FREQUENCY ESTIMATION BASED ON ADJACENT DFT BINS

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1 FREQUENCY ESTIMATION BASED ON ADJACENT DFT BINS Michaël Betser, Patrice Collen, France Telecom R&D 3551 Cesson-Sévigné, France Gaël Richar. Telecom Paris Paris, France ABSTRACT This paper presents a metho to erive efficient frequency estimators from the Discrete Fourier Transform (DFT) of the signal. These estimators are very similar to the phase-base Discrete Fourier Spectrum (DFS) interpolators but have the avantage to allow any type of analysis winow (an especially non-rectangular winows). As a consequence, it leas to better estimations in the case of a complex tone (cisoi) perturbe by other cisois. Overall, our best estimator leas to results similar to those of phase vocoer an reassignment estimators but at a lower complexity, since it is base on a single Fast Fourier Transform (FFT) computation. 1. INTRODUCTION Sinusoial moeling [1] is a very popular an efficient representation for speech an music signals. It has le to numerous applications such as auio coing, analysis, synthesis an soun transformation. However, to be eligible for such applications, these moels require accurate parameter estimations an, in particular, accurate frequency estimation. Many frequency estimators use the Short Time Fourier Transform (STFT) as a starting point. Such estimators can be classifie into three main categories: namely, time methos where the time of the STFT varies as in the phase vocoer [] an the erivative metho [1], winow methos where the winow varies as in the spectral reassignment [3], an frequency methos where the frequency varies as in the amplitue spectrum interpolation [4] or the phase-base DFS interpolators [5, 6]. A comparison of the state-of-the-art frequency methos can be foun in [7, 8]. The latter methos have the avantage to require only one FFT computation since they only use ajacent frequency bins of this single FFT. This article presents a new frequency estimator that is rather similar to the phase-base DFS interpolator. However, the metho use to erive the frequency estimator is original an presents the avantage to allow the use of winowing, which was not the case in [5]. Similarly to previous work, it is suppose in this paper that the stuie signal is a complex sinusoi with quasi-constant amplitue an frequency, in the neighborhoo of a time t. Such a signal can be written as: x(t + τ) Ãe jβ τ +w(t + τ) (1.1) where à Ae jα is the complex amplitue of the sinusoi, (A is the real amplitue an α the constant phase), an β is the pulsation, both for the time t. τ is the local time in the neighborhoo of the time t. This sinusoi can be perturbe by a complex noise w, which is suppose to be zero-mean white Gaussian. The asymptotic properties of the estimator presente in section 4 also hols uner weaker noise properties escribe in [9, 6]. The analysis is base on the Short Time Fourier Transform (STFT) of the partial, efine as: X(t,ω k ;h) N/ n= N/ x(τ n +t) h(τ n )exp( j τ n ω k ) (1.) where N is the size in samples of the winow support h, F is the sampling frequency, k is the frequency bin, τ n = n/f is the time in secons of the corresponing sample number (n is an integer). Finally, ω k = πkf N is the pulsation of the bin k. Here N is suppose to be o. This STFT correspons to the centere form of the FT, which is sometimes calle zero-phase FT, because the phase spectrum response of a symmetric winow h has a phase equal to zero in the neighborhoo of the frequency zero. This efinition is preferre here, because it will simplify the evelopments presente in this article. If N is even, a centere form of the STFT is also possible, but will be slightly ifferent as the sum will not be on integer values anymore. The STFT of the signal (1.1) can be put uner the form: X(t,ω k ;h) = ÃΓ(ω k β;h) +W(t,ω k ;h) (1.3) where Γ(ω; h) is the iscrete time FT, using the efinition (1.), of the winow h for the pulsation ω, an W is the STFT of the noise. In practice, the efinition of the FT use is often the linear-phase FT (i.e. the sum in the FT is one from 0 to N 