Performance Evaluation of Mean Square Error of Butterworth and Chebyshev1 Filter with Matlab

Performance Evaluation of Mean Square Error of Butterworth and Chebyshev1 Filter with Matlab

Mamta Katiar Anju

Associate professor Lecturer, Kalpana Chawala Govt. Mahararishi Markandeshwer University , Mullana Polytechnic for Women, Ambala city,

Haryana,India. Haryana, India

Abstract

A signal is any physical phenomenon which conveys information of any k ind from one place or person to another. In communication system, during the processing of signal, some noise is added in the signal and signal becomes noisy. This is now mandatory to extract the signal buried under noise and periodic interference. In this paper, a signal is denoised by Butterworth filter and Chebyshev1 filter and calculating mean square error and signal to noise ratio from reconstructed signal at receiver and then compare the Butterworth and Chebyshev1 filter to find the best results. For this evaluation, all data is coded in the MATLAB.

Keywords

Butterworth filter, Chebyshev1 filter, Mean square error, Signal to noise ratio.

1.Introduction

The Digita l Filtering is one of the most powerful tools of DSP. The d igital filters consist of software and hardware. The input and output signals in the digital filter is digital or discrete time variant. The procedure for designing dig ital filters involves the determination of a set of filter coeffic ients to meet a set of design specifications . Digital filters co me in two flavours: FIR and IIR. As the terminology suggest, these classifications re fer to the filters impulse response. By vary ing the we ight of the coeffic ients and number of filter taps , virtually any frequency response characteristics can be realised with an FIR filter. FIR filters have a very useful property: they can exhibit linear phase shift for all frequencies. IIR filters have infin ite impulse

response. IIR filters have much better frequency response than FIR filters of the same error. In IIR filters their phase characteristics is not linear , wh ich can cause a problem to the systems which need phase linearity but in MATLAB software data processing is commonly performed offline, i.e. the entire data sequence is available prior to filtering[1]. This a llows for a non causal, zero phase filtering approach (via the filtfilt function), which e liminates the non linear phase distortion of an IIR filters.IIR filters can achieve the same level of attenuation as FIR filters but with far fewe r coefficients. Therefore, an IIR filter can provide a significantly faster and most efficient filtering operation than an FIR filter. This paper considers two IIR filters: Butterworth and Chebyshev1.

1. Butterworth Filter

   The butterworth filter has a ma ximally flat response, i.e., no passband ripple and roll-off of minus 20db per pole. Another name for it is flat ma xima lly magnitude filters at the frequency of = 0, as the first 2N – 1 derivatives of the transfer function when = 0 a re equal to zero. [2]. The Butterworth filters achieve its flatness at the expense of a relat ively wide transition region fro m passband to stopband with average transient characteristics. This filter is complete ly defined mathe matica lly by two parameters i.e . cut of frequency and number of poles. Co mpared to chebyshev filter, the phase linearity of buttorworth filter is better. In other words, the group delay (derivative of phase with respect to frequency) is more constant with respect to frequency. This means that the waveform d istortion of the butterworth filte r is lower. Th is Butterworth filters have the following characteristics [3].

   The magnitude response is nearly constant (equal to 1) at lower frequencies. That means pass band is ma xima lly flat.

   The response is monotonically decreasing fro m the specified cut off frequencies.

   The ma ximu m ga in occurs at = 0 and it is

   |H(0)|= 1.

   Half power frequency, or 3db down frequency, that corresponds to the specified cut off frequencies.

   The magnitude squared response of low pass Butterworth filter is given by

   H( )|=1/1+(/c)2N (1)

   This equation is also expressed as

   |H()|2=1/1+ C2(/p)2N (2)

   Here |H()|=Magnitude of analog low pass filter.

   c=Cut-off frequency (-3db frequency) p=Pass band edge frequency.

   C=Para mete r re lated to ripples in pass band.

   N=Order of the filter.

   The order o f filter means the nu mber o f stages used in the design of filter. As the order of filter N increases, the response of filter is more c lose to the ideal response as shown in Fig.1.

   |H()|

   Fig.1.1- Effect of N on frequency response characteristics.

2. Chebyshev Type1 Filter

   Chebyshev1 filters have a narrowe r transition region between the passband and the stopband. The sharp transition between the passband and the stopband of a chebyshev filter produces smaller absolute errors and faster execution speeds than a butterworth filter. The poles of chebyshev filter lies on an e llipse. ripple increase (band), the roll-o ff becomes sharper(good). The chebyshev filte r is complete ly defined by three parameters-cut-off frequency, number o f poles and passband ripples. The chebyshev response is a mathe matica l strategy for achieving a faster ro ll off by allo wing ripple in the frequency response. The chebyshev response is an optimal trade-off between these two parameters. The magnitude squared frequency response is given by

   |H()|2=1/1+ C2CN2(/p) (3)

   Here |H()|=Magnitude of analog low pass filter.

