4.9 Sampling of SC Processes

In this section, a sampling theorem for SC processes is stated to link the Loève bifrequency spectrum of the sampled discrete-time process with that of the continuous-time process. An analogous result is established in terms of densities. Furthermore, in the case of strictly band-limited continuous-time SC processes, sufficient conditions on the sampling frequency to avoid aliasing effects are provided (Napolitano 2011).

For a WSS process, two equivalent conditions describe the absence of aliasing in the Loève bifrequency spectrum of the sampled signal. The former is the non overlap of replicas in the power spectrum of the sampled signal (Lloyd 1959). The latter is that the mappings ν_i = f_i/f_s, i = 1, 2, between frequencies f_i [− f_s/2, f_s/2] of the density of Loève bifrequency spectrum of the continuous-time process and frequencies ν_i of the density of Loève bifrequency spectrum of the discrete-time process hold ∀ν₁ [− 1/2, 1/2], on the main diagonal ν₂ = ν₁. In fact, on the main diagonal the density of Loève bifrequency spectrum is coincident with the power spectrum. It is well known that both conditions are satisfied provided that the sampling frequency is equal to or greater than twice the process bandwidth. In this section it is shown that for SC processes, unlike the case of WSS processes, a sampling frequency equal to twice the signal bandwidth is not sufficient to assure that the mappings ν_i = f_i/f_s hold for the density on every support curve ∀ν₁ [− 1/2, 1/2] (or ∀ν₂ [− 1/2, 1/2]). A sufficient condition is derived on the sampling frequency to assure that these mappings hold and known results for ACS processes (Napolitano 1995), (Izzo and Napolitano 1996), (Gardner et al. 2006) are obtained as special cases.

It is worth underlining that the problem of avoiding aliasing in the spectral correlation density of the sampled process should not be confused with the problem of signal reconstruction from its samples. This latter problem is addressed, for example, in (Lloyd 1959) for stationary processes and in (Gardner 1972), (Lee 1978) for nonstationary processes. In (Lloyd 1959), for stationary processes it is shown that the lack of overlap of replicas in the bifrequency Loève spectrum of the sampled process is a necessary and sufficient condition to allow the continuous-time process reconstruction with appropriate sense of convergence. In the case of base-band signals, this condition is the classical Shannon condition on the sampling frequency. In (Lee 1978), it is shown that for nonstationary processes such a condition is only sufficient. Finally, note that the class of processes considered here, namely the SC and ACS processes, have sample paths that are finite-power functions. Thus, known results for functions (Garcia et al. 2001), (Song et al. 2007) or functions cannot be exploited.

4.9.1 Band-Limitedness Property

It is well known that for WSS processes the strictly band-limitedness condition allows one to avoid aliasing after uniform sampling, provided that the sampling frequency exceeds twice the process bandwidth. Nonstationary processes should be carefully handled. In particular, not every nonstationary structure is compatible with the band-limitedness property. ACS processes can be strictly band limited (Gardner et al. 2006). In contrast, in Sections 2.2.2 and 2.2.3 (see also (Napolitano 2007a) and (Napolitano 2009)) it is shown that GACS processes cannot be strictly band limited. In the sequel it is proved that SC processes can be strictly band limited (Napolitano 2011). For this purpose, a definition of strict band limitedness is given for the general case of nonstationary processes. Specifically, a nonstationary process is said to be strictly band limited when it passes unaltered through an ideal strictly band-limited filter.

Let h^(b)(t) be the impulse response function of the ideal low-pass filter with harmonic response H^(b)(f) = rect(f/2b).

Definition 4.9.1 A continuous-time nonstationary stochastic process x_a(t) is said to be (almost-surely) strictly band limited with bandwidth B if B is the infimum of the values of b such that

(4.204)

almost surely (a.s.) ∀t, where ⊗ denotes convolution.

If x_a(t) is a second-order process, then (4.204) holding a.s. ∀t is equivalent to

(4.205)

The condition of Definition 4.9.1 can be relaxed by requiring that (4.204) holds for almost all t. If (4.204) is satisfied, it follows that (see (5.179) in Section 5.10)

(4.206)

which is the band-limitedness condition for nonstationary processes considered in (Gardner 1972). Such a condition can also be obtained by following (Lii and Rosenblatt 2002) in terms of the spectral measure which is a set function (Hurd and Miamee 2007). Let A be a Borel subset of . The process x_a(t) is strictly band limited in [− B, B] if

(4.207)

If (4.206) holds, in (Gardner 1972) it is shown that for a sampling period T_s < 1/(2B) the process x_a(t) admits the mean-square equivalent “sample representation”

(4.208)

where sinc(t) sin(πt)/(πt). Further band-limitedness definitions for nonstationary processes are considered in (Zakai 1965), (Cambanis and Masry 1976).

