are summarized in Table 11.4. These results demonstrate the improvement in the frequency estimate resulting from the second pass through the FD as discussed earlier.

11.3.2.1 Joint Frequency and Symbol Time Estimation Using Discriminator

The approach described in this section to estimate the frequency and symbol or bit timing is based on the FD discussed in Sections 11.2.2.4 and 11.2.2.5. The preamble synchronization segment in this analysis is composed of the 1100 repeated bit pattern and, because the coarse frequency estimate has been removed, the frequency error is a fraction of the bit rate. The detection processing is based on a parallel repetition of the complex discriminator function shown in Figure 11.20a and depicted in Figure 11.32 as a lag‐correlator²⁴ (LC).

The concept involves computing LC outputs for K − 1 hypotheses of the optimum bit timing of the sampled 1100 repeated bit pattern as shown in Figure 11.33. The N_s samples per bit, corresponding to each timing hypothesis, are then used to compute K − 1 lag‐correlator outputs as shown in Figure 11.34. The LC output, corresponding to the optimum bit timing delay relative to the receiver time base, is associated with the imaginary part having the highest mean value and lowest standard deviation or simply having the highest ratio , such that,

(11.39)

Referring to Figure 11.33, the bits are sampled at intervals of T_s with N_s samples per bit. Relative to the local demodulator time base, the bipolar data is denoted as d_n = d_n+1 = 1 and d_n+2 = d_n+3 = −1 corresponds to the repeated binary bit pattern b_n = (1100). The demodulator time base can be defined to start anywhere over the sampled data pattern without affecting the detection of the optimum timing delay. Processing the samples over the entire data pattern as shown results in a discriminator delay τ = 2T and, from (11.28), the corresponding LC frequency error is (1 − η)/4T Hz. In this case, the LC spans 4T seconds and the sign of the samples corresponding to the negative data must be changed. This is easily accomplished from the knowledge of the hypothesized timing. Defining the parameter N as the number of sample intervals between each bit time hypothesis, the resulting timing error is

(11.40)

and the number of hypotheses is given by

(11.41)

For example, with N_s = 8 and the requirement T_ε = ±T/8 results in N = 2 and K = 16 hypotheses. The timing offsets relative to the demodulator time reference are computed as kNT_s where the parameter k represents the k‐th timing hypothesis over the range 0 ≤ k ≤ K − 1.

Referring to Figure 11.34 the inputs into the k‐th LC, denoted as LC_k, are computed as

(11.42)

and

(11.43)

The minus sign in (11.43) results from the sign of the data bits d_n+2 and d_n+3. Using (11.42) and (11.43) in the context of Figure 11.32, the output of LC_k is computed as

(11.44)

where denotes the time average that is implemented as a LPF. Using the bit timing error estimate k′NT_s, the local demodulator time base is brought into alignment with the repeated data pattern. The fine‐frequency estimate is determined using the imaginary part of as described by (11.23) through (11.30). The fine‐frequency estimate is removed from the baseband signal prior to the carrier acquisition and tracking. Following these corrections the bit and frequency acquisition and tracking are executed as described in Section 11.3.5. If the preamble samples have been stored, the acquisition and tracking functions are applied to the time and frequency corrected samples resulting in the most efficient use to the preamble. Reusing the corrected preamble samples results in a shorter preamble and lower message overhead.

11.3.2.2 Signal Detection and Parameter Estimation Using Signal Spectrum

In this section the coarse‐frequency estimate of the CW preamble segment is refined in the frequency domain for subsequent acquisition and tracking by the PLL. However, as indicated in the introduction, the signal spectrum can also be examined over a wide frequency range with additional signal processing. The focus in this section corresponds to the symbol synchronization segment involving a known data pattern that is used for both fine‐frequency and symbol rate estimation. In this case, the data pattern is uniquely designed to enhance the spectral characteristics for determination of the frequency error and the modulation symbol rate. The spectrum also contains information regarding synchronization to the data pattern; however, the unknown signal phase is problematic in making an unambiguous data pattern synchronization decision. In view of these remarks, the entire preamble is often sampled and stored for processing in the frequency domain and then revisited and processed in the time domain for symbol time synchronization and then revisited a third time for frequency, phase, and symbol tracking. Although, this approach requires more memory and more intense processing to maintain real‐time throughput, the preambles are typically shorter resulting in lower message overhead. Furthermore, in many network applications, the frequency and timing information is stored during network entry allowing for shorter preambles in subsequent transmissions.

