Bessel Samp 3

8/13/2019 Bessel Samp 3

http://slidepdf.com/reader/full/bessel-samp-3 1/5

5082 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 10, OCTOBER 2011

manipulate such matrices. The good agreement between the theoreti-

cally-predicted and the empirically-obtained separation performance

was demonstrated in simulation.

A remaining key-question is, given the sources’ covariance matrices

and our analytic expressions for the performance prediction, whether

(and if so, how) the association-matrices can be chosen so as to opti-

mize the performance (in some sense). This question will be addressed

in future work.

REFERENCES

[1] A. Yeredor, “On optimal selection of correlation matrices for matrix-

pencil-based separation,” in Proc. 8th Int. Conf. Independent Compo-

nent Anal. Source Separation (ICA), 2009, pp. 187–194.

[2] L. Tong, V. C. Soon, Y.-F. Huang, and R. Liu, “AMUSE: A new blindidentification algorithm,” in Proc. ISCAS , 1990, pp. 1784–1787.

[3] L. Tong and R. Liu, “Blind estimation of correlated source signals,” inProc. 24th Asilomar Conf., 1990, pp. 258–262.

[4] J. F. Cardoso, “Source separation using higher-order moments,” inProc. IEEE Int. Conf. Acoust., Speech, Signal Process. (ICASSP),1989, pp. 2109–2112.

[5] A. Yeredor, “Blind source separation via the second characteristic func-tion,” Signal Process., vol. 80, no. 5, pp. 897–902, 2000.

[6] A. M. Tomé, “The generalized eigendecomposition approach to the

blind source separation problem,” Digit. Signal Process., vol. 16, no.

3, pp. 288–302, 2006.[7] L. Parra and P. Sajda, “Blind source separation via generalized eigen-

value decomposition,” J. Mach. Learn. Res., vol. 4, pp. 1261–1269,2003.

[8] E. Ollila, H. Oja, and V. Koivunen, “Complex-valued ICA based on a

pair of generalized covariance matrices,” Comput. Statist. Data Anal.,

vol. 52, pp. 3789–3805, 2008.[9] J. F. Cardoso and A. Souloumiac, “Blind beamforming for non

Gaussian signals,” Proc. Inst. Electr. Eng.—F , vol. 140, no. 6, pp.

362–370, 1993.

[10] L. Fêty and J.-P. V. Uffelen, “New methods for signal separation,” inProc. 14th Conf. HF Radio Syst. Tech., 1988, pp. 226–230.

[11] L. Parra and C. Spence, “Convolutive blind source separation of non-stationary sources,” IEEE Trans. Speech Audio Process., pp. 320–327,2000.

[12] A. Belouchrani, K. Abed-Meraim, J.-F. Cardoso, and E. Moulines, “A

blind source separation technique using second-order statistics,” IEEE

Trans. Signal Process., vol. 45, no. 2, pp. 434–444, 1997.

[13] D.-T. Pham and J.-F. Cardoso, “Blind separation of instantaneous mix-tures of nonstationary sources,” IEEE Trans. Signal Process., vol. 49,

no. 9, pp. 1837–1848, 2001.

[14] A. Belouchrani and M. G. Amin, “Blind source separation based ontime-frequency signal representations,” IEEE Trans. Signal Process..,

vol. 46, no. 11, pp. 2888–2897, 1998.

[15] K. Abed-Meraim, Y. Xiang, J. H. Manton, and Y. Hua, “Blind sourceseparation using second-order cyclostationary statistics,” IEEE Trans.

Signal Process., vol. 49, no. 4, pp. 694–701, 2001.

[16] A. Yeredor, “On using exact joint diagonalization for non-iterative ap-proximate joint diagonalization,” IEEE Signal Process. Lett., vol. 12,

no. 9, pp. 645–648, 2005.[17] J.-F. Cardoso and B. Laheld, “Equivariant adaptive source separation,”

IEEE Trans. Signal Process., vol. 44, no. 12, pp. 3017–3030, 1996.

A Class of Scaled Bessel Sampling Theorems

Luc Knockaert

Abstract—Sampling theorems for a class of scaled Bessel unitary trans-forms are presented. The derivations are based on the properties of the

generalized Laguerre functions. This class of scaled Bessel unitary trans-formsincludesthe classical sine andcosine transforms, but also novel chirpsine and modified Hankel transforms. The results for the sine and cosinetransform can also be utilized to yield a sampling theorem, different from

Shannon’s, for the Fourier transform.

