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^ = 1=Ns independent. Using (1) and R show, after some algebra, that

f

3 E R R

RRg

=E

1 N

Ns

n=1

1 N

Ns

= N1

s

E

k=1

6Nn=1 x(n)xH (n), we can

3 T x n x n

( ) ( ) 0 R3

( ) H (k) 0 R

x k x

[2] F. Li and R. Vaccaro, “Unified analysis for DOA estimation algorithms in array signal processing,” Signal Process., vol. 25, pp. 147–169, Nov. 1991. [3] A. Graham, Kronecker Products and Matrix Calculus: with Applications. London, U.K.: Ellis Horwood, 1981. [4] B. Porat and B. Friedlander, “Performance analysis of parameter estimation algorithms based on high-order moments,” Int. J. Adapt. Contr. Signal Process., vol. 3, pp. 191–229, 1989. [5] D. H. Brandwood. “A complex gradient operator and its application in adaptive array theory,” Proc. Inst. Elect. Eng., pp. 11–16. Feb. 1983.

f(x3 (n)xT (n)) (x(n)xH (n))g

0 N1s R3 R

and

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(32)

High Resolution Spectral Analysis of Images Using the Pseudo-Wigner Distribution

f RRg = N1s E f(x(n)xH (n))

E R R

(x(n)xH (n))g 0 N1s

Javier Hormigo and Gabriel Crist´obal R

R:

(33)

We can use (32) to compute 61 of (23). In other words, the elements of 61 are the elements of (32). However, the ordering the the two matrices are different so that a reordering of (32) is needed to compute 61 : Similarly, we can use (33) to compute 62 of (23). The two matrices of fourth-order moments E f(x3 (n)xT (n))

(x(n)xH (n))g and E f(x(n)xH (n)) (x(n)xH (n))g are needed to compute (32) and (33), respectively. If we assume that n is a vector of zero mean circularly symmetric complex noise and the signals are distributed in such a way that the odd moments are zero, then we can show E

f(xxH )3 (xxH )g = (W 3 W )E fs3sT ssH g(W T W H ) + (W S W H )3 Rn + (W 3 W ) vec (S ) vec (Rn )H + vec (Rn ) vec (S )H (W T W H ) + Rn3 W S W H + E f(n3 n)(n3 n)H g

(34)

f(xxH ) (xxH )g = (W W )E fssH ssH g(W W )H + W S W H Rn + (W I )E fsnH nsH g(I W H ) + (I W )E fsnH nsH gH (W H I ) + Rn W S W H + E f(n n)(n n)H g:

(35)

and E

Index Terms— Aliasing, analytic signal, multicomponent signals, Wigner distribution.

I. INTRODUCTION

The matrix R n is the covariance matrix of the noise. The matrices E s3sT ssH ; E ssH ssH ; E n3 n n3 n H ; E n n n n H and E snH ns H have elements that depend on the

f

)(

g

g f f

Abstract—Several methods for the computation of the discrete Wigner Distribution (DWD) through the use of two-dimensional (2-D) analytic signals have been proposed, depending of the direction of the phase shift. Most of the methods cope the problem of aliasing of the DWD by lowpass prefiltering the spectrum but reducing spatial frequency support. In this correspondence, a new method for 2-D DWD free of aliasing and simultaneously increasing the spatial frequency support is proposed through the use of a new analytic signal. In this way, local spectral analysis for small window sizes can be accomplished improving the resolution but reducing cross terms and other interferences inherent in the DWD computation. For the three analytic images used in the comparison, the method proposed here also performs better than the other two for such tasks.

g f( )( ) g f(

g

second- and fourth-order moments of the signals and noise. These moments depend on the particular probability distributions of the signals and noise. Given these distributions, (34) and (35) can then be computed. REFERENCES [1] H. Liu and G. Xu, “A subspace method for signature waveform estimation in synchronous CDMA systems,” IEEE Trans. Commun., vol. 44, pp. 1346–1354, Oct. 1996.

Over the last 20 years, the joint space-spatial frequency representations have received special attention in the fields of image processing, vision, and pattern recognition. They could be more useful than the general techniques of spatial filtering of images, and therefore, the possibilities of information filtering would be improved. Some researchers have considered the Wigner distribution (WD) as a master distribution from which the rest of distributions can be derived [1], [19]. One of the most salient WD properties is its high time–frequency concentration, but it may suffer from aliasing and cross terms that in principle can reduce the scope of its applications. We refer to several thorough reviews about the WD for one- and two–dimensional (1D and 2-D) signals [6], [9]. To cope with the problem of cross terms, some different modifications of the original definition have been proposed, e.g., a kernel-type WD [3], analytic signals [30], [31], image processing techniques [23], power type L-Wigner [20], Manuscript received March 31, 1997; revised January 6, 1998. This work was supported in part by NATO Collaborative Grants Program, Spanish Commission for Science and Technology Grant TIC94-0849. J. Hormigo was supported in part by a Spanish Ministry of Education and Culture fellowship. The associate editor coordinating the review of this paper and approving it for publication was Dr. Phillip A. Regalia. J. Hormigo is with the Departmento de Arquitectura de Computadores, Universidad de Malaga, Malaga, Spain (e-mail: [email protected]). G. Crist´obal is with the Instituto de Optica (CSIC), Madrid, Spain (e-mail: [email protected]). Publisher Item Identifier S 1053-587X(98)04032-X.

1053–587X/98$10.00 1998 IEEE

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or multidimensional WD in the frequency domain [22]. For a recent review about the reduced interference distributions, see [27]. The motivation of this correspondence is twofold. The first is to obtain a pseudo-Wigner distribution (PWD) with high resolution through the use of a new 2-D analytical signal but simultaneously eliminating aliasing and reducing as much as possible the presence of cross terms. The PWD and the spectrogram can be considered to be particular cases of filtering. In most situations, signal processing filters are used to create a sparse sampling of the space/spatial-frequency domain. However, there exist some applications, e.g., shape from texture, that require a more dense sampling provided by the WD or the spectrogram [16], [17]. Second, the reduction of cross terms will be considered on the basis of providing a feasible quantitative measure of the PWD concentration or “peakedness.” In Section II, a brief review of the continuous WD and the discrete PWD is presented. In Section III, a new 2-D analytical image is proposed for computing the PWD and simultaneously eliminating aliasing and reducing cross terms. In Section IV, we will evaluate and compare the results of computing the PWD using the new analytic signal using some synthetic and natural image tests.

Fig. 1. WD computation through three different analytic images. (a) Analytic image proposed here. (b) “One fourth domain method” proposed in [25]. (c) Analytic image proposed in [30].

II. THE WIGNER DISTRIBUTION The Wigner distribution was introduced by [26] as a phase space representation in quantum mechanics and gives a simultaneous representation of a signal in space and spatial frequency variables. Subsequently, Ville derived in the area of signal processing the same distribution that Wigner proposed in quantum mechanics [25]. Many other authors later use “Wigner–Ville” or “Ville distribution” to denote the same distribution. The symmetric definition of the Wigner distribution (WD) corresponding to a continuous and integrable 1-D signal is given by Wf (x; ) =

1 01

f x+

2

f3 x

0 2 e0j d

(1)

where x and are spatial variables, is the spatial frequency variable, and f 3 (1) means the complex conjugate of f (1). The WD can be considered as a particular occurrence of a complex spectrogram, where the function f (x) is chosen as the shifting window function. It belongs to a large class of bilinear distributions known as the Cohen’s class in which each member can be obtained choosing different kernels in the generalized bilinear distribution [5]. The WD presents a set of desirable mathematical properties, as was formulated by [4], including high signal concentration (maximum autocomponent concentration in space/spatial frequency). However, the presence of cross terms has limited its use in some practical applications. Cross terms can have a peak value as high has twice that of the autocomponents [15]. It is generally accepted that smoothing the WD will reduce the presence of cross terms but at the expense of smearing the autocomponent concentration. This effect has been recently demonstrated in [21]. In numerical implementations, we can consider the so-called pseudo Wigner distribution (PWD) by using two smoothing windows: a spatial averaging window g (l) and a spatial-frequency averaging window h(k) [2], [18], [19].1 This double smoothing operation will produce a further improvement in the cross term reduction. The 1 Although the term Pseudo-Wigner distribution is used when smoothing is done in the frequency domain only, we use the same term when the smoothing operation is done in both domains, following [18] and [19]. Windowing and averaging are dual operations. Therefore, in the following sections, we refer to g (1) and h(1) as the frequency and the spatial window, respectively.