The signal parameters involving unique data patterns are examined in the frequency domain relative to the demodulator local time reference as shown in Figure 11.35. Denoting the received symbols in terms of the binary data suggests that binary PSK modulation is used; however, using the symbol phase notation ϕ_k,K = kπ/K to denote higher order modulations with a unique data sequence increases the notational complexity that tends to obscure the synchronization concepts. Therefore, the focus in this description is on BPSK modulation.

The analysis that follows uses the analytic baseband description of the received signal expressed as

(11.45)

where τ is the signal delay relative to the demodulator time reference, ω_ε is the frequency error in radians per second, and φ is a constant phase error. The notation rep_To(x) is Woodward’s repetition function [39] with period T_o = MT, so the data sequence shown in Figure 11.35 is repeated every M symbols as defined in this segment of the preamble. As described in (11.45), the signal phase function is outside of the rep_To(x) brackets, so the received signal description corresponds to the phase of the carrier frequency.

Equation (11.45) is a general description of the signal and applies to any repeated data pattern; however, to simplify the analysis, the commonly used preamble synchronization bit pattern of 1100… bits is examined for which M = 4, T_o = 4T, and d₀ = d₁ = 1, d₂ = d₃ = −1 where the unipolar bit to bipolar data translation is d_i = 2b_i − 1. The analysis of the signal spectrum uses the Fourier transform of periodic functions discussed in Section 1.2 for which the spectrum is evaluated as

(11.46)

where f_o = 1/T_o and the C_n are complex coefficients. In the context of (11.45) and the repeated data pattern 1100…, C_n and f(t) are computed as

(11.47)

and

(11.48)

Substituting (11.48) into (11.47) and performing the appropriate integrations, combining of terms, some simplifications, including f_oT = T/T_o = 1/4, and then substituting the expression for C_n into (11.46), the spectrum for the repeated data sequence is expressed as

(11.49)

where . Equation (11.49) is shown in Figure 11.36 for a zero frequency error, that is, f_εT = 0. The spectrum is simply shifted to the right or left depending on the frequency error. The discrete spectral lines at f ± 1/4T are of interest and have magnitudes of 2A/π corresponding to a loss of 3.9 dB; the nearest neighbors at f ± 3/4T have magnitudes of 2A/3π resulting in a loss of 13.46 or 9.56 dB below the spectral lines of interest.

The spectral phase function is defined as the phase of F(f) and is expressed as

(11.50)

Upon using λ_nT, as given earlier, and f_oT = 1/4, (11.50) is evaluated as

(11.51)

The spectral phase function has a considerable amount of structure, being dependent upon each of the three parameters: τ/T, f_εT, and φ. Noting that the bit rate is R_s = 1/T, the frequency tones of interest occur at f_u = f_ε + R_s/4 and f_l = f_ε − R_s/4 corresponding to n = +1 and −1, respectively. For example, these tones rotate linearly in opposite directions by (π/2)τ/T radians, so that over the interval of one data pattern (1100) the phase rotates by 2π radians. These delay‐dependent phase rotations occur independently of the frequency error; however, an additional frequency‐dependent phase of 4πf_εT radians is encountered. The unknown signal frequency error f_ε and phase error φ are problematic in determining the signal delay from the spectral phase information contained in (11.49). However, by determining f_u and f_l corresponding to the two largest spectral lines, the estimates of the frequency error and bit rate are determined as

(11.52)

and

(11.53)

In many cases, the symbol rate is known and is used to advantage in locating f_u and f_l when the received signal includes channel and receiver noise. Upon removing the frequency error from (11.51) the signal phase φ remains an impediment to unambiguously determining the signal delay τ/T from the spectral phase function. If, however, the signal phase were successfully removed, the delay estimate can be established to any degree of accuracy with increasing signal‐to‐noise ratio in the measurement bandwidth. For example, for a symbol timing error , the corresponding accuracy of the spectral phase function is radians. Considering an ℓ‐sigma measurement requirement, the required signal‐to‐noise ratio (γ_Bm) in the measurement bandwidth (B_m) is computed as²⁵

(11.54)

and using ℓ = 3 the required signal‐to‐noise ratio for the 3‐sigma delay estimate is 14.6 dB. In the following case study, the FFT is used to determine the spectrum of the repeated 1100 preamble sequence and the probability of correctly estimating the symbol rate²⁶ and frequency error are examined.

