4.12 Numerical Results

In this section, simulation results are reported to corroborate the theoretical results of the previous sections on the spectral cross-correlation density estimation.

4.12.1 Simulation Setup

The spectral cross-correlation density function between complex-envelope signals x(t) and y(t) is measured, where x(t) and y(t) are the input and output signals, respectively, of a linear time-variant system that models the channel between transmitter and receiver in relative motion with constant relative radial speed, that is (Section 7.3)

(4.304)

In (4.304), a is the (possibly complex) scaling amplitude, d the delay, s the time-scale factor, and ν the frequency shift.

For an ACS input process x(t) the Loève bifrequency cross-spectrum between y(t) and x(t) is given by (see (7.276))

(4.305)

where and are the cyclic spectra and the set of cycle frequencies, respectively, of x(t). Therefore, if s ≠ 1, x(t) and y(t) are not jointly ACS (even if they are singularly ACS) but are jointly SC and exhibit joint spectral correlation on curves f₂ = Ψ⁽ⁿ⁾(f₁) which are lines with slope 1/s in the bifrequency plane.

In the simulation experiments, both time and frequency are discretized with sampling increments T_s = T/N and ΔF = 1/T, respectively, where T is the data-record length and N the number of samples. The channel parameters are fixed at a = 1, d = 0, ν = 0, and s = 0.99. The input signal x(t) is a binary PAM signal with full-duty-cycle rectangular pulse and bit rate 1/T_p = 1/8T_s. Thus, x(t) has an approximate bandwidth B = 1/8T_s.

4.12.2 Unknown Support Curves

In this section, the support curves of the Loève bifrequency cross-spectrum are assumed to be unknown. Hence, the time-smoothed cross periodogram is adopted as estimator of the bifrequency spectral cross-correlation density (4.5).

The time-smoothed cross-periodogram (4.123) (with (*) present) is obtained by taking B_Δf(f) and a_T(t) both rectangular windows (see discussion in Section 4.7.3). Its performance is evaluated by the sample mean and the sample standard deviation computed on the basis of 100 Monte Carlo trials. Two experiments are conducted with increasing data-record length aimed at illustrating the asymptotic behavior of the estimator. In both the experiments, Δf = 1/2⁶T_s and the time-smoothed cross-periodogram is computed for (f₁, f₂) [− 0.025/T_s, 0.025/T_s] × [− 0.025/T_s, 0.025/T_s]. Moreover, to better highlight slopes of the support lines close to unity, all the graphs are reported as functions of (f₁, α), with α f₁ − f₂. A support line with slope 1/s in the (f₁, f₂) plane becomes a line with slope 1 − 1/s in the (f₁, α) plane.

In the first experiment, the number of samples is N = 2¹¹. Therefore, and, hence, s cannot be approximated by 1 (Section 7.5.1). Moreover, TΔf = 32 and, hence, , , that is, condition (4.135) of Corollary 4.5.8 is practically verified. In Figure 4.24, (a) graph and (b) “checkerboard” plot of the magnitude of the sample mean of the time-smoothed periodogram of the ACS input signal x(t) are reported as functions of (f₁, α). A portion of the power spectral density (α = 0) is evident. In Figure 4.25, (a) graph and (b) “checkerboard” plot of the sample standard deviation of are shown as functions of (f₁, α). In Figure 4.26, (a) graph and (b) “checkerboard” plot of the magnitude of the sample mean of the time-smoothed cross-periodogram are reported. It is evident that y(t) and x(t) are jointly SC processes and exhibit joint spectral correlation along the line α = (1 − 1/s)f₁ (Figure 4.26(b)). In Figure 4.27, (a) graph and (b) “checkerboard” plot of the sample standard deviation of are drawn.

Figure 4.24 (a) Graph and (b) “checkerboard” plot, as functions of (f₁, α), of the magnitude of the sample mean of the time-smoothed periodogram of x(t), for a data-record length N = 2¹¹.

Source: (Napolitano 2003) © IEEE

Figure 4.25 (a) Graph and (b) “checkerboard” plot, as functions of (f₁, α), of the sample standard deviation of the time-smoothed periodogram of x(t), for a data-record length N = 2¹¹.

Source: (Napolitano 2003) © IEEE

Figure 4.26 (a) Graph and (b) “checkerboard” plot, as functions of (f₁, α), of the magnitude of the sample mean of the time-smoothed cross-periodogram of y(t) and x(t), for a data-record length N = 2¹¹.