In Sections 2.2.2 and 2.2.3 it is shown that GACS processes cannot be strictly band limited (see also (Izzo and Napolitano 2002a) and (Izzo and Napolitano 2002b)). In contrast, for the SC processes the following results hold.

Let x_a,i(t), (i = 1, 2), be continuous-time jointly SC processes with Loève bifrequency cross-spectrum

(4.209a)

(4.209b)

where .

Theorem 4.9.2 Strictly Band-Limited Spectrally Correlated Processes (Napolitano 2011, Theorem 2.1). Let x_a,i(t), (i = 1, 2), be jointly SC processes with Loève bifrequency cross-spectrum (4.209a), (4.209b). If the processes are strictly band-limited with bandwidths B_i and if the points of the intersections of the support curves are non-dense (in the meaning assumed in the proof), the result is that

(4.210a)

(4.210b)

Moreover, the functions and can be suitably modified so that (4.210a) and (4.210b) hold everywhere since with this modification the cross-correlation (4.24b) (with and (4.209a) or (4.209b) substituted into) remains unaltered. In addition, and can be assumed to be zero for f₁6 [− B₁, B₁] and f₂6 [− B₂, B₂], respectively. In this case, is considered to be invertible from the support of to the support of for each .

Conversely, if at least one of (4.210a) or (4.210b) holds, then also the other one holds and the processes x_a,i(t) are strictly band-limited with bandwidth B_i.

Proof: See Section 5.10.

Corollary 4.9.3 Support of the Spectral Cross-Correlation Density Functions (Napolitano 2011, Corollary 2.1). Let x_a,i(t), (i = 1, 2), be strictly band-limited jointly SC processes with bandwidths B_i. In the expressions of the Loève bifrequency cross-spectrum (4.209a), (4.209b) it results that

(4.211a)

(4.211b)

where supp means support.

4.9.2 Sampling Theorems

Theorem 4.9.4 Loève Bifrequency Cross-Spectrum of Sampled Jointly Spectrally Correlated Processes (Napolitano 2011, Theorem 4.1). Let x_a,i(t), (i = 1, 2), be continuous-time jointly SC processes with Loève bifrequency cross-spectrum (4.209a), (4.209b) and let

(4.212)

be the discrete-time processes obtained by uniformly sampling x_a,i(t) with sampling periods T_si 1/f_si. The processes x_i(n) are discrete-time jointly SC with Loève bifrequency cross-spectrum given by

(4.213)

and the bounded variation condition (4.183) specializes to

(4.214)

Proof: See Section 5.10.

By comparing (4.213) with (4.209a) it follows that the Loève bifrequency cross-spectrum of the discrete-time signals is constituted by replicas of that of the continuous-time signals. In addition, according to Theorem 4.9.19, it follows that the discrete-time processes x_i(n) are jointly SC. In fact, their Loève bifrequency cross-spectrum exhibits spectral masses concentrated on a countable set of support curves and is periodic with period 1 in both variables ν₁ and ν₂, even if in general the functions and of Theorem 4.8.17 cannot be easily expressed in terms of and . Furthermore, at every point (ν₁, ν₂) [− 1/2, 1/2)², nonzero contribution to the Loève bifrequency cross-spectrum is given by the terms in (4.213) such that

(4.215)

and

(4.216)

for some . Specifically, the support curves in the bifrequency principal domain (ν₁, ν₂) [− 1/2, 1/2)² are described by the set

(4.217)

In general, aliasing effects are present. The set can contain clusters of curves or can be dense in the square [− 1/2, 1/2)² if, for some ν₁ [− 1/2, 1/2), f_s2 is incommensurate with a countable infinity of values of , , , and in correspondence of these values the spectral densities are nonzero. In such a case, is non identically zero for |f₁| > B for some B.

In the special case of support lines , single sampling frequency f_s1 = f_s2 = f_s, and if frequencies ν₁ are evaluated on the bins of a N-point discrete Fourier transform (DFT), that is, ν₁ = n₁/N, n₁ = − N/2, …, (N − 1)/2, (N even), then

(4.218)

Thus, if is infinite and a^(k) is irrational and/or b^(k)/f_s is irrational for some k, then the restriction of to ν₁ = n₁/N is dense (in ν₂) due to the presence of clusters of lines in , provided that the spectral density is not identically zero for |f₁| > B, for some B. In (Lii and Rosenblatt 2002), only condition a^(k) irrational is considered since the special case of finite is treated, so that only the term a^(k)p₁f_s can make the set dense when a^(k) is irrational and .