1), as for many implementations of the FFT for example. A practical way to zero-phase the FFT is to perform a (N 1)/ sample circular permutation of the winowe signal before computing the FFT [1]. In the remainer of this article, all the FT will be zero-phase.. PROPOSED METHOD The propose metho combines Fourier Transforms (FT) compute for two ifferent frequencies ω 1 an ω. The winow h is suppose to be symmetric, real an positive. In this part, the noise is not consiere. Let s introuce the following FT ratio: H X(t,ω 1;h) X(t,ω ;h) X(t,ω 1 ;h) + X(t,ω ;h) = Γ(ω 1 β;h) Γ(ω β;h) Γ(ω 1 β;h) + Γ(ω β;h) (.1) (.) Because of its particularity, one can show that this ratio can be unerstoo as a ratio of two FT iffering in winows. Let ω ω ω 1, ω b ω 1+ω an δ ω b β, then H can

2 be written as: H = j Γ(δ;h s) Γ(δ;h c ) where h s an h c are new analysis winows efine by: (.3) Han Ham Rec Bla Gau b 1 1.5e-4.1e- 3.9e-6 5.4e-3.4e- b 1.3e-1 1.4e-1.5e-1 1.0e-1 1.3e-1 Error boun(hz).6e-3 3.8e-1 8.3e-5 9.4e- 4.3e-1 h s (τ) sin( ωτ)h(τ), h c (τ) cos( ωτ)h(τ) Table 1: Boun values for ifferent analysis winows. If h is even, then h s is o an h c is even. Remark that H is necessarily real. In fact, as h c is symmetric, Γ(ω;h c ) is purely real, an as h s is anti-symmetric, Γ(ω;h s ) is purely imaginary. It will now be shown that an estimator can be efine from (.3) an the parity hypothesis on h. Taylor expansions aroun the frequency zero will be one. The frequency erivative property of the FT states that: i Γ ω i (ω;h) = ( j)i Γ(ω;τ i.h) (.4) Let Γ(h) = Γ(0;h). As Γ(τ i.h s ) = 0 if i is even, an Γ(τ i.h c ) = 0 if i is o, the upper part of (.3) will be expane to an orer 1 an the lower part to an orer 0. c 1 an c in [0,δ] exist such that: H = Γ(τ.h s)δ Γ(c 1 ;τ 3.h s ) δ 3 6 Γ(c ;τ h c ) δ = δ Γ(τ.h s) ( 1 P ) where P an Q are the Lagrange remainers, P Γ(c 1;τ 3.h s ) δ (.5) ( 1 Q ) (.6) 6, Q Γ(c ;τ.h c ) δ (.7) The values of Γ(τ i.h s ) an Γ(τ i.h c ) epen only on the analysis winow h an are known in avance. So if the corrective terms P an Q are small compare to 1 (cf section 3), an estimation of the frequency can be obtaine as: ˆβ = ω b R(H ) Γ(h c) (.8) In practice, H is never exactly real, this is why the real part (R()) of H is taken. When using an FFT, formula (.8) can be applie to the two most energetic bins of the cisoi: the maximum DFS bin k an the highest DFS bin between k + 1 an k 1. In this case, ω is the half frequency resolution of the FFT an ω b is the mile of the two bins selecte. Algorithm: 1. Initialization: compute an for the winow h use in the FFT.. Compute the zero-phase FFTs for a time t 3. Select the maximum bin k = argmax i X(t,ω i ;h) an the secon maximum k = argmax i {k+1,k 1} X(t,ω i ;h) 4. Compute the ratio H = X(t,ω k;h) X(t,ω k ;h) X(t,ω k ;h)+x(t,ω k ;h) 5. Compute the estimate frequency: ˆβ = ω b R(H ) Γ(h c), where ω b = (ω k + ω k )/ 3. ERROR BOUND In this section the performances of the algorithm without noise are stuie. Using equation (.6) an the efinition of the estimator (.8), the error between β an the estimation ˆβ can be rewritten as: β ˆβ = (Q P) (1 Q) δ (3.1) Since β is insie [ω 1,ω ], δ is lower than the half frequency resolution of the DFT, R = πf/n. We will first try to boun P an Q. The FT of a real symmetric an positive winow reaches its maximum in ω = 0 an is ecreasing for ω [0;R]. Therefore P an Q are positive, an tight bouns on P an Q are: P Γ(τ3.h s ) R 6, Q Γ(τ.h c ) R (3.) If no aitional hypotheses on the winow h are mae, P π /4 an Q π /8. It means that Q coul be equal to 1. Let s now suppose that h verifies the following property: n N/ h(n) N/4 n N/ h(n) (3.3) For all the usual winows, the energy of the center is superior to the energy of the eges. Consequently all the usual winows verify (3.3). With this hypothesis, the boun on Q becomes: Q 5π /64 < 1, which proves that 1/(1 Q) is O(1) 1. As P is also O(1), an δ is O(N 1 ), then, from (3.1), the estimate error β ˆβ is O(N 1 ), for all the winows verifying property (3.3). For some winows, the corrective terms P an Q will be of the same orer, leaing to smaller orer of error: it has been shown that for the rectangular winow, the estimator is O(N ) [6]. In orer to fin the boun on P Q, the Lagrange remainers P an Q will be rewritten as : P = Q = δ i ( 1) i Γ(τi+1.h s ). (i + 1)! ( 1) i Γ(τi.h c ). δ i (i)! (3.4) (3.5) 1 Let n be an integer variable which tens to infinity, let g(n) be a positive function an f (n) any function. Then f = O(g) means that f A.g for some constant A an all values of n. [10]

