1

SIGNAL THEORY AND ANALYSIS

A signal, in general, refers to an electrical waveform whose amplitude varies with time. Signals can be fully described in either the time or frequency domain. This chapter discusses the characteristics of signals and identifies the main tools used for signal processing. Some functions widely used in signal processing are described in Section 1.1. A quick review of the linear system and convolution theory is covered in Section 1.2. Fourier series representation of periodic signals is discussed in Section 1.3. Fourier transform of nonperiodic signals and periodic signals are covered in Sections 1.4 and 1.5, respectively. Section 1.6 describes sampling theory together with signal interpolation. Some advanced sampling and interpolation techniques are reviewed in Section 1.7.

1.1 SPECIAL FUNCTIONS USED IN SIGNAL PROCESSING

1.1.1 Delta or Impulse Function δ(t)

The delta function or impulse function δ(t) is defined as

and

On the basis of this definition, one can obtain

and

1.1.2 Sampling or Interpolation Function sinc (t)

The function sinc (t) is defined as

and

A sinc (t) function for t = −4 to 4 is depicted in Fig. 1.1. Notice that sinc (t) = 0 for all integers of t, and its local maxima corresponds to its intersection with the cos (πt).

FIGURE 1.1 A sinc (t) function.

1.2 LINEAR SYSTEM AND CONVOLUTION

A linear system, as shown in Fig. 1.2, can be represented as a box with input , output and a system operator H that defines the relationship between and . Both and can be a set of components.

FIGURE 1.2 A linear system.

A system is linear if and only if

where a and b are constants, is the system's input signal, and is the output signal.

In addition, a linear system having the fixed input–output relation

is time-invariant if and only if

for any (t) and any τ. In the following discussion, only the linear and time-invariant system is considered.

Let p_τ(t) be a pulse with amplitude 1/Δτ and duration Δτ; then any function f(t) can be represented as

Figure 1.3 illustrates the relationship between p_τ(t) and the function f(t). Figure 1.3a shows a rectangular polygon with amplitude 1/Δ τ and duration Δ τ; Fig. 1.3b displays how a function f(t) can be approximated by a series of delayed rectangular polygon p_τ(t− n Δ τ) with amplitude f(n Δ τ) Δτ.

FIGURE 1.3 Graphical representations of a function in terms of pulses.

As Δτ → 0, n Δτ → τ. Therefore

p_τ(t) → δ(t)

and

p_τ(t − n Δ τ) → δ(t − τ).

The summation of Eq. (1.4a) then becomes

Let h(t) be the impulse response of a system:

H δ(t) = h(t).

Then, for any input function x(t), the output function y(t) can be expressed as

where the asterisk (symbol*) refers to convolution. If x(t) = δ(t), then

Equation (1.5) states the relationship between the input function x(t), the impulse response or system function h(t), and the output function y(t). It serves as a fundamental equation and is widely used in linear and time-invariant systems. A simple block diagram that illustrates this relationship is shown in Fig. 1.4

FIGURE 1.4 A time-invariant linear system.

1.2.1 Key Properties of Convolution

1.2.1.1 Commutative

By letting λ = t − τ, Eq. (1.5) becomes

Therefore

1.2.1.2 Associative

If

y(t) = [x(t) * h(t)] * z(t),

then

1.2.1.3 Distributive

If

y(t) = [x(t) * h(t)] + x(t) * z(t),

then

1.2.1.4 Timeshift

If

y(t) = x(t) * h(t),

then

1.3 FOURIER SERIES REPRESENTATION OF PERIODIC SIGNALS

A signal g_p(t) is called a periodic signal with period T₀ if it remains unchanged after it has been shifted forward or backward by T₀, that is

g_p(t) = g_p(t +/− T₀),

where T₀ = 2π/ω₀.

There are three different Fourier series representations for a periodic signal. The first two representations are in terms of trigonometric functions, while the third is in exponential form. The three Fourier series representations of a periodic signal g_p(t) are described below.

