Adaptive STFT-like Time-Frequency analysis from arbitrary distributed signal samples

Transcription

1 Adaptive STFT-like Time-Frequency analysis from arbitrary distributed signal samples Modris Greitāns Institute of Electronics and Computer Science, University of Latvia, Latvia modris Abstract The paper concentrates on the consequences of time-frequency representation of signals. Shorttime Fourier transform, wavelet transform and joint time-frequency distribution are the classical approaches used to analyze non-stationary signals. An enhancement of non-stationary signal processing is developed, which is based on the adaptation of transformation functions to the local statistical characteristics of signals. The main advantages of the proposed approach are increased resolution, suppressed side-loops and cross-terms as well as applicability to non-uniform sampling cases with sampling density less than the Nyquist rate. 1 Introduction Spectral estimation methods typically assume stationarity of signals and equidistantly spaced samples at Nyquist sampling rate. Statistical characteristics of signals of practical interest often change with time [1]. The non-stationarity of real data is usually accommodated by windowing methods. The lack of uniformly spaced samples and insufficient sampling density are typically addressed by methods that fill in the data in some way. It has been quit difficult to satisfactorily handle nonstationary signals using conceptualizations based on stationarity as well as to satisfactorily process signals sampled with density lower than Nyquist using conceptualizations based on Shannon theorem. This paper presents an advanced approach to both of these problems: to signal statistical characteristics adaptive transformation that works regardless of how the signal samples are spaced. Time-frequency analysis often deals with signals for which the local bandwidth of signal is considerably narrower than the whole bandwidth of analysis [2]. As examples can be quoted chirps, Doppler signals, frequency tracking etc. The Fourier transform of a signal takes into account its spectral components at all times and the required sampling density is defined from such a point of view. It is obvious that sample signals at their global Nyquist rate not always is cost-effective way for time-frequency analysis. There is a basic rule for equidistantly spaced samples: if we sample below the Nyquist rate we get the spectral analysis results, which have corrupting artifacts - so called aliases. A dilemma arises: on the one hand the global signal bandwidth defines sampling frequency according to Nyquist, while on the other hand the signal regions where the bandwidth is narrower accommodate a lower sampling density. The settlement of this problem can be based on the use of a non-uniform sampling technique. The proper application of it suppresses the frequency aliasing and allows to process samples with density below the Nyquist rate [3]. 2 Analysis of non-stationary signals In practice the time-frequency representation (TFR) of signal is characterized by points on a time-frequency gram with a finite duration time axis and finite bandwidth frequency axis [4]. The classical method for analyzing non-stationary signals is short-time Fourier transform (STFT). It was

2 proposed by Gabor in STFT is based on the well known Fourier transformation. The basic idea of STFT is to introduce the time window, which is moved along the signal, and in such a way time indexed spectrum can be calculated. In general there is no strict restriction regarding to the distribution of signal samples. Short-time Fourier transform for signal sampled at time instants t k can be calculated as ST F T (ω, τ) = where summation involves the samples which fall within selected time-window w. The problem with the STFT lies in the inverse relation between the time and frequency resolutions. Extension of the window s length improves the frequency resolution but deteriorates the temporal selectivity [5]. To overcome the difficulties with the resolution limitation of short time Fourier transform, a several alternative time-frequencies analysis approaches have been developed. The two most popular of them are a wavelet transform (WT) and a joint (quadratic) time-frequency distributions (for example, Wigner-Ville function (WVF)) [5],[6],[7]. The continuous wavelet transform of a signal x(t) is defined as W T (a, τ) = 1 a ( ) t τ x(t)h dt, a where a is the scaling factor and h(t) is the so-called analyzing wavelet. The time-frequency version is obtained by making the substitution a = ω 0 /ω. The analysis can be viewed as a filter bank comprising bandpass filters with bandwidths proportional to frequency. The multiresolution nature of wavelet analysis leads to some limitations. Wavelet transform techniques use a scaling profile such that frequency resolution decreases at high frequencies, while temporal resolution decreases at low frequencies. While this choice of scaling leads to nice mathematical structures and algorithms, there is no physical reason to assume that it corresponds to natural structure behavior. In addition, the timeand scale-sampling grid should usually be considerably oversampled, in order to get the best performance of WT analysis. This oversampling introduces redundancy in the time-scale representation. Wigner-Ville function provides high-resolution representation in time and in frequency for monocomponent signals. However, if the signal consists of several subcomponents, additional interference or cross-terms appears [5],[7]. A discrete form of the WVF can be expressed as W V F (ω, τ) = 2 k= x(τ+t k )x (τ t k ) exp( j2ωt k ), Note that the necessity to know signal values at x(t k )w time instants τ + t (t k τ) exp( jωt k ), k and τ t k for all k leads to the WVF application only for equidistantly spaced k:(t k τ) T w samples. Moreover, to avoid the distortion due to frequency aliasing, the signal has to be sampled at twice the Nyquist frequency for real valued signal. The appearance of cross-terms are due to the quadratic kernel of WVF and nonlinearity property of it. In order to mitigate the deleterious effects of cross-terms, a variety of modified kernels have been introduced. One way to remove these cross-terms is by smoothing the time-frequency plane, but this will be at the expense of decreased resolution in both time and frequency [7]. A promising approach how to suppress cross-terms and improve resolution is the use of signal-depended kernels considered by several authors [8]. 3 Proposed approach The ambition of developed approach is the keeping of the valuable features of classical algorithms and the minimization of the impact of its drawbacks. The basic idea is to combine the STFT-like approach with high-resolution spectral estimation method using signal dependent transformation [9]. The STFT transformation is based on windowed exponential functions w (t k τ)exp( jωt k ), which are unrelated to the spectral nature of the signal. The approach featured there suggests adaptation of transformation functions to the local spectrum of the signal. In general form the adaptive STFT-like transformation can be expressed as: AST F T (ω, τ) = x(t k )a (τ) k (ω), k:(t k τ) T w where {a (τ) k (ω)} is a set of transformation functions for time moment of analysis τ. For {a (τ) k (ω)} calculation the minimum variance (MV) filter algorithm [10] is exploited. The basic idea of the MV