   C=Para meter re lated to ripples in pass band. CN(x)=Chebyshev polynomia l of order N

   The chebyshev1 polynomia ls are determined by using the equations

   CN+1(x)=2x CN(x)- CN-1(x) (4)

   with C0(x)=1 and C1(x)=x

   The following figure shows the frequency response of a lo wpass Chebyshev1 filter.

   Fig.1.2- Effect of N on Chebyshev1 filter characteristics

   Chebyshev

3. Mean Square Error

   The Mean Square Error(MSE) has been the dominant quantitive performance matric in the field of signal

   processing. It is the standard criterion fo r the assessment of signal quality fidelity[4]. It is the method of choice for comparing competing signal processing methods of systems. It is one of the best choices of design engineers seeking to optimize signal processing algorithms.

   The difference between the orig inal signal & the reconstructed signal is Error signal which is denoted as err. Mean squre error is calculated by taking the average of the err. The value of MSE should be as low as possible. The formu la for MSE is given by

   MSE= [ err2]/M (5)

   where M is the length of signal.

   The MSE has many attractive features: MSE is simp le.

   It is para meter free and ine xpensive to

   compute, with a comp le xity of only one mu ltip ly and two additions per sample.

   It is also me mory less the squared error

   can be evaluated at each sample, independent of other samples.

   It has a clear physical mean ingit is the

   natural way the energy of the error signal.

   The MSE is an e xce llent metric in the context of optimization.

4. Signal to Noise Ratio

Signal to noise ratio (SNR) is a para meter use to quantify and compare the performance of a lgorith ms and also determine the noise level in an reconstructed signal. The e xpression used to calculate signal to noise ratio is given by

SNR= 10log10[variance(So)/varience(So-Sf)]

Where So= original signal and Sf = filtered signal.

2. METHOD

The transmitted signal is easily corrupted by noises such as Gaussin noise, Power line interference and so on. The process of adding noise to original noise is mathemat ically shown as

F(n)= X(n)+D(n), (6) n=1,2,3………..N

X(n) is the orig inal signal

D(n) is the Random No ise signal. F(n) is the Signal+Noise

The F(n) signal is then filtered one by one at receiver by butterworth filter and chebyshev1 filter.

Fl ow c hart for signal extr acti on burie d in noise.

Ste ps for Calculating Mean Square Error:

1. In itia lly set the passband frequency (wp), stopband frequency (ws), passband ripples(rp) and stopband ripples(rs).

2. Determine the order and coefficients of filters .In MATLAB, use the command buttord() and cheb1ord() for butterworth filter and chebyshev1 filter respectively.

   [n,wn] = buttord(wp,ws,rp,rs)

   Where n is order of filter and wn is a cut off frequency.

3. Applying the command butter() to find the filter coeffic ients of butterworth filter .

   [b,a] = butter (n, wn, ftype)

   In case of chebyshev1 filter, use command cheby1().

   [b,a] = cheby1(n,wn,rp,ftype)

   This function designs a highpass , lowpass or bandstop filter, where the string ftype is high,

   low, or stop. It returns the filter coeffic ients in

   length n+1 ro w vectors b and a, with coefficients in descending powers of z.

   H(z)=[b (1)+b(2)z-1+—-+b(n +1)z-n]/[1+a(2)z-1+——-

   -+a(n +1)z-n] (7)

4. Applying the same noisy signal as an input on the Butterworth filter and Chebyshev1 filter and plotting the graph.

5. Ca lculate the mean square error and signal to noise ratio.

Table 3.3

Table 3.4

3. RESULTS

Specifications taken for the design of Butterworth and Chebyshev1 filters are:

Sa mpling frequency=2000Hz. Passband ripples=3db Stopband ripples=43db

By giv ing different values of cut off frequency to Butterworth filter and chebyshev1 filter, we get the parameters as shown belo w in Table 3.1, 3.2, 3.3 and 3.4.

Table 3.1

Table 3.2

The results showed in the tables states that as compare to chebyshev1 filter, the butterworth filters have better MSE and SNR va lues. The Order of butterworth filter is observed to be more than chebyshev1 filter at sa me cut off frequency. The following plots had been generated at a cut -off frequency of 200Hz.

Fig 3.1- Orig inal Signal at Trans mitter

Fig 3.2- Graph of channel No ise

Fig 3.3- Signal over channel with noise

Fig 3.4- Graph of MSE and SNR for Chebyshev1 filter

Fig 3.5- Graph of MSE and SNR for Butterworth filter

References

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