1.3.1 Trigonometric Fourier Series

A periodic signal g_p(t) can be represented as

where a_n and b_n for n = 1, 2, ... can be computed as

1.3.2 Compact Trigonometric Fourier Series

Alternatively, a periodic signal g_p(t) can be represented as

where

1.3.3 Exponential Fourier Series

A periodic signal g_p(t) can also be represented as

where

or

Example 1.1 Figure 1.5 shows a periodic signal g_p(t), which is expressed as

with

FIGURE 1.5 A periodic signal g_p(t).

The Fourier series coefficients of g_p(t) in terms of these three representations can be computed as the following equations show:

1. From Eq. (1.10)

2. From Eq. (1.11)

3. From Eq. (1.12)

Figure 1.6 displays the G_n for the case when A = 1 and τ = T₀ / 2. Notice that the Fourier series coefficients of {G_n} are discrete and the dashed line represents the envelope of {G_n}.

FIGURE 1.6 Fourier series coefficients of a periodic pulse.

Example 1.2 Let the periodic signal g_p(t) in Example 1.1 be modified with τ = 0 and A = ∞, such that Aτ = 1:

The three Fourier series representations of a periodic impulse train can be computed as follows:

1. From Eq. (1.10)

Therefore

2. From Eq. (1.11)

Therefore

3. From Eq. (1.12)

Therefore

1.4 NONPERIODIC SIGNAL REPRESENTATION BY FOURIER TRANSFORM

A periodic signal g_p(t) can always be represented in one of the three Fourier series forms described in the previous section. Consider the signal based on exponential representation as shown in Eq. (1.12a), that is

where G_n can be derived as

or

A nonperiodic signal g(t) can be considered as a periodic signal g_p(t) with the period T₀ → ∞:

Notice that T₀ → ∞ implies ω₀ = (2π/T₀) → 0.

Let ω₀ Δω; then

From the integration shown above, it can be seen that T₀G_n is a function of nΔω; therefore, one can define

T₀G_n = G(n Δ ω),

and

Now

Similarly

Equations (1.13) and (1.14) are referred to as the Fourier transform pair. G(ω) in Eq. (1.14) is considered as the direct Fourier transform of g(t), while g(t) in Eq. (1.13) is the inverse Fourier transform of G(ω). The transform pair can also be expressed as

and

The Fourier transform pair can also be expressed symbolically as

g(t) G(ω).

Some key properties of the Fourier transform are listed below:

Example 1.3 Let g(t) be defined as

The Fourier transform of g(t) can be computed as

Figure 1.7 displays the time domain function g(t) and its Fourier transform G(ω).

FIGURE 1.7 A single pulse g(t) and its Fourier transform G(ω).

By comparing Figs. 1.6 and 1.7, one can see that the Fourier series coefficients of a periodic pulse train is the discrete version of the Fourier transform of a single pulse.

Example 1.4 Let G(ω) be defined as

The inverse Fourier transform of G(ω) can be computed as

Figure 1.8 displays the frequency domain function G(ω) and its inverse Fourier transform g(t).

FIGURE 1.8 A single-pulse frequency spectrum G(ω) and its inverse Fourier transform g(t).

Example 1.5 Let g(t) = δ(t), the Fourier transform of g(t) can be computed as

Therefore, δ(t) and 1 are a Fourier transform pair:

δ(t) 1.

Similarly, if G(ω) = δ(ω), then

Therefore, 1 and 2πδ(ω) are a Fourier transform pair:

1 2πδ(ω).

1.5 FOURIER TRANSFORM OF A PERIODIC SIGNAL

Although the Fourier transform was derived from the nonperiodic signal, it can also be used to represent the periodic signal. The Fourier transform of a periodic signal can be computed by first representing the periodic signal in terms of a Fourier series expression, then transforming each Fourier series coefficient (represented in exponential form) into the frequency domain. The following examples illustrate the Fourier transform of periodic signals.

