Sampling in the light of Wigner distribution

360

J. Opt. Soc. Am. A / Vol. 21, No. 3 / March 2004

A. Stern and B. Javidi

Sampling in the light of Wigner distribution Adrian Stern and Bahram Javidi Department of Electrical and Computer Engineering, University of Connecticut, Storrs, Connecticut 06269-1157 Received July 25, 2003; revised manuscript received November 17, 2003; accepted November 20, 2003 We propose a new method for analysis of the sampling and reconstruction conditions of real and complex signals by use of the Wigner domain. It is shown that the Wigner domain may provide a better understanding of the sampling process than the traditional Fourier domain. For example, it explains how certain nonbandlimited complex functions can be sampled and perfectly reconstructed. On the basis of observations in the Wigner domain, we derive a generalization to the Nyquist sampling criterion. By using this criterion, we demonstrate simple preprocessing operations that can adapt a signal that does not fulfill the Nyquist sampling criterion. The preprocessing operations demonstrated can be easily implemented by optical means. © 2004 Optical Society of America OCIS codes: 350.6980, 070.2590, 070.2580, 350.5500.

1. INTRODUCTION 1

The classical Shannon sampling theorem is best understood in the Fourier domain. Here we propose a more powerful field for understanding sampling: the Wigner chart.2–5 We show that the Wigner chart provides an appropriate field to explain the classical sampling theorem and also provides additional insights that are not observed with Fourier domain. The Wigner distribution2,4 (WD) is a joint space– spatial frequency representation of real and complex signals. We are motivated to use the WD because it reflects the interaction between a reciprocal pair (space and spatial frequency). We will explain by the following examples. Consider a space-limited (finite support) signal. Its Fourier spectrum is infinitely wide; therefore, the signal cannot be precisely reconstructed from its samples according to the Shannon–Whittaker theorem.1,6 However, the Fourier transform of the same signal can be sampled and completely reconstructed because it is band limited in the reciprocal (space) domain. Thus, we see that the finite bandwidth required in the classical sampling theorem and the finite spatial size limitation play similar roles. The sampling–analysis domain should reflect this duality. Another less trivial example is the Fresnel transform.7 The Fresnel transform has an all-pass transfer function7; therefore, the bandwidth of the signal is maintained by the transform. Consequently, if the bandwidth of the transformed signal is finite the Fresnel transform can be sampled according to the Shannon– Whittaker theorem. But finite bandwidth is not the only condition for sampling. It has been shown8–10 that the Fresnel transform of any signal with a finite support (therefore infinite bandwidth) can be completely reconstructed from its samples. Thus, we see again the conjugation between the reciprocal spaces’ implying that for a better understanding of the sampling process, it should be examined in a joint space–spatial frequency domain. The tool we choose for our investigation is the WD because it provides a compact representation of the signal in the phase space (space–spatial frequency).11 1084-7529/2004/030360-07$15.00

The main result from our Wigner-space-based sampling analysis is a generalized sampling criterion. This sampling criterion can be viewed as a generalization of the Nyquist12 sampling criterion. It differs from the Nyquist criterion in that it imposes a requirement on the local bandwidth (defined in Section 2) of the signal rather than the classical bandwidth (total bandwidth). The generalized sampling criterion is applicable to real and complex signals. The sampling analysis and the generalized sampling criterion are given in Section 2. The Wigner space provides a convenient field for signal processing. In Section 3, we demonstrate with the Wigner chart a few preprocessing methods of signal adaptation for sampling. The proposed preprocessing operations enable us to sample and reconstruct real signals that do not fulfill the Nyquist criterion. A numerical example of a signal reconstructed from the samples of the transformed signal at a rate three times less than the Nyquist rate is presented. All the preprocessing operations demonstrated can be implemented by optical systems.

