Modeling of fractional-order signals from their ramp ... - IEEE Xplore

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 50, NO. 9, SEPTEMBER 2003

657

Modeling of Fractional-Order Signals From Their Ramp Cepstra

sequence) is generated. This sequence of numbers may be looked upon as a discrete-time signal. Let the sequence g [n] be defined as

Dean J. Schmidlin

g[n] = rn Cn (cos !0 )u[n]

Abstract—A method is presented which computes the ramp cepstrum of a fractional-order signal and models the ramp cepstrum as a signal having a rational -transform. A partial fraction expansion of the resulting model identifies the exponents of the fractional-order signal. An example is presented to illustrate the method.

where u [n] is a step function, and r , !0 , and are real numbers (0 ; 0 < jrj 1). The signal defined by (4) will be called a Gegenbauer signal. A second-order difference equation for the Gegenbauer signal can be formulated by letting w = cos !0 in (1), replacing n by n 0 1, and then multiplying both sides of (1) by rn . The result is

!0

Index Terms—Fractional-order signals, modeling, ramp cepstrum.

an g[n] = bn g[n 0 1] + cn g[n 0 2];

I. INTRODUCTION

n2

(5)

where

Fractional-order signals and systems are utilized to model physical processes whose magnitude spectra do not have slopes that are multiples of 20 dB per decade. Examples of such processes are relaxation behavior of polarized impedances in dielectrics and interfaces [1], [2], cardiac rhythm [3], transmission line impedance [4], viscoelasticity [5], and colored noise [6], [7]. This paper is concerned with the modeling of discrete-time fractional-order signals and makes use of the property that their ramp cepstra have rational z -transforms. Section II defines a “prototypal” fractional-order signal and shows that this signal can be represented in terms of a sequence of Gegenbauer polynomials. Expressions are derived for the fractional-order signal, its ramp cepstrum, and their z -transforms. In Section III, a method is presented for modeling a fractional-order signal from its samples. The ramp cepstrum of the signal is determined and modeled as a rational z -transform using any of the well-established techniques (Pade approximations, AR and ARMA modeling, Prony’s algorithm, etc.) [8], [9]. From the partial fraction expansion of the resulting model, the exponents of the fractional-order signal are obtained. An example is presented to illustrate the basic steps of the method. Section IV consists of summary remarks and a direction for future work. II. GEGENBAUER SIGNALS The Gegenbauer polynomials Cn (w) satisfy the recurrence formula [10]

(4)

an =n bn =2( + n 0 1)r cos !0 cn = 0 r2 (2 + n 0 2):

The values of the Gegenbauer signal at n = 0 and n = 1 are determined with the aid of (2) as g [0] = 1 and g [1] = 2r cos !0 . The z -transform of the Gegenbauer signal is obtained by letting w = cos !0 in (3) and replacing z by rz 01 . Doing this yields

G (z ) =

1

(1

0 2r cos !0 z01 + r2 z02 )

0(n + 2 0 1)Cn01 (w);

n 1 (1)

;

jzj > jrj :

(7)

For = 6p=q, G (z ) has branch points of order q 0 1 at z = re6j! and z = 0, and g [n] is a fractional-order signal. Samples of this signal can be computed from (5) and (6). Let g^ [n] and g~ [n] = ng^ [n] denote the cepstrum and ramp cepstrum of g [n], respectively. Their z -transforms are related as

G^ (z ) = loge G (z ) dG^ (z ) : G~ (z ) = 0 z dz

(8) (9)

The substitution of (7) into (8) and the result of the substitution into (9) yield

(n + 1)Cn+1 (w ) = 2(n + )wCn (w )

(6)

G~ (z ) = 2

1

0 r cos !0 z01

0 2r cos !0 z01 + r2 z02 0 1 1

:

(10)

The inverse z -transform of (10) is

where

C0 (w) = 1; C1 (w) = 2w:

(2)

They are often defined in terms of their generating function [10]

1 n=0

Cn (w) z n =

1

(1

0 2wz + z2 )

; jz j < 1:

(3)

g~ [n] = ng^ [n] = 2rn cos n!0 u [n 0 1] :

Equations (11) and (10) reveal that the ramp cepstrum g~ [n] of the fractional-order signal (4) is a damped sinusoid and the z -transform of the ramp cepstrum is a rational function of z . An important special case of (4) occurs when !0 = 0 and is replaced by =2. Equations (4), (7), (10), and (11) become

The Gegenbauer polynomials are a sequence of polynomial functions of the continuous variable w . By fixing the value of w , a sequence of numbers (the values of the various Gegenbauer polynomials in the Manuscript received June 18, 2001; revised September 23, 2002. This paper was recommended by Associate Editor R. Rao. The author is with the Department of Electrical and Computer Engineering, University of Massachusetts-Dartmouth, North Dartmouth, MA 02747-2300 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSII.2003.816930

(11)

G (z ) =

1

(1

0 rz01 )

; jz j > jrj

g [n] =rn Cn=2 (1) u [n]

0 rz01 0 1 ; jzj > jrj g~ [n] =rn u [n 0 1] :

G~ (z ) =

1

1

The ramp cepstrum simplifies to a decaying exponential.