Example 1.6 Let g_p(t) be a periodic impulse train, expressed as

The Fourier transform of g_p(t) can be computed by first expressing the periodic signal g_p(t) in terms of the Fourier series in the exponential form. Thus, from Example 1.2, we obtain

where ω₀ = 2π/T₀. From Example 1.5, the Fourier transform of g_p(t) can then be computed as

Figure 1.9 displays the time-domain impulse train g_p(t) and its Fourier transform G(ω). Both g_p(t) and G(ω) appear to be impulse trains. Notice that the amplitude of the impulse train in the frequency domain is ω₀ and its spectrum repeated at ± nω₀ with n = 1, 2, ... and ω₀ = 2π/T₀.

FIGURE 1.9 A periodic impulse train and its Fourier transform.

Example 1.7 Consider the signal g_p(t) shown in Example 1.1, expressed as

where

The Fourier transform of the periodic pulse train g_p(t) can be computed in two steps:

Step 1 A periodic signal should first be expressed in terms of the Fourier series in exponential form:

The Fourier series coefficients G_n can be computed as

Therefore, g_p(t) can be expressed as

Step 2 The Fourier transform of g_p(t) can then be computed as

Example 1.5 has shown that

1 2πδ(ω).

Therefore

If τ = T₀/2 and A = 1, then

FIGURE 1.10 Fourier transform of a periodic pulse train.

Notice that G(ω) is a discrete signal that exists when ω = nω₀ with n equal to an integer. Figure 1.10 demonstrates the frequency spectrum of G(ω) for n = −8 to 8, and its envelope is shown as sinc (n/2).

Note that from Figs. 1.6 and 1.10, for the same periodic sequence g_p(t), the amplitude of Fourier series coefficients G_n and the amplitude of Fourier transform G(ω) are the same with a scaling difference of only 2π.

1.6 SAMPLING THEORY AND INTERPOLATION

The sampling theory states that any signal that is frequency band-limited to f_m can be reconstructed from samples taken at a uniform time interval of T_s ≤ 1/(2f_m). The time interval T_s = 1/(2f_m) is called the Nyquist interval, and the corresponding sampling rate is known as the Nyquist rate. The sampling theory can be derived as explained below.

Consider a signal x(t) with its Fourier transform as X(ω) and its frequency spectrum band-limited to f_m. Let g_p(t) be a unit impulse train as described in Example 1.6. Multiplication of x(t) with g_p(t) yields the sampled signal x_s(t):

As shown in Example 1.6, a periodic pulse train can be expressed in terms of the Fourier series; that is, with ω_s = 2π/T_s, are obtains

FIGURE 1.11 Graphical representations of the sampling theory.

By taking the Fourier transform of x_s(t), one obtains

This equation states that after multiplication of x(t) by the unit impulse train g_p(t), the new frequency spectra X_s(ω) consists of X(ω), plus replica located at ω = ± nω_s, for n = 0, 1, 2, .... The amplitude of X_s(ω) is attenuated by a factor of 1/T_s.

Figure 1.11 illustrates the sampling theory. The original signal x(t) and its analog frequency spectrum |X(ω)| are shown in Figs. 1.11a and 1.11b. A periodic impulse train g_p(t) and its spectra are shown in Figs. 1.11c and 1.11d. By multiplying x(t) with g_p(t), one can then display the resultant x_s(t) in Fig. 1.11e with the corresponding spectra shown in Fig. 1.11f.

From Fig. 1.11f, one can see that to prevent overlap between the neighboring spectra, the sampling frequency ω_s = 2π/T_s must satisfy the requirement that ω_s ≥ (2 × 2πf_m) = 2ω_m.

To reconstruct the original signal x(t) from the digitized signal x_s(t), one needs to filter out the spectrum X(ω), as shown in Fig. 1.11b, from the spectra X_s(ω), as shown in Fig. 2.11f. A lowpass filter (LPF) with a cutoff frequency at 2πf_m and a gain equal to T_s meets the filtering requirement. By passing the x_s(t) through this lowpass filter, one can reconstruct the original signal x(t).

