Fractional Power Spectrum

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 20084199Fractional Power SpectrumRan Tao, Senior Member, IEEE, Feng Zhang, and Yue WangAbstract—In this paper, by investigating the definitions of the fractional power spectrum and the fractional correlation for the deterministic process, we consider the case associated with the random process in an explicit manner. The fractional power spectral relations for the fractional Fourier domain filter are derived, and the expression for the fractional power spectrum in terms of the fractional correlation is obtained. In addition, the definitions and the properties of the fractional white noise and the chirp-stationary process are presented. Simulation results verify the theoretical derivations and demonstrate the potential applications, such as detection and parameter estimation of chirp signals, fractional power spectral estimation and system identification in the fractional Fourier domain. Index Terms—Fractional correlation function, fractional Fourier transform, fractional power spectrum, fractional white noise.I. INTRODUCTION T is desirable to estimate the statistical characteristics of a random process in various applications such as radar Doppler processing, adaptive filtering, speech processing, underwater acoustics, spectral estimation, pattern recognition, and array processing [1]. Based on the Fourier transform (FT), the response of a linear time-invariant (LTI) system to a deterministic function, and the statistics, such as the correlation function and the spectral density, at the output of a LTI system in response to random inputs have been analyzed [1]–[3]. In recent years, the fractional Fourier transform (FRFT) has attracted attention in many areas such as optics, quantum mechanics, pattern recognition, time-frequency representation, signal processing [4]–[8], etc. The FRFT can be interpreted as a rotation of signals in the continuous time-frequency plane and serves as an orthonormal signal representation for chirp signals. Besides being a generalization of the FT, the FRFT has been proved to relate to other time-varying signal analysis tools, such as Wigner distribution, short-time Fourier transform, wavelet transform [9]–[14], etc. Several properties of the FRFT have been investigated in the literature [4]–[6], [15]–[21]. Compared with the FT, the FRFT is more flexible and suitable for processing nonstationary signals due to an additional degree of freedom. Therefore, it has been widely applied in signal processing, such as filtering in the fractional Fourier domain (especially removal of the chirp noise), chirp signal detec-IManuscript received June 30, 2007; revised April 7, 2008. Published August 13, 2008. This work was supported in part by the National Science Foundation of China for Distinguished Young Scholars by Grant 60625104 and by the National Science Foundation of China by Grants 60232010 and 60572094. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Mats Bengtsson. The authors are with the Department of Electronic Engineering, Beijing Institute of Technology, Beijing 100081, China (e-mail: rantao@; oUo@). Digital Object Identifier 10.1109/TSP.2008.925579tion and estimation, analysis of long range dependence in time series [8], [11], [22]–[28], etc. Some linear time-variant (LTV) operations can be easily performed by multiplying with a filter function in the fractional Fourier domain [11], [16], [20]: Such as swept-frequency filters [4], optimal filtering in the fractional Fourier domains [8], and repeated filtering in consecutive fractional Fourier domains [24], which can be interpreted as a generalization of multiplicative filtering in the conventional Fourier domain. Furthermore, these improvements can be implemented as efficiently as the ordinary FT since the FRFT has fast digital algorithms much like the FT [5], [10], [29], [30]. Power spectral density (PSD) and correlation function are fundamental tools in random signal analysis and processing. The definitions and properties of the fractional power spectrum and the fractional correlation proposed in the literature are only for the deterministic signal [4], [6], [11], [14], [17], [20]. Therefore it is natural to ask the following questions: How to define the fractional power spectrum and the fractional correlation for the random signal? What is the fractional power spectral relationship between the input and output of the fractional Fourier domain filter? What is the relationship between the fractional power spectrum and the fractional correlation? In this paper, our main motivation is to answer these questions, and explore the important properties of the fractional power spectrum and the fractional correlation. We also discuss the applications of these results and use them to illustrate some special random processes. Fractional order signal processing is becoming an active research area in recent years, due to the nonstationary and non-Gaussian model in signal processing [31]–[33]. The fractional power spectrum and the fractional correlation function presented in this paper are the second-order statistics in the FRFT domain, which is different from the fractional lower-order statistics in the literature [32], [33]. The outline of this paper is as follows. In the next section, we review the definitions of the FRFT and the fractional Fourier domain filters briefly. In Section III, we derive the fractional power spectral relations between the input and the output of the fractional Fourier domain filter, and then obtain the relationship between the fractional power spectrum and the fractional correlation. In Section IV, the definitions and properties of the fractional white noise and the chirp-stationary random process are presented. In Section V, two practical applications involving detection and parameter estimation of chirp signals, and the system identification in the fractional Fourier domain are presented. We conclude the paper in Section VI. II. PRELIMINARIES A. Definition of FRFT The FRFT is a generalization of the FT, and can be interpreted as a rotation of the signal to any angles in the time-frequency1053-587X/$25.00 © 2008 IEEE4200IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008Similarly, the th fractional cross-power spectrum of can be expressed asandFig. 