Index Terms—Bessel functions,chirp transform, Hankel transform, sam-

pling theorems.

I. INTRODUCTION

The classical Shannon [1] sampling theorem is omnipresent in large

areas of signal processing. Numerous generalizations have been de-

rived [2]–[5], while extensions of the sampling theorem have arisen in

connection with wavelets [6], the fractional Fourier and quasi-Fourier

transforms [7]–[10]. In this correspondence, we develop sampling the-orems for a class of scaled Bessel unitary transforms in a constructive

way, based on the generalized Laguerre functions. This class includes

the sine and cosine transforms, but also novel chirp sine and modified

Hankel transforms. The results for the sine and cosine transform can

also be utilized to yield a sampling theorem, different from Shannon’s,

for the Fourier transform. Novel asymptotic truncation error expres-

sions are derived and pertinent examples are presented.

II. MAIN RESULTS

We begin by stating the following result due to Kramer [2], [4], [5].

Theorem 1: Let be a real unitary operator over

with

product kernel such that the set

forms a complete or-

thogonal basis in

for some countable set of strictly increasingpositive constants

. Suppose further that

is -bandlimited to

, i.e., the transform

is in

such that

for . Then, admits the

sampling representation

(1)

where the sampling kernel is

and

.

Proof: See [2], [4], and [5].

Manuscript received February 15, 2011; revised June14, 2011; accepted June14, 2011. Date of publication June 27, 2011; date of current version September14, 2011. The associate editor coordinating the review of this manuscript andapproving it for publication was Prof. Xiang-Gen Xia. This work was supportedby a grant of the Research Foundation-Flanders (FWO-Vlaanderen).

The author is with the INTEC-IBCN-IBBT, Ghent University, B-9050 Gent,Belgium (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2011.2160634

1053-587X/$26.00 © 2011 IEEE



IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 10, OCTOBER 2011 5083

Note that, by virtue of the unitarity of , the functions

are

orthogonal in

.

Next we need to show that there exist product kernels with the re-

quired properties. In fact there are many of them. We have

Theorem 2: The class of scaled Bessel product kernels

0

0

with

0

satisfy the requirements of Theorem 1.

Proof: Consider the generalized Laguerre functions

0

0

0

where

are the Laguerre polynomials. It is shown in [11] that the

generalized Laguerre functions

form a complete orthonormal

basis for

. Next consider the operator with kernel

0

Clearly, the operator is real and unitary with eigenfunctions

and eigenvalues 0

. Moreover, is a product kernel since

we have

This follows from the well-established formula [12, p. 189]

0

0

0

0

0

0

0

valid for

, and letting tend to 0

from inside the unit circle.

From the complete orthonormal basis

, we can construct another

one by non-linear scaling

0

leading to

0

0

We areleft with one last, butmajor, requirement :

should

form a complete orthogonal basis in

. After some calculations,

we find that

0

(2)

For orthogonality, expression (2) must vanish when

. This is re-

mindful of the Fourier-Bessel [12] relationships, which can be written

as

and where

are the positive zeroes of

. Hence, we must take

0

(3)

In other words, we have proved that the functions

are orthogonal in

. Completeness is guaranteed by a theorem

of Szász [13, p. 282].

The sampling kernel

can be explicitly written as

0

2

0

(4)

It can be verified that

0

, implying the characteristic

interpolation property

. The sampling representation (1)

can be written as

, where

is the truncated

sampling expression and

is the truncation error

The asymptotic behavior of the truncation error for 0

is the sub-

ject of the two following corollaries.

Corollary 1: Let the -bandlimited function be

0

for

0

with . Then, for fixed

0 0 for 0

and

0

for 0

Proof: See Appendix I.The next corollary proves that there are numerous -bandlimited

functions which satisfy the requirements of Corollary 1. We have

Corollary 2: Let

be in

such that

for

0 with

0 . Then

0

for 0

Proof: See Appendix II.

Note that when , Corollary 1 yields

0

and also

0

.

III. EXAMPLES

A. Sine Transform

Here, we take and . The product kernel is

and

. The sampling kernel is

0

0

2

The sampling scheme is equispaced with sampling step1

, but

unlike the usual Fourier sinc sampling kernel—see the sequel—which

falls off as

0

at infinity, the sine sampling kernel falls off fasterat a

0

rate.