PWD can be defined by

0

N 1

PWD(n; ) =

0

2

2

0j 2k

[h(k )] e

k= N +1 M 1

0

0

g [l]f [n + l + k]f 3 [n + l

l= M +1

0 k]

(2)

where n and = 2=N are the spatial and frequency variables, respectively, and M and N are the spatial and spatial-frequency window size, respectively. The PWD for 2-D signals will be given by PWD(n1 ; n2 ; 1 ; 2 ) N

=

0

01

N

0

01

hN N (k; l)

k= N +1 l= N +1 M 1 M 1

2

0

0

gM M (r; s)f (n1 + r + k; n2 + s + l)

0 0 2 f 3(n1 + r 0 k; n2 + s 0 l)e0j2(k r= M +1 s= M +1

+l ):

(3)

III. ANALYTIC SIGNAL PROPOSED The concept of the 1-D analytic signal was introduced by Gabor as a complex signal that has a spectrum identical to that of the real signal for positive frequencies and zero for negative frequencies [7], [11]. The extension of the analytic signal concept for multidimensional signals has been done recently by Hahn [12], but for dimensions bigger that one, the definition of the multidimensional analytic signal is not unique. In practice, the 2-D DWD of the analytic signal (which is also called the Wigner–Ville distribution) is used to overcome the principal discretization problems: aliasing and cross terms [28]. Aliasing or frequency foldover in discrete signals is due to the use of an insufficient sampling rate (below Nyquist limit, i.e., double of highest frequency content). To prevent aliasing, two methods arise to reduce this problem: a) oversampling (or interpolating) by a factor of 2 and b) frequency filtering the regions that cause aliasing (lowpass filtering). The latter can substantially suppress artifacts but at the cost of reducing significantly the spatial-frequency support. One of the problems associated with the 2-D DWD is that its periodicity is instead of the 2 of the Fourier transform [4], [8]. This fact implies that signals for which 2-D DFT does not introduce aliasing may be distorted with aliasing in the 2-D PWD due to a wraparound frequency phenomenon for high frequencies [16], [24]. As stated before, the definition of an analytic image for 2-D signals is not unique [24], [29]. Several 2-D analytic signals have been proposed elsewhere with the aim to reduce aliasing artifacts but retaining the main WD properties. However, most of the methods fail to produce a substantial reduction of the aliasing and cross terms. The analytic image proposed here can be considered as a combination of two previous definitions [24], [29]. The region of support of the analytic image here [Fig. 1(a)] is the same that the proposed in [29] [Fig. 1(c)]

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 6, JUNE 1998

Fig. 2. WD computation through the new analytic image. (a) Region of support corresponding to low frequencies in the vertical direction. (b) WD corresponding to the analytic image in (a). (c) Region of support corresponding to high frequencies in the vertical direction. (d) WD corresponding to the analytic image in (c). (f) WD composite associated with (b) and (d).

but separating the region considered into two parts [depicted in Fig. 1(a) by two different shadings]. Although the present method requires the computation of two WD corresponding to each region, this is not in detriment of a substantial computational cost reduction in comparison with the interpolation-based methods. The latter one can be easily implemented by a zero-padding operation and, therefore, with higher computational cost by computing a PWD with four times the number of samples.2 Here we, consider a new 2-D analytic signal that might avoid most of the shortcomings of the previous definitions. This method is based in zeroing out the negative side of the spectrum as in Fig. 1(c).3 However, here, we split the positive region in two subregions [see Fig. 1(a)] and take a PWD from each subregion. Fig. 2 depicts a two-step procedure for obtaining the new analytic image proposed here. Fig. 2(a) and (c) represent by shading the signal’s frequency components retained in the analytic image proposed here. The first PWD is taken from the region shown in Fig. 2(a), producing a PWD given in Fig. 2(b). This spectrum can be rearranged in the regions A and B depicted in Fig. 2(f) according to the following regions of support.4 PWD1 =

Region A Region B

0

1 < =2; 0=2 2 < =2 : 1 < ; 0=2 2 < =2

=2

(4)

With the mask represented in Fig. 2(b). we filter out the higher frequencies that can be arranged according to the following four regions [see Fig. 2(f)].

PWD2 =

Region Region Region Region

C D E F

1 < ; 0 2 < 0=2 1 < =2; 0 2 < 0=2 =2 1 < ; =2 2 < 0 1 < =2; =2 2 > : =2 0

(5)

Regions A, D, and F in Fig. 2(b) and (d) correspond to positive frequencies, and regions B, C, E correspond to negative frequencies. The final PWD will be the sum of PWD1 + PWD2 . This procedure 2 The interpolation-based methods will require a window h(1) of double size for obtaining the same PWD resolution. 3 Any other criteria that preserve the vanishing halfplane property can be considered as in [29]. 4 This rearrangement leads to a controlled frequency foldover.

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obviously produces a PWD with higher resolution but in a range 0 to and, therefore, doubles the number of points along each coordinate. If the original image is real, the PWD presents a property of symmetry with respect to the origin of the spatial frequency domain. Fig. 2(f) depicts such property by using identical shadings between symmetrical regions. It must be observed here that some potential drawbacks of the analytic signal defined here are the introduction of additional sharp filter boundaries and boundary effects that might lead to ringing, besides the creation of a frequency variant PWD. Ringing has been ameliorated by smoothing all the sharp transitions with a raised cosine (Hanning) function. Fig. 3 illustrates the described ringing phenomenon for the analytic image represented in Fig. 1(c). Fig. 3(a) shows the original test image composed by different patches of 2D sinusoids in different orientations. Fig. 3(b) shows the magnitude of the analytic signal associated with Fig. 3(a) by considering the analytic image of Fig. 1(c). It can be observed how the vertical sinusoid appears as an artifact in the diagonal region [Fig. 3(b), top right] due to the sharp transition between the “positive” and “negative” regions. Fig. 3(c) shows the result after smoothing the edge between both regions. With regard to the cross terms issue, it must be noted that the removal of the negative frequencies of the analytic signal also eliminates the cross terms between positive and negative frequencies. However, in addition to that, cross terms have oscillatory terms of high relative frequencies that can often be reduced by PWD lowpass filtering at the expense of autoterm broadening [15]. The present method outperforms traditional ones by removing crossterms between higher and lower spatial-frequency regions without autoterm broadening. IV. RESULTS A. PWD Computation of Image Tests In this section, we will evaluate and compare the performance of the new PWD for some test images. Fig. 4(a) and (b) show two examples of a synthetic image “zone plate” cos(r2 ) with variable local frequency along the two main diagonals. Fig. 4(c) shows the multicomponent test image considered here composed by the sum of Figs. 4(a) and (b) plus a small dc term. Fig. 5(a) depicts the results of the PWD computation corresponding to the new analytic image proposed here [Fig. 5(a[e–h])] in comparison with two other alternative analytic signals showed in Fig. 1(b) and (c) [Fig. 5(a[i–l]) and Fig. 5(a[m–p]), respectively]. The four points of interest have been shown in [Fig. 5(a[a–d])] by the corresponding line intersections. For the first point [Fig. 5(a[a])] corresponding to a low to mid-range frequency of Fig. 4(c), the three methods provide similar results, but for the three other points corresponding to a higher frequency [Fig. 5(a[b–d])], the analytic signal method depicted in Fig. 1(b) simply fails to represent such range [Fig. 5(a[j–l])], and the analytic signal depicted in Fig. 1(c) produces aliasing [Fig. 5(a[n–p])]. Another example is shown in Fig. 5(b) for the four points shown in Fig. 5(b[a–d]). This example illustrates that the problem of the sharp transitions between the different regions is still present in all methods. However, we can observe that the method proposed here performs better than the other two methods. As the frequency increases, the presence of the border attenuates the frequencies for some points [Fig. 5(b[f–g])], but the other two methods either fail to represent such range [Fig. 5(b[k–l])] or introduce aliasing [Fig. 5(b[n–p])]. A substantial reduction of the PWD cross terms can be obtained by spatial and spatial-frequency smoothing. Fig. 6(a) shows a multicomponent image composed of two pure tones located in adjacent spatial positions. It is well known that the WD will produce

