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ˆ K  X(N)E N (1) S Xx  Directly from Signal Being Estimated X(N) International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.1, Issue.2, pp-247-251 ISSN: 2249-6645 Performance Evaluation of Filter-bank based Spectrum Estimator M.Venakatanarayana1, Dr.T.Jayachandra Prasad 2 1Assoc . Prof , Dept. of ECE, KSRM College of Engg., Kadapa. 2Professor, Dept. of ECE, RGMCET, Nandyal, Abstract— signal. Estimator based on parametric method provides higher In this paper on attempt has been made to study the degree of detail. performance of Filter-bank based nonparametric spectral The disadvantage of parametric method is that if the signal is estimation. Several methods are available to estimate non not sufficiently and accurately described by the model, the parametric power spectrum. The band pass filter, which result is less meaningful. Non Parametric methods, on the sweeps through the frequency interval of interest, is main other hand, do not have any assumption about the shape of the element in power spectrum estimation setup. The filter- bank based spectrum estimation is developed and is power spectrum and try to find acceptable estimate of the applied to multi tone signal. The spectrum estimated based power spectrum without prior knowledge about the underlying on filter-bank approach has been compared with stochastic approach. The following sub-sections give review conventional nonparametric spectrum estimation on some of the spectrum estimation methods. techniques such as Periodogram, Welch and Blackman- Tukey. It is observed that the filter-bank method gives A. Periodogram better frequency resolution and low statistical variability. The most commonly known spectrum estimation technique is It is also found there is a tradeoff between resolution and periodogram, which is classified as a non parametric estimator. statistical variability. The procedure starts by calculating the Discrete Fourier Transform (DFT) of the random signal being estimated, Index Terms— Correlogram, Filter-bank, Frequency resolution, Periodogram, Spectral leakage etc., followed by taking the square of it and then dividing the result with the number of samples N. I. INTRODUCTION There are five common nonparametric PSE available in the literature: the periodogram, the modified periodogram, In Signal processing, the nonparametric spectrum estimation Bartlett’s method, Blackman–Tukey, and Welch’s method. plays an important role in determining perioditicity in random However, all these nonparametric PSE, s are modifications of signals and thus a comprehensive elaboration of filter-bank the classical periodogram method introduced by Schuster. based spectrum estimation techniques has been presented. In general, spectrum estimation can be categorized into direct and The periodogram is defined as [2] 2jkn 2 indirect methods. In direct method (usually recognized as 1 N 1 frequency domain approach), the power spectrum is estimated ˆ k x(n)e N (1) S xx directly from signal being estimated x(n) . On the other N n0 hand, in indirect method, also known as time domain It is known that the periodogram is asymptotically unbiased approach, the autocorrelation function of the signal being but inconsistent because the variance does not tend to zero for estimated is calculated. From this autocorrelation large record lengths. One can show [10], that the Variance on value, the power spectrum density can be found by applying the periodogram Sˆ (k) of an ergodic weakly stationary the Discrete Fourier Transform on . Another way to xx categorize spectrum estimation methods is by classifying them signal x(n) for n 0: N 1is asymptotically proportional into parametric or non-parametric methods. Parametric method 2 to S xx(k), the square of the true power at frequency bin. is basically model based approach [10]. In this method, a signal is modeled by Auto Regressive (AR), Moving Average The periodogram uses a rectangular time-window, a weighting (MA) or Auto Regressive Moving Average (ARMA) process. function to restrict the infinite time signal to a finite time Once the signal is modeled, all parameters of the underlying horizon, the modified periodogram uses a nonrectangular time model can be estimated from the observed window [3]. A way to enforce a decrease of the variance is averaging. Bartlett’s method divides the signal of length www.ijmer.com 247 | P a g e International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.1, Issue.2, pp-247-251 ISSN: 2249-6645 N N into K segments of length L each. The Prototype filter K (0th band) H ( f ) periodogram method is then applied to each of the K ith 1st 2nd segments. The average of the resulting estimated power band band band spectra