power spectrum estimation using welch method for various window techniques

International Journal of Scientific Research Engineering & Technology (IJSRET)Volume 2 Issue 6 pp 389-392 September 2013 www.ijsret.org ISSN 2278 – 0882

IJSRET @ 2013

Power Spectrum Estimation using Welch Method for variousWindow Techniques

Hansa Rani Gupta Rajesh MehraNITTTR, Sector-26, Chandigarh, India Electronics & Communication DepartmentSushila Batan NITTTR, Sector-26, Chandigarh, IndiaNITTTR, Sector-26, Chandigarh, India

ABSTRACTA good resolution in PSE may be achieved by usingoptimum size of data sample. In this paper PSE has beenperformed for variable data length using Rectangular andHamming window with Welch Method. Welch methodis nonparametric method that include the periodogramthat have the advantage of possible implementationusing the fast Fourier Transform. The periodogramtechnique based on Welch method is capable ofproviding good resolution if data length samples areselected optimally. The PSE based on both Rectangularas well as Hamming window has been designed andsimulated using MATLAB. It can be observed that theRectangular and Hamming give better results than otherwindows like Bartlett, Hanning and Blackman window.

Keywords- FT, FIR, MAC, PSE, PSD, Welch Method

I. INTRODUCTION

In this paper, we show the effect of data length on powerspectral density by the help of Welch method inrectangular and hamming window. The nonparametricWelch method in which the power of any input isguesstimation at different frequencies [1]. It is animprovement on the periodogram (a method ofestimating the autocorrelation of finite length of asignal) spectrum estimation method where signal tonoise ratio is high and reduces noise in the estimatedpower spectra in exchange for reducing the frequencyresolution. Periodogram is easy to compute and havelimited ability to produce accurate power spectrumestimation. It is biased when dealing with finitewindows. As data length increase, the rate of fluctuationin this is also increase. The rectangular window hasgood resolution characteristics for sinusoids ofcomparable strength. The hamming window is use tominimize the maximum nearest side lobes. TheRectangular window has admirable resolutioncharacteristics for sinusoids of comparable strength.

Bartlett methods provide a method to reduce thevariance of the periodogram in exchange for a reductionof resolution in compared to standard periodograms.Blackman method is use to correlation samples prior totransformation to reduce variance of the estimator. Thevariance of the periodogram is reduced by applying awindow to the autocorrelation estimate to decrease thecontribution of unreliable estimates to the periodogram.The Welch Method used a modified version of Bartlettmethod in which the portion of the series contributing toeach periodogram are allowed to overlap [2]. Resolutionis the ability to discriminate spectral feature and is a keyconcept on the analysis of spectral estimatorperformance.

II. POWER SPECTRUM ESTIMATION

PSE is most important application area in Digital SignalProcessing. There are mainly two types of powerspectrum estimation (PSE) method: Parametric andnonparametric. Parametric or non-classical methods ananalyzed process is eplace by an appropriate model withknown spectrum. Non-parametric do not make anyassumption on the data generating process. It is start byestimating autocorrelation sequence from a given data.The power spectrum then is estimated via FT of anestimated autocorrelation sequence. Window function isa mathematical function that is zero valued outside ofsome chosen period. When another unction or waveformor data sequence is multiplied by a window function, theproduct is also zero-valued outside the period; all that isleft is the part where they overlap the observationthrough the window. It is simple to apply andunderstand. In this method the frequency of a filter,HD(w) and the corresponding impulse response,hD(n),are related by the inverse Fourier transform:

(1)

International Journal of Scientific Research Engineering & Technology (IJSRET)Volume 2 Issue 6 pp 389-392 September 2013 www.ijsret.org ISSN 2278 – 0882

IJSRET @ 2013

Where HD(w) is frequency response of a filter andhD(n) is corresponding impulse response. The subscriptD is used to make a difference between the ideal andpractical responses. Here, HD(w) can be obtained fromhD(n) by evaluating the inverse Fourier transform. Thetruncation of hD(n) to a length M-1 is equivalent tomultiplying hD(n)by a rectangular window [2] definedas