3 0.5 Normalized frequency DFT Normalized time Figure 1: Frequency traces of test-signal Normalized frequency Figure 2: Discrete Fourier transform of test-signal. filter is to pass the frequency ω 0, for which the filter is designed, through the filter without distortion and to minimize the variance of the output signal for all others frequencies. The frequency response of such a filter adapts to the spectral components of the input signal on each frequency of interest. In fact, the output of MV filter can be interpreted similarly to the output of selective Fourier filter ŝ F (f 0 ) = xe(ω 0 ), e i (ω 0 ) = exp(j2πω 0 t i ) namely, as an estimate of a complex spectral value of signal x(t) at the frequency ω 0. It is shown in [11] that the coefficients of the MV filter for the frequency ω 0 are determined as i a k (ω) = (R 1 ) i,k exp( jω 0 t i ) k,i (R 1 ) i,k exp( jω 0 (t k t i ), where (R 1 ) i,k is an element of inverse autocorrelation matrix R ik = R(t i t k ). The complex spectrum values s of signal can be calculated as s(ω 0 ) = xa(ω 0 ) and the the power spectral density of signal can be estimated as P SD(ω) = s(ω)s (ω). The proposed approach assumes that frequency band of spectral analysis is covered by a set of MV filters. To obtain a high resolution spectral estimation, it is reasonable to select the frequency step several times smaller than Fourier frequency step f = 1 Θ, where Θ is the length of the signal observation to be analyzed. In general, the frequencies of these filters can be chosen arbitrarily, while the particular case is if the filter frequencies are located equidistantly. In this case the signal reconstruction can be done by inverse discrete Fourier transform. The adaptation of transformation functions to the statistical characteristics of signal is organized by involving autocorrelation matrix into calculations. That provides the possibility to get high resolution spectrum from short time series of signal. In absence of a priori knowledge of signal autocorrelation function, the adaptation of transformation to the spectral characteristics of signal can be managed by iterative updates of local autocorrelation function [12],[13]. 4 Simulation results The performance of the proposed signal dependent time-frequency transformation has been simulated using special test-signal. The test-signal consists of two components: one is a descending chirp, which diminish from high frequency (normalized frequency ) to low frequency (normalized frequency ) and the second is a frequency modulated signal in middle frequency region (sin mod-

4 a) b) c) d) Figure 3: Time-frequency representations of test-signal sampled at Nyquist rate: (a) STFT; (b) Wavelet transform; (c) Wigner-Ville distribution; (d) Proposed approach. ulation with period 256, central frequency 0.25 and modulation range from 0.05 to 0.45 of normalized frequency). The amplitude of chirp component is changing in conformity with a Gauss window function (maximum value is equal to one unit), while the amplitude of frequency modulated signal is a constant one unit. The traces of subcomponents frequencies are shown in Figure 1. The Discrete Fourier transform (DFT) of testsignal sampled uniformly at Nyquist rate is illustrated in Figire 2. The DFT result demonstrates that from global point of view the signal occupies the whole bandwidth, while from simulation point of view we know that at the each time moment there should be only two sinusoidal components. To compare the proposed time-frequency analysis with classical approaches the 512 test-signal samples at Nyquist rate has been used. The timefrequency representations obtained by the proposed and three classical approaches are demonstrated in Figure 3. For STFT analysis a Hamming window with length of 81 samples is used. STFT (Figure 3a) clearly identifies both subcomponents of test-signal, but with low resolution. The Figure 3b illustrates the limited temporal resolution of wavelet transform at low frequencies and limited frequency resolution at high frequencies. Note, that the frequency axis in this plot is not linear due to multiresolution nature of WT. The pseudo (frequency smoothing Hamming window is used) Wigner distribution provides good resolution (Figure 3c), but significant cross-term appears in timefrequency representation additionally to the true test-signal components. The time-frequency representation obtained by suggested approach is shown in Figure 3d. It demonstrates high temporal and spectral resolution without cross-terms. The proposed algorithm of time-frequency anal-

