Instantaneous Frequency Rate Estimation for High-Order Polynomial-Phase Signals

Transcription

1 TIME-FREQUENCY SIGNAL ANALYSIS 1069 Instantaneous Frequency Rate Estimation for High-Order Polynomial-Phase Signals Pu Wang, Hongbin Li, Igor Djurović, and Braham Himed Abstract Instantaneous frequency rate(ifr) estimation for high-order polynomial phase signals (PPSs) is considered. Specifically, an IFR estimator with only a second-order nonlinearity is proposed. The asymptotic mean-squared error (MSE) of the proposed IFR estimator is obtained via a multivariate first-order perturbation analysis. Our results show that the proposed estimator yields a smaller MSE and a lower signal-to-noise ratio (SNR) threshold than a popular IFR estimator involving higher nonlinearity. The proposed IFR estimator is also extended to estimate the phase parameters of a PPS. Numerical studies are presented to illustrate the performance of the proposed estimator. I. I Instantaneous frequency rate (IFR) reveals the rate-of-change of the frequency, which is proportional to the acceleration of a moving target1]. Consider a polynomial-phase signal (PPS): s(n)ae jφ(n) Ae j P i0 a in i, (1) wherep istheorderofthepps,aistheamplitude, φ(n) the instantaneous phase(ip) and a i } P i0 the phase parameters, respectively. TheIFRΩ(n)isdefinedasthesecondderivativeoftheIP]: Ω(n) d φ(n) dt P i(i 1)a i n i. () i TheproblemofinterestistoestimatetheIFR from noisy observations of s(n). Three cases are of interest: 1. P (linear frequency-modulated (FM) signals): the IFR is often referred to as the IEEE Signal Processing Letters, Vol. 16, No. 9, Sept chirp-rate, which can be estimated by wellestablished methods (e.g., 3] and references therein);. P 3(quadratic FM signals): the IFR is linearly proportional to the time n (cf. ()), and can be estimated using the cubic phase function(cpf)]; 3. P >3(high-orderPPSs): theifrisanonlinearfunctionofn,andcanbeestimatedusing the high-order phase function(hpf)4]. For the high-order PPS, a qth-order HPF is defined as4] q/ m Ml1 H q (n,ω) s(n+d l m)s(n d l m)] (rl) e jωm, where M+1 is the lengthof a lag window, d d 1,d,,d q/ } denotes a set of lagcoefficients, r r 1,r,,r q/ } is used to imposecomplexconjugationifr l 1ornone ifr l 1,andωdenotestheindexintheIFR domain. TheHPForderqandcoefficientsets dandrarechosensuchthatthehpfiscenteredalongtheifrofthesignal5,proposition1]. NotethattheHPFwithq,d 1 1 andr 1 1reducestotheCPF]. For high-order PPSs, the HPF often involves high-order nonlinearity. For example, a fourth-order PPS requires q 6. Such high-order nonlinearity results in a large meansquarederror(mse)andahighsnrthreshold in IFR estimation. We herein propose an IFR estimator with only a second-order nonlinearity. Analytical results via a multivariate first-order perturbation analysis show that the proposed IFR estimator is asymptotically unbiasedandprovidesasmallermseandalower SNR threshold than the HPF-based approach.

2 1070 TIME-FREQUENCY SIGNAL ANALYSIS II. P IFRE A. IFR Estimation for High-Order PPSs Consider the phase of a bilinear transformations(n+m)s(n m)forapthorderpps: args(n+m) s(n m)} φ (l) (n)m l φ(n)+ (l)! l1 φ(n)+ω(n)m + φ (l) (n)m l, (l)! l where L P/. Note that the resulting phase is a polynomial in m with even orders only, and each even order term is associated with a corresponding even-order derivative of the IP. In particular, the coefficient of the second-ordertermm isthe IFRΩ(n)ofthe signal. In order to obtain these phase derivatives, a multidimensional matched filter can be applied: A L,M (n,ψ) m M s(n+m)s(n m)e j l1ω l m l, (3) whereψ ω 1,ω,,ω L ] T denotestheindices of the L even-order phase derivatives. Note that ω 1 is the IFR index for the proposedfunctionin(3). WhenL1, the proposedfunctionreducestothecpf.intheabsenceofnoise,thesquared-magnitudeofa L,M is maximized along the L phase derivatives, i.e.,ψ 0 Ω(n), φ(4) (n) 4!,, φ(l) (n) (L)! ] T. Now consider a noisy PPS x(n) s(n)+ v(n), where v(n) is a complex white Gaussian noise with zero mean and variance σ. The proposed estimator is given by ˆΩ(n), ˆφ (4) (n),, ˆφ (L) ]T (n) 4! (L)! argmax Ψ B L,M(n,Ψ), (4) where B L,M (n,ψ) is defined similarly as A L,M (n,ψ)in(3)withs(n)replacedbyx(n). Remark 1: The multidimensional matched filter in (3) is related to the local polynomial Fourier transform (LPFT) and the local polynomial Wigner distribution (LPWD) 6, 7], but there are several notable differences. First, the LPFT and the LPWD were used to estimate the instantaneous frequency (IF) while our focus is IFR estimation. Second, the LPFT uses a P-dimensional matched filter while the proposed function(3) involves only a P/ -dimensional matched filter. Third, the LPWD estimates the odd-order phase derivatives, while the proposed function estimates the even-order phase derivatives. The proposed function(3) is similar to the CPF 4] in that both use the same secondorder moment of the observed signal. As shown in Section II-C, the proposed function reducestothecpfforappswithorderless than 4. For higher-order PPSs, however, the CPF becomes inapplicable, whereas the proposed function can still offer statistically consistent IFR estimation. B. Asymptotic Bias and MSE The asymptotic bias and MSE of the estimator(4) are obtained. The detailed analysis is presented in the Appendix, which leads to the following result. Proposition 1: ForaPth-ordernoisyPPS, the L phase-derivative estimates obtained in (4) are all asymptotically unbiased and their asymptotic MSEs are given by: E (δω l ) } ( ) M 4l+1 SNR 1 ],l1,,l, (5) ll where is an L L matrix with the lk-th element ] lk lk (l+1)(k+1)(l+k+1), (6) andthesnrisdefinedasa /σ. From Proposition 1, the MSEs of the L estimates are independent of the phase parameters a i } P i0 of the PPS. At high SNR,