(2)And unit impulse response

(3)Frequency domain function in representation of windowfunction is,

(4)The Rectangular window has admirable resolutioncharacteristics for sinusoids of comparable strength.When the signal is harmonically related to the windowlength in first harmonic frequency corresponding to arecord length, the signal appears periodic and infiniteand is faithfully reproduced even by the rectangularwindow. The individuality of it play a significant role inestablishment of the resulting frequency response of thefinite impulse response filter obtained by truncationhD(n) to length M. The undesirable effects are bestalleviated by the use of window that do not containabrupt discontinuities in their time domaincharacteristics and have likewise low side lobes in theirfrequency domain characteristics. For the [2] same valueof M for both Rectangular and Hamming window orother windows, the width of the main lobe is also widerfor these windows compared to the rectangular window.The Fourier transform of rectangular window:

(6)The Hamming window function in time domain decreasemore gently towards zero on either side and in frequencydomain, the amplitude of the main lobes is widerapproximately double than that of rectangular window,but side lobes are lesser relative to the main lobe about40 dB down the main lobes, compared with 14 dB forthe rectangular window. Hamming window lead to afilter with wider transition width but higher stopbandattenuation.

(7)Where Y(n) is output signalx(n) is input signal, and w(n) [1] is window function.

(8)Transition width, (9)Where N is filter length, and ∆f is normalized transitionwidthWindow Length,

(10)Where n is window size.Bartlett Method in which the final estimate of thespectrum at a given frequency is obtained by averagingthe estimates from the periodograms resultant from anon-overlapping portion of the original series. In this,the n point sequence is subdivided in to K non-overlapping segment where each segment has length L.this result in K data segment. Power spectrum estimationfor Bartlett method [3]:

(11)In the Blackmen-Tukey, we use auto-correlation and FTto get PSD in effect smooth out the periodogram, it hasbetter varience and better precision than Bartlett method.Power spectrum estimation for Blackman-Tukeymethod [4][5]:

(12)Table 1. Feature of window function [1]

International Journal of Scientific Research Engineering & Technology (IJSRET)Volume 2 Issue 6 pp 389-392 September 2013 www.ijsret.org ISSN 2278 – 0882

IJSRET @ 2013

III. WELCH METHOD

PSE is most important application area in Digital SignalWelch method have two basic modification to theBartlett method. These are allowed the data length tooverlap. The data segment can be represented as

(13)Where iD is the starting point for the ith sequence. If D= M, the segment do not overlap and the L of datasequence is identical to the data segment of Bartlettmethod.The second change in Welch method is to window thedata segments prior to computing the periodogram.

(14)Where U is a normalization factor for the power

(15)The Welch power spectrum estimate is the average ofmodified periodgram, is

(16)Mean value of Welch estimate

(17)The resolution of estimated power estimation isdetermine by the spectral resolution of each segmentwhich is of length L. it is window dependent.

IV. RESULT

In this paper we are trying to show the Data length effecton resolution.For this we are considering the data sequence of 301samples and fs = 1000 in Rectangular window andHamming window:

Data sequence of 301 samples and fs = 1000 in Hanningwindow and Bartlett window:

Data sequence of 301 samples and fs = 1000 inRectangular window and Bartlett window:

Data sequence of 301 samples and fs = 1000 in Bartlettwindow and Blackman window:

International Journal of Scientific Research Engineering & Technology (IJSRET)Volume 2 Issue 6 pp 389-392 September 2013 www.ijsret.org ISSN 2278 – 0882

IJSRET @ 2013

V. CONCLUSION

In this paper, the Welch method gives the result onpower spectrum estimation with different windows inwindow method. In DSP, Welch method is used to findthe PSD of a signal with reducing the effect of noise. Inour paper the special techniques like window functionand hamming, Hanning, Bartlett, and rectangularwindow to extract the unwanted noise from the signal.The sampling frequency 1000Hz fixed and the numberof sample are 301. The graph shows the variationaccording to the different window. We can see thedifference in all windows. The purposed algorithmsoperate in frequency domain, where the calculation ofsamples is done from the frequency domain using cosinewaveforms. The quality of the estimate increase as thelength N of the data increase, which means that theconsistence. When data length is short Blackman tuckeymethod is better than Welch method but as the datalength increase Welch Method gives excellent results.The rectangular and Bartlett window have the clear peakin the graph showing the power spectrum estimation.

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[5] ELEN 5301 Adv. DSP and Modeling, Summer II2008, p7-8, 25-26

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