1. Multiplicative filtering in the fractional Fourier domain.(4) where the superscript denotes the complex conjugate. Next, we introduce the input-output relationships of the fractional power spectrum for the fractional filter. and be the input and the output of Theorem 1: Let , and be the th the th fractional filter and , respectively, fractional autopower spectrum of be the th fractional cross-power spectrum, then we have (5)plane [4]. The continuous FRFT with angle [5], [34], [35]is defined as [4],(1) where the transform kernel is given by(2) . For a deeper understanding of and the FRFT, the reader is referred to [5], [36]. B. Multiplicative Filtering in the Fractional Fourier Domain Multiplicative filtering in the th fractional Fourier domain is as shown in Fig. 1. In this configuration, we first obtain the th FRFT of the input signal , and apply multiplicative filter in this fractional Fourier domain; the signal is transformed to obtain the output signal in the time with the angle domain [24]. In [5], [8], [11], [26], and [27], it is shown that the additional degree of freedom in the angle of the FRFT provides improved performance in a variety of circumstances [24]. Furthermore, these improvements can be implemented at no additional cost since the FRFT has fast digital algorithms much like the FT [5], [8], [10], [24], [29], [30]. III. FRACTIONAL POWER SPECTRUM Estimation of the PSD based on the FT has been used for applications in radar, sonar, imaging, speech processing, array processing [1], etc. Since the FRFT generalizes the FT into arbitrary angle of the time-frequency plane, it is worthwhile to consider the fractional counterpart of the conventional PSD and the correlation function. denote the truncation function in Definition 1: Let of the sample function of the random process , the th FRFT of , then, the th fractional and can be defined as autopower spectrum of (3)and (6) Proof: By utilizing the fractional convolution theorem [20], [22] to the fractional filter shown in Fig. 1, the relationship and can be expressed as between (7) where . By employing (7), the cross-correlation function can be expressed as(8) where of . Let , and denotes the autocorrelation(9) as follows from (8) and (9) (see (10) at the bottom of the page). Comparing (10) to (7), we can write (11)(10)TAO et al.: FRACTIONAL POWER SPECTRUM4201Noting that, the fractional power spectrum can be expressed as shown in (12) at the bottom of the page. From (9), and the definition of the FRFT, (12) can be written as (13) Similarly, (14) Thus, by substituting (13) and (14) into (11), we can obtain (5). Similar to (8), the output autocorrelation function can be expressed as shown in (15) at the bottom of the page. Let (16) then substitute (15) into (16)of the output to the relating the power spectrum of the input . This well-known power spectrum method of system identification is based on the assumption that the input random process is stationary, and the input-output relation is the conventional convolution. Considering the random process that may be nonstationary to the input of the fractional filter, the input-output relation results in the fractional convolution theorem for the FRFT. Theorem 1 describes the input-output relationships of the fractional power spectrum for the fractional filter, which has similar results as the classical . The theorem, and reduces to (21) and (22) when amplitude and the phase of the unknown fractional system can be determined in terms of the transfer function fractional power spectrum by employing (5) and (6). Note that, though the fractional filter is a linear time-variant system, the fractional spectral relation for the fractional filter is still simple and similar to that of the conventional results. A. Corollary From the proof of Theorem 1, we can obtain the following corollary directly. Corollary 1: The “ th fractional correlation function,” can be represented as two equivalent forms, i.e., and , in which denotes time-averaging. Moreover, (13) and (19) express the fractional power spectrum in terms of the fractional correlation functions. , (13) and (19) reduce to the conventional reWhen sult, i.e., the power spectrum of a random process is the FT of the time average of the correlation function, which becomes the Wiener-Khinchine theorem for the case of stationary random process. Therefore, (13) and (19) are a generalization of the Wiener-Khinchine theorem in the fractional Fourier domain, which provides an efficient alternative method directly for the detection and parameter estimation of chirp signal corrupted by noise, and also has other applications such as fractional power , (9) will bespectral estimation, etc. Furthermore, when come(17) By utilizing the properties of the FRFT [4] and comparing (17) with (7), we can obtain (18) To establish (6), notice that, the fractional power spectrum can also be expressed as (19) we