11.3.2.3 Case Study: Detection, Bit Rate, and Frequency Estimation Using BPSK‐Modulated Waveform

In this case study, the FFT is used to determine the spectrum of the repeated 1100 preamble data and the resulting probability of correctly estimating the frequency error is examined when the bit rate is unknown.²⁷ It is assumed that the coarse frequency error of the received signal has been resolved during the CW preamble segment to an accuracy of . The following description establishes the sampling frequency (f_s), the frequency resolution (Δf), the size (N_fft) of the FFT, and the samples per bit (N_s) for the fine‐frequency estimation. Given and referring to Figure 11.36, the Nyquist band is chosen as B_N = 2R_b so the sampling frequency must satisfy the Nyquist condition f_s > 2R_b; the value f_s = 8R_b is selected.²⁸ The frequency resolution is taken to be one‐fourth of the spectral tone at R_b/4, that is, Δf = R_b/16 from which the size of the FFT is determined as . In the time domain, these parameters result in an estimation interval of or 16 bits corresponding to four 1100 data pattern repetitions with N_s = 8 samples per bit. If a wider frequency range is desired while maintaining a fixed resolution bandwidth, the sampling rate and antialiasing filter bandwidth can be increased with a commensurate increase in the size of the FFT; in this case, the estimation interval remains the same. On the other hand, if a finer frequency resolution is necessary, the size of the FFT can simply be increased while maintaining the sampling frequency.

With a uniformly weight FFT, that is, without windowing, the selection of Δf corresponds to the measurement bandwidth, B_m = Δf, so that a specified received signal‐to‐noise ratio of γ_b = E_b/N_o corresponds to a signal‐to‐noise ratio in the measurement bandwidth of

(11.55)

This results in a signal‐to‐noise ratio improvement of 12 dB. However, as indicated in Figure 11.36, the signal‐to‐noise ratio in the FFT cells at f_ε ± R_b/4 is 3.9 dB lower than given by (11.55) resulting in an improvement of 8.1 dB at each tone frequency of interest. The selection of a smaller Δf will allow a more accurate estimation of the frequency error and bit rate as determined using (11.52) and (11.53).

The simulated spectrums under the foregoing conditions and a normalized frequency error of f_εT = 1 are shown in Figure 11.37 for received signal‐to‐noise ratios of E_b/N_o = 9.6 and 4.1 dB. These signal‐to‐noise ratios correspond, respectively, to uncoded and rate 1/2, K = 7 convolutional coded antipodal modulation performance at P_be = 10⁻⁵. The simulated signal‐to‐noise ratio of each spectral tone, corresponding to n = ±1, is evaluated by establishing the received signal‐to‐noise ratio, given the signal amplitude A, as expressed in (11.45) and then zeroing the signal into the demodulator and computing the total noise power as the sum of the noise variances on the quadrature rails of the complex spectrum. The resulting simulated signal‐to‐noise ratio, averaged over two independent FFT records, is 17.2 and 12.14 dB for the respective E_b/N_o values in Figure 11.37. These signal‐to‐noise ratios are simulated in the measurement bandwidth equal to Δf and compare favorably with the theoretical value computed using (11.55) when adjusted by the 8.1 dB loss at the spectral tones at n = ±1. This simple evaluation, or test, goes a long way in confirming the validity simulation.

The results of this case study are presented in normalized form, that is, the frequencies are normalized by the bit rate as f/R_b or fT. Because of the frequency normalization and the AWGN channel being considered, the following performance results can be applied to any user bit rate of interest; however, the bit rate must be considered when channels with memory are evaluated. The simulation includes a 145 tap transversal antialiasing filter with a normalized sampling frequency of f_sT = 8 and normalized 3 dB cutoff and transition frequencies of f_cT = 2 and f_TT = 1.5 with a band‐reject attenuation of 50 dB. Figure 11.38 shows the noise‐free spectrums and the antialiasing filter with and 1.5. The relative magnitudes of the spectral tones at odd integer, when compensated for the 3.9 dB normalizing level used in Figure 11.38, compare favorably with the theoretical values computed using (11.49).