1057-7130/03$17.00 © 2003 IEEE

(12) (13) (14) (15)

658


Fig. 1. A signal, its complex cepstrum, and its ramp cepstrum.

A fractional-order signal y [n] can be represented as a convolution of Gegenbauer signals of types (4) and (13). The z -transform Y (z ) has the form K

Y (z ) =

1

k=1 (1

0 rpk z01 ) 2

K

1

k=1 1

2 z 02 0 2rgk cos !0k z01 + rgk

Y~ (z ) =

: (16)

Equation (16) shows the positions of the individual factors when all of the ’s are positive. If a particular is negative, then the associated factor appears in the numerator rather than the denominator. The z -transform Y~ (z ) = 0z (d loge Y (z ))=dz of the ramp cepstrum of y[n] is

Y~ (z ) =

K

0 rpk z01 0 1 k=1 K 01 1 0 rgk cos !0k z + 2gk 2 z 02 0 1 1 0 2rgk cos !0k z 01 + rgk k=1 pk

1

(19)

The partial fraction expansion of (19) is

Y~ (z ) = 00:2

1

0

1

+ 1:6 0:75z 01 1

From (17), we see that p1 = has the z -transform

00:2 and g1 = 0:8. The signal y[n] 1

(1

0 0:4z01

4: 0 0:8z01 + 0:64z02 0 1:(20) 1

0 0:75z01

0:2

0 0:8z01 + 0:64z02 )0:8

(21)

which corresponds to the z -transform of the simulated signal.

: (17)

Note that Y~ (z ) is a rational z -transform having the property that ~ (z ) = y lim Y ~ [0] = 0:

01 0 1:384z 02 + 0:672z 03 : 1 0 1:55z 01 + 1:24z 02 0 0:48z 03 0:49z

Y (z ) =

1

z !1

= 0, = 00:2. The two component signals were computed by means of (5) and (6). The ramp cepstrum y~ [n] was computed by employing MATLAB’s m-file “cceps” to determine the cepstrum y^ [n] and multiplying by n. The signals y [n], y^ [n], and y~[n] are shown in Fig. 1. Applying MATLAB’s m-file “prony” (Prony’s discrete filter fit to time response) to the ramp cepstrum resulted in the z -transform

!0

III. MODELING OF FRACTIONAL-ORDER SIGNALS

(18)

The rational form of (17) suggests the following method for modeling a given fractional-order signal y [n]: 1) compute the ramp cepstrum y~ [n] of y [n]; 2) from the ramp cepstrum determine a rational realization of Y~ (z ); and 3) extract the parameters of the fractional-order model from Y~ (z ). The poles of Y~ (z ) determine the quantities rpk , rgk , and !0k and a partial fraction expansion yields the values of pk and gk . In order to test this method, a fractional-order signal y [n] was simulated by generating and convolving two prototypal fractional-order signals, one with r = 0:8, !0 = =3, = 0:8 and the other with r = 0:75,

IV. CONCLUSION Fractional-order signals have been expressed in terms of Gegenbauer polynomials. This allows for the possibility that known properties of Gegenbauer polynomials may lead to a deeper understanding and a better facility with fractional-order signals. For example, the property that the Gegenbauer polynomials form an orthogonal set allows one to quickly compute samples of rudimentary fractional-order signals by means of linear second-order difference equations. It has also been shown in this paper that the ramp cepstra of fractional-order signals have rational z -transforms. From this property evolved a method for modeling a fractional-order signal whereby one computes its ramp cepstrum, uses any one of the well-known techniques for synthesizing rational functions, and then extracts the parameters of the fractional-order model. The method was applied successfully to a simulated fractional-


order signal. Future work needs to test the method on noisy simulated signals and real physical signals. REFERENCES [1] H. Sun and B. Onaral, “A unified approach to represent metal electrode interface,” IEEE Trans. Biomed. Eng., vol. BME-31, pp. 309–406, July 1984. [2] D. W. Davidson and R. H. Cole, “Dielectric relaxation in glycerine,” Chem. Phys., vol. 18, p. 1414, 1950. [3] A. L. Goldberger, V. Bhargava, B. J. West, and A. J. Mandell, “On the mechanism of cardiac electrical stability,” Biophys. J., vol. 48, Sept. 1985. [4] J. C. Wang, “Realizations of the generalized Warburg impedance with RC ladder networks and transmission lines,” J. Electrochem. Soc., vol. 134, no. 8, pp. 1915–1940, 1987. [5] R. C. Koeller, “Application of fractional calculus to the theory of viscoelasticity,” J. Appl. Mech., vol. 51, June 1984. [6] B. Mandelbrot, “Some noises with 1=f spectrum, a bridge between direct current and white noise,” IEEE Trans. Inform. Theory, vol. IT-13, pp. 289–298, Apr. 1967. [7] M. Deriche and A. H. Tewfik, “Signal modeling with filtered discrete fractional noise processes,” IEEE Trans. Signal Processing, vol. 41, pp. 2839–2849, Sept. 1993. [8] L. Ljung, Systems Indentification: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall, 1987. [9] S. L. Marple, Digital Spectral Analysis with Applications. Englewood Cliffs, NJ: Prentice-Hall, 1987. [10] A. Erdelyi, Higher Transcendental Functions. Melbourne, FL: Krieger, 1981, vol. II.