can obtain (20) Finally, we obtainRemarks: An important application of spectral estimation is the identification of a system by a random process. The underlying method is based on the well-known equations [2] (21) (22)(23)(12)(15)4202IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008which denotes the ensemble energy average; and (13) will reduced towhere(24) deserves its name in that it can be which suggests that interpreted as the fractional spectral density of power. B. Special Cases and From (9) and (16), if can be written as functions of , i.e., and (25) and . In this then we have becomes a “ th chirp-stationary case, the random process random signal,” which is presented in Section IV. C. Discrete Counterpart of the Fractional Power Spectrum and Correlation The discrete counterpart definitions of the fractional power spectrum, the fractional correlation function, and their relationship can follow the discrete analogy of the above definitions and theorem. The discrete-time FRFT (DTFRFT) is defined as [22]denote the correlation function of the , and can be defined as The fractional power spectrum of. (31) (32)and the discrete time version of the Theorem 1 as (33) (34) IV. SOME SPECIAL CASES OF FRACTIONAL SPECTRUM In this section, some special cases of nonstationary random processes often encountered in random signal analysis associated with the FRFT are presented. A. Fractional White Noise Definition 2: Let be a random process with continuous time parameter . If the th fractional power spectrum of the is a constant, i.e., (35) where is a constant, we call the th fractional white noise. Theorem 2: The th fractional white noise is also the th fractional white noise, and the fractional white noise has the same fractional power spectral density in different FRFT domain. Proof: The result can be derived by utilizing (10) and the properties of the FRFT [17], [22]. A direct application of Theorem 2 is the identification of the fractional filter driven by the fractional white noise which is a non-stationary noise. In this case, if the input is a fractional white noise, the system function of the fractional filter can be identified by using (5) or (6). When the th fractional bandlimited white noise is considered, i.e. (36) the fractional correlation function can then be expressed as(26) where is the sampling period. , then the forward and inverse forms of (26) can Let be written as(27) and(28) Similar to Corollary 1 in the continuous-time case, the two equivalent forms of the fractional correlation function in the discrete-time case can be defined as follows: B. Chirp Stationarity(37)(29)(30)Definition 3: For a random signal , if is a stationary random signal, we the th chirp-stationary random signal. call According to Definitions 1 and 3, we can obtain the relationship for the conventional power spectrum of the chirped staand the fractional power spectrum tionary random signal . of the th chirp-stationary random signal Theorem 3: The relationship between the th fractional power spectrum of the th chirp-stationary random signalTAO et al.: FRACTIONAL POWER SPECTRUM4203and the conventional power spectrum of the stationary random is signal(38) in which is the conventional power spectrum of , and the th fractional power spectrum of . Proof: This relationship can be deduced by using (25) and Corollary 1. Equation (38) shows that, the th fractional power spectrum and the convenof the th chirp-stationary random signal have the tional power spectrum of stationary random signal same spectral shape. Note that a conventional white noise and a stationary random process become a fractional white noise and a chirp-stationary random process respectively in the fractional Fourier domain. Thus they are the generalization of the conventional results and are useful in the conventional sense. V. SIMULATIONS A. Fractional Spectral Estimation Consider the noise-corrupted multicomponent chirp signals which consist of two chirps with chirp rate and the initial fre, , , and quency values of , respectively; is zero mean complex Gaussian white noise; and the signal-to-noise ratio (SNR) is 0 dB. is According to (9), the fractional correlation function of of 500 Monte Carlo simulations. Since Fig. 2(a) is three-dimensional, we plot its projection along the axis in Fig. 2(b); this will present the estimation effect of and clearly. The fractional power spectral estimation via the proposed fractional correlation method is as shown in Fig. 3(a) and this is also based on 500 Monte Carlo simulations. According to (29), both the time average and the ensemble average are in the fractional correlation function, and there are some time average estimators which yield unbiased and consistent estimates of the correlation function by using a single record of the nonstationary processes. However, the fractional correlation function estimate used here , is which is a asymptotically unbiased estimator. Fig. 4 shows the fractional correlation estimate : Fig. 4(a) is the real part ; Fig. 4(b) is the projection of Fig. 4(a), (c), (d) are of in the special angle and the respectively. The fractional power spectrum can be estimated via , where is a preselected lag window function. Here, the window function used is a rectangular window which has the : , . Simsame length as ilar to Fig. 2(b), the Fig. 3(b) is the projection of Fig. 2(a) along the axis. B. Fractional Fourier Domain Identification of Transfer Function In this part, we explore the system identification in the fractional Fourier domain. The received signal is modeled as in the frachaving been obtained via filtering of the input tional Fourier domain, followed by the addition of a zero-mean complex Gaussian noiseFig. 