Source: (Napolitano 2003) © IEEE

Figure 4.27 (a) Graph and (b) “checkerboard” plot, as functions of (f₁, α), of the sample standard deviation of the time-smoothed cross-periodogram of y(t) and x(t), for a data-record length N = 2¹¹.

Source: (Napolitano 2003) © IEEE

A second experiment with N = 2¹³ is conducted, aimed at showing that, unlike the case of WSS and ACS processes, for (jointly) SC processes augmenting the data-record length does not necessarily imply a better performance of the spectral cross-correlation density function estimator. Figure 4.28a shows the graph of the magnitude of the sample mean and Figure 4.28b the graph of the sample standard deviation of the time-smoothed periodogram of x(t). By comparing Figures 4.28a and 4.28b with Figures 4.24a and 4.25a, respectively, the result is that both bias and variance of the estimate have decreased in accordance with the well known consistency properties of the time-smoothed cross-periodogram of ACS signals. For N = 2¹³, it results in BT 1024 > 1/|1 − s| and, hence, also in this experiment, s cannot be approximated by 1. Moreover, TΔf = 128 so that , , that is, (4.135) is not verified. Consequently, a significant bias of the estimate is expected. In Figures 4.29a and 4.29b, the graph of the magnitude of the sample mean and the graph of the sample standard deviation of the time-smoothed cross-periodogram of y(t) and x(t) are reported. By comparing Figures 4.29a and 4.29b with Figures 4.26a and 4.27a, respectively, it is evident that, according with the theoretical results of Theorems 4.5.7 and 4.5.9 and Corollary 4.5.8, even if the variance has decreased, the bias has grown.

Figure 4.28 Graph of (a) the magnitude of the sample mean and (b) the sample standard deviation of the time-smoothed periodogram of x(t), as function of (f₁, α), for a data-record length N = 2¹³.

Source: (Napolitano 2003) © IEEE

Figure 4.29 Graph of (a) the magnitude of the sample mean and (b) the sample standard deviation of the time-smoothed cross-periodogram of y(t) and x(t), as function of (f₁, α), for a data-record length N = 2¹³.

Source: (Napolitano 2003) © IEEE

4.12.3 Known Support Curves

In this section, the location of one support curve of the Loève bifrequency cross-spectrum is assumed to be known. Hence, the frequency-smoothed cross-periodogram along this curve is adopted as estimator of the spectral cross-correlation density on this curve (Section 4.7).

The performance of the frequency-smoothed cross-periodogram , defined in (4.147), along the known support curve

(4.306)

with is evaluated by the sample mean and the sample standard deviation of both amplitude and phase of . A rectangular frequency-smoothing window A_Δf(f) with window-width Δf = 1/(2⁶T_s) is used. By increasing the data-record length T = NT_s, both bias and standard deviation decrease, according to Theorems 4.7.5 and 4.7.7 where the mean-square consistency of the estimator is proved (see Figure 4.30 for N = 2⁷, Figure 4.31 for N = 2⁹, and Figure 4.32 for N = 2¹¹).

Figure 4.30 (Top) Magnitude and (bottom) phase of the frequency smoothed cross-periodogram along the known support curve f₂ = f₁/s − (α_p + ν/s), with α_p = 1/T_p, estimated by N = 2⁷ samples. Thin line: Actual value; Thick line: Sample mean; Shaded area: Sample standard deviation

Figure 4.31 (Top) Magnitude and (bottom) phase of the frequency smoothed cross-periodogram along the known support curve f₂ = f₁/s − (α_p + ν/s), with α_p = 1/T_p, estimated by N = 2⁹ samples. Thin line: Actual value; Thick line: Sample mean; Shaded area: Sample standard deviation

Figure 4.32 (Top) Magnitude and (bottom) phase of the frequency smoothed cross-periodogram along the known support curve f₂ = f₁/s − (α_p + ν/s), with α_p = 1/T_p, estimated by N = 2¹¹ samples. Thin line: Actual value; Thick line: Sample mean; Shaded area: Sample standard deviation

Although it would be correct to represent mean plus/minus standard deviation for real and imaginary parts of the frequency-smoothed cross-periodogram, in Figures 4.30–4.32 mean plus/minus standard deviation of magnitude and phase are reported to show the accuracy of the phase estimate when the magnitude is small.