The following simulated synchronization performance results are based on 500 Monte Carlo trials for each signal‐to‐noise ratio with the random number generators reinitialized for each signal to noise. For each trail the normalized frequency error is uniformly distributed over the range . Because the bit rate and frequency error are unknown, the two largest spectral magnitudes found in the bandwidth of the antialiasing filer are used in the estimation process. The frequency estimation for each trial is based on the normalized form of (11.52) where f_uT or f_lT are computed for x = {u,l} as

(11.56)

The FFT array has been left shifted by N_fft/2 so that the zero frequency location corresponds to ℓ = N_fft/2 + 1 and the locations ℓ_l and ℓ_u are determined from the locations of the two largest spectral magnitudes. Using (11.56) the normalized frequency error estimate, expressed in terms of locations ℓ_l and ℓ_u, becomes

(11.57)

Similarly, the bit rate estimate expressed, in terms of locations ℓ_l and ℓ_u, is evaluated as

(11.58)

Because the bit rate is unknown, it is essential to estimate the bit rate correctly to establish the two spectral tones for determining the frequency error. The probability that the normalized bit rate is estimated to within 12.5%, that is, , is shown in Figure 11.39a as a function of the signal‐to‐noise ratio; this resolution corresponds to two frequency resolution cells or 2Δf. Unfortunately, the bit rate estimation is four times more sensitive to errors in the locations ℓ_l and ℓ_u than is the frequency estimation (see Problem 13). This is evident from the normalized frequency and bit rate estimation errors shown in Figure 11.39b and c as a function of the signal‐to‐noise ratio.²⁹ The irregularity in the maximum and minimum errors at low signal‐to‐noise ratios occurs because they represent the relatively small sample size of outliers in the 500 samples at each signal‐to‐noise ratio; the random generator seeds are reset for each signal‐to‐noise ratio. The maximum and minimum values for signal‐to‐noise ratio ≥8 dB correspond to a variation of one frequency resolution cell for the frequency and bit rate estimation errors.

The performance in this case study represents the use of fundamental algorithms and the performance can be improved by using more processing intense techniques. Some examples are as follows:

• Zero padding the FFT will improve the accuracy of the frequency estimation, although the resolution bandwidth remains unchanged.
• Increasing the measurement window, with a commensurate increase in the FFT size. The resolution bandwidth is improved with an accompanying improvement in the measurement signal‐to‐noise ratio.
• Using linear or parabolic interpolation to refine the frequency resolution measurement.
• Using an FFT window to reduce the spectrum leakage into adjacent FFT cells when the signal tones at fT ± n/4 do not fall at the center of an FFT cell.
• Using a higher bit rate estimation tolerance for declaring a successful bit rate acquisition, for example, increasing the 12.5–25%. However, increasing the bit rate estimation tolerance is limited by the acquisition and tracking algorithm requirement discussed in Section 11.3.5.
• Using multiple FFTs and an m‐of‐n decision criterion for declaring a successful bit rate detection.

In many applications the bit rate or a number of possible bit rates are known and the issue becomes one of finding the ℓ_l and ℓ_u pairs corresponding to the received bit rate. This can be accomplished by searching through pairs of ℓ_l and ℓ_u corresponding to combinations of FFT cells with successively lower magnitudes until the bit rate is found or a limit is reached and a missed bit rate is declared. When the bit rate is declared the frequency estimation is performed using (11.52). The performance improvements listed earlier also apply to the known bit rate case; however, the use of multiple FFTs and the m‐of‐n decision criterion is a major consideration for both detection and confirmation before proceeding to the symbol acquisition and tracking processing.

11.3.2.4 Detection, Symbol Rate, and Frequency Estimation Using MSK‐Modulated Waveform

In this section, the characteristics of the MSK‐modulated waveform spectrum are examined for detection and estimation of the frequency and symbol rate. In this regard, the objectives are identical to those discussed in Section 11.3.2.2; however, the focus is on the unique spectral characteristics of the MSK waveform. The preamble for the multi‐h CPM waveform, discussed in Chapter 9, uses MSK with a symbol rate equal to that of the multi‐h‐modulated user data.