In (Lii and Rosenblatt 2002), it is explained that the aliasing problem (in the sense of spectral mass folding onto [− 1/2, 1/2]²) is a consequence of the relationship (see also (5.185))

(4.219)

This aliasing effect is illustrated in the numerical experiments in (Lii and Rosenblatt 2002), (Napolitano 2010b) and Section 4.9.3.

Theorem 4.9.5 Loève Bifrequency Cross-Spectrum of Sampled Strictly Band-Limited Jointly SC Processes (Napolitano 2011, Theorem 4.2). Let x_a,i(t), i = 1, 2, be continuous-time jointly SC processes with Loève bifrequency cross-spectrum (4.209a), (4.209b) strictly band-limited with bandwidths B_i and functions invertible. If f_si ≥ 2B_i, then the processes x_i(n) defined in (4.212) are discrete-time jointly SC having Loève bifrequency cross-spectrum (4.190a) and (4.190b) with

(4.220a)

(4.220b)

(4.221a)

(4.221b)

and all periodic functions (4.220a)–(4.221b) are sums of nonoverlapping replicas. In such a case, the bounded variation condition (4.214) is equivalent to the bounded variation condition for the continuous-time processes (4.18). Moreover, if and are also differentiable, then the assumptions for Theorem 4.8.3 are verified and (4.191a) and (4.191b) hold.

Proof: See Section 5.10.

The support function and the corresponding spectral correlation density are said to be exactly identifiable starting from the samples (4.212) if and only if the equalities

(4.222)

(4.223)

hold ∀ν₁ [− 1/2, 1/2]. Under the assumption f_si ≥ 2B_i of Theorem 4.9.20, the relationships (4.222) and (4.223) hold only for which, in general, is a proper subset of [− 1/2, 1/2], where suppc means “support curve of”. Analogously,

(4.224)

(4.225)

hold only for which, in general, is a proper subset of [− 1/2, 1/2]. Thus, for (jointly) SC processes, (4.222), (4.223) defined by the mapping ν₁ = f₁/f_s1 do not hold for every support curve ∀ν₁ [− 1/2, 1/2] and (4.224), (4.225) defined by the mapping ν₂ = f₂/f_s2 do not hold for every support curve ∀ν₂ [− 1/2, 1/2]. In contrast, for (jointly) WSS processes, spectral masses are present only on the main diagonal of the bifrequency plane and condition f_si ≥ 2B_i assures that (4.222)–(4.225) defined by the mappings ν_i = f_i/f_si hold for the unique support line and the corresponding density ∀ν_i [− 1/2, 1/2], i = 1, 2.

By considering a more stringent condition on the sampling frequencies, (4.222)–(4.225) determined by the ν_i ↔ f_i mappings can be made valid for every support curve ∀ν_i [− 1/2, 1/2], i = 1, 2. We have the following result.

Theorem 4.9.6 Density of Loève Bifrequency Cross-Spectrum of Sampled Strictly Band-Limited Jointly SC Processes (Napolitano 2011, Theorem 4.3). Let x_a,i(t), i = 1, 2, be continuous-time jointly SC processes with Loève bifrequency cross-spectrum (4.209a), (4.209b) and let the processes be strictly band limited with bandwidths B_i. If

(4.226)

then conditions (4.222) and (4.223) hold ∀ν₁ [− 1/2, 1/2]. Analogously, if

(4.227)

then conditions (4.225) and (4.225) hold ∀ν₂ [− 1/2, 1/2].

Proof: See Section 5.10.

For and strictly increasing or decreasing functions, the sup over ν₁ and ν₂ in (4.226) and (4.227) are achieved for ν_i equal to 1/2 or −1/2.

If the sufficient conditions (4.226) and (4.227) are satisfied then, accounting for (4.222)–(4.225), support curves and cross-spectral densities of the continuous-time signals can be reconstructed from those of the discrete-time signals by the frequency rescalings ν₁ = f₁/f_s1 and ν₂ = f₂/f_s2 as follows.

(4.228)

(4.229)

(4.230)

(4.231)

where the second equality in (4.228) and (4.230) holds in all points where support curves do not intersect and and are the bifrequency spectral cross-correlation densities of the Loève bifrequency cross-spectrum (4.190a), (4.190b)

(4.232a)

(4.232b)

defined analogously to their counterparts (4.20a) and (4.20b) for continuous-time processes.