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(a)

(a)

(b)

(b)

(c) Fig. 4. Multicomponent test image. (a) Circular chirp “zone plate” 1 (frequency variation upwards). (b) Zone plate 2(frequency variation downwards). (c) Image sum of (a) plus (b) plus a small dc term (10% of the maximum).

(c) Fig. 3. Ringing artifacts with the definition of a 2-D analytic image of Fig. 1(c). (a) Original image. (b) Absolute value of analytic image of Fig. 1(c). (c) Same as (b) but smoothing the edge with a raised-cosine function.

cross terms in the transition between the two regions [7], [8]. Fig. 6(b) and (c) (from top left to bottom right) shows the PWD

associated with points along a horizontal line crossing the border of separation between the two regions for two different window sizes. For Fig. 6(b) and (c), we can observe the effect of cross term reduction by frequency smoothing. Fig. 6(b) shows the PWD computation of several points in the perpendicular direction across the border. Each line represents the PWD corresponding to the same points along the border between the two regions for a frequency window size of M = 0 (no frequency window), 3; 7; and 15 respectively. By increasing the frequency window size [Fig. 6(b), rows from top to bottom], a reduction in the cross terms can be obtained, but simultaneously, a reduction of the spatial resolution is observed (number of border points). Fig. 6(c) shows the PWD for the same points considered in Fig. 6(a) but increasing the window size. Each line represents the PWD corresponding to points along the border between the two regions for a frequency window size of 0 (no frequency window), 3; 7; and 15; respectively. In summary, Fig. 6(b) and (c) shows in a qualitative manner how increasing the

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(c)

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(a)

(a)

(b)

(b)

(d)

Fig. 5. (a) [See (d) for legend] PWD computation (spatial raised-cosine window size = 15 15; frequency window size = 5 5) of a composite of two cos(r2 ) image “zone plates” (image size = 128 128 b/pixel) at several points of interest (indicated by the line intersection). The second column corresponds to the analytic signal proposed here [Fig. 1(a)]. Note either the presence of aliasing or the absence of spectrum terms in the other two analytic signals (third and forth column, respectively). (b) [See (c) for legend] Same as (a), but the second row corresponds to the analytic signal proposed here [Fig. 1(a)]. Same remarks as in (a) hold here related with third and fourth rows, respectively. (c) Legend for Fig. 5(b). (d) Legend for Fig. 5(a).

2

2 2

(c)

spatial window size reduces the spatial resolution but increases the frequency concentration. B. Concentration Measures In this section, we will introduce a quantitative measure of signal’s concentration. Several measures of space-spatial frequency concen-

Fig. 6. (a) Image test composed by two sinusoidal tones with different. (b) PWD computation of (a) along points near the border of the two regions for a spatial window size N = 15 15 for different frequency window sizes (M = 0; 3; 7; and 15 from top row to bottom row, respectively). (c) PWD computation of (a) along points near the border of the two regions for a spatial window size 31 31 for different frequency window sizes (M = 0; 3; 7; and 15 from top row to bottom row, respectively).

2

2

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the frequency window size produces a reduction in the concentration measure (as well as a reduction of its value at the transition region). V. CONCLUSION We proposed a 2-D PWD through the use of a new analytic image with high resolution and frequency support. In addition to that, a substantial reduction of cross terms has been obtained by spatial and spatial-frequency smoothing. A quantitative measure of peakedness has been computed for 2-D PWD with some synthetic and textured images for evaluating the cross term reduction. The present scheme for 2-D PWD computation has been successfully applied for texture segmentation and classification through eigenanalysis in the PWD domain and will appear in a forthcoming publication. ACKNOWLEDGMENT The authors would like to thank the reviewers for their very helpful comments and constructive suggestions for improving the manuscript. (a)

REFERENCES

(b) Fig. 7. Evaluation of a concentration measure corresponding to PWD of Fig. 6(a). (a) Different spatial window sizes have been considered here (see legend) for a frequency window size (g ) = 9 9. (b) Different frequency windows influence of this spatial smoothing operation in the reduction of cross terms (see legend) for a raised-cosine spatial window of size (h) = 15 15.

2

2

tration has been proposed in the literature, but most of them has been defined for single-component signals. To give a quantative figure of merit of the reduction of cross terms in the case of multicomponent signals, we have computed a measure of peakedness in a similar manner to the defined by [14]. This measure is analogous to the kurtosis used in statistics and in other fields as vision modeling [10] and is defined for 1-D signals by C=

N01 0N +1 N01 0N +1

N01 jPWD(n; )j4 0N +1 N01 jPWD(n; j2 )2 0N +1

(6)

the extension for 2-D signals being straightforward. Fig. 7(a) depicts the results of the evaluation of the peakedness measure C for several points across a direction perpendicular to the border of a synthetic multicomponent image [Fig. 6(a)]. In this plot, we can observe the influence of changing the spatial window size for a fixed value of the frequency window size (g ) = 9 2 9. Increasing the spatial window size produces an increase in the concentration measure C (and a reduction in its value at the transition region). On the contrary, Fig. 7(b) shows how a progressive increment in