Letting H(ω) be such a rectangular LPF, one can compute the time-domain function h(t) as

Let the sampling frequency f_s = 2f_m; then

In the time domain, passing the signal x_s(t) through the filter h(t) is equivalent to having x_s(t) convolved with h(t):

Equation (1.25) states that x(t) can be reconstructed from its discrete samples x(nT_s) and the interpolation function sinc (2f_mt–n). Notice that x(nT_s) are all equally spaced with time interval T_s for integers n = −∞ to ∞, and also that x(t) is a continuous function in the time domain. Thus, Eq. (1.25) can be used to find x(t₁), for t₁ = nT_s + ΔT_s with Δ < 1, based on the discrete values of x(nT_s). The new discrete time sequence x(t₁) can be considered as a resampling of x(t), or interpolated from x(nT_s).

In practical applications, the resampling process that utilizes the interpolation filter sinc (2f_mt) is simplified by two approximations. First, the interpolation filter is chosen with a finite number of sidelobes. For radar image processing, an 8-tap sinc filter is normally used to generate a new interpolated sample. The second approximation involves choosing a finite number of interpolation intervals (or fractional shifts). This means that the value of Δ < 1 is discrete and finite. A set of 16 sinc filters with a minimum of interpolation sample intervals, or Δ = , serves well for radar image processing. This set of 16 sinc filters provides a fractional sample shift from , , ..., . The = 1 is not considered a sample shift.

FIGURE 1.12 Interpolation filters.

Figure 1.12 displays the waveforms of 16 sets of interpolation filters with each having 8-tap coefficients. Figure 1.12a shows a reference sinc (t) function with 5 sidelobes around the mainlobe. The window function is normally applied to the interpolation filter, yet no window or rectangular window is used here for simplification. Figure 1.12b displays four interpolation filters with delays equal to , , , and sample intervals with respect to the top reference function, respectively. Figure 1.12c displays the digitized version of all 16 interpolation filters, and each one corresponds to a sample delay from each other.

Table 1.1 lists the coefficients of the 16 interpolation filters, with each one having 8 coefficients. The first row of filters has a shift of sample interval, while the last one has = 1 or no sample shift. From Table 1.1 and Fig. 1.12c, one can see that the interpolation filters are symmetric; that is, filter coefficients of row 1 with sample shift are the same as that of row 15 with sample shift, except that they are time-reversed. The filter coefficients of rows 1, 5, 9, and 13 of Table 1.1 correspond to the four sinc filters shown in Fig. 1.12b.

To illustrate the principle of resampling based on an interpolation filter, consider a signal x(n) that consists of three normalized frequencies, f₁ = 0.15, f₂ = 0.25, and f₃ = 0.45, and is expressed as

x(n) = 0.35 cos (2πnf₁) + 0.2 sin (2πnf₂) − 0.4 cos (2πnf₃),

where n = 1, 2, ..., 32.

TABLE 1.1 16 sets of 8-tap interpolation filters

To resample x(n) to obtain a new set of digitized samples with a 0.5 sample shift, row 8 of the interpolation filter from Table 1.1 is chosen. The new set of data x(n + 0.5) is computed from Eq. (1.25) by convolving the eight old data samples of x(n) with the interpolation filter. Figure 1.13 displays the results of the resampling process. Only the effective samples from samples 4–28 of the convolution output are displayed for comparison.

FIGURE 1.13 Original and half-sample-shifted digital signals.

Figure 1.13a shows the digitized signal x(n), marked by hollow circles (), together with the original analog signal x(t). Figure 1.13b shows the interpolated (or half-sample-shifted) signal x′ (n), marked by asterisks (*), together with the original analog signal x(t). As can be seen, the interpolated and digitized signal closely matches the original signal.