2. ANALYSIS OF SAMPLING AND RECONSTRUCTION OF SIGNALS IN WIGNER SPACE A. Sampling Analysis in Wigner Space A function f(x) can be described indirectly and uniquely by its WD in the space–spatial frequency domain2,4,13: W f 共 x, ␯ 兲 ⫽

冕 冉 ⬁

⫺⬁

f x⫹

x⬘ 2

冊冉

f* x ⫺

x⬘ 2

冊

exp共 ⫺j2 ␲ x ⬘ ␯ 兲 dx ⬘ , (1)

or, equivalently, W f 共 x, ␯ 兲 ⫽ W F 共 ␯ , x 兲 ⫽

冕 冉 ⬁

⫺⬁

F ␯⫹

␯⬘ 2

冊 冉

F* ␯ ⫺

⫻ exp共 ⫺j2 ␲ x ␯ ⬘ 兲 d␯ ⬘ , © 2004 Optical Society of America

␯⬘ 2

冊 (2)

A. Stern and B. Javidi

Vol. 21, No. 3 / March 2004 / J. Opt. Soc. Am. A

361

where F( ␯ ) is the Fourier transform of f(x):

冕

F共 ␯ 兲 ⫽

⬁

⫺⬁

f 共 x 兲 exp共 ⫺j2 ␲ x ␯ 兲 dx.

(3)

The integrand in Eq. (1) can be obtained by an inverse Fourier transform of the WD: f 共 x 1 兲 f *共 x 2 兲 ⫽

冕 冉 ⬁

⫺⬁

Wf

x1 ⫹ x2 2

冊

, ␯ exp关 j2 ␲ 共 x 1 ⫺ x 2 兲 ␯ 兴 d␯ ,

(4)

where we used the variable transformation x 1 ⫽ x ⫹ x ⬘ /2, x 2 ⫽ x ⫺ x ⬘ /2. Consider an ideal sampled version of f(x) at constant intervals ⌬: f s共 x 兲 ⫽ f 共 x 兲

兺 ␦ 共 x ⫺ n⌬ 兲 ,

(5)

n

where ␦ (•) is the delta distribution. By substituting Eq. (5) in Eq. (1) and using Eq. (4) we find the WD of f s (x): W f s 共 x, ␯ 兲 ⫽

冕 冉 ⬁

⫺⬁

⫻

f x⫹

兺␦ n

⫻

兺␦ l

⫽

冕冕 ⬁

⬁

⫺⬁

⫺⬁

⫻

冉 冉

x⬘ 2

x⫹

x⫺

冊冉

x⬘

f* x ⫺

x⬘ 2 x⬘ 2

2

冊 冊

冊

⫺ n⌬

⫺ l⌬ exp共 ⫺j2 ␲ x ⬘ ␯ 兲 dx ⬘

W f 共 x, ␯ ⬘ 兲 exp共 ⫺j2 ␲ x ⬘ ␯ ⬘ 兲 d␯ ⬘

兺 ␦ 共 x ⫺ n⌬ 兲 兺 ␦ 共 x ⬘ n

l

⫺ l⌬ 兲 exp共 ⫺j2 ␲ x ⬘ ␯ 兲 dx ⬘

⫽

兺 ␦ 共 x ⫺ n⌬ 兲 冕

⬁

⫺⬁

n

W f 共 x, ␯ ⬘ 兲

冕

⬁

兺 ␦ 共x⬘

⫺⬁ l

⫺ l⌬ 兲 exp关 j2 ␲ x ⬘ 共 ␯ ⫺ ␯ ⬘ 兲兴 d␯ ⬘ dx ⬘ ⫽

兺 ␦ 共 x ⫺ n⌬ 兲 冕 n

⫻

兺␦ k

⫽

冉

⬁

⫺⬁

W f 共 x, ␯ ⬘ 兲

␯ ⫺ ␯⬘ ⫺ k

1

1 ⌬

兺 ␦ 共 x ⫺ n⌬ 兲 兺 W ⌬ n

k

冊 冉

f

1 ⌬

d␯ ⬘ x, ␯ ⫺ k

1 ⌬

冊

.