[1] K. H. Bartelt, H. O. Brenner, and A. W. Lohmann, “The Wigner distribution and its optical production,” Opt. Commun., vol. 32, no. 1, pp. 32–38, 1980. [2] M. Basseville, P. Flandrin, and N. Martin, “M´ethodes temps-fr´equence,” Traitment du Signal, vol. 9, pp. 79–112, 1992. [3] H. Choi and W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 862–871, 1989. [4] T. A. C. M. Claasen and W. F. G. Mecklenbrauker, “The Wigner distribution, a tool for time-frequency signal analysis. Part I. Continuous-time signals,” Philips J. Res., vol. 35, pp. 217–250, 1980. [5] L. Cohen, “Generalized phase-espace distribution functions,” J. Math. Phys., vol. 7, pp. 781–786, 1966. , “Time-frequency distributions, a review,” Proc. IEEE, vol. 33, [6] pp. 31–37, 1989. , Time-Frequency Analysis Englewood Cliffs, NJ: Prentice-Hall, [7] 1995. [8] G. Cristobal, J. Bescos, and J. Santamaria, “Image analysis through the Wigner distribution function,” Appl. Opt., vol. 28, no. 2, pp. 262–271, 1989. [9] G. Cristobal, C. Gonzalo, and J. Bescos, “Image filtering and analysis through the Wigner distribution function,” in Advances in Electronics and Electron Physics, P. W. Hawkes, Ed. Boston, MA: Academic, 1991, vol. 80, pp. 309–397. [10] D. J. Field, “What is the goal of sensory coding?,” Neural Comput., vol. 6, no. 4, pp. 559–601, 1994. [11] D. Gabor, “Theory of communication,” J. Inst. Elect. Eng., vol. 93, pp. 429–457, 1946. [12] S. L. Hahn, “Multidimensional complex signals with single-orthant spectra,” Proc. IEEE, vol. 80, pp. 1287–1300, 1992. [13] L. Jacobson and H. Wechsler, “Joint spatial/spatial-frequency representation,” Signal Process., vol. 14, pp. 37–68, 1988. [14] D. L. Jones and T. W. Parks, “A high resolution data-adaptive timefrequency representation,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 38, pp. 2127–2135, 1990. [15] D. L. Jones and T. W. Parks, “A resolution comparison of several timefrequency representations,” IEEE Trans. Signal Processing, vol. 40, pp. 413–420, 1992. [16] J. Krumm, “Space frequency shape inference and segmentation of 3D surfaces,” Ph.D. dissertation, Carnegie Mellon Univ., Pittsburgh, PA, 1993. [17] J. Malik and R. Rosenholtz, “Computing local surface orientation and shape from texture for curved surfaces,” Tech. Rep. UCB/CSD 93/775, Univ. Calif. Berkeley, Comput. Sci. Division, 1994. [18] W. Martin and P. Flandrin, “Detection of changes of signal structure by using the Wigner–Ville spectrum,” Signal Process., vol. 8, pp. 215–233, 1985. [19] T. Reed and H. Wechsler, “Segmentation of textured images and Gestalt organization using spatial/spatial-frequency representations,” IEEE Trans. Pattern Anal. Machine Intell., vol. 12, pp. 1–12, 1990. [20] L. Stankovic, “A method for improved concentration in the timefrequency analysis of multicomponent signals using the L-Wigner

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distribution,” IEEE Trans. Signal Processing, vol. 43, pp. 1262–1268, May 1995. , “Autoterm representation by the reduced interference distributions: A procedure for kernel design,” IEEE Trans. Signal Processing, vol. 44, pp. 1557–1564, June 1996. S. Stankovic, L. Stankovic, and Z. Uskokovic, “On the local frequency, group shift and cross-terms in some multidimensional time-frequency distribution: A method for multidimensional time-frequency analysis,” IEEE Trans. Signal Processing, vol. 45, pp. 1719–1725, July 1997. M. Sun, C. C. Li, L. N. Sekhar, and R. J. Sclabassi, “Elimination of cross-components of discrete pseudo Wigner distribution via image processing,” in Proc. IEEE Int. Conf. Acoust. Speech Signal Process., 1989, pp. 2230–2233. H. Suzuki and F. Kobayashi, “A method of two-dimensional spectral analysis using the Wigner distribution,” Electron. Commun. Jpn., vol. 75, no. 1, pp. 1006–1013, 1992. J. Ville, “Theorie et applications de la notion de signal analitique,” Cables et Transmission, vol. 2A, pp. 61–74, 1948. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev., vol. 40, pp. 749–759, 1932. W. J. Williams, “Reduced interference distributions: Biological applications and interpretations,” Proc. IEEE, vol. 84, pp. 1264–1280, 1996. Y. M. Zhu and R. Goutte, “Analysis and comparison of space/spatialfrequency and multiscale methods for texture segmentation,” Opt. Eng., vol. 34, no. 1, pp. 269–282, 1995. Y. M. Zhu, R. Goutte, and M. Amiel, “On the use of a two-dimensional Wigner-Ville distribution for texture segmentation,” Signal Process., vol. 30, pp. 205–220, 1993. Y. M. Zhu, R. Goutte, and F. Peyrin, “The use of a two-dimensional Hilbert transform for Wigner analysis of 2-dimensional real signals,” Signal Process., vol. 19, pp. 205–220, 1990. Y. M. Zhu, F. Peyrin, and R. Goutte, “Equivalence between the twodimensional real and analytic signal Wigner distribution,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 1631–1634, Oct. 1989.

Finite-Precision Analysis of the Pipelined Strength-Reduced Adaptive Filter Manish Goel and Naresh R. Shanbhag Abstract— In this correspondence, we compare the finite-precision requirements of the traditional cross-coupled (CC) and a low-power strength-reduced (SR) architectures. It is shown that the filter block (F block) coefficients in the SR architecture require 0.3 bits more than the corresponding block in the CC architecture. Similarly, the weight-update (WUD) block in the SR architecture is shown to require 0.5 bits fewer than the corresponding block in the CC architecture. This finite-precision architecture is then used as a near-end crosstalk (NEXT) canceller for 155.52 Mb/s ATM-LAN over unshielded twisted pair (UTP) category-3 cable. Simulation results are presented in support of the analysis.

I. INTRODUCTION Strength reduction is an algebraic transformation that has been proposed [4] to trade off multipliers with adders in a complex multiplication, thereby achieving power reduction. In [7], we proposed the application of strength reduction transformation at the Manuscript received November 12, 1996; revised December 11, 1997. This work was supported by the NSF CAREER Award MIP-9623737. The associate editor coordinating the review of this paper and approving it for publication was Dr. Konstantin Konstantinides. The authors are with the Coordinated Science Laboratory and Electrical and Computer Engineering Department, University of Illinois at UrbanaChampaign, Urbana, IL 61801 USA (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 1053-587X(98)03944-0.

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algorithmic level to adaptive systems involving complex signals and filters. It was shown in [7] that the strength-reduced (SR) filter enables power savings of 21–25% over the traditional cross-coupled (CC) filter with no loss in performance. However, the application of strength reduction increases the critical path, and, hence, an inherently pipelined SR (PIPSR) architecture was also presented. Furthermore, by trading the throughput gained through pipelining with power supply scaling [4], it was demonstrated that additional power savings of 40–69% are feasible. In this correspondence, we compare the finite-precision requirements of the SR and the PIPSR architectures developed in [7] with that of the CC architecture. It is shown that the precision requirements of the SR and PIPSR architectures are similar to those of the CC architecture. This makes the SR and the PIPSR architectures attractive alternatives to the traditional CC architecture for high bit-rate communications and digital signal processing applications. In this correspondence, a linear model is employed for coefficient quantization noise. The filter (F) block precision BF is chosen such that the signal-to-quantization-noise-ratio (SQNR) is greater than the desired signal-to-noise ratio SNRo . The coefficient precision for weight-update (WUD) block BWUD is determined by applying the stopping criterion [3], [5], which puts a lower limit upon the correction term being added to the weight update. This criterion is given by