The sampling theory discussed so far is based on a unit impulse train. In practice, the impulse train is replaced with a finite-time-duration pulse train. Assuming, that all conditions are the same except that the narrow pulse train with duration τ is used, the only difference will be that the amplitude of the Fourier series coefficient 1/T_s is replaced with sin (nπτ/T_s)/nπ. This, in turn, causes Eq. (1.23) to become

The lowpass filter defined in Eq. (1.24) can then be applied to the digitized signal x_s(t). Accordingly, the x(t) can again be reconstructed from its digitized samples x_s(t) by passing the x_s(t) through a lowpass filter h(t). The new reconstruction equation, based on a pulse train with duration τ, can be obtained by modifying Eq. (1.25) to become

1.7 ADVANCED SAMPLING TECHNIQUES

1.7.1 Sampling with Bandpass Signal

The sampling theory discussed in Section 1.6 generally applies to baseband signals with a maximum frequency component f_m. In the case where a signal is band-limited with f_l ≤ f ≤ f_m and its center frequency is greater than zero, the Nyquist sampling period T_s ≤ 1/(2f_m) must be modified to reduce the sample rate for signal processing.

Figure 1.14 shows the frequency spectra |G(ω)| of a bandpass signal g(t). Figure 1.14a shows the frequency spectra |G(ω)| of g(t), while Fig. 1.14b shows the frequency spectra |G_s(ω)| of a sampled signal g_s(t). From the sampling theory, the spectrum of the sampled signal will consist of the attenuated spectrum from the original signal and its replica located at ± nω_s, where n = 0, 1, 2, .... The two-sided spectra, a and b, shown in Fig. 1.14a are repeated around ± ω_s, ± 2ω_s ... in Fig. 1.14b. The spectra na and nb are the repeated spectra around ± nω_s.

From Fig. 1.14, it can be seen that no spectrum is overlapped as long as the conditions of f_s = 2 (f_m − f_l) and f_l = kf_s with k as an integer are satisfied. If the latter condition is not true, then a higher sampling frequency of f_s > 2 (f_m − f_l) will be needed.

FIGURE 1.14 Frequency spectra of a digitized bandpass signal.

Resampling is necessary for some applications, such as digital communication or audio/video processing. Resampling is used to reduce the computation complexity for signals with a finite bandwidth. Some resampling techniques will now be discussed.

1.7.2 Resampling by Evenly Spaced Decimation

Evenly spaced sample decimation can be used to reduce the sample rate by an integral factor. Consider the case of downsampling on a sequence x(n) by an integer M = f_old/f_new, where f_old and f_new are the old and new sampling rates, respectively. Such a downsampling process can simply retain every Mth sample and discard the others if the frequency aliasing does not occur after downsampling. However, frequency overlap indeed occurs in most downsampling processes. Therefore, a lowpass Finite Impulse Response (FIR) filter is normally used to limit the bandwidth of the signal at a new sampling rate. Both the input sequence x(n) and the FIR filter function h(n) operate at an old sample rate f_old, but the output sequence y(m) operates at a new sample rate f_new. Figure 1.15a shows the block diagram of downsampling with a FIR filtering. Figures 1.15b and 1.15c show the signal frequency spectra before and after the decimation for the case when M = 2. As can be seen from Fig. 1.15c, the signal frequency spectra must be band-limited with f_new ≥ 2f_m to avoid the aliasing problem.

1.7.3 Resampling by Evenly Spaced Interpolation

Evenly spaced sample interpolation can be used to increase the sample rate by an integral factor. Consider the case of upsampling on a sequence x(n) by an integer M = f _new/f _old, where f _old and f _new are the old and new sampling rates, respectively. Upsampling by M can be implemented by inserting M−1 zeros for every sample of x(n), and then passing it through a lowpass filter operating at the f _new sampling rate. The lowpass filter is required to filter out the frequency spectra at sampling rate f_new.

FIGURE 1.15 Downsampling with a FIR filter.

Figure 1.16 illustrates the process of interpolation with the sample rate increased by M = 3. The overall system diagram is shown in Fig. 1.16a, with x(n) as the input and y(m) as the output. Figures 1.16b and 1.16c show the original signal x(n) and its corresponding spectra |X(ω)| with sample rate f _old. By inserting two zeros for every sample of x(n), one obtains a new signal sequence x′(m). Figures 1.16d and 1.16e show the new sequence x′(m) and its corresponding spectra |X′(ω)| with sample rate f _new. After passing x′(m) through a lowpass filter (LPF), one obtains the interpolated signal y(m) as shown in Fig. 1.16f, and its frequency spectra are shown in Fig. 1.16g.