3 INSTANTANEOUS FREQUENCY RATE ESTIMATION FOR HIGH-ORDER the MSEs are all approximately proportional to SNR 1, while at low SNR they are proportional to SNR. Moreover, the ith estimate, i.e., ω i, has an MSE inversely proportional to M 4i+1. Hence, the larger the window length, the lower the MSE. As such, for a given SNR and time n, the asymptotic MSEs of the estimates are minimized by using the maximum window length given by 1) M N 1 N 1 n intheasymmetricsampling case with n 0,,N 1}; and ) M N 1 n in the symmetric sampling casewithn } N 1 N 1,,,whereN is the number of samples and we assume N is odd. C. IllustrativeExamplesofL1andL C.1 ThePPSwithorderP andp 3 Since L P/ 1, B L,M (n,ψ) reduces to B 1,M (n,ω 1 ) m x(n+m)x(n m)e jω1m, whichisthecpfin]. Inthiscase,thematrix reducestoascalar1/45. Accordingto Proposition 1 in the symmetric sampling case, theminimummseof the IFRestimate fora givensnrandtimenis E (δω 1 ) } 45 ( ( N 1 n ) 5 SNR. (7) whichagreeswiththeresultderivedin(40)of 4]. C. ThePPSwithorderP 4andP 5 Inthiscase,B L,M (n,ψ)withlis m B,M (n,ω 1,ω ) x(n+m)x(n m)e j(ω1m +ω m 4 ). Theestimatesbasedon B,M areˆω 1 ˆΩ(n) and ˆω ˆφ (4) (n)/1. According to Proposition1,theMSEsofbothω 1 andω estimates are ( ) E (δω 1 ) } , (8) M5 SNR ( ) E (δω ) } (9) M9 SNR ) For comparison, the MSEs of the HPF-based IFRestimatorareproportionaltoSNR 6 due to a sixth-order nonlinearity(q 6)4, Section III],whereastheMSEsoftheproposedIFRestimatorareproportionaltoSNR (cf. (8))at low SNR. This implies that the proposed IFR estimator exhibits a much lower SNR threshold than the HPF-based method, which will be numerically verified in Section IV. III. E O P P As a by-product, the proposed estimator(4) can be utilized to estimate some phase parameters of a PPS. For example, consider a fourth-orderppswithphaseφ(n)a 0 +a 1 n+ a n +a 3 n 3 +a 4 n 4. The B,M (n,ω 1,ω ) in Section II-C. can be used to obtain two estimates, namely ˆΩ(n) in the ω 1 domain and ˆω ˆφ (4) (n)/1â 4 intheω domain. The latter can be used to estimate a 4. The MSE ofthe a 4 estimatecaneasilybederivedfrom (9): E (δa 4 ) } E (δω ) } 4 M N N 9 SNR, (10) at high SNR. Meanwhile, the frequently used techniquefora 4 estimationemploysthehighorder ambiguity function(haf)8]. The MSE ofthehaf-baseda 4 estimatoris E (δa 4 ) } HAF N 9 SNR, (11) athighsnr8]. Acomparisonbetween(10) and (11) shows that our estimator provides a much lower MSE at high SNR. Moreover, since the Cramér-Rao bound (CRB) for a 4 is CRBa 4 } 050/(N 9 SNR) 9], the proposed a 4 estimator is asymptotically efficient athighsnr. IV. N E Consider a fourth-order PPS with parameters A 1, (a 0,a 1,a,a 3,a 4 ) (,0.0,10 4,10 6,10 8),andN 19. The IFRisestimatedatn64,whichisthemiddle point of the observations. Fig. 1(a) shows