Solutions of inequalities (4.226) and (4.227) can be easily found in the special case where and are linear functions with unit slope. Specifically, for a single ACS process x_a,i(t) = x_a(t), f_si = f_s, i = 1, 2, and (*) present, one has . Thus, , and (4.226) becomes f_s ≥ B + f_s/2 + 2B, that is, f_s ≥ 6B. This condition, however, can be relaxed to f_s ≥ 4B (Section 1.3.9) by exploiting the fact that the support curves are parallel lines (with unit slope) (Gardner et al. 2006), (Izzo and Napolitano 1996) and by using bounds for supports of cyclic spectra.

As a concluding remark, let us consider GACS processes, which are the other class of processes that, as the class of the SC processes, extends that of the ACS processes. The discrete-time analysis of GACS processes differs from that of SC processes. In fact, in Section 2.5 it is shown that continuous-time GACS processes, unlike WSS, ACS, and SC processes, do not have a discrete-time counterpart and uniformly sampling a continuous-time GACS process leads to a discrete-time ACS process. In addition, in Sections 2.2.2 and 2.2.3 it is shown that a GACS process cannot be strictly band limited and is not harmonizable unless in the special case it reduces to an ACS process (Izzo and Napolitano 2002b; Napolitano 2007a).

4.9.3 Illustrative Examples

In this section, illustrative examples of the results of Section 4.9 are presented. These results which are expressed in terms of Loève bifrequency spectra, can be equivalently formulated in terms of spectral cross-correlation densities (4.20a), (4.20b), (4.232a), and (4.232b) by replacing Dirac with Kronecker deltas.

In the first experiment, a pulse-amplitude-modulated (PAM) signal x_a(t) is considered with stationary white modulating sequence, raised cosine pulse with excess bandwidth η, and symbol period T_p. It is second-order cyclostationary with period T_p and strictly band limited with bandwidth (1 + η)/(2T_p). Thus, an upper bound for the bandwidth is B = 1/T_p and the signal has three cycle frequencies α_h = h/T_p, h {0, ± 1} (Gardner et al. 2006). A frequency-warped version of x_a(t), namely

(4.233)

is also considered, with frequency-warping function ψ(f) = B_mtan⁻¹(f/B_s). This kind of frequency warping is used in spectral analysis with nonuniform frequency spacing on the frequency axis (Oppenheim et al. 1971), (Oppenheim and Johnson 1972), (Braccini and Oppenheim 1974). Transformation (4.233) preserves in y_a(t) the possible wide-sense stationarity of x_a(t) (Franaszek 1967), (Franaszek and Liu 1967), (Liu and Franaszek 1969). In contrast, if x_a(t) is ACS, then y_a(t) is SC (Section 4.13). Moreover, the signals y_a(t) and x_a(t) turn out to be jointly SC with Loève bifrequency cross-spectrum having support curves (Section 4.13)

(4.234)

Computing the bifrequency SCD (4.20a) between y_a(t) and x_a(t) leads to

(4.235)

where denote the cyclic spectra of the PAM signal x_a(t).

In the figures, supports of impulsive functions are drawn as “checkerboard” plots with gray levels representing the magnitude of the density of the impulsive functions. In addition, for notation simplicity, the bifrequency SCD of signals a and b is denoted by .

In Figure 4.6, (top) magnitude and (bottom) support of the bifrequency SCD of the continuous-time signal x_a(t) are reported as functions of f₁/f_r and f₂/f_r, where f_r is a fixed reference frequency. In Figure 4.7, (top) magnitude and (bottom) support of the bifrequency SCD in (4.235) of the continuous-time signals y_a(t) and x_a(t) are reported as functions of f₁/f_r and f₂/f_r. In the experiment, η = 0.85, T_p = 4/f_r, B_m = 0.21 f_r and B_s = 0.1 f_r.