The following descriptions of the MSK‐modulated signal are based on unique data patterns that result in distinct spectral lines at harmonics related to the symbol rate. These spectral characteristics are exploited to estimate the signal presence, carrier frequency, and symbol rates during the synchronization segment of the preamble. The data patterns examined are the repeated sequences 1100… and 10… where the number of repetitions commensurate with the detection and signal‐to‐noise specification based on the system application. For example, the repeated 1100… data is specified for the multi‐h CPM‐modulated waveform preamble and is repeated for a total of 192 bits corresponding to the bit rate of the 2‐h CPM modulation. When mark‐ or space‐hold data is applied to the MSK modulator the transmitted signal is a CW tone corresponding to upper MSK tone f_c + R_s/2 and when alternate mark‐space data is applied the lower MSK tone at f_c − R_s/2 is transmitted; if required, these tones can serve as a CW preamble. However, when a randomly modulated MSK waveform is squared, the resulting spectrum contains discrete spectral tones, at 2f_c ± R_s. These tones can be used for signal detection and synchronization without the necessity of having a CW preamble. The downside of the signal squaring is a decrease in the signal‐to‐noise ratio resulting from the squaring loss and a frequency doubling that requires a higher demodulator sampling frequency. The noise‐free spectrums of the squared MSK‐modulated signal with random and the repeated 1010… data patterns are shown, respectively, in Figure 11.40a and b. The characteristics of the signal processing and the 1024‐point FFT are summarized in Table 11.10. In both of these cases, the spectral lines of interest occur at f_c ± R_s.

Figure 11.41a shows the noise‐free spectrum of a randomly modulated MSK waveform and is provided as a point of reference for the spectrum of the MSK‐modulated preamble data 11001100… shown in Figure 11.41b; the preamble pattern is repeated for 192 bits. These spectrums are normalized by the maximum level and are based on computer simulations using a 1024‐point FFT as detailed in Table 11.10. The efficiency is a measure of the duration of a single FFT relative to the synchronization preamble length of 192 bits. The 1024‐point FFT occupies 66.7% of the synchronization interval so, to ensure that at least one FFT captures the interval; the FFTs must be overlapped by 33.3%. On the other hand, the 768‐point FFT ensures that at least one FFT captures the interval and, with 50% overlapping, at least two FFTs occupy the synchronization interval. Another important parameter is the signal‐to‐noise ratio improvement, relative to γ_b = E_b/N_o, expressed as

(11.59)

Figure 11.41c and d show the spectrum of the MSK synchronization preamble segment using the 1024‐point FFT for the signal‐to‐noise ratios γ_b = 6 and 3 dB, respectively. In these cases, the normalized frequency error of the received signal is f_εT_b = 1. The plots with additive channel noise use different noise seeds so the results are statistically independent. The −23.1 and −29.9 dB spectral lines are buried in the noise; however, spectral lines at f_c ± R_s/2 and f_c ± R_s exhibit a sufficient signal‐to‐noise ratio, although the variations resulting from the additive noise in the FFT resolution cells are evident.

Figure 11.42a and b are similar to those in Figure 11.41c and d except that the 768‐point FFT is used. The increase in the noise floor is evident; however, there is still a healthy signal‐to‐noise ratio in the cells containing the spectral lines of interest. Referring to Section 9.3.4.1, the minimum γ_b for the multi‐h CPM modes at P_be = 10⁻⁵ is about 5.3 dB so the synchronization processing for the 3 dB condition using either FFT provides a degree of acquisition robustness. The processing of correctly estimating the frequency error and symbol rate is enhanced by using a weighted sliding window smoothing function across the FFT spectrum cells and selecting the optimum frequency error and symbol rate corresponding to the minimum weighted mean‐square error (MSE) between the sliding window and the magnitudes of the FFT cells. In this case, the weights of the smoothing function are chosen as the magnitude of the three or five symmetrical cells from Figure 11.41b; this approach is similar to the parabolic interpolation discussed in Appendix 2C.