B. Cosine Transform

Here, we take

0 and . The product kernel is

0

0

and

0 . The sampling kernel is

0

0

2

Note that the sampling kernels for the cosine and sine transforms are

the same, except for a translation 0

in the

coefficients.

C. Fourier Transform

Although the Fourier transform

0

admits the usual

0

sampling theorem

0

1

0 1

where 1 , it is possible to utilize the sine and cosine sampling

kernels. If we define the even and odd components of a function as

0

0

0

the Fourier transform can be written as

0

0

0

Hence, signal reconstruction with

0

behavior at infinity is pos-

sible by utilizing the sine sampling kernel for the odd component of the

function, and the cosine sampling kernel for the even component of the

function.

We compared the bandlimited function

0

0

with its truncated sine plus cosine sampling reconstruction

for and over the range

. The result is shown in

Fig. 1.

D. Hankel Transform

We will restrict ourselves to the modified Hankel transform of order

0, in which case and The product kernel is

and

with

the positive zeroes of

. The sampling

kernel is given by

2

0

Fig. 1. versus for the Fourier transform.

Fig. 2. versus for the Hankel transform.

Fig. 3. versus for the Hankel transform.

We compared the -bandlimited function

with its truncated sampling representation

for and

over the range

. The result is shown in Fig. 2. With the

same sampling conditions as above, another example is

0

0

For this example, we have

0

and hence we may ex-

pect a better truncated sampling approximation as compared to the firstexample. The result is shown in Fig. 3.



IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 10, OCTOBER 2011 5085

Fig. 4. versus for the chirp sine transform.

E. Chirp Sine Transform

Here, we take and . The product kernel is

and

. The sampling kernel is

0

0

2

We compared the -bandlimited function

where is the Fresnel integral

, with its

truncated sampling representation

for and over

the range

The result is shown in Fig. 4.

IV. CONCLUSION

Sampling theorems for a class of scaled Bessel unitary transforms

have been presented. This class of scaled Bessel unitary transforms in-

cludesthe classical sine andcosine transforms, butalso novel chirp sine

and modified Hankel transforms. Novel asymptotic truncation error ex-

pressions were derived and pertinent examples presented.

APPENDIX I

PROOF OF COROLLARY 1

Corollary 1: Let the -bandlimited function

be

0

for 0

with . Then for fixed

0 0

for 0

and

0

for 0

Proof: The asymptotic McMahon expansion [14] of the zeros of

Bessel functions guarantees that

. This implies

0

0

From formula (4) we deduce that

0

0

We do not have to worry about

or for that case

since

these values are asymptotically located on the 6

envelope of the

Bessel function. Hence

0 0

0 0

For the squared truncation error, we utilize the orthogonality of the

functions

. We have

From (3), we deduce that

0 0

leading to

0

APPENDIX II

PROOF OF COROLLARY 2

Corollary 2: Let

be in

such that

for 0 with

0 . Then

0

for 0

Proof: Without loss of generality, we can take . Consider

the integral [15, p. 684]

0 0

0

0 0

0

0

Here,

0 is a constant and

is the Lommel

function. Since

0

for

0

, we readily obtain that

0 0

0

(5)

Now let

be a strictly increasing sequence of real numbers greater

than 0 and take

0 . Then Müntz’s theorem [16]

states that the system of powers

is complete in

if and

only if

(6)

Condition (6) is not difficult to satisfy, i.e., taking

for

is just fine. Completeness implies that

can be written as




and this leads to

For any of the exponents

, we can write

0

By construction

and hence (5) implies that

0

0

0

for

0

Since consists of an infinite sum of elements with identical be-

haviour at infinity, itself must exhibit the same behavior at infinity

and the proof is complete.

REFERENCES

[1] C. E. Shannon, “Communication in the presence of noise,” in Proc.

IRE , 1949, vol. 37, pp. 10–21.[2] H. P. Kramer, “A generalized sampling theorem,” J. Math. Phys., vol.

38, pp. 68–72, 1959.[3] A. J. Jerri, “Some applications for Kramer’s generalized sampling the-

orem,” J. Eng. Math., vol. 3, no. 2, pp. 103–105, 1969.[4] A. J. Jerri, “The Shannon sampling theorem—its various exten-

sions and applications: A tutorial review,” Proc. IEEE , vol. 11, pp.1565–1596, 1977.