4 MSE (db) MSE (db) 107 TIME-FREQUENCY SIGNAL ANALYSIS N19, n64 HPF (simulation, M3) Proposed (simulation, M3) HPF (simulation, M64) Proposed (simulation, M64) HPF (analysis, M3) Proposed (analysis, M3) HPF (analysis, M64) Proposed (analysis, M64) CRB N19 HAF (simulation) Proposed (simulation) HAF (analysis) Proposed (analysis) CRB SNR (db) (a) SNR (db) Fig.1. MSEsofparameterestimatesforafourth-orderPPS,(a)IFRestimate;(b)Phaseparametera 4 estimate. (b) the MSEs obtained by analysis and simulation as a function of SNR for the proposed and, respectively, the HPF-based IFR estimators with two window sizes M 3 and M 64 (i.e., the maximum length at n 64). We have the following observations: 1) At high SNR, the simulation agrees with the analysis forthemseoftheproposedestimatorforboth M 3 and M 64; ) The MSE of the proposed estimator with M 64 attains the CRBathighSNR;3)Witheitherwindowsize, the MSE of the proposed estimator is generally lower than that of the HPF-based method at high SNR; 4) The proposed estimator gives a lower SNRthreshold(ofabout6dB lower) than the HPF-based estimator. Wenowconsiderthephaseparametera 4 estimation for the fourth-order PPS as explained insectioniii. TheMSEofthea 4 estimateis obtained at n64. The proposed estimator usesb,m (n,ω 1,ω )withm 64. TheMSEs obtained by analysis and simulation are shown infig.1(b). Itisseenthat,forSNRabove1 db, the MSE obtained by simulation for the proposeda 4 estimatoragreeswithitstheoreticalresultin(10). TheMSEoftheproposed estimator also reaches the CRB at high SNR. In addition, the SNR threshold is 11 db for the HAF-based a 4 estimator, whereas it is 1 dbfortheproposeda 4 estimator. V. C We have proposed an IFR estimator with a second-order nonlinearity. The asymptotic bias and MSE of the proposed estimator for appswithanarbitraryorderhavebeenobtained by using a multivariate first-order perturbation analysis. We have also discussed howtousetheproposedestimatorfortheestimationofotherphaseparametersofapps.in particular,weshowedthatthea 4 estimatorfor a fourth-order PPS is asymptotically efficient athighsnr. A Asymptotic Bias and Variance] The asymptoticbiasandvarianceoftheestimator(4)are obtained using a multivariate first-order perturbation analysis which extends the univariate first-order perturbation analysis in 10]. For a noisy PPS x(n) s(n)+v(n), the B L,M (n,ψ) can be decomposed into a noisefreecomponentc s (n,ψ)andanoisycomponentc vs (n,ψ): m M C s (n,ψ) s(n+n)s(n m)e j l1ω l m l,

5 INSTANTANEOUS FREQUENCY RATE ESTIMATION FOR HIGH-ORDER C vs (n,ψ) m M z vs (n,m)e j l1ω l m l, where z vs (n,m) s(n + m)v(n m) + s(n m)v(n+m)+v(n+m)v(n m). Let f L,M (n,ψ) denote B L,M (n,ψ), i.e., the objective function in (4). The f L,M (n,ψ) canfurtherbedecomposedintof s (n,ψ)and f vs (n,ψ) within a first-order approximation 10]: f s (n,ψ)c s (n,ψ)c s(n,ψ) (1) f vs (n,ψ) RC s (n,ψ)c vs(n,ψ)}, (13) where R } denotes the real part of }. On one hand, from (3), the noise-free term f s (n,ψ) is maximized at Ψ 0 Ω(n), φ(4) (n) 4!,, φ(l) (n) (L)! ] T. Ontheother hand, the noisy termf vs (n,ψ), actinglike a random perturbation, moves the global maximum from Ψ 0 to Ψ 0 +δψ, where δψ δω 1,,δω L ] T isassumedtobesmallinan asymptotic sense. Our purpose here is to obtainthebiasandvarianceofthelestimates. Noting that Ψ 0 and Ψ 0 +δψ are, respectively,themaximaoff s (n,ψ)andf s (n,ψ)+ f vs (n,ψ),wehave f s (n,ψ 0 ) } Cs (n,ψ 0 ) R C ω s(n,ψ 0 ) 0, (14) l f s (n,ψ 0 +δψ) + f vs(n,ψ 0 +δψ) 0, (15) wherel1,,l. Afirst-orderTaylorseries expansionof(15)aroundψ 0 leadsto f s (n,ψ 0 ) + f vs(n,ψ 0 ) + Dueto(14),thefirsttermof(16)iszeroand, asaresult, wecanrepresentthe above equationsinamatrixform f vs +F s δψ0 L 1 (17) where C s (n,ψ 0 ) F s ] lk R C ω s(n,ψ 0 ) k + C s(n,ψ 0 ) C } s(n,ψ 0 ) ω k (a) 3A 4 M (l+k+) lk (l+1)(k+1)(l+k+1), (18) and(a) here follows from the following result C s(n,ψ 0 ) ζa M, C s (n,ψ 0 ) jζ A m M m l jζ A M(l+1) (l+1), C s (n,ψ 0 ) ζ A m l+k ω k m ζ A M(l+k+1) (l+k+1), with ζ e jφ(n), and the approximation M m M mk M (k+1) /k+1 ifm k. Furthermore, Cs (n,ψ 0 ) f vs ] l R Cvs ω (n,ψ 0) l +C s (n,ψ 0 ) C vs(n,ψ 0 ) (a) 4A MIΓ(l)}, (19) wherei }denotestheimaginarypartof }, Γ(l) } k1 f s (n,ψ 0 ) ω k δω k 0. (16) ζ m ) (m l Ml (l+1) z vse j i1 φ (i) (n) (i)! m i,