3 E{(β ˆβ) (Q P) } = δ (1 Q) + σ A Γ(δ;h c ) E{(β ˆβ) } R b 1 (1 b ) + σ A Γ(R;h c ) [δ Γ(h (1 P) c ) (1 Q) + Γ(h c) Γ(h s [ R Γ(h c ) (1 b 1 ) + Γ(h c) Γ(h ] s ) ) ] + O(N 3.5 ln(n) 1.5 ) (4.4) (4.5) A boun on P Q is therefore the infinite sum: P Q R i Γ(τi+1.h s ) (n + 1) Γ(τi.h c ) (i + 1)! (3.6) Alembert s rule shows that this boun is a convergent series, an as R is usually small, this series converges fast. The first terms give a goo approximation of this boun. Let s note b 1 an b the bouns on P Q (3.6) an Q (3.) respectively. Values of these bouns for ifferent typical winows are given in table 1. As b < 1, one can conclue that a boun on the error is: β ˆβ b 1 1 b δ (3.7) Theorem 1 If the winow h is real, symmetric, positive, an verifies the property (3.3), then the error of the estimator (.8) without consiering the noise influence is at least O(N 1 b ) an is boune by 1 1 b R, where b 1 an b are the bouns in equations (3.6) an (3.) respectively, an R is the half Fourier resolution. In the last line of table 1, the bouns are given in Hz for F = an N = 51, an for various analysis winows. The small values of the boun b1 show that the two error terms P an Q compensate each other, especially for the rectangular winow. This is why the first orer Taylor expansion of equation (.5), which seems a bit rough at first, can nevertheless give goo results, epening on the winow use. It can be note that superior orer Taylor expansions of (.3) can lea to more precise estimators, but at the cost of an increase in complexity. In this case the frequency estimation is now one of the roots of a polynomial which has the same orer as the expansion orer. 4. STATISTICAL PROPERTIES OF THE ESTIMATOR The noise influence on the performance of the estimator is iscusse in this section. The analysis will be very similar to the one given in [6], as they consier almost the same estimator but only in the case of a rectangular winow. It also follows the strategy aopte by Quinn in [9, 10]. Let Z j Γ(δ ;h s) Γ(δ ;h c ) an ÃΓ(δ;h c ). From equation (.3), if no noise is present, there is ientity between Z an H. From the efinition of H in equation (.1), an using equation (1.3), the ratio H has the form: H = Z + (W 1 W ) 1 + (W 1+W ) (4.1) where W 1 an W are the STFT of the noise for the frequencies ω 1 an ω respectively. In [11], it is prove that if w is white Gaussian, then W(t,ω;h) is O( N ln(n)) almost surely. This property is still true for more general assumptions on the noise, which are escribe in [11, 9]. If the function use to construct the iscrete winow h is continuous, positive, an normalize, i.e. 1 on the interval of efinition, then n h n will be O(N) an 1/( n h n ) will be O(N 1 ). This property will be useful to etermine the function orers. As Γ(δ;h c ) is O(N), then (W 1 +W )/ is O(N 1/ ln(n) 1/ ). H = ( Z + (W 1 W ))( (W 1 +W ) 1 + (W 1 +W ) + O(N 1.5 ln(n) 1.5 ) ) (4.) The expansion has been one to an orer, because orer 1 terms will be cancele when consiering the expectation. What interests us is the expectation of the square error between the true frequency an the estimate frequency, which correspons to the Mean Square Error (MSE) in section 5: { ( E{(β ˆβ) Γ(hc ) ) } } = E R(H ) δ (4.3) Substituting equation (4.) insie eq (4.3) leas us to an asymptotic evelopment of the MSE. After some simplifications, one can foun that this evelopment is given by equation (4.4). If N is large enough, a variance estimate, or at least a tight boun on the variance, coul be compute for each value of δ. We have chosen to present only the worst estimation case boun which is given by equation (4.5). This boun has been compute for