Modification of the Discrete Polynomial Transform - Signal Processing

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... ON SIGNAL PROCESSING, VOL. 46, NO. 5, MAY 1998. A Modification of the Discrete Polynomial Transform ... duced recently as a computationally efficient algorithm for estimating the ... using a fast Fourier transform (FFT) calculation. The original DPT ... purely imaginary. The case of signals with time-varying amplitudes.

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 5, MAY 1998

A Modification of the Discrete Polynomial Transform Stuart Golden and Benjamin Friedlander

Abstract— The discrete polynomial transform (DPT) has been introduced recently as a computationally efficient algorithm for estimating the phase parameters of constant-amplitude polynomial phase signals. In this correspondence, we present a modification of the DPT, which improves the estimation accuracy. We show by a perturbation analysis that the mean-squared error of the estimates is reduced when the order of the polynomial is three or greater. Index Terms—Chirp, nonstationary, parameter estimation.

I. INTRODUCTION In this correspondence, we are concerned with the estimation of the polynomial phase coefficients of a polynomial phase signal. Special cases of the signal model used in this correspondence arise in various applications. Some of these applications are addressed in [8]. First, pulse compression radar systems use signals that have polynomial phase. Typical examples of such systems would be signals with linear or quadratic phase. If the target for the radar is in motion, then the received signal will have different polynomial phase coefficients than the polynomial phase coefficients of the transmitted signal. Depending on the relative motion of the target, the received signal may be modeled as a polynomial phase signal of higher order than the last transmitted signal. Yet another proposed application for polynomial phase signals is for modeling certain animal sounds and is proposed in [8]. Earlier results on estimating polynomial phase coefficients include the method proposed by Kumaresan and Verma [5]. Their method can be obtained as a special case of the approach given in this correspondence. Specifically, their method was to take finite differences of the phase by using the method presented in this correspondence with all of the delay parameters equal to unity. Another approach was proposed by Kitchen [4]. That approach was to first finite-difference the phase and then use a weighted least squares estimation algorithm. The approach taken in this correspondence is a modification of the discrete polynomial transform (DPT). The DPT was recently presented in [7]–[10] as a computationally efficient algorithm for estimating the phase coefficients of a constant-amplitude polynomial phase signal. The DPT algorithm estimates each of the phase coefficients in a successive manner from the highest order phase coefficient to the lowest order. Each phase coefficient is estimated by solving a nonlinear optimization problem in a computationally efficient manner. Specifically, a solution to the optimization problem is obtained by using a fast Fourier transform (FFT) calculation. The original DPT algorithm uses a single delay parameter that is chosen to minimize the mean-square error (MSE) of the estimates. In this correspondence, we present a variation of the DPT algorithm that involves M 0 1 delay parameters when the order of the polynomial is M: Further, we choose all of the M 0 1 delay parameters to minimize the MSE of the estimate. The original DPT algorithm corresponds to Manuscript received March 5, 1996; revised July 9, 1997. This work was supported by the Office of Naval Research under contracts N00014-91-J-1602 and N00014-95-1-0912 and by the National Science Foundation under grant NSF MIP-90-17221. The associate editor coordinating the review of this paper and approving it for publication was Dr. Petar M. Djuric. S. Golden is with Torrey Science Corporation, San Diego, CA 92121 USA. B. Friedlander is with the Univeristy of California, Davis, CA 95616 USA. Publisher Item Identifier S 1053-587X(98)03265-6.

constraining all of the M 01 delay parameters to be equal. By relaxing this constraint, we obtain an algorithm that has a smaller MSE of the estimate at a high SNR when the order of the polynomial is of three or greater. The original and modified versions of the DPT algorithm have similar computational requirements. In [11], an alternative name for the DPT algorithm was taken to be the “higher order ambiguity function” since the DPT can be considered to be a generalization of the ambiguity function. The signal model considered here can be expressed in the following manner. We observe a complex signal sn in identically distributed circular white Gaussian noise wn : That is, we observe

yn

=

sn + wn

(1)

where n ranges from 1; 2; 1 1 1 ; N: We will assume that the logarithm of the complex signal is exactly representable by a finite-order polynomial. Specifically, we consider signals that can be expressed as

sn

n2

a0 + a 1 n + a 2

= exp

2!

+

111

+

aM

nM M!

(2)

where the coefficients of the polynomial are unknown parameters. In this correspondence, we consider only constant-amplitude polynomial phase signals. That is, a0 is complex, and a1 ; a2 ; 1 1 1 ; aM are purely imaginary. The case of signals with time-varying amplitudes (complex am ) is studied in [3]. II. THE DPT ALGORITHM The DPT algorithm is an iterative algorithm. The first stage of the algorithm estimates the highest order coefficient of (2). Then, the lower order coefficients can be estimated by repeating the estimation procedure after the estimated terms have been demodulated from the observation. We therefore examine the estimation of the highest order coefficient. This estimate is given as the solution of the optimization problem [7], [8] as defined by

! ^

= arg max

!

F (!)

where ! is used to represent the Im(aM ), and

F (! ) =

N n=

+1

gnM 01 (y )hnM 01 (y 3 ) exp(0j! n)

(3)

2 (4)

is the objective function we are maximizing. The function F (! ) is M 01 3 (y ): maximized by taking the FFT of the quantity gnM 01 (y )hn The maximizing frequency is found by searching close to the bin M 01 (y ) with maximum magnitude. The functions gnM 01 (y ) and hn are products of delayed versions of the observation. The purpose of M 01 3 (y ) is to transform the observation into computing gnM 01 (y )hn a single sinusoid. These functions are obtained recursively where the recursion is given by

gnm (y ) = gnm01 (y )hnm001 (y )

hnm (y ) = hnm01 (y )gnm001 (y )

and m varies from 1; 2; 1 1 1 ; M initialized by letting

0

(5) (6)

: These recursive equations are

1

gn0 (y ) = yn hn0 (y ) = 1:

(7) (8)

In (4), we define s to be the sum of all of the delay parameters 1 ; 2 ; 1 1 1 ; M 01 and to be the product of these delay parameters.

