Where has IIR filters are digital filters with infinite impulse response. Unlike FIR filters, they have the feedback (a recursive a part of a filter) and are mentioned as recursive digital filters, therefore. For this reason, IIR filters have far better frequency responses than FIR filters of an equivalent order. Unlike FIR filters, their phase characteristic isn’t linear which may cause a drag to the systems which require phase linearity. For this reason, it isn’t preferable to use IIR filters in digital signal processing when the phase is of the essence. Otherwise, when the linear phase characteristic isn’t important, that is typical of IIR filters only. FIR filters don’t have such a drag as they are doing not have the feedback. For this reason, it’s always necessary to see after the planning process whether the resulting IIR filter is stable or not.

IIR FILTER DESIGN
For the given specifications to style a digital IIR filter, first we’d like to style analog filter (Butterworth or chebyshev). The resultant analog filter is transformed to digital filter by using either “Bilinear transformation or Impulse Invariant transformation”

Steps for performing IIR filter

Step I: while enter the pass band ripple (rp) and stop band ripple (rs).

Step II: Let’s enter the pass band frequency (wp) and stop band frequency (ws).

Step III: After we get the sampling frequency (fs).

Step IV: Next calculate normalized pass band frequency, and normalized stop band frequency w1 and w2 respectively.

Step V :And  make use of the following function to calculate order of filter Butterworth filter order [n,wn]=buttord(w1,w2,rp,rs ) Chebyshev filter order [n,wn]=cheb1ord(w1,w2,rp,rs)

Step VI : Let’s Design an nth order digital low pass Butterworth or Chebyshev filter using the following statements. Butterworth filter [b, a]=butter (n, wn) Chebyshev filter [b,a]=cheby1 (n, 0.5, wn)

Step VII : To find the digital frequency response of the filter by using ‘freqz( )’ function

Step VIII : Next calculate the magnitude of the frequency response in decibels (dB) mag=20*log10 (abs (H))

Step IX : To plot the magnitude response [magnitude in dB Vs normalized frequency]

Step X : And calculate the phase response using angle (H) S

Step XI : To plot the phase response [phase in radians Vs normalized frequency (Hz)].

Program:

clc;

clear all;

close all;

disp(‘enter the IIR filter design specifications’);

rp=input(‘enter the passband ripple’);

rs=input(‘enter the stopband ripple’);

wp=input(‘enter the passband freq’);

ws=input(‘enter the stopband freq’);

fs=input(‘enter the sampling freq’);

w1=2*wp/fs;w2=2*ws/fs;

[n,wn]=buttord(w1,w2,rp,rs,’s’);

disp(‘Frequency response of IIR HPF is:’);

[b,a]=butter(n,wn,’high’,’s’);

w=0:.01:pi;

[h,om]=freqs(b,a,w);

m=20*log10(abs(h));

an=angle(h);

figure, subplot(2,1,1);plot(om/pi,m);

title(‘magnitude response of IIR filter is:’);

xlabel(‘(a) Normalized freq. –>’);

ylabel(‘Gain in dB–>’);

subplot(2,1,2);plot(om/pi,an);

title(‘phase response of IIR filter is:’);

xlabel(‘(b) Normalized freq. –>’);

ylabel(‘Phase in radians–>’);

output:

enter the IIR filter design specifications

enter the passband ripple13

enter the stopband ripple50

enter the passband freq1200

enter the stopband freq250

enter the sampling freq6000

POWER SPECTRUM DETERMINATION OF GIVEN SIGNAL

THEORY: The power spectrum describes the distribution of signal power over a frequency spectrum. The most frequently used way of generating a power spectrum is with using a discrete Fourier transform, but also other techniques such as the maximum entropy method can also be used. The power spectrum also can be defined because the Fourier transform of auto correlation function.

Steps involved in performing power spectrum operation for a given signal

Step I: let’s consider input sequence x

Step II: And the given sampling frequency, input frequency and length of the spectrum.

Step III: In order to find power spectrum of input sequence by using matlab command spectrum.

Step IV: And plot power spectrum using specplot.

Program:

clc;

clear all;

close all;

N=1024;

fs=8000;

f=input(‘enter the frequency[1 to 5000]:’);

n=0:N-1;

x=sin(2*pi*(f/fs)*n)

pxx=spectrum(x,N);

specplot(pxx,fs);

grid on

xlabel(‘freq(hz)’);

ylabel(‘magnitude(db)’);

title(‘power spectrum of x(n)’);

Enter the frequency [1 to 5000]:2000

LOWPASS FIR FILTER IMPLEMENTATION

THEORY:

Where the FIR filters are also known as digital filters with finite impulse response. They are also referred as non-recursive digital filters because they do not have the feedback.