ifferent typical winows, represente in figure 1. The boun is compose of two terms: one corresponing to the eterministic error, an another to the error cause by the noise. If we consier this boun as a function of the SNR: σ /A = 10 ( SNR/10), the eterministic error will be constant, an the noise error will be a linear function of the SNR in a log scale. Therefore, when the eterministic error is ominant - i.e. for high SNRs - the estimator error will be constant, an when the noise error becomes ominant, for low SNRs, the error will be linear (in log scale). This explains the shape of the curves, in two parts, of the figure 1. Theorem If the function use to construct the winow h is real, continuous, positive, normalize, symmetric an verifies the property (3.3), then the estimator efine in (.8) is asymptotically unbiase an, when N is large enough, a worst case boun on the variance is given by equation (4.5). 5. PERFORMANCE COMPARISON The purpose of this section is to compare the behavior of the new estimators to the classical ones. As stuies comparing

4 10 10 F Gau F Bla F Ham F Rec Error Variance (hz ) Error Variance (hz ) MacLeo Vocoer (a) Single cisoi comparison (b) Winow comparison Error Variance (hz ) 10 Error Variance (hz ) MacLeo Vocoer MacLeo Vocoer (c) Two cisois with 100Hz separation () Two cisois with 1000Hz separation Figure : Performance comparison Error Variance (hz ) F Gau F Bla F Ham F Rec Figure 1: Theoretical winow comparison the classical frequency estimation methos have been one more than once [7], only the algorithms giving the best results will be consiere, namely the classical phase vocoer ( Vocoer ), the reassignment metho ( ), an another interesting metho, Macleo s 3 samples interpolator ( Macleo ). All the methos are summarize in table. The new metho is name F followe by the first three letters of the winow use. In orer to achieve a frequency estimation, peak etection is neee, but as our purpose is to compare the frequency estimators, it will be assume in all experiments that the correct maximum bins are known. The secon maximum bin is still suppose unknown.the classical Cramer-Rao Boun () framework is use to compare the estimators for ifferent Signal to Noise Ratios (SNR) [5]. The is represente by a ashe line. The experiments are presente for F = an N = 51. The error between the true an estimate values is base on an average of the possible causes of error: on noise, using K inepenent realizations, on frequency values, using ranomly picke frequency in regularly space interval (10Hz size) over all the spectrum, an on the initial phase an amplitue, using ranom values between [0, π[ an [0.1, 0.9] respectively. The noise variance is compute from the current amplitue value. In the experiments (c) an (), the secon sinusoi has the same amplitue as the first one. Figure (b) shows the raw performances of the F estimator on a single cisoi, for ifferent analysis winows. As the SNR increases, the noise becomes negligible, an the inherent bias ue to approximation (.6) appears, as explaine in section 4. The theoretical curves in figure 1 have the same shape an the same performance relations as the experimental curve. They have also approximately the same magnitue orer as the experimental MSE. The shift between the theoretical curves an the experimental MSE is explaine by the fact that the boun (4.5) correspons to the worst estimation

5 Name Estimation Winow Vocoer ˆβ = F.( X(t + 1/F,ω;h) X(t,ω;h)) Han ˆβ = ω + I( X(t,ω;h ) X(t,ω;τ.h) ) Han Macleo ˆβ = ω + ω ( 1+8δ 1) 4δ, Rec where δ = R((X(t,ω ω;h) X(t,ω+ ω;h))x (t,ω;h)) R((X(t,ω;h)+X(t,ω ω;h)+x(t,ω+ ω;h))x (t,ω;h)) Table : Summary of the ifferent methos compare. case. The F estimator is compare to the classical methos escribe in table. For clarity, only two ifferent versions of the F estimator have been retaine: F-Rec an F-Han. In the case of a single cisoi estimation, all methos give similar values an perform quite well as all the MSE are containe within 1b of the. F-Han, because of its inherent bias, oes not perform as well as the other estimators. The best results are