1.7.4 Resampling by Fractional Rate Interpolation

The interpolation process described in the previous section increases the sample rate with an integral factor. In cases where the sample rate change factor M is not an integer, interpolation followed by decimation can be used to obtain any sample rate change. Let the new sample rate change factor be M = I + F, where I is an integer and F is a fractional value. Then M = U/D, where U and D correspond to the upsampling and downsampling rates, respectively.

FIGURE 1.16 Interpolation with zero insertion and FIR LPF.

Consider the case of M = 2.5 with U = 5 and D = 2. Let the original digitized signal and sampling frequency be x(n) and f_s, respectively. To implement the interpolation at a resampling rate of 2.5, one can first upsample x(n) with sampling frequency equal to 5f_s by inserting four zeros between each sample of x(n). The new sequence of samples x′(n) are then lowpass-filtered (operating at 5f_s) and followed by downsampling with sampling frequency equal to 2f_s. Figure 1.17a illustrates the overall process of interpolation with U = 5 and D = 2. Figure 1.17b shows the original signal spectra |X(ω)| with sample rate f_s. By inserting four zeros for every sample of x(n) followed with a lowpass FIR filter operating at new sampling rate f_new = 5f_s, one obtains a new sequence x′(m). Figure 1.17c shows the spectra of |X′(ω)|. The new sequence x′(m) is then decimated by dumping one out of every sample of x′(m). The spectra of interpolated signal y(n′) running at 2.5f_s is shown in Fig. 1.17d. Alternatively, one can just compute one FIR filter output for every two samples shifted into the FIR filter.

FIGURE 1.17 Fractional rate interpolation with M = 2.5.

1.7.5 Resampling from Unevenly Spaced Data

Before discussing the resampling method based on unevenly spaced data, the Jacobian of transformation will be explained. This process is used extensively during the transformation between different coordinate systems.

1.7.5.1 Jacobian of Transformation.

A signal can be transformed from one coordinate system into a different coordinate system. Variables inevitably change during a transformation from one system to another. The transformation can be linear or nonlinear. The transformation from one system to another may therefore cause an extra factor to appear. This extra factor is called the Jacobian of transformation.

Consider the case where u = g(v), then (du/dv) = (dg(v)/dv) = g′(v) and

The extra factor g′(v) in this integration is called the Jacobian of transformation.

In general, let x = g(u,v) and y = h(u,v) be a transformation from (x,y) domain to (u,v) domain. Then, the Jacobian of transformation, or simply Jacobian, is defined as

Let f(x,y) be a function in the R(x,y) region, and a transformation of x = g(u,v) and y = h(u,v) transforms f(x,y) from R(x,y) into S(u,v) region. If g(u,v) and h(u,v) have continuous partial derivatives such that the Jacobian is never zero, then

As an example, let x = g (r, θ) = r cos θ and y = h(r, θ) = r sin θ. From Eq. (1.28), the Jacobian of transformation is

Therefore, the integration shown above becomes

This is the well-known rectangular-to-polar coordinate transformation.

The operation of integration in the analog domain is equivalent to summation in the digital domain. The principle of the Jacobian of transformation can therefore be applied to the interpolation or resampling in the digital signal processing field.

FIGURE 1.18 Interpolation on evenly spaced data.

The sampling theory states that given a set of evenly spaced data x(n) with sampling frequency f_s = 2f_m, one can interpolate x(n) with another set of evenly spaced data x(k). Assume that the sampling frequency is normalized to one, that is, t = kT_s = k. Then Eq. (1.24) becomes

The interpolated data x(k) and the original data x(n) can be considered as in k and n coordinate systems, respectively, and they are related as

where Δ is a fractional number and 0 < Δ < 1. The summation of Eq. (1.29) is the discrete convolution of x(n) with sinc (n), and the data sequence x(n) is transformed from n space to k space. Figure 1.18 shows the interpolation of evenly spaced samples from the view point of transformation between coordinate systems.

