A new zoom algorithm and its use in frequency estimation

Transcription

1 Waves Wavelets Fractals Adv. Anal. 5; :7 Research Article Open Access Manuel D. Ortigueira, António S. Serralheiro, and J. A. Tenreiro Machado A new zoom algorithm and its use in requency estimation DOI.55/wwaa-5- Submitted Aug 6,5; accepted Sep 6, 5 Abstract: This paper presents a novel zoom transorm algorithm or a more reliable requency estimation. In act, in many signal processing problems exact determination o the requency o a signal is o paramount importance. Some techniques derived rom the Fast Fourier Transorm (FFT), just pad the signal with enough zeros in order to better sample its Discrete-Time Fourier Transorm. The proposed algorithm is based on the FFT and avoids the problems observed in the standard heuristic approaches. The analytic ormulation o the novel approach is presented and illustrated by means o simulations over short-time based signals. The presented examples demonstrate that the method gives rise to precise and deterministic results. Keywords: Zoom algorithm; requency estimation; FFT Introduction Frequency estimation usually involves the use o FFTbased estimators, either directly, as it is the case o periodogram, or indirectly, as in the MEM, MUSIC and other similar methods. In all cases, we are looking or the exact position o the spectral peaks in a given spectral estimate S( ), < (where denotes requency). Nevertheless, when using the FFT, S( ) is sampled over an uniorm grid and, as a consequence, it is very unlikely that we are successul in obtaining the true peak positions. A better es- timation o the positions o the spectra peaks can be obtained using large zero padding, thus leading to very large FFT lengths. In [3] a warped discrete Fourier transorm is used and its perormance is compared with several other procedures, namely: Dichotomous-search, Tretter s linear regression, Kay s phase dierence and chirp Z-transorm methods. Further readings and applications o this technique can be ound in [, 4, 5, 8 ]. We have an interpolation problem in the requency domain, but it is very special case since we know the interpolating unction, which is the Fourier transorm. The real problem appears because we are using a DFT implementation. As this deines completely the Fourier transorm, we can approach this problem rom a quite dierent point o view: the zooming o a small portion o the spectrum that includes the peak position. To perorm the spectral zoom, two dierent methods o interpolation have been proposed and usually reerred as the zoom transorm []. However, since these methods imply a return to time, modulation and iltering, they are not very useul when dealing with short-time signals. An alternative and simpler algorithm was proposed in [6] that merely explores the act that the FFT (DFT) is a sampling o the Fourier Transorm and, so, it has the whole inormation we need. This paper urther details this topic and is organized as ollows. In section we present a general ormulation that allows us to choose the requency search grid. Furthermore, it is also shown that the proposed algorithm is useul in converting a spectral representation rom a linear to logarithm scale. In section 3 we present some numerical results. Finally section 4 outlines the main conclusions. Manuel D. Ortigueira: CTS-UNINOVA, Department o Electrical Engineering, Faculty o Sciences and Technology, Universidade Nova de Lisboa, Portugal; mdo@ct.unl.pt; Tel.: ; Fax: António S. Serralheiro: Academia Militar and INESC, R. Alves Redol, 9, º, -9, Lisbon, Portugal; antonio.serralheiro@inesc-id.pt; Tel.: ; Fax: J. A. Tenreiro Machado: Institute o Engineering, Polytechnic o Porto, Dept. Electrical Engineering, Rua Dr. António Bernardino de Almeida, 43, Porto, Portugal; jtm@isep.ipp.pt; Fax: The Zoom Algorithm Let x(n), n =,, L, denote an L-length sample sequence. Every N L point DFT sequence represents samples o the Discrete-Time Fourier Transorm (DTFT): ( X e jω) L = x (n) e jωn. () n= 5 M. D. Ortigueira et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3. License.

2 8 M. D. Ortigueira et al Figure : Zoom in two dierent bands o a spectrum. This relation deines an ininite number o DFTs, each one characterized by one sampling grid. We do not need any additional inormation to pass rom one to another. Let X N (k) denote the DFT o x(n), corresponding to sampling X ( e jω) at a uniorm grid: X N (k) = DFT [x (n)] = X k =,, N, N L. ( e j π N k), () Accordingly to what we said we have many DFTs: one or each grid that can be deined by N. Each o these DFTs has the same inverse given by: x (n) = N X N N (k) e j π N kn, n =,, N. (3) k= Substituting equation (3) into equation () results in: ( X e jw) = N X N N (k) G (ω, k), k =,, N, (4) k= where G (ω, k) is given by G (ω, k) = e j (ω π N k )L e j (ω π N k ), (5) or ω π and k < N. This relation allows us to compute the Fourier coeicients corresponding to one grid rom the ones o another grid. It is straightorward to show that: ] G (ω, k) = sinc [ ( ω π where sinc (x) = sin(x) x N k) L sinc [ ( ω π N k) ] e j (ω π N k ) L, (6). Since ω = π, we obtain: ) ] L G (, k) = sinc [ ( k N sinc ( ) e jπ ( k k N )(L ). (7) N Figure : a) Estimated requency as a unction o the SNR by zooming showing on red the exact requency value (top) and the mean square error in ( db) with the Cramer-Rao bound in blue (bottom), b) Periodogram and zoom o the peak or a sinusoidal signal, =.76 Hz. So, equations (4) and either (6) or (7) allows us to zoom into the requency region o interest. O course, we are not interested in zooming the whole spectrum¹, just a given band (see Fig. ). 3 Simulation results 3. Frequency Estimation In this section, we present some simulation results obtained with a sinusoidal signal o angular requency ω =.43 ( =.76 Hz) or dierent values o the signal-to-noise ratio (SNR), spanning rom up to +5 db. The Cramer-Rao bound is included or reerence But we can do it, i we ind it useul.