is taken as the estimated power spectrum. One can show that the variance is reduced by a factor K , but the spectral resolution is also decreased by a factor K , [2]. The Welch method eliminates the tradeoff between spectral resolution and variance in the Bartlett method by allowing the segments to overlap [4]. B. Blackman-Tukey method (Windowed Correlogram) Figure.1 The filter bank concept. f Blackman-Tukey method is a variant of correlogram that computes the approximated autocorrelation and later A. Periodogram spectral estimator realization through applies a suitable window function w[k]. The power spectra filter banks density is then obtained by computing the Fourier Transform of windowed auto-correlation sequence [10]. Spectrum estimation is about finding the power spectrum Blackman-Tukey method is generally described as follows density (PSD) of a finite sample set (2) {x(n),n 1,2,..........N} for frequency . The And its frequency domain representation is given by classical approach to spectrum estimation is to use Fourier (3) transforms to obtain a Periodogram, given as [17]: from equation (3) the Blackman-Tukey method can actually be N 2 viewed as a process of smoothing the correlogram by p j2f 1 j2fn S (e ) x(n)e (4) convolving the correlogram with the kernel of selected xx N n1 window. This smoothing process plays an important role to reduce the bias of estimated PSD but this convolution process for any given frequency f i , (4) is written as: would reduce the frequency resolution. The amount of 2 1 N frequency resolution reduction is strongly related to the size of S p (e j2fi ) x(n)e j2fin the main lobe of the window kernel. Section II talks about xx N n1 filter bank approach to spectrum estimation technique. Section (5) N 2 III focuses the performance analysis of the proposed 1 j2f (N n) techniques in comparison with the conventional techniques. x(n)e i N n1 II. SPECTRUM ESTIMATION AS A FILTER BANK ANALYSIS From the perspective of spectrum estimation, a filter bank can it should be noted that (5) is possible be considered as an array of band pass filters that separates the e( j2fi N) 1 input signal into several frequency components, each one since . By introducing new carrying a single frequency sub-band [6]. The filter banks are variable k N n , equation (5) is written as: usually implemented based on single prototype filter, which is N 1 2 a low pass filter. This low pass filter is normally used to p j2f 1 j2f n S (e i ) x(N k)e i realize the zero-th band of the filter bank while filters in the xx N k 0 other bands are formed through the modulation of the (6) 2 prototype filter [9]. Figure 2.6 illustrates the main idea of filter 1 N 1 h (k)x(N k) bank concept. This section basically tries to explore the filter N i bank paradigm in spectrum estimation. k 0 1 where h (k) e( j2f i k ) for k 0,1,2,........N 1 i N by observing the summation within the magnitude operation in (6) and the summation expressed as: N 1 y(N) hi (k)x(N k) (7) k0 (7) is actually the truncated convolution sum at particular point N, which is again written as general convolution sum at the same point associated with a linear causal system by padding www.ijmer.com 248 | P a g e International Journal of Modern Engineering Research (IJMER) www.ijmer.com Vol.1, Issue.2, pp-247-251 ISSN: 2249-6645 hi (k) with zeros[43]. Then (7) is rewritten as y (N) h (k)x(N k) (8) a i k0 with ( j2fi\k) w(k)e fork 0,1,2,...........N 1 hi (k) (9) 0 otherwise Figure.2 The demodulation of received signal and window function w(k) 1/ N . It is clear that (8) From the above, the implementation of a spectrum estimator represents as passing N samples through a filter having using filter bank for signal analysis is clear, namely by passing an input signal through a bank of filters. The output power of impulse response hi (k) and then taking only single sample of each filter is a measure of the estimated power over the the filtered signal at point N. Based on this perspective, the corresponding sub-band. Hence the power spectral density frequency response of the linear filter having impulse response (PSD) estimate of i-th sub band of the filter bank is is represented as [16]: N 1 (12) jk 1 j(i ) Hi () hi (k)e e k0 N k0 In (12), avg [] describes time average operator while y (n) is (10) i 1 e j(i )N 1 th the output signal of i sub band filter. N e j(i ) 1 finally gives III. RESULTS AND ANALYSIS Consider a signal having two closed frequencies embedded in sin[N(i ) / 2] N 1 Hi () exp[ j( )(i )] (11) white noise i.e. , N sin[i ) / 2] 2 where f1 0.1, f 2 0.12 and is white noise having If w(k) in (9) is taken to be a prototype FIR (Finite Impulse zero mean and unit variance.
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