2. Fractional power spectral estimation by squared modulus of the FRFT. (a) Three dimensional representation. (b) The projection of (a) along z axis.Fig. 3. Fractional power spectral estimation by the FRFT of the fractional correlation function. (a) Three-dimensional representation. (b) The projection of (a) along z axis.otherwise (39) denotes the Dirac function. where From (13), the fractional power spectrum can be written asotherwise. (40) In fact, the fractional power spectrum is the function of two variables and , thus (40) can be written as (41) The plots of the fractional power spectra estimated by the modulus square of the FRFT and the FRFT of the proposed fractional correlation function are as shown in Figs. 2 and 3. In the observations of the time series simulations, the is obtained by sampling the signal with the sampling pein . The discrete step size of riod the angle is . In Fig. 2(a), the estimation of the fractional power spectrum is obtained by the modulus square of the FRFT , which is plotted based on an average4204IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008Fig. 6. System identification using the fractional power spectrum. Fig. 4. Fractional correlation function. (a) Three-dimensional representation. (b) The projection of (a) along z axis. (c) The fractional correlation function at = and (d) = . the angle = 8 20= 9 20is the cross-power spectral estimate between where and , and the power spectral estimate of the input. Here, we consider a fractional filter with the impulse response of finite duration (with ) in our numerical example, where the number 9 is a randomly chosen value. The input signal is a zero-mean Gaussian white noise, with a ranrandomly generated. In domly chosen length of 1000 and Fig. 5, we have plotted the magnitude and phase of the original and estimated fractional transfer function in the fractional Fourier domain using 200 Monte Carlo trials where the original SNR is 15 dB. Fig. 6 illustrates the fractional spectral SNR versus the original SNR in different fractional Fourier domain, where at each level of the SNR 200 Monte Carlo simulations are used. The fractional spectral SNR which is an analytical measure of estimation accuracy here has the formFig. 5. System identification in the th = fractional Fourier domain. Original fractional transfer function. (a) Magnitude. (b) Phase; identified frac. (c) Magnitude. (d) Phase. tional transfer function with= SNR = 15 dB5and can be transformed to the time domain by using the Parsevals theoremwhere the fractional filter output is described by the fractional convolution . The original SNR for the received signal is calculated viaBased on Theorem 1, the fractional system transfer function can be identified by using the fractional power spectrum methodAs we can see, the proposed identified method is valid in different fractional Fourier domain. Note that, the identified fractional filter is a time-variant system, and no noise reduction techniques before the system identification have been applied. It should be noted that, the system identification model considered here is the classical model. Here we emphasize on Theorem 1 and Corollary 1, and provide a framework for implementing a system identification in the fractional Fourier domain by using the fractional power spectrum.TAO et al.: FRACTIONAL POWER SPECTRUM4205VI. CONCLUSION AND FUTURE RESEARCH In this paper, we have considered properties of the fractional power spectrum and the fractional correlation function for the random case. It is shown that, the fractional power spectrum can be expressed in terms of the fractional correlation function, and the fractional filter has the simple input-output fractional power spectral relation. These results can be regarded as the generalizations of the conventional theorems and can be used in the detection and parameter estimation of chirp signals, system identification in the fractional Fourier domain, etc. We have also introduced definitions and properties of the fractional noise and the chirp-stationary random process, which are useful in the random signal analysis based on the FRFT. It should be noted that, this paper presents a framework for the random signal analysis in the fractional Fourier domain, and a lot more remain to be done. We can evaluate the statistical performance of the fractional spectral estimation technique in terms of the SNR, record length, etc., which impacts the parameter estimation precision of the chirp signals. 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Rev., vol. 1, pp. 15–16, 1994. [36] Y. Chen, Dr YangQuan Chen’s FRFT (Fractional Fourier Transform) Web Pages [Online]. Available: /foc/ FRFT/.Ran Tao (M’00–SM’04) received the Ph.D. degree in electrical engineering from Harbin Institute of Technology, Harbin, China, in 1993. He has been a Professor with the Beijing Institute of Technology, Beijing, China, since 1999. From March 2001 to April 2002, he was a visiting scholar with the University of Michigan, Ann Arbor. His research interests are in the areas of the fractional Fourier transform, time-frequency signal processing, and signal processing for radar and communications. Dr. Tao was the recipient of National Science Foundation of China for Distinguished Young Scholars in 2006, and of the Teaching and Research Award for Outstanding Young Teachers in Higher Education Institutions of MOE, P.R.C., in 2000. He is the Vice-President of Chinese Radar Industry Association and a Fellow of Chinese Institute of Electronics.。