6 1074 TIME-FREQUENCY SIGNAL ANALYSIS and(a) here follows from the following result C vs(n,ψ 0 ) m C vs(n,ψ 0 ) j m z vse j i1 m l z vse j φ (i) (n) (i)! m i, i1 φ (i) (n) (i)! m i. Asaresult,theerrorvectorcanbeexpressed as δψ F 1 s f vs. (0) Taking the expectation on both sides of (0) yields EδΨ} F 1 s Ef vs}0 L 1, (1) sinceez vs}s(n+m)ev(n m)}+s(n m)ev(n+m)}+ev(n+m)v(n m)} 0 for any n and m 8] which results in EΓ(l)}0. Therefore,from(1),allLestimates are asymptotically unbiased. The covariance matrix of the estimate from (0) is E δψδψ T} F 1 s ΞF 1 s, () where the diagonal elements give the variance ofthelestimates,andthelk-thelementofξ is Ξ lk Ef vs ] l f vs ] k } (a) 8A 4 M REΓ(l)Γ (k)] EΓ(l)Γ(k)]}, (b) 18A4( A σ +σ 4) M l+k+3 lk (l+1)(k+1)(l+k+1), (3) where (a) follows from the fact that EIx]Iy]} 0.5RExy } Exy}], and(b)isdueto EΓ(l)Γ (k)} ( 4A σ +σ 4) ) (m l Ml (l+1) m ) (m k Mk, (4) (k+1) EΓ(l)Γ(k)}0. (5) Eqn.(4)canbeverifiedbyusing Ez vs(n,m 1 )z vs (n,m )} ( A σ +σ 4) δ(m 1 +m ) + ( A σ σ 4) δ(m 1 m ) where δ( ) denotes the Kronecker delta function. Finally, combining (18), () and (3) yields(5). R 1] D. R. Wehner, High-Resolution Radar. Norwood, MA: Artech House, ] P. O Shea, A new technique for estimating instantaneous frequency rate, IEEE Signal Processing Lett., vol. 9, no. 8, pp. 51 5, August 00. 3] T. Abotzoglou, Fast maximum likelihood joint estimation of frequency and frequency rate, IEEE Trans. Aerosp. Electron. Syst., vol., pp , November ] P. O Shea, A fast algorithm for estimating the parameters of a quadratic FM signal, IEEE Trans. Signal Processing, vol. 5, no., pp , February ] P. Wang, I. Djurović, and J. Yang, Generalized high-order phase function for parameter estimation of polynomial phase signal, IEEE Trans. Signal Processing, vol. 56, no. 7, pp , July ] V. Katkovnik, Nonparametric estimation of instantaneous frequency, IEEE Trans. Information Theory, vol. 43, no. 1, pp , January ] L. Stanković, Local polynomial Wigner distribution, Signal Processing, vol. 59, pp , ] B. Porat and B. Friedlander, Asymptotic statistical analysis of the high-order ambiguity function for parameter estimation of polynomial-phase signals, IEEE Trans. Info. Theory, vol. 4, no. 3, pp , May ] S. Peleg and B. Porat, The Cramer-Rao lower bound for signals with constant amplitude and polynomial phase, IEEE Trans. Signal Processing,vol.39,no.3,pp ,March ], Linear FM signal parameter estimation from discrete-time observations, IEEE Trans. Aerosp. Electron. Syst., vol. 7, no. 4, pp , July 1991.