[5] A. I. Zayed , Advances in Shannon’s Sampling Theory. Boca Raton,

FL: CRC Press, 1993.[6] I. W. Selesnick, “Interpolating multiwavelet bases and the sampling

theorem,” IEEE Trans. Signal Process., vol. 47, no. 6, pp. 1615–1621,1999.

[7] X. G. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signa l Process. Lett., vol. 3, no. 3, pp. 72–74, 1996.

[8] T. Erseghe, P. Kraniauskas, and G. Cariolaro, “Unified fractionalFourier transform and sampling theorem,” IEEE Trans. Signal

Process., vol. 47, no. 12, pp. 3419–3423, 1999.[9] Q. Wang and L. Wu, “A sampling theorem associated with

quasi-Fourier transform,” IEEE Trans. Signal Process., vol. 48,no. 3, pp. 895–895, 2000.

[10] D. Y. Wei, Q. W. Ran, and Y. M. Li, “Generalized sampling expansionfor bandlimited signals associated with the fractional Fourier trans-form,” IEEE Signal Process. Lett., vol. 17, no. 6, pp. 595–598, 2010.

[11] G. Szego , Orthogonal Polynomials. New York: Amer. Math. Soc.Coll., 1939.

[12] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi , Higher Transcendental Functions. New York: McGraw-Hill, 1953, vol. 2.

[13] P. J. Davis , Interpolation and Approximation. New York: Dover,1975.

[14] M. Abramowitz and I. Stegun , Handbook of Mathematical Func-

tions. New York: Dover, 1965.[15] I. S. Gradshteyn and I. Ryzhik , Table of Integrals, Series and Prod-

ucts. New York: Academic, 1965.[16] V. Operstein, “Full Müntztheoremin ,” J. Approx. Theory, vol.

85, pp. 233–235, 1996.

Sparse Approximation Property and Stable Recovery of

Sparse Signals From Noisy Measurements

Qiyu Sun

Abstract—In this correspondence, we introduce a sparse approximationproperty of order for a measurement matrix

:

for all where is the best -sparse approximation of the vector

in ,

is the -sparse approximation error of the vector

in , and and are positive constants. The sparse approximationproperty for a measurement matrix can be thought of as a weaker versionof its restricted isometry property and a stronger version of its null spaceproperty. In this correspondence, we show that the sparse approximationproperty is an appropriate condition on a measurement matrix to considerstable recovery of any compressible signal from its noisy measurements.In particular, we show that any compressible signal can be stably recov-ered from its noisy measurements via solving an -minimization problem

if the measurement matrix has the sparse approximation property with , and conversely the measurement matrix has the sparse ap-

proximation property with

if any compressible signal can be

stably recovered from its noisy measurements via solving an -minimiza-tion problem.

Index Terms—Additive noise, approximation methods, compressedsensing, signal reconstruction.

I. INTRODUCTION

Given positive integers and with and a measurement

matrix of size 2 , we consider the problem of compressive sam-

pling in recovering a compressible signal

from its noisy mea-

surements via solving the following

-minimization

problem:

0

(I.1)

where , , , and the measurement noise

satisfies

[1]–[8]. Here, 1

, , stands for the

“

-norm” on the Euclidean space.

Given a subset and a vector

, denote by

the vector whose components on are the same as those of the vector

and vanish on the complement

. A vector

is said to be

-

sparse if

for some subset with its cardinality

less than or equal to , where . Denote by 6

the set of

all -sparse vectors. Given a vector , its best -sparse approximation

vector

in

is an

-sparse vector which has minimal distance to

in

; i.e., 0

6

0

. For ,

we use

instead of

for brevity.

In this correspondence, we introduce a new property of a measure-

ment matrix : there exist positive constants and such that

0

for all

(I.2)

Manuscript received June 06, 2010; revised October 26, 2010 and April 13,2011; accepted June 28, 2011. Dateof publication July 12, 2011; date of currentversion September 14, 2011. The associate editor coordinating the review of thismanuscript and approving it for publication was Prof. Namrata Vaswani. Part of this work was done when the author visited Ecole Polytechnique Federale deLausanne.

The author is with the Department of Mathematics, University of CentralFlorida, Orlando, FL 32816 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2011.2161470

1053-587X/$26.00 © 2011 IEEE

Bessel Samp 3

Documents

Transcript of Bessel Samp 3