(6)

It can be seen that the WD of f s (x) has nonzero values only at x ⫽ n⌬, which is clearly understood as a result of the sampling process. Also it can be seen that the WD is formed from infinite replicas of the WD of f(x) in the frequency ␯ direction, similar to the way that the Fourier spectrum of sampled signals behaves. The distance between the replicas is 1/⌬. The effect of sampling in the Wigner domain is illustrated in Fig. 1. The rectangle in Fig. 1(a) denotes the space–bandwidth product

Fig. 1. Effect of sampling in the Wigner space. (a) The space– bandwidth product of a typical real function. (b) The space– bandwidth product of the sampled function. The WD of the sampled function has nonzero values at discrete location x and is formed from infinite replicas of the WD frequency direction ␯. The reconstruction is carried out by filtering the WD within the dashed horizontal lines.

关 SW(x, ␯ ) 兴 according to one of the definitions in Refs. 14 and 15. Basically, it is the region for which the WD is essentially nonzero. In Fig. 1(b), we show the Wigner chart of the sampled signal.

B. Aliasing and Reconstruction Condition First, we shall describe the classical sampling theorem by WD. Then we will generalize it in Subsection 2.C. According to the classical Shannon1,16 theorem, if the function f(x) contains no frequencies higher than B ␯ /2 (commonly referred to as the Nyquist frequency12), it is completely determined by providing its ordinates at a series of samples spaced 1/B ␯ apart. The reconstruction from the samples is obtained by passing f(x) through an ideal low-pass filter with a cutoff frequency B ␯ /2. This is understood through the Wigner space representation in Fig. 1. If the distance between the replicas 1/⌬ (commonly referred to as the Nyquist rate12) is smaller than the bandwidth B ␯ then the replicas of Eq. (6) overlap, leading to aliasing. The information in the overlapped area does not represent a one-to-one relation to the continuous signal. Therefore, the WD of the continuous function can not be retrieved from the WD of the sampled one. On the other hand, if the distance between the replicas 1/⌬ is larger than the bandwidth B ␯ then aliasing can be avoided. A horizontal strip can be filtered out from the WD of the sampled signal [Fig. 1(b)] to yield the zero-order zone from the WD [the zone given by k ⫽ 0 in Eq. (6)] of the sampled signal. We will show that this zone has information similar to that of the WD of the continuous signal. The width of the strip must be between B ␯ and 2/⌬ ⫺ B ␯ . In the case of Fig. 1 the object of masking a strip in the WD of the sampled function is obtained by using a low-pass filter that has a cutoff frequency of half the desired strip width. The resulting WD W˜f s (x, ␯ ) obtained by filtering out the zero-order zone from the WD [the zone given by k ⫽ 0 in Eq. (6)] of the sampled signal contains only one replica of W f s (x, ␯ ) in Eq. 6: W˜f s 共 x, ␯ 兲 ⫽

1 ⌬

兺 ␦ 共 x ⫺ n⌬ 兲 W 共 x, ␯ 兲 . f

n

(7)

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J. Opt. Soc. Am. A / Vol. 21, No. 3 / March 2004