2 E [je(n)j2 ]x2 202B

(1.1)

where

E [je(n)j2 ] x2 BWUD

step-size; mean-squared error; power of the received signal x(n); precision (including sign-bit) of the coefficients in the WUD block. A more accurate nonlinear analysis presented in [1] and [2] can be employed to provide a tighter bound on BWUD . However, the purpose of this paper is to compare the precision requirements for CC and SR architectures, and hence, we employ the analysis in [3]. This analysis provides useful design guidelines for applications such as those in digital subscriber loops where the final step sizes are reasonably large. We demonstrate an application of the finite-precision SR architecture as a near-end crosstalk (NEXT) canceller for 155.52 Mb/s [6] ATM-LAN over 100 m of unshielded twisted pair category-3 (UTP3) cable employing 64-CAP (carrierless amplitude/phase) modulation scheme. We present the simulation results for this application in order to determine the precision requirements of various signals and to support the analytical results presented in the correspondence. The organization of the paper is as follows. In Section II, we present PIPSR adaptive filter architecture. In Section III, we determine the finite-precision requirements of CC, SR, and PIPSR architectures. Finally, in Section IV, the finite-precision architectures are employed as a near-end crosstalk (NEXT) canceller for 155.52 Mb/s ATM-LAN. II. A PIPELINED STRENGTH-REDUCED (PIPSR) ADAPTIVE FILTER In this section, we review the strength reduction transformation and development of the PIPSR architecture [7] from the CC architecture. The product of two complex numbers (a + |b) and (c + |d) is given by (a + |b)(c + |d) = (ac

1053–587X/98$10.00 1998 IEEE

0 bd) + |(ad + bc):

^ = 1=Ns independent. Using (1) and R show, after some algebra, that

f

3 E R R

RRg

=E

1 N

Ns

n=1

1 N

Ns

= N1

s

E

k=1

6Nn=1 x(n)xH (n), we can

3 T x n x n

( ) ( ) 0 R3

( ) H (k) 0 R

x k x

[2] F. Li and R. Vaccaro, “Unified analysis for DOA estimation algorithms in array signal processing,” Signal Process., vol. 25, pp. 147–169, Nov. 1991. [3] A. Graham, Kronecker Products and Matrix Calculus: with Applications. London, U.K.: Ellis Horwood, 1981. [4] B. Porat and B. Friedlander, “Performance analysis of parameter estimation algorithms based on high-order moments,” Int. J. Adapt. Contr. Signal Process., vol. 3, pp. 191–229, 1989. [5] D. H. Brandwood. “A complex gradient operator and its application in adaptive array theory,” Proc. Inst. Elect. Eng., pp. 11–16. Feb. 1983.

f(x3 (n)xT (n)) (x(n)xH (n))g

0 N1s R3 R

and

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(32)

High Resolution Spectral Analysis of Images Using the Pseudo-Wigner Distribution

f RRg = N1s E f(x(n)xH (n))

E R R

(x(n)xH (n))g 0 N1s

Javier Hormigo and Gabriel Crist´obal R

R:

(33)

We can use (32) to compute 61 of (23). In other words, the elements of 61 are the elements of (32). However, the ordering the the two matrices are different so that a reordering of (32) is needed to compute 61 : Similarly, we can use (33) to compute 62 of (23). The two matrices of fourth-order moments E f(x3 (n)xT (n))

(x(n)xH (n))g and E f(x(n)xH (n)) (x(n)xH (n))g are needed to compute (32) and (33), respectively. If we assume that n is a vector of zero mean circularly symmetric complex noise and the signals are distributed in such a way that the odd moments are zero, then we can show E

f(xxH )3 (xxH )g = (W 3 W )E fs3sT ssH g(W T W H ) + (W S W H )3 Rn + (W 3 W ) vec (S ) vec (Rn )H + vec (Rn ) vec (S )H (W T W H ) + Rn3 W S W H + E f(n3 n)(n3 n)H g

(34)

f(xxH ) (xxH )g = (W W )E fssH ssH g(W W )H + W S W H Rn + (W I )E fsnH nsH g(I W H ) + (I W )E fsnH nsH gH (W H I ) + Rn W S W H + E f(n n)(n n)H g:

(35)

and E

Index Terms— Aliasing, analytic signal, multicomponent signals, Wigner distribution.

I. INTRODUCTION

The matrix R n is the covariance matrix of the noise. The matrices E s3sT ssH ; E ssH ssH ; E n3 n n3 n H ; E n n n n H and E snH ns H have elements that depend on the

f

)(

g

g f f

Abstract—Several methods for the computation of the discrete Wigner Distribution (DWD) through the use of two-dimensional (2-D) analytic signals have been proposed, depending of the direction of the phase shift. Most of the methods cope the problem of aliasing of the DWD by lowpass prefiltering the spectrum but reducing spatial frequency support. In this correspondence, a new method for 2-D DWD free of aliasing and simultaneously increasing the spatial frequency support is proposed through the use of a new analytic signal. In this way, local spectral analysis for small window sizes can be accomplished improving the resolution but reducing cross terms and other interferences inherent in the DWD computation. For the three analytic images used in the comparison, the method proposed here also performs better than the other two for such tasks.

g f( )( ) g f(

g

second- and fourth-order moments of the signals and noise. These moments depend on the particular probability distributions of the signals and noise. Given these distributions, (34) and (35) can then be computed. REFERENCES [1] H. Liu and G. Xu, “A subspace method for signature waveform estimation in synchronous CDMA systems,” IEEE Trans. Commun., vol. 44, pp. 1346–1354, Oct. 1996.

Over the last 20 years, the joint space-spatial frequency representations have received special attention in the fields of image processing, vision, and pattern recognition. They could be more useful than the general techniques of spatial filtering of images, and therefore, the possibilities of information filtering would be improved. Some researchers have considered the Wigner distribution (WD) as a master distribution from which the rest of distributions can be derived [1], [19]. One of the most salient WD properties is its high time–frequency concentration, but it may suffer from aliasing and cross terms that in principle can reduce the scope of its applications. We refer to several thorough reviews about the WD for one- and two–dimensional (1D and 2-D) signals [6], [9]. To cope with the problem of cross terms, some different modifications of the original definition have been proposed, e.g., a kernel-type WD [3], analytic signals [30], [31], image processing techniques [23], power type L-Wigner [20], Manuscript received March 31, 1997; revised January 6, 1998. This work was supported in part by NATO Collaborative Grants Program, Spanish Commission for Science and Technology Grant TIC94-0849. J. Hormigo was supported in part by a Spanish Ministry of Education and Culture fellowship. The associate editor coordinating the review of this paper and approving it for publication was Dr. Phillip A. Regalia. J. Hormigo is with the Departmento de Arquitectura de Computadores, Universidad de Malaga, Malaga, Spain (e-mail: [email protected]). G. Crist´obal is with the Instituto de Optica (CSIC), Madrid, Spain (e-mail: [email protected]). Publisher Item Identifier S 1053-587X(98)04032-X.

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or multidimensional WD in the frequency domain [22]. For a recent review about the reduced interference distributions, see [27]. The motivation of this correspondence is twofold. The first is to obtain a pseudo-Wigner distribution (PWD) with high resolution through the use of a new 2-D analytical signal but simultaneously eliminating aliasing and reducing as much as possible the presence of cross terms. The PWD and the spectrogram can be considered to be particular cases of filtering. In most situations, signal processing filters are used to create a sparse sampling of the space/spatial-frequency domain. However, there exist some applications, e.g., shape from texture, that require a more dense sampling provided by the WD or the spectrogram [16], [17]. Second, the reduction of cross terms will be considered on the basis of providing a feasible quantitative measure of the PWD concentration or “peakedness.” In Section II, a brief review of the continuous WD and the discrete PWD is presented. In Section III, a new 2-D analytical image is proposed for computing the PWD and simultaneously eliminating aliasing and reducing cross terms. In Section IV, we will evaluate and compare the results of computing the PWD using the new analytic signal using some synthetic and natural image tests.

Fig. 1. WD computation through three different analytic images. (a) Analytic image proposed here. (b) “One fourth domain method” proposed in [25]. (c) Analytic image proposed in [30].