obtaine with Macleo s estimator, but F-Rec is very close. These two methos use the rectangular winow, an the way Macleo erive his in [5] makes them very similar. If one takes a closer look to the formula of Macleo s estimator (table ), one can see that it is the solution of an orer polynomial. As it has been mentione in the previous section, the error mae with approximation (.6) may be reuce by using higher Taylor expansion orers, which is what is one in Macleo s estimator. The increase in performance is nonetheless quite small. For low SNR (-0b), the performances of F-Rec an F-Han rop faster than for other estimators. This is cause by the asymmetry of the metho: the estimation is best one when using the maximum bin an the secon highest bin, but for this noise level the secon highest bin is har to fin. A solution, as for Quinn s estimator, is to compute the estimation for both bins aroun the maximum, an to use a test to etermine which estimation is best [5]. But this error appears only in a failure area where all estimators perform baly. Figures (c) an () represent the errors for an estimation perturbe by a secon cisoi which is, respectively, at 100Hz an 1000Hz from the first cisoi. The methos using rectangular winows now give worse results than the others, except when the perturbation ue to the secon sinusoi becomes smaller than the noise perturbation. The other methos perform better because they use a Hann winow which has a better sie lobe attenuation. The F-Han metho appears to be a goo compromise between sie lobe attenuation an single cisoi precision, comparable to the phase-vocoer an the reassignment. 6. CONCLUSION This paper has presente a new frequency estimator base on frequency variations of the FFT. This estimator is very similar to the estimation calle DFS interpolator using the phase, but the metho presente allows the use of non-rectangular winows, which was not possible before. The avantage of using non-rectangular winows is to keep goo performances even if the estimation is perturbe with close cisois, which was the main efault of the DFS interpolator using the phase. The results using a Hann winow are similar to the performances of the classical phase vocoer an the reassignment metho. In future work, we believe that using superior orer expansion an testing ifferent ratio filters H will lea to estimators performing better than the classical ones in all the scenarii. REFERENCES [1] Sylvain Marchan, Soun moels for computer music (analysis, transformation, synthesis), Ph.D. thesis, University of Boreaux, 000. [] Eric Moulines an Jean Laroche, Non-parametric techniques for pitch-scale an time-scale moification of speech, Speech Communication, [3] François Auger an Patrick Flanrin, Improving the reaability of time-frequency an time-scale representation by the reassignment metho, IEEE Transaction on Signal Processing, vol. 43, no. 5, May [4] Mototsugu Abe an Julius O. Smith III, Design criteria for simple sinusoial parameter estimation base on quaratic interpolation of fft magnitue peaks, Auio Engineering Society 117th Convention, 004. [5] M. Macleo, Fast nearly ml estimation of the parameters of real or complex single tones or resolve multiple tones, IEEE Transaction on Signal Processing, vol. 46, no. 1, pp , Jan [6] Elias Aboutanios an Bernar Mulgrew, Iterative frequency estimation by interpolation on fourier coefficients, IEEE Transactions on Signal Processing, vol. 53, no. 4, pp , Apr 005. [7] Konra Hofbauer, Estimating frequency an amplitue of sinusois in harmonic signals, Tech. Rep., Graz University of Technology, Apr 004. [8] Stephen Hainsworth an Malcolm Macleo, On sinusoial parameter estimation, Proc. of the 6th Int. Conf. on Digital Auio Effects (DAFx), 003. [9] Barry G. Quinn, Estimating frequency by interpolation using fourier coefficients, IEEE Transactions on Signal Processing, vol. 4, no. 5, pp , May [10] B. G. Quinn an E. J. Hannan, The estimation an tracking of frequency, Cambrige Univ. Press, 001. [11] Z.-G. Chen H.-Z. An an E. J. Hannan, The maximum of the perioogram, J. Multivariate Anal., vol. 13, pp , 1983.