Clearly, W˜f s (x, ␯ ) has the same SW(x, ␯ ) as the WD of the original signal [Fig. 1(b)]. From Eq. (7) we see that W˜f s (x, ␯ ) is only a sampled version of W f (x, ␯ ) in the x direction. However, we will show that the sampled version W˜f s (x, ␯ ) determines uniquely the continuous signal f(x). Theorem 1: Let f(x) be a signal of finite energy ( 兰兩 f(x) 兩 2 dx ⬍ ⬁) and finite width B x . There is a one-toone relation between the function f(x) and the samples of the WD in the x direction separated at distance ⌬ ⬍ B x /2. The proof of Theorem 1 is given in Appendix A. Thus, by Theorem 1, if the central part (zero-order zone) of the WD of the sampled function 关 k ⫽ 0 in Eq. (6)] can be masked out without aliasing, the resulting distribution (Eq. 7) uniquely determines the continuous signal and its WD. C. Generalized Sampling Criterion From the previous discussion we conclude that to reconstruct a signal from its samples, we require that the shifted replicas of the WD do not overlap. In the particular case that the maximum frequency component B ␯ /2 and minimum frequency component ⫺B ␯ /2 are at the same location x in the Wigner chart (as for the example in Fig. 1) then the Nyquist sampling criterion is obtained (1/⌬ ⭓ B ␯ ). But other cases are possible as illustrated in Fig. 2. In Fig. 2(a), the SW(x, ␯ ) of a signal with large bandwidth B ␯ is shown. Examples of functions encountered in optics with such kind of SW(x, ␯ ) will be given in Section 3. Figure 2(b) illustrates the SW(x, ␯ ) of the sampled function at a sampling rate 1/⌬ which is much lower than B ␯ . Nevertheless, since the replicas do not overlap, the signal can be reconstructed precisely by applying an appropriate filter.4 In the following we will present a generalization of the Nyquist criterion that includes such cases. Definition: Local bandwidth B ␯ (x 0 ) at location x 0 of the function f(x) is defined as the support of W f (x 0 , ␯ )

A. Stern and B. Javidi

along the frequency ␯ axis for which W f (x 0 , ␯ ) is essentially nonzero. Practically, it can be measured as the width of the space–bandwidth product14,15 at location x 0 along the ␯ coordinate. In Fig. 2(a), the local bandwidths at two locations x 0 and x 1 are illustrated. Theorem 2: A real or complex signal can be completely recovered from its samples if the sampling rate 1/⌬ is larger than its maximum local bandwidth (MLB) 兵 max关B␯(x)兴其 and larger than 2/B x . Proof: The proof of Theorem 2 is a direct result of the previous discussion. If the sampling rate 1/⌬ is larger than the MLB, then there is no overlapping of the replicas of the WD described by Eq. (6). As a result, the central part of the WD of the sampled signal 关 k ⫽ 0 in Eq. (6)] can be filtered out. The resulting distribution is given by Eq. (7), which according to Theorem 1 is a one-to-one mapping of the continuous signal f(x) if ⌬ ⬍ B x /2. The principal condition for sampling according to Theorem 2 is that the sampling rate should be larger than the MLB. The second condition, that the sampling rate is to be larger than 2/B x , is an almost trivial requirement. It states that there should be at least two samples in the support of f(x). It can be seen that Theorem 2 can be viewed as a generalization of the Nyquist criterion in which the classical ‘‘bandwidth’’ is replaced by the MLB. In particular cases where the MLB is the same as the bandwidth B ␯ , as for example in Fig. 1, Theorem 2 reduces to the classical Shannon–Whittaker theorem.

3. SIGNAL ADAPTATION FOR SAMPLING Consider a real or complex signal with the WD chart shown in Fig. 3(a) and assume that we intend to sample it below the Nyquist sampling rate, that is 1/⌬ ⬍ B ␯ . Clearly, direct undersampling of the signal causes overlapping of the replicas in the Wigner space [Fig. 3(b)], preventing perfect reconstruction. However, the signal can be transformed to meet the sampling requirement of Theorem 2. In the following, we give several examples of such kinds of transformations encountered in optical systems. A. Scaling of x Assume that a signal g(x) is obtained by f 共 x 兲 → g 共 x 兲 ⫽ f 共 ax 兲 ,

(8)

where a is a real constant. In optics, g(x) is simply obtained by a change of aspect ratio (magnification). It is easy to verify that in such a case W f 共 x, ␯ 兲 → W g 共 x, ␯ 兲 ⫽ W f 共 ax, ␯ /a 兲 .