Optimal Design of Multichannel Transmultiplexers With Stopband Energy and Passband Magnitude Constraints Yang Shi and Tongwen Chen

Abstract—Optimal design incorporating filter stopband energy and passband magnitude constraints is studied for multichannel, uniform, and nonuniform transmultiplexers. Central to the development is a reformulation of the design problem as a quadratic programming problem with quadratic constraints and semi-infinite constraints while the composite distortion measure for reconstruction is minimized. An iterative optimal design procedure for finite-impulse-response synthesis and analysis filters is developed and applied to two design examples, one for the uniform case and another for the nonuniform case based on general building blocks. Index Terms—Multirate signal processing, multirate systems, optimal design, transmultiplexers.

I. INTRODUCTION Multirate signal processing has recently received much attention and has been widely applied to communication systems [14], speech processing, and image processing [12]. It provides a more efficient way in Manuscript received March 7, 2001; revised May 9, 2003. This work was supported by the National Sciences and Engineering Research Council of Canada. Please note that the review process for this paper was handled exclusively by the TCAS-I editorial office and that the paper has been published in TCAS-II for logistical reasons. This paper was recommended by Associate Editor P. P. Vaidyanathan. The authors are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2G7, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TCSII.2003.816917

659

processing signals via resampling (upsampling or downsampling) of the original signals by an appropriate factor. In general, multirate systems have many different structures; multirate filter banks and transmultiplexers seem to be the most important ones. Much research over the past decades has dealt with the signal reconstruction problem for these multirate systems (see [2], [10], [12], [15], and the references therein). Transmultiplexers are systems that convert time-division multiplexed (TDM) signals into frequency-division multiplexed (FDM) signals and vice versa [12]. The FDM format is often used for long-distance transmission, whereas the TDM format is more convenient for digital switching. Generally, according to the sampling rates of the input signals, transmultiplexers can be divided into two types: uniform and nonuniform ones. In order to achieve perfect reconstruction for nonuniform filter banks, general building blocks—some linear dual-rate systems [2]—were proposed to allow more design freedom. Similarly, we can use such building blocks to construct nonuniform transmultiplexers. In [7], a measure J based on the two-norm of transfer matrices was proposed to quantify the degree of closeness to perfect reconstruction for uniform transmultiplexers; connections of this error measure to the traditional crosstalk, magnitude, and phase distortions were established. Further in [8], optimal design of nonuniform transmultiplexers was considered. In typical applications, transmultiplexers are required with two properties: First, the reconstruction performance should be good, i.e., J should be as small as possible; second, the filters involved should have certain frequency selectivity. In [7], optimal design based on minimizing J was formulated as a least squares problem. This captures the first requirement well; but there is no direct control of filter frequency characteristics. In [8], penalties on the stopband energy in the synthesis filters were added to the cost function; however, combining the penalties with the cost function casts only a type of “soft” constraints on the stopband energy, while it is more desirable to have direct control over stopband energy and passband magnitude of the filter(s) involved. The goal in this brief is to consider both requirements in design by imposing “hard” constraints on stopband energy and passband magnitude. The improvement over the results in [8] lies in the following two aspects: 1) the “soft” constraints only take the frequency characteristics into account to some restrictive extent, and the final results will depend on the tradeoff parameters that can be chosen only by trial and error; in contrast, the “hard” constraints give a more realistic mathematical description of the frequency requirements and 2) the optimal design problem is formulated as a quadratic programming problem with quadratic constraints and semi-infinite constraints, which can be solved numerically. This brief is organized as follows. Section II formulates the optimal design problem and constraints. Section III presents an iterative design procedure and applies it to some examples. Section IV provides some concluding remarks. II. PROBLEM FORMULATION A. Stopband Energy Constraints The frequency selectivity of filters can be taken into account by considering the stopband energy constraints. To see this, take the constraint(s) for Hi in the M -channel analysis filter bank as an example; we then have

jHi (!)j2 d! i ;

i

= 0; 1; . . . ; M 0 1

(1)

where H defines the stopband frequency interval(s) for Hi , and i is a pre-specified level (positive).

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