II. THE WIGNER DISTRIBUTION The Wigner distribution was introduced by [26] as a phase space representation in quantum mechanics and gives a simultaneous representation of a signal in space and spatial frequency variables. Subsequently, Ville derived in the area of signal processing the same distribution that Wigner proposed in quantum mechanics [25]. Many other authors later use “Wigner–Ville” or “Ville distribution” to denote the same distribution. The symmetric definition of the Wigner distribution (WD) corresponding to a continuous and integrable 1-D signal is given by Wf (x; ) =

1 01

f x+

2

f3 x

0 2 e0j d

(1)

where x and are spatial variables, is the spatial frequency variable, and f 3 (1) means the complex conjugate of f (1). The WD can be considered as a particular occurrence of a complex spectrogram, where the function f (x) is chosen as the shifting window function. It belongs to a large class of bilinear distributions known as the Cohen’s class in which each member can be obtained choosing different kernels in the generalized bilinear distribution [5]. The WD presents a set of desirable mathematical properties, as was formulated by [4], including high signal concentration (maximum autocomponent concentration in space/spatial frequency). However, the presence of cross terms has limited its use in some practical applications. Cross terms can have a peak value as high has twice that of the autocomponents [15]. It is generally accepted that smoothing the WD will reduce the presence of cross terms but at the expense of smearing the autocomponent concentration. This effect has been recently demonstrated in [21]. In numerical implementations, we can consider the so-called pseudo Wigner distribution (PWD) by using two smoothing windows: a spatial averaging window g (l) and a spatial-frequency averaging window h(k) [2], [18], [19].1 This double smoothing operation will produce a further improvement in the cross term reduction. The 1 Although the term Pseudo-Wigner distribution is used when smoothing is done in the frequency domain only, we use the same term when the smoothing operation is done in both domains, following [18] and [19]. Windowing and averaging are dual operations. Therefore, in the following sections, we refer to g (1) and h(1) as the frequency and the spatial window, respectively.

PWD can be defined by

0

N 1

PWD(n; ) =

0

2

2

0j 2k

[h(k )] e

k= N +1 M 1

0

0

g [l]f [n + l + k]f 3 [n + l

l= M +1

0 k]

(2)

where n and = 2=N are the spatial and frequency variables, respectively, and M and N are the spatial and spatial-frequency window size, respectively. The PWD for 2-D signals will be given by PWD(n1 ; n2 ; 1 ; 2 ) N

=

0

01

N

0

01

hN N (k; l)

k= N +1 l= N +1 M 1 M 1

2

0

0

gM M (r; s)f (n1 + r + k; n2 + s + l)

0 0 2 f 3(n1 + r 0 k; n2 + s 0 l)e0j2(k r= M +1 s= M +1

+l ):

(3)

III. ANALYTIC SIGNAL PROPOSED The concept of the 1-D analytic signal was introduced by Gabor as a complex signal that has a spectrum identical to that of the real signal for positive frequencies and zero for negative frequencies [7], [11]. The extension of the analytic signal concept for multidimensional signals has been done recently by Hahn [12], but for dimensions bigger that one, the definition of the multidimensional analytic signal is not unique. In practice, the 2-D DWD of the analytic signal (which is also called the Wigner–Ville distribution) is used to overcome the principal discretization problems: aliasing and cross terms [28]. Aliasing or frequency foldover in discrete signals is due to the use of an insufficient sampling rate (below Nyquist limit, i.e., double of highest frequency content). To prevent aliasing, two methods arise to reduce this problem: a) oversampling (or interpolating) by a factor of 2 and b) frequency filtering the regions that cause aliasing (lowpass filtering). The latter can substantially suppress artifacts but at the cost of reducing significantly the spatial-frequency support. One of the problems associated with the 2-D DWD is that its periodicity is instead of the 2 of the Fourier transform [4], [8]. This fact implies that signals for which 2-D DFT does not introduce aliasing may be distorted with aliasing in the 2-D PWD due to a wraparound frequency phenomenon for high frequencies [16], [24]. As stated before, the definition of an analytic image for 2-D signals is not unique [24], [29]. Several 2-D analytic signals have been proposed elsewhere with the aim to reduce aliasing artifacts but retaining the main WD properties. However, most of the methods fail to produce a substantial reduction of the aliasing and cross terms. The analytic image proposed here can be considered as a combination of two previous definitions [24], [29]. The region of support of the analytic image here [Fig. 1(a)] is the same that the proposed in [29] [Fig. 1(c)]

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 6, JUNE 1998

Fig. 2. WD computation through the new analytic image. (a) Region of support corresponding to low frequencies in the vertical direction. (b) WD corresponding to the analytic image in (a). (c) Region of support corresponding to high frequencies in the vertical direction. (d) WD corresponding to the analytic image in (c). (f) WD composite associated with (b) and (d).

but separating the region considered into two parts [depicted in Fig. 1(a) by two different shadings]. Although the present method requires the computation of two WD corresponding to each region, this is not in detriment of a substantial computational cost reduction in comparison with the interpolation-based methods. The latter one can be easily implemented by a zero-padding operation and, therefore, with higher computational cost by computing a PWD with four times the number of samples.2 Here we, consider a new 2-D analytic signal that might avoid most of the shortcomings of the previous definitions. This method is based in zeroing out the negative side of the spectrum as in Fig. 1(c).3 However, here, we split the positive region in two subregions [see Fig. 1(a)] and take a PWD from each subregion. Fig. 2 depicts a two-step procedure for obtaining the new analytic image proposed here. Fig. 2(a) and (c) represent by shading the signal’s frequency components retained in the analytic image proposed here. The first PWD is taken from the region shown in Fig. 2(a), producing a PWD given in Fig. 2(b). This spectrum can be rearranged in the regions A and B depicted in Fig. 2(f) according to the following regions of support.4 PWD1 =

Region A Region B

0

1 < =2; 0=2 2 < =2 : 1 < ; 0=2 2 < =2

=2

(4)

With the mask represented in Fig. 2(b). we filter out the higher frequencies that can be arranged according to the following four regions [see Fig. 2(f)].

PWD2 =

Region Region Region Region

C D E F

1 < ; 0 2 < 0=2 1 < =2; 0 2 < 0=2 =2 1 < ; =2 2 < 0 1 < =2; =2 2 > : =2 0

(5)

Regions A, D, and F in Fig. 2(b) and (d) correspond to positive frequencies, and regions B, C, E correspond to negative frequencies. The final PWD will be the sum of PWD1 + PWD2 . This procedure 2 The interpolation-based methods will require a window h(1) of double size for obtaining the same PWD resolution. 3 Any other criteria that preserve the vanishing halfplane property can be considered as in [29]. 4 This rearrangement leads to a controlled frequency foldover.