A unique sampling frequency f_s = 1/T_s = f_s1 = f_s2 is used for both signals. In Figures 4.8 and 4.9, the bifrequency SCD of the discrete-time signals and , as a function of ν₁ and ν₂ is reported for two values of the sampling frequency f_s. Specifically, in Figure 4.8 (top) magnitude and (bottom) support of the bifrequency SCD for f_s = 2B are reported and in Figure 4.9 (top) magnitude and (bottom) support of the bifrequency SCD for f_s = 4B are reported. According to the results of Theorem 4.9.20, condition f_s = 2B assures non-overlapping replicas (Figure 4.8). However, the SCD of the discrete-time signals along a support curve is not always a scaled version of the SCD of the continuous-time signals along the corresponding support curve ∀ν₁ [− 1/2, 1/2]. Specifically, in Figure 4.10, the slice of the magnitude of the SCD of the continuous-time signals y_a(t) and x_a(t) (thin solid line) along for α_k = 1/T_p and of the rescaled (ν₁ = f₁/f_s) SCD of the discrete-time signals and (thick dashed line) along ν₂ = [B_m tan ⁻¹(ν₁f_s/B_s) − (1/T_p)]/f_s ≡ Ψ^(k)(ν₁), as a function of f₁/f_r is reported for (top) f_s = 2B and (bottom) f_s = 4B. In case f_s = 2B, the left thick-line replica R₁ is exactly overlapped to the thin-line curve but a further aliasing thick-line replica R₂ is present. The two replicas correspond to the supports with the same labels R₁ and R₂ in Figure 4.8 (bottom). In contrast, in case f_s = 4B only one thick-line replica (labeled R₁ in Figure 4.9 (bottom)) is present which exactly overlaps the thin-line curve. In fact, according to Theorem 4.9.21, condition f_s = 4B = f_r is sufficient to prevent the presence of aliasing replicas so that (4.228) holds (with f_s1 = f_s). Specifically, from (4.226) with f_s1 = f_s2 = f_s and B = 1/T_p we have

(4.236)

where the last inclusion follows from inequality 0 ≤ tan ⁻¹(x) < π/2 for x ≥ 0.

Note that for f_s = 2B f_r/2 (Figure 4.10 (top)) the mapping ν₁ = f₁/f_s1 between the arguments of the SCDs of the continuous-and discrete-time signals holds only for f₁ [− f_s/2, 0] and ν₁ [− 1/2, 0] whereas for f_s = 4B f_r (Figure 4.10 (bottom)) such mapping holds for f₁ [− f_s/2, f_s/2] and ν₁ [− 1/2, 1/2].

In the second experiment, a PAM signal x_a(t) with full-duty-cycle rectangular pulse and baud-rate 1/T_p is considered. Such a signal is second-order cyclostationary with period T_p but is not strictly band limited. It has approximate bandwidth B = 1/T_p and presents cycle frequencies α_h = h/T_p, . When it passes through a multipath Doppler channel, the output signal y_a(t) given in (4.56) is SC with Loève bifrequency spectrum given by (4.58) (see also Section 7.7.2). Since the number of cycle frequencies of the continuous-time signal is infinite and the cyclic spectra are not identically zero outside some frequency interval, the discrete-time signal is SC with infinite support curves in the principal spectral domain. However, in the experiment, a finite number N_c of support curves is considered by taking only those curves for which the SCD is significantly different from zero. This number N_c depends on the (finite) number N_r of replicas used to evaluate the aliased bifrequency SCD function. In the experiment, N_r = 5 and N_c = 25 are used and T_p = 4/f_r. The multipath Doppler channel is characterized by K = 2, a₁ = 1, d₁ = 2.5T_s, ν₁ = 0.05/Ts, s₁ = 0.85, a₂ = 0.7, d₂ = 6.8T_s, ν₂ = − 0.03/T_s, and s₂ = 1.25.

In Figures 4.11 and 4.12, for the continuous-time input signal x_a(t) and output signal y_a(t), respectively, (top) magnitude and (bottom) support of the bifrequency SCD are reported as functions of f₁/f_r and f₂/f_r. In Figures 4.13 and 4.14, (top) magnitude and (bottom) support for f_s = 2B and f_s = 5B, respectively, of the bifrequency SCD of the discrete-time signal , as functions of ν₁ and ν₂ are reported. In Figure 4.15, the slice of the magnitude of the SCD of the continuous-time signal y_a(t) along (thin solid line) and of the rescaled (ν₁ = f₁/f_s) SCD of the discrete-time signal along ν₂ = ν₁/s₁ − 1/(T_pf_s) ≡ Ψ^(k)(ν₁) (thick dashed line), as functions of f₁/f_r are reported for (a) f_s = 2B 0.685f_r and (b) f_s = 5B 1.71f_r. In this experiment, the sufficient condition f_s = 2B of Theorem 4.9.20 does not assure a complete non overlap of replicas since the signal x_a(t) is not strictly band limited. For f_s = 5B replicas turn out to be sufficiently separate and also (4.228) is practically verified.