An FIR filter has two most important advantages when compared with an IIR design:

Firstly, there’s no feedback circuit within the structure of an FIR filter. Due to not having a feedback circuit, an FIR filter is inherently stable. Meanwhile, for an IIR filter, we’d like to see the steadiness.
Secondly, it also provide a linear-phase response. As a matter of fact, a linear-phase response is that the most frequently used advantage of an FIR filter when compared with an IIR design otherwise, for the same filtering specifications; an IIR filter will lead to a lower order.

FIR FILTER DESIGN

An FIR filter is meant by finding the coefficients and filter order that meet certain specifications, which may be within the time-domain (e.g. a matched filter) and while in the frequency domain (most common). Matched filters perform a cross-correlation between the input and a known pulse-shape. The FIR convolution may be a cross-correlation between the input and a time-reversed copy of the impulse-response. When a specific frequency response is desired, several different design methods are common:

1. Window design method

2. Frequency sampling method

3. Weighted least squares design

WINDOW DESIGN METHOD

In the window design method, one first designs a perfect IIR filter then truncates the infinite impulse response by multiplying it with a finite length window function. The result’s a finite impulse response filter whose frequency response is modified from that of the IIR filter.

Steps involved in low pass FIR filter design:

Step I: Let’s enter the pass band frequency (fp) and stop band frequency (fq).

Step II: Get the sampling frequency (fs), length of window (n) for the low pass FIR filter.

Step III: Let’s calculate the cut off frequency, fn

Step IV: Using boxcar, hamming, blackman Commands to design window.

Step V: Design filter by using above parameters.

Step VI: To find frequency response of the filter by using matlab command freqz.

Step VII: And plot the magnitude response and phase response of the filter.

Program:

clc;

clear all;

close all;

n=20;

fp=200;

fq=300;

fs=1000;

fn=2*fp/fs;

window=blackman(n+1);

b=fir1(n,fn,window);

[H W]=freqz(b,1,128);

subplot(2,1,1);

plot(W/pi,abs(H));

title(‘magnitude response of lpf’);

ylabel(‘gain in db——–>’);

xlabel(‘normalized frequency——>’);

subplot(2,1,2);

plot(W/pi,angle(H));

title(‘phase response of lpf’);

ylabel(‘angle——–>’);

xlabel(‘normalized frequency——>’);

Theory:

FFT is a most efficient algorithm used in speech processing, frequency estimation and communication. FFT utilizes full advantage of two symmetry properties in order to provide efficient algorithm for DFT. while considering a divide and conquer approach, a computationally efficient algorithm can be developed. This methods depends on the decomposition of an N-point DFT into successively smaller size then DFTs. While in the N-point sequence, the term N can be expressed as N = r₁r₂r₃, …, r_m, then N = r ^m, can be decimated into r-point sequences For each r point sequence, r-point DFT are often computed. Hence the DFT is of size r. if the number r is called the radix of the FFT algorithm while the number m indicates the number of stages in computation. If the value of r = 2, it is called radix-2 FFT.

Steps involved in finding the FFT of the given sequence:

• Let’s consider an input sequence as x[n].
• Let’s find the length for the input sequence using length command.
• In order to find FFT and IFFT by using the matlab commands fft and ifft.
• Let’s plot the magnitude and phase response for the given sequence.
• And finally display the results.

Program:

clc;

clear all;

close all;

x=input(‘Enter the sequence : ‘)

N=length(x)

xK=fft(x,N)

xn=ifft(xK)

n=0:N-1;

subplot (2,2,1);

stem(n,x);

xlabel(‘n—->’);

ylabel(‘amplitude’);

title(‘input sequence’);

subplot (2,2,2);

stem(n,abs(xK));

xlabel(‘n—->’);

ylabel(‘magnitude’);

title(‘magnitude response’);

subplot (2,2,3);

stem(n,angle(xK));

xlabel(‘n—->’);

ylabel(‘phase’);

title(‘Phase response’);

subplot (2,2,4);

stem(n,xn);

xlabel(‘n—->’);

ylabel(‘amplitude’);

title(‘IFFT’);

Output:

Enter the sequence: [3 2 4 5 1]

x = 3     2     4     5     1

N =5

xK =15.0000 + 0.0000i  -3.3541 – 0.3633i   3.3541 – 1.5388i   3.3541 + 1.5388i  -3.3541 + 0.3633i

xn =3.0000    2.0000    4.0000    5.0000    1.0000