(9)

The Wigner chart of g(x) is illustrated in Fig. 3(c). If a ⭐ B ␯ /⌬, the MLB of g is smaller than 1/⌬ and no overlapping occurs, as shown in Fig. 3(d). Fig. 2. Space–bandwidth product of a function for which the maximum frequency B ␯ /2 and the minimum frequency ⫺B ␯ /2 do not appear at the same location x. (b) The space–bandwidth product of the sampled function. A precise reconstruction can be carried out by filtering the area within the dashed rectangle even though the Nyquist criterion is not fulfilled (B ␯ Ⰷ 1/⌬).

B. Fourier Transform Assume that g(x) is the Fourier transform of f(x): f共 x 兲 → g共 x 兲 ⫽

冕

⬁

⫺⬁

f 共 x 兲 exp共 ⫺2 ␲ x ␯ 兲 dx.

(10)

ERRATA Sampling in the light of Wigner distribution: errata Adrian Stern Electro-Optical Department, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel

Bahram Javidi Department of Electrical and Computer Engineering, University of Connecticut, Storrs, Connecticut 06269-2157 © 2004 Optical Society of America OCIS codes: 750.6980, 070.2590, 070.2580, 350.5500.

The result of Eq. (6) in Ref. 1 should be corrected:

W f s 共 x, ␯ 兲 ⫽ ⫽

冕 冉 冕冕 ⬁

⫺⬁ ⬁

⬁

⫺⬁

⫺⬁

⫹ ⫽

f x⫹

冊 冉

x⬘ x⬘ f* x ⫺ 2 2

冊兺 冉 n

␦ x⫹

W f 共 x, ␯ ⬘ 兲 exp共 j2 ␲ x ⬘ ␯ ⬘ 兲 d␯ ⬘

that the sampling rate must be larger than the MLB as

x⬘ ⫺ n⌬ 2

冋兺 n

冊兺 冉

␦ x⫺

l

␦ 共 x ⫺ n⌬ 兲

冊

x⬘ ⫺ l⌬ exp共 ⫺j2 ␲ x ⬘ ␯ 兲 dx ⬘ 2

兺 ␦ 共 x ⬘ ⫺ 2l⌬ 兲 l

册

兺 ␦ 冠x ⫺ 冉 n ⫹ 2 冊 ⌬ 冡 兺 ␦ 冠x ⬘ ⫺ 2 冉 l ⫹ 2 冊 ⌬ 冡 exp共 ⫺j2 ␲ x ⬘ ␯ 兲 dx ⬘ 1

1

n

l

1 2⌬

兺 ␦ 共 x ⫺ n⌬ 兲 兺 W 冉 x, ␯ ⫺ k 2⌬ 冊 ⫹ 2⌬ 兺 ␦ 冠x ⫺ 冉 n ⫹ 2 冊 ⌬ 冡

⫻

兺

1

n

k

k

冉

1

f

exp共 j ␲ k 兲 W f x, ␯ ⫺ k

1

n

冊

1 . 2⌬

Equation (1) is periodic with respect to the spatial frequency ␯ with a period that is twice smaller than that in Eq. (6) in Ref. 1. Therefore, using the arguments in Ref. 1, a sampling condition that is twice more relaxed than that presented in Ref. 1 can be obtained; the sampling rate (1/⌬) must be at least twice larger than the maximum local bandwidth (MLB). However as the example in Ref. 1 demonstrates, signals sampled at sampling rates lower than twice the MLB can be reconstructed (e.g., 1/⌬ ⬇ 1.1MLB in Fig. 4 in Ref. 1). Indeed, in Ref. 2 we prove

1084-7529/2004/102038-01$15.00

(1)

stated in Theorem 2 in Ref. 1, thus validating all the results of Ref. 1.

REFERENCES 1. 2.

A. Stern and B. Javidi, ‘‘Sampling in the light of Wigner distribution,’’ J. Opt. Soc. Am. A 21, 360–366 (2004). A. Stern and B. Javidi, ‘‘Generalized sampling theorem and application to digital holography,’’ in Optical Information Systems II, B. Javidi and D. Psaltis, eds., Proc. SPIE 5557 (2004).

© 2004 Optical Society of America