5 a) b) c) d) Figure 4: TFR of test-signal from sparsely non-uniformly sampled data: (a) Proposed approach - Nyquist sampling density; (b) STFT - sampling density 50% of Nyquist rate; (c) Proposed approach - sampling density 50% of Nyquist rate; (d) Proposed approach - sampling density 25% of Nyquist rate. ysis is applicable to the arbitrary distribution of signal samples. The benefit gained from that feature is the possibility of using sampling point flows with a density below the Nyquist rate. To investigate this feature the test signal has been sampled sparsely and obtained results are demonstrated in Figure 4. In general the developed signal dependent algorithm assumes the a priori knowledge of signal autocorrelation function, but in practice often only signal samples are known and the set of signal dependent transformation functions is calculated by iterative updates. The Figure 4 illustrates such case, and the test-signal sampled with different densities is analyzed. For comparison purposes Figure 4b demonstrates the result obtained by discrete short-time Fourier transform when sampling density is 50% of Nyquist rate. Considerable appearance of artifacts is the basic impact in TFR of STFT from insufficient sampling density. The proposed signal dependent transformation avoids this drawback at this sampling density (Figure 4c). The shortage of samples influences the magnitudes of frequency peaks, while the resolution of representation and ability to determine precise frequency tracking remain. The artifacts in time-frequency representation appear when sampling density is decreased at the level 25% of Nyquist rate (Figure 4d). 5 Conclusions The main advantage of the proposed approach is the increased resolution in comparison with STFT. It is achieved by adapting the transformation functions to the local spectral characteristics of the sig-

6 nal. Simulation results have shown that the developed method provides narrow frequency peaks, permitting more precise frequency identification enhancing the ability to determine frequency changes at any time instant. The proposed method has no problems with side-loops and cross-terms as well as it preserves the relative amplitudes of multicomponent signals thereby overcoming drawback of autoregressive model based methods [10]. The applicability for arbitrarily distributed samples makes the developed algorithm useful for processing non-uniformly sampled signals in the cases when the sampling density is less than the Nyquist rate. The shortage of samples influences the magnitudes of frequency peaks, while the resolution of time-frequency representation and ability to determine frequency tracking remain. The similar nature of the result of the proposed transformation with the short time Fourier transform provides the simplicity of signal reconstruction - the inverse STFT can be used for that. References [1] M. Akay, Ed., Time frequency and wavelets in biomedical signal processing, IEEE Press, [2] N. N. Brueller, N. Peterfreund, and M. Porat, Non-stationary signals: optimal sampling and instantaneous bandwidth estimation, in Proc. of the IEEE-SP International Symposium on Time-Frequency and Time- Scale Analysis, Pittsburgh, USA, Oct. 1998, pp [3] I. Bilinskis and A. Mikelsons, Randomized Signal Processing, Prentice-Hall, [6] C. K. Chui, Wavelet Analysis and its Applications, Boston, MA: Academic Press, [7] L. Cohen, Time-Frequency Distributions-A review, in Proc. Of the IEEE, vol 77, no. 7, pp , [8] R. G. Baraniuk and D. L. Jones, A signaldependent time-frequency representation: Optimal kernel design, IEEE Trans. Signal Proc., vol. 41, no. 4, pp , April [9] M. Greitans, Spectral analysis based on signal dependent transformation, in the 2005 International Workshop on Spectral Methods and Multirate Signal Processing SMMSP 2005, Riga, Latvia, June 20 22, [10] S. L. Marple Jr., Digital spectral analysis with applications, Prentice-Hall, [11] R. N. McDonough, Application of the maximum-likelihood method and the maximum-entropy method to array processing, in Nonlinear Methods of Spectral Analysis, S. Haykin, Ed., chapter 6. Springer- Verlag, New York, 2 edition, [12] M. Greitans, Iterative reconstruction of lost samples using updating of autocorrelation matrix, in Proceedings of the International Workshop SampTA 97, Aveiro, Portugal, Jun. 1997, pp [13] M. Greitans, Multiband signal processing by using nonuniform sampling and iterative updating of autocorrelation matrix, in Proceedings of the 2001 International Conference on Sampling Theory and Application, Orlando, Florida, USA, May 2001, pp [4] M. Greitans, Advanced processing of nonuniformly sampled non-stationary signals, Electronics and electrical engineering - Kaunas: Technologija, ISSN , vol. 59, no. 3, pp , [5] F. Hlawatsch, G. F. Boudreaux-Bartels, Linear and quadratic time-frequency signal representations, in IEEE Signal Proc. Mag., vol 9, pp , April 1992.