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obviously produces a PWD with higher resolution but in a range 0 to and, therefore, doubles the number of points along each coordinate. If the original image is real, the PWD presents a property of symmetry with respect to the origin of the spatial frequency domain. Fig. 2(f) depicts such property by using identical shadings between symmetrical regions. It must be observed here that some potential drawbacks of the analytic signal defined here are the introduction of additional sharp filter boundaries and boundary effects that might lead to ringing, besides the creation of a frequency variant PWD. Ringing has been ameliorated by smoothing all the sharp transitions with a raised cosine (Hanning) function. Fig. 3 illustrates the described ringing phenomenon for the analytic image represented in Fig. 1(c). Fig. 3(a) shows the original test image composed by different patches of 2D sinusoids in different orientations. Fig. 3(b) shows the magnitude of the analytic signal associated with Fig. 3(a) by considering the analytic image of Fig. 1(c). It can be observed how the vertical sinusoid appears as an artifact in the diagonal region [Fig. 3(b), top right] due to the sharp transition between the “positive” and “negative” regions. Fig. 3(c) shows the result after smoothing the edge between both regions. With regard to the cross terms issue, it must be noted that the removal of the negative frequencies of the analytic signal also eliminates the cross terms between positive and negative frequencies. However, in addition to that, cross terms have oscillatory terms of high relative frequencies that can often be reduced by PWD lowpass filtering at the expense of autoterm broadening [15]. The present method outperforms traditional ones by removing crossterms between higher and lower spatial-frequency regions without autoterm broadening. IV. RESULTS A. PWD Computation of Image Tests In this section, we will evaluate and compare the performance of the new PWD for some test images. Fig. 4(a) and (b) show two examples of a synthetic image “zone plate” cos(r2 ) with variable local frequency along the two main diagonals. Fig. 4(c) shows the multicomponent test image considered here composed by the sum of Figs. 4(a) and (b) plus a small dc term. Fig. 5(a) depicts the results of the PWD computation corresponding to the new analytic image proposed here [Fig. 5(a[e–h])] in comparison with two other alternative analytic signals showed in Fig. 1(b) and (c) [Fig. 5(a[i–l]) and Fig. 5(a[m–p]), respectively]. The four points of interest have been shown in [Fig. 5(a[a–d])] by the corresponding line intersections. For the first point [Fig. 5(a[a])] corresponding to a low to mid-range frequency of Fig. 4(c), the three methods provide similar results, but for the three other points corresponding to a higher frequency [Fig. 5(a[b–d])], the analytic signal method depicted in Fig. 1(b) simply fails to represent such range [Fig. 5(a[j–l])], and the analytic signal depicted in Fig. 1(c) produces aliasing [Fig. 5(a[n–p])]. Another example is shown in Fig. 5(b) for the four points shown in Fig. 5(b[a–d]). This example illustrates that the problem of the sharp transitions between the different regions is still present in all methods. However, we can observe that the method proposed here performs better than the other two methods. As the frequency increases, the presence of the border attenuates the frequencies for some points [Fig. 5(b[f–g])], but the other two methods either fail to represent such range [Fig. 5(b[k–l])] or introduce aliasing [Fig. 5(b[n–p])]. A substantial reduction of the PWD cross terms can be obtained by spatial and spatial-frequency smoothing. Fig. 6(a) shows a multicomponent image composed of two pure tones located in adjacent spatial positions. It is well known that the WD will produce

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(a)

(a)

(b)

(b)

(c) Fig. 4. Multicomponent test image. (a) Circular chirp “zone plate” 1 (frequency variation upwards). (b) Zone plate 2(frequency variation downwards). (c) Image sum of (a) plus (b) plus a small dc term (10% of the maximum).

(c) Fig. 3. Ringing artifacts with the definition of a 2-D analytic image of Fig. 1(c). (a) Original image. (b) Absolute value of analytic image of Fig. 1(c). (c) Same as (b) but smoothing the edge with a raised-cosine function.

cross terms in the transition between the two regions [7], [8]. Fig. 6(b) and (c) (from top left to bottom right) shows the PWD

associated with points along a horizontal line crossing the border of separation between the two regions for two different window sizes. For Fig. 6(b) and (c), we can observe the effect of cross term reduction by frequency smoothing. Fig. 6(b) shows the PWD computation of several points in the perpendicular direction across the border. Each line represents the PWD corresponding to the same points along the border between the two regions for a frequency window size of M = 0 (no frequency window), 3; 7; and 15 respectively. By increasing the frequency window size [Fig. 6(b), rows from top to bottom], a reduction in the cross terms can be obtained, but simultaneously, a reduction of the spatial resolution is observed (number of border points). Fig. 6(c) shows the PWD for the same points considered in Fig. 6(a) but increasing the window size. Each line represents the PWD corresponding to points along the border between the two regions for a frequency window size of 0 (no frequency window), 3; 7; and 15; respectively. In summary, Fig. 6(b) and (c) shows in a qualitative manner how increasing the

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 6, JUNE 1998

(c)

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(a)

(a)

(b)

(b)

(d)

Fig. 5. (a) [See (d) for legend] PWD computation (spatial raised-cosine window size = 15 15; frequency window size = 5 5) of a composite of two cos(r2 ) image “zone plates” (image size = 128 128 b/pixel) at several points of interest (indicated by the line intersection). The second column corresponds to the analytic signal proposed here [Fig. 1(a)]. Note either the presence of aliasing or the absence of spectrum terms in the other two analytic signals (third and forth column, respectively). (b) [See (c) for legend] Same as (a), but the second row corresponds to the analytic signal proposed here [Fig. 1(a)]. Same remarks as in (a) hold here related with third and fourth rows, respectively. (c) Legend for Fig. 5(b). (d) Legend for Fig. 5(a).

2

2 2

(c)

spatial window size reduces the spatial resolution but increases the frequency concentration. B. Concentration Measures In this section, we will introduce a quantitative measure of signal’s concentration. Several measures of space-spatial frequency concen-

Fig. 6. (a) Image test composed by two sinusoidal tones with different. (b) PWD computation of (a) along points near the border of the two regions for a spatial window size N = 15 15 for different frequency window sizes (M = 0; 3; 7; and 15 from top row to bottom row, respectively). (c) PWD computation of (a) along points near the border of the two regions for a spatial window size 31 31 for different frequency window sizes (M = 0; 3; 7; and 15 from top row to bottom row, respectively).

2

2

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the frequency window size produces a reduction in the concentration measure (as well as a reduction of its value at the transition region). V. CONCLUSION We proposed a 2-D PWD through the use of a new analytic image with high resolution and frequency support. In addition to that, a substantial reduction of cross terms has been obtained by spatial and spatial-frequency smoothing. A quantitative measure of peakedness has been computed for 2-D PWD with some synthetic and textured images for evaluating the cross term reduction. The present scheme for 2-D PWD computation has been successfully applied for texture segmentation and classification through eigenanalysis in the PWD domain and will appear in a forthcoming publication. ACKNOWLEDGMENT The authors would like to thank the reviewers for their very helpful comments and constructive suggestions for improving the manuscript. (a)

REFERENCES

(b) Fig. 7. Evaluation of a concentration measure corresponding to PWD of Fig. 6(a). (a) Different spatial window sizes have been considered here (see legend) for a frequency window size (g ) = 9 9. (b) Different frequency windows influence of this spatial smoothing operation in the reduction of cross terms (see legend) for a raised-cosine spatial window of size (h) = 15 15.