The Jacobian of transformation is then

Therefore, Eq. (1.29) holds without any extra factor for transformation from an evenly spaced system to another evenly spaced system.

FIGURE 1.19 Generation of unevenly spaced data index.

Now, consider the case where one tries to resample a set of evenly spaced data f(k) from a set of unevenly spaced data f(n) with no aliasing. Before the resampling process, the unevenly spaced signal f(n) must be defined and generated. As an example, the index n of data f(n) is generated nonlinearly through the transformation g(k) as

where k is an evenly spaced variable and equal to half of the radius of a circle with origin at (0, 0). The symbol s is a constant and s < 2k.

Figure 1.19 shows the mapping relation of Eq. (1.30), with the unevenly spaced data index n obtained from Eq. (1.30) when k = 1, 1.25, 1.5, ..., 3, and s = s_i < 1. The evenly spaced data index k is the intersection point between the circle with radius r = 2k and the horizontal axis (abscissa), which is labeled as 1, 2, 3, .... The unevenly spaced data index n is obtained by mapping the intersection point between the circle of radius r = 2k and the line s = s_i to the horizontal axis, which is labeled as a, b, c...

The Jacobian of transformation from the n space to k space is therefore

This Jacobian is then used to transform the data f(n) in n domain to f(k) in k domain. The interpolation equation for evenly spaced data shown in Eq. (1.29) therefore becomes

Notice that in this equation, the variable n shown in summation serves as an index of the unevenly spaced sample. Figure 1.20 shows, as an example, how the unevenly spaced data f(n) and the evenly spaced data f(k) were transformed from the data indices n and k. Given the unevenly spaced data f(n), shown as dots (•), the interpolated and evenly spaced data f(k) can therefore be generated from Eq. (1.31), which are denoted by small hollow circles ().

FIGURE 1.20 Relationship between evenly and unevenly spaced data.

FIGURE 1.21 Interpolation on unevenly spaced data.

The evenly spaced signal f(k) shown in Fig. 1.13 will now be used to illustrate the interpolation on the basis of the unevenly spaced samples. The signal f(k) is expressed as

f(k) = 0.35 cos(2πkf₁) + 0.2 sin(2πkf₂) − 0.4 cos (2πkf₃),

where f₁ = 0.25, f₂ = 0.45, and f₃ = 0.15 are all normalized frequencies. We will reverse the process by generating an unevenly spaced data f(n) from f(k) first, then try to interpolate from f(n) to obtain f(k).

The unevenly spaced samples f(n) are generated by computing the unevenly spaced index n for various k from Eq. (1.30). Once the index n is computed, the value of f(n) can be obtained.

Consider the case of k ranging from 0.5 to 6.5 with Δk = 0.25 and s = 0.75; the corresponding Jacobian can be computed as

The 8-tap filters shown in Table 1.1 will be used in this example, and Eq. (1.31) then becomes

Figure 1.21a shows the plot of unevenly spaced f(n) with 2k – Δ_k and k = 0.5, 0.75, ..., 6.5, 0 < Δ_k = (0.75)²/(4k) < 1. A total of 25 sinc (x) functions are generated, each corresponding to different Δ_k. Figure 1.21b displays the interpolated plot of evenly spaced samples f(k), k = 0.5, 0.75, ..., 6.5. For comparison purposes, the original evenly spaced signal is displayed in Fig. 1.21c. Notice that Fig. 1.21a is plotted based on the index n′ = 2n, Figs. 1.21b and 1.21c are plotted based on the index k′ = 4k −1. Notice also that f(n) appears to be unevenly spaced at the left part of the display, and becomes approximately evenly spaced at the right end of the plot as expected. Just like the original digitized samples shown in Fig. 1.21c, the interpolated signal is evenly spaced and matches quite well with the bottom signal. Only a few samples at both ends differ from the original, because of the edge effect of convolution, which is covered in the next chapter.