3 A new zoom algorithm and its use in requency estimation 9 Figure 3: Frequency estimation as a unction o the SNR o a sinusoidal signal ( =.76 Hz) corrupted by a sinusoid with identical amplitude, or =. and noise, showing, a) rom top to bottom: estimated requency, inverse o mean square error (db), b) rom top to bottom: periodogram and zoom o the peak. Figure 4: Frequency estimation as a unction o the SNR o a sinusoidal signal ( =.76 Hz) corrupted by a sinusoid with identical amplitude, or =.9 and noise, showing a) rom top to bottom: estimated requency, inverse o mean square error (db), b) rom top to bottom: periodogram and zoom o the peak. as in [3] and the reciprocal value o the variance in db is also shown. For the calculations the values L = 4, N = 64 are adopted. Furthermore, the band o requencies to be zoomed is [.,.5] Hz, where points are computed in a total o runs (Figure a)). Figure b) depicts the periodogram and the corresponding peak zoom or the same simulations o Figure a). Other numerical calculations are perormed and their results are presented in Figures 3 and 4. The values L = 4, N = 64 are considered, and the band o requencies to be zoomed consists o [.,.5], where points are computed in a total o runs. In both cases, a sinusoid with identical amplitude is now added to the original signal, plus noise as beore. The new sinusoid has a requency equal to. (Fig. 3) and equal to.9 (Fig. 4). We veriy that the presence o the second sinusoid produces a bias in the estimated requencies and, as expected, when the two requencies are closer, the bias becomes higher. The saturation eect we observe in Figure 3 is due to the numerical errors. 3. Comparision to FFT Zoom In this section we present several comparison experiments o the proposed zoom algorithm against a classical FFT based zoom: a non-destructive zoom Fast Fourier transorm o a time history [7]. We ran both algorithms in the same circumstances or N = 3, N = 64 and N = 8 as pictured in Figures 5, 6 and 7, respectively. The increased perormance o the proposed algorithm both in terms o the requency estimation and its error variance is evident. However, as the time sequence length increases, both algorithms approach their perormances.

4 M. D. Ortigueira et al. estimated and correct requencies estimated and correct requencies Figure 5: Frequency estimation as a unction o the SNR o a sinusoidal signal ( =.76 Hz) corrupted white noise, showing rom algorithm, or N = 3. Figure 7: Frequency estimation as a unction o the SNR o a sinusoidal signal ( =.76 Hz) corrupted white noise, showing rom algorithm, or N = 8. estimated and correct requencies Figure 6: Frequency estimation as a unction o the SNR o a sinusoidal signal ( =.76 Hz) corrupted white noise, showing rom algorithm, or N = Frequency Scale Conversion Another application o this algorithm is the conversion o a linear requency scale into a logarithmic one. In Figure 8 we use the algorithm or converting the transer unction o a th-order low-pass FIR ilter, with a bandwidth o.5 Hz, rom a linear (blue plot) to log requency scale (green plot). This conversion can be easily done using equation (7) as an interpolation actor or the ilter requency response to the new requency scale. Figure 8: Example o linear ( o and blue line) to log ( + and green line) requency conversion o a th-order low-pass FIR ilter with a cut-o requency o.5 Hz. 4 Conclusions A zoom algorithm was presented, with applications ranging rom requency estimation to requency scale conversion. Oten the requency estimation problem involves the usage o FFT-based estimators, either directly (e.g., periodogram), or indirectly (e.g., MEM, MUSIC and similar methods). In all the cases, we are looking or the exact position o the peaks in a given spectral estimate. The underlying idea o several existing algorithms is to reine the sampling o the DFT, but with the expense o large computational requirements. Other methodologies [] try to solve this problem by zooming a small portion o the spectrum where the peak position is located. A irst drat o the algorithm was proposed in [6] and was now urther developed and explored. The scheme was applied to spectral estimation with good results.

5 A new zoom algorithm and its use in requency estimation Reerences [] Burrus C.S., McClellan J.H., Oppenheim A.V., Parks T.W., Schaer R., Schuessler H., Computer-Based Exercices or Signal Processing using MatLab, Prentice-Hall, USA (994) [] Chen W.J., Spectrum magniier: Zooming into local details in the requency domain. In: 3 IEEE China Summit & International Conerence on Signal and Inormation Processing, pp. 8 85, Beijing, China (3) [3] Franz S., Mitra S.K., Doblinger G., Frequency estimation using warped discrete Fourier transorm, Signal Processing 83(8), (3) [4] Hossen A., Heute U., A novel simple adaptive spectral analysis zoom or narrow-band signals. In: th European Signal Processing Conerence, pp. 4, Toulouse, France () [5] Murugan K., Electrocardiogram signal analysis using zoom FFT. In: ISSNIP Biosignals and Biorobotics Conerence, pp. 4. Manaus, Brazil () [6] Ortigueira M.D., Matos C.J.C., Piedade M.S., Fractional discretetime signal processing: Scale conversion and linear prediction, Nonlinear Dynamics 9( 4), 73 9 () [7] Randall R.B., Frequency Analysis, Brüell and Kjær, Denmark (987) [8] wei Wang L., liu Zhao J., ying Wang Z., na Pang L., Application o zoom FFT technique to detecting EM signal o SLF/ELF, Acta Seismologica Sinica (), 63 7 (7) [9] Yanhua J., Gun L., Kaiyu Q., Study o a novel zoom spectrum analysis approach or wireless communication analyzer. In: International Conerence on Communications and Mobile Computing, vol., pp Yunnan, China (9) [] Zolnai Z., Juranić N., Markley J.L., Macura S., Zooming, a practical strategy or improving the quality o multidimensional NMR spectra, Journal o Magnetic Resonance, Series A 9(), 5364 (996)