2

2

tration has been proposed in the literature, but most of them has been defined for single-component signals. To give a quantative figure of merit of the reduction of cross terms in the case of multicomponent signals, we have computed a measure of peakedness in a similar manner to the defined by [14]. This measure is analogous to the kurtosis used in statistics and in other fields as vision modeling [10] and is defined for 1-D signals by C=

N01 0N +1 N01 0N +1

N01 jPWD(n; )j4 0N +1 N01 jPWD(n; j2 )2 0N +1

(6)

the extension for 2-D signals being straightforward. Fig. 7(a) depicts the results of the evaluation of the peakedness measure C for several points across a direction perpendicular to the border of a synthetic multicomponent image [Fig. 6(a)]. In this plot, we can observe the influence of changing the spatial window size for a fixed value of the frequency window size (g ) = 9 2 9. Increasing the spatial window size produces an increase in the concentration measure C (and a reduction in its value at the transition region). On the contrary, Fig. 7(b) shows how a progressive increment in

[1] K. H. Bartelt, H. O. Brenner, and A. W. Lohmann, “The Wigner distribution and its optical production,” Opt. Commun., vol. 32, no. 1, pp. 32–38, 1980. [2] M. Basseville, P. Flandrin, and N. Martin, “M´ethodes temps-fr´equence,” Traitment du Signal, vol. 9, pp. 79–112, 1992. [3] H. Choi and W. J. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 862–871, 1989. [4] T. A. C. M. Claasen and W. F. G. Mecklenbrauker, “The Wigner distribution, a tool for time-frequency signal analysis. Part I. Continuous-time signals,” Philips J. Res., vol. 35, pp. 217–250, 1980. [5] L. Cohen, “Generalized phase-espace distribution functions,” J. Math. Phys., vol. 7, pp. 781–786, 1966. , “Time-frequency distributions, a review,” Proc. IEEE, vol. 33, [6] pp. 31–37, 1989. , Time-Frequency Analysis Englewood Cliffs, NJ: Prentice-Hall, [7] 1995. [8] G. Cristobal, J. Bescos, and J. Santamaria, “Image analysis through the Wigner distribution function,” Appl. Opt., vol. 28, no. 2, pp. 262–271, 1989. [9] G. Cristobal, C. Gonzalo, and J. Bescos, “Image filtering and analysis through the Wigner distribution function,” in Advances in Electronics and Electron Physics, P. W. Hawkes, Ed. Boston, MA: Academic, 1991, vol. 80, pp. 309–397. [10] D. J. Field, “What is the goal of sensory coding?,” Neural Comput., vol. 6, no. 4, pp. 559–601, 1994. [11] D. Gabor, “Theory of communication,” J. Inst. Elect. Eng., vol. 93, pp. 429–457, 1946. [12] S. L. Hahn, “Multidimensional complex signals with single-orthant spectra,” Proc. IEEE, vol. 80, pp. 1287–1300, 1992. [13] L. Jacobson and H. Wechsler, “Joint spatial/spatial-frequency representation,” Signal Process., vol. 14, pp. 37–68, 1988. [14] D. L. Jones and T. W. Parks, “A high resolution data-adaptive timefrequency representation,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 38, pp. 2127–2135, 1990. [15] D. L. Jones and T. W. Parks, “A resolution comparison of several timefrequency representations,” IEEE Trans. Signal Processing, vol. 40, pp. 413–420, 1992. [16] J. Krumm, “Space frequency shape inference and segmentation of 3D surfaces,” Ph.D. dissertation, Carnegie Mellon Univ., Pittsburgh, PA, 1993. [17] J. Malik and R. Rosenholtz, “Computing local surface orientation and shape from texture for curved surfaces,” Tech. Rep. UCB/CSD 93/775, Univ. Calif. Berkeley, Comput. Sci. Division, 1994. [18] W. Martin and P. Flandrin, “Detection of changes of signal structure by using the Wigner–Ville spectrum,” Signal Process., vol. 8, pp. 215–233, 1985. [19] T. Reed and H. Wechsler, “Segmentation of textured images and Gestalt organization using spatial/spatial-frequency representations,” IEEE Trans. Pattern Anal. Machine Intell., vol. 12, pp. 1–12, 1990. [20] L. Stankovic, “A method for improved concentration in the timefrequency analysis of multicomponent signals using the L-Wigner

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[21] [22]

[23]

[24] [25] [26] [27] [28] [29] [30] [31]

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Finite-Precision Analysis of the Pipelined Strength-Reduced Adaptive Filter Manish Goel and Naresh R. Shanbhag Abstract— In this correspondence, we compare the finite-precision requirements of the traditional cross-coupled (CC) and a low-power strength-reduced (SR) architectures. It is shown that the filter block (F block) coefficients in the SR architecture require 0.3 bits more than the corresponding block in the CC architecture. Similarly, the weight-update (WUD) block in the SR architecture is shown to require 0.5 bits fewer than the corresponding block in the CC architecture. This finite-precision architecture is then used as a near-end crosstalk (NEXT) canceller for 155.52 Mb/s ATM-LAN over unshielded twisted pair (UTP) category-3 cable. Simulation results are presented in support of the analysis.

I. INTRODUCTION Strength reduction is an algebraic transformation that has been proposed [4] to trade off multipliers with adders in a complex multiplication, thereby achieving power reduction. In [7], we proposed the application of strength reduction transformation at the Manuscript received November 12, 1996; revised December 11, 1997. This work was supported by the NSF CAREER Award MIP-9623737. The associate editor coordinating the review of this paper and approving it for publication was Dr. Konstantin Konstantinides. The authors are with the Coordinated Science Laboratory and Electrical and Computer Engineering Department, University of Illinois at UrbanaChampaign, Urbana, IL 61801 USA (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 1053-587X(98)03944-0.

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algorithmic level to adaptive systems involving complex signals and filters. It was shown in [7] that the strength-reduced (SR) filter enables power savings of 21–25% over the traditional cross-coupled (CC) filter with no loss in performance. However, the application of strength reduction increases the critical path, and, hence, an inherently pipelined SR (PIPSR) architecture was also presented. Furthermore, by trading the throughput gained through pipelining with power supply scaling [4], it was demonstrated that additional power savings of 40–69% are feasible. In this correspondence, we compare the finite-precision requirements of the SR and the PIPSR architectures developed in [7] with that of the CC architecture. It is shown that the precision requirements of the SR and PIPSR architectures are similar to those of the CC architecture. This makes the SR and the PIPSR architectures attractive alternatives to the traditional CC architecture for high bit-rate communications and digital signal processing applications. In this correspondence, a linear model is employed for coefficient quantization noise. The filter (F) block precision BF is chosen such that the signal-to-quantization-noise-ratio (SQNR) is greater than the desired signal-to-noise ratio SNRo . The coefficient precision for weight-update (WUD) block BWUD is determined by applying the stopping criterion [3], [5], which puts a lower limit upon the correction term being added to the weight update. This criterion is given by

2 E [je(n)j2 ]x2 202B

(1.1)

where

E [je(n)j2 ] x2 BWUD

step-size; mean-squared error; power of the received signal x(n); precision (including sign-bit) of the coefficients in the WUD block. A more accurate nonlinear analysis presented in [1] and [2] can be employed to provide a tighter bound on BWUD . However, the purpose of this paper is to compare the precision requirements for CC and SR architectures, and hence, we employ the analysis in [3]. This analysis provides useful design guidelines for applications such as those in digital subscriber loops where the final step sizes are reasonably large. We demonstrate an application of the finite-precision SR architecture as a near-end crosstalk (NEXT) canceller for 155.52 Mb/s [6] ATM-LAN over 100 m of unshielded twisted pair category-3 (UTP3) cable employing 64-CAP (carrierless amplitude/phase) modulation scheme. We present the simulation results for this application in order to determine the precision requirements of various signals and to support the analytical results presented in the correspondence. The organization of the paper is as follows. In Section II, we present PIPSR adaptive filter architecture. In Section III, we determine the finite-precision requirements of CC, SR, and PIPSR architectures. Finally, in Section IV, the finite-precision architectures are employed as a near-end crosstalk (NEXT) canceller for 155.52 Mb/s ATM-LAN. II. A PIPELINED STRENGTH-REDUCED (PIPSR) ADAPTIVE FILTER In this section, we review the strength reduction transformation and development of the PIPSR architecture [7] from the CC architecture. The product of two complex numbers (a + |b) and (c + |d) is given by (a + |b)(c + |d) = (ac

1053–587X/98$10.00 1998 IEEE

0 bd) + |(ad + bc):