1053–587X/98$10.00  1998 IEEE

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 5, MAY 1998

Further, we let y represent the vector of the received observations and s represent the vector of the noise-free signal. Although (3)–(8) describe the modified DPT, the original DPT can be obtained from these equations as a special case. Specifically, to obtain the original DPT from the above definitions, simply let  = 1 = 2 = 1 1 1 = M 01 : Like the original DPT, the delay parameters for the modified DPT algorithm should be chosen such that the solution is unambiguous. Specifically, we take j! j < : For the case M = 2, the signal model is a chirp (the frequency is linearly increasing over time). For this example, the recursive 1 (y ) = yn0 : Thus, equations imply that gn1 (y ) = yn and that hn the objective function becomes

F (!) =

N n= +1

yn yn3 0

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and

@ 2 F (!) @!2

=

N

0

2

gnM 01 (s) =

0j!1n) :

exp(

is essentially a single tone. The exp() term attempts to demodulate the tone to baseband so that the accumulation process will achieve maximum energy gain. III. STATISTICAL ANALYSIS To determine the appropriate values of the delay parameters, we begin by determining an approximation of the MSE of the estimate as a function of all of the delay parameters. This approximation is valid when we assume that the signal-to-noise ratio is sufficiently high. A first-order approximation of the estimate can be obtained by differentiating the estimate with respect to the observation. Since the observation is taken to be complex, we will make use of the complex analysis theory described by Brandwood [2]. Thus, the first-order approximation of the estimate can be stated as

!^ 

N

@ !^ @ !^ 3 3 (yn 0 sn ) + (yn 0 sn ): 3 @y @y n y =s n=1 n=1 n y =s

(9)

Therefore, by using the definition of the MSE of a random variable presented in [6], we can compute the MSE of the first-order approximation of the estimate to be

!)  2 MSE(^

@ !^ @yy y =s

2

+

2 @ !^

@yy3 y =s

2

:

(10)

To determine the derivative of the estimate with respect to the observation, we begin by using a necessary but not sufficient condition to determine the global maximum of (3), that is

@F (^!) @!

= 0:

(11)

To determine the desired derivative of (10), we differentiate (11) with respect to the observation, that is

@ 2 F (^!) + @ 2 F (^! ) @ !^ @[email protected]! @!2 @yy

= 0:

(12)

By solving for the derivative of the estimate, (12) becomes

@ !^ @yy

=

2 (^!) @ 2 F (^!) 01 0 @@[email protected]! @!2

(13)

where the required derivatives of the objective function can be expressed as

@ 2 F (^!) @[email protected]!

N =

n;m= +1

j (m 0 n)

M 01 M 01 1 gnM 01 (y) @[email protected] (y ) + @[email protected] (y ) hmM 01(y ) 1 hnM 01(y 3 )gmM 01 (y 3 ) exp(j !^ (m 0 n)) (14)

0 n)]2 exp(j! (m 0 n))

functions that are required to determine (14). Although the derivatives of these functions can be determined by the recursive definitions of these functions, we find it beneficial to differentiate their nonrecursive definitions. Specifically, note that we can express these functions as

hnM 01(s) =

The sequence yn yn3 0

N

[ (m

n;m= +1 1 3gnM 01(y )hnM 01(y3 )gmM 01 (y 3 )hmM 01(y): (15) M 01 (y ) We now examine the derivatives of the gnM 01 (y ) and hn

L l=1 L l=1

sn0

(16)

sn0

(17)

where L = 2M 02 and the constants l and l are determined recursively from the delay parameters. The recursion is initialized by the M = 2 case. For the M = 2 case, 1 = 0, and 1 = 1 : When M > 2, we use the recursive equations

2 2

with

1; 2;

+k = k + k+1 +k = k + k+1

(18) (19)

m varying from 2; 3; 1 1 1 ; M 0 1 and k varying 1 1 1 ; 2m02 : By differentiating (16) and (17), we obtain @gnM 01 (y ) @yi @hnM 01 (y ) @yi

from

gnM 01 (y ) L  yi k=1 n0 0i hnM 01(y ) L  = : yi k=1 n0 0i =

(20)

(21)

Next, we substitute these equations into (14) and evaluate the derivatives when the observation vector y is equal to the noise-free signal vector s: A substantial simplification results by noting that

gnM 01 (s)hnM 01(s3 ) = exp(j! n + c)

(22)

where c is a complex constant and not a function of n: Making these substitutions, the derivative of the estimate becomes

N

(m

@ !^ n;m= +1 =0 @yi y =s

0 n)

si

L

(n0i0 + m0i0 )

k=1 N

(m

n;m= +1

0 n)2

:

(23)

By substituting (23) into (10) and using the fact that the derivative of the estimate with respect to the conjugate of the observation is simply the complex conjugate of (23), the first-order approximation of the MSE of the estimate can easily be expressed for any choice of the delay parameters without any additional assumptions. Specifically, we can state the MSE of the estimate as MSE(^ !) =

2 K 2 M +1 N exp(2

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