Question

Discrete Time Signal Processing Question 1. Consider an IIR filter A(1-2-1 cos ω0) 1-2cos ω02-1+2 I. Compute its impulse response using the difference equation with an impulse signal δ(n) as the input. Use trigonometric identities to simplify the result as much as you can 2. Draw the diagram showing the implementation of this filter in terms of adders, delays and multipliers Note: The IIR filter above generates a cosinusoidal signal when an impulse signal is applied at its input.] Question 2. Using the "butter" function in MATLAB, design a fourth-order Butterworth IIR lowpass filter H(z) = 1+a121+a222+a3+az4 The 3-dB cutoff frequency of lowpass filter should be 2.5 kHz. Assume the sampling frequency is 8 kHz. Plot its magnitude function |H() in linear as well as dB scales. List the filter coefficients. Plot the pole-zero plot. Plot the phase response and comment on the group delay of the Butterworth IIR filter.

Answer 1

%BUTTERFILTFILT Design and apply Butterworth filter.

%

% INPUTS

% ------

% x - Data to be filtered.

% Filter operates on first non-singleton dimension.

%

% fc - Cutoffs frequencies (one frequency value for types 'lowpass'

% and 'highpass' and two frequency values for types 'bandpass'

% and 'bandstop').

%

% fs - Data sampling frequency.

%

% order - Order of the Butterworth filter (default: order = 6)

%

% type - Filtering type option. Can be either 'lowpass', 'highpass',

% 'bandpass' or 'bandstop'.

%

% direction - Direction to filter in

% 'forward' | 'backward' | {'both'}

% NB: if this is 'both' the output has zero phase filtering,

% and order must be an even number.

%

% NB: If a bandpass is requested with filter bounds [0 F], a lowpass

% filter is performed instead. If a bandpass is requested with filter

% upper bound Inf or the Nyquist freq, a highpass filter is performed.

%  

%

% OUTPUTS

% -------

% x - Filtered data

%

% A - Filter coefficient matrix (denominator)

%

% B - Filter coefficient matrix (numerator)

function [x, A, B] = butterfiltfilt(x, fc, fs, order, type, direction)

% DEFAULT INPUTS ----------------------------------------------------------

if nargin<6 || isempty(direction)

direction = 'both';

end

if nargin<5 || isempty(type)

type = 'bandpass';

end

if nargin<4 || isempty(order)

order = 6;

end

if nargin<3 || isempty(fc)

if all(fc>=0) && all(fc<=1)

fs = 1;

else

error('Sampling frequency not given');

end

end

% INPUT HANDLING ----------------------------------------------------------

if nargin<2; error('Insufficient number of inputs'); end

if ~isnumeric(x); error('Data must be numeric'); end;

if ~isnumeric(fc); error('Filter bounds must be numeric'); end

if ~isnumeric(fs) || ~isscalar(fs); error('Sampling frequency must be a scalar'); end

if ~isnumeric(order) || ~isscalar(order); error('Order must be a scalar'); end

if ~ischar(type); error('Type must be a string'); end

if ~ischar(direction); error('Direction must be a string'); end

% MAIN --------------------------------------------------------------------

% If bidirectional filter, halve the order as filter is done twice

if strcmp(direction,'both')

if mod(order,2)~=0;

error('Filter order must be even if filter is bidirectional');

end

order = order/2;

end

% Change type to lowercase

type = lower(type);

% Move to band notation

switch type

% Lowpass

case {'low','lowpass'}

if numel(fc)~=1; error('Too many bounds given'); end;

fc = [0 fc];

  

% Highpass

case {'high','highpass'}

if numel(fc)~=1; error('Too many bounds given'); end;

fc = [fc Inf];

  

% Bandpass

case {'band','bandpass'}

% Do nothing

  

% Bandstop

case {'bandstop','stop'}

type = 'stop'; % Cannonical

  

otherwise

error('Type %s not found',type);

  

end

% Make bounds be relative to Nyquist frequency

fc = fc*2/fs;

% Design the filter -------------------------------------------------------

if numel(fc)~=2

error('Filter requires two cutoff frequencies');

  

elseif strcmp(type,'stop')

% Do bandstop filter

[B,A] = butter(order, fc, 'stop');

% If not bandstop, must be bandpass.

  

elseif (fc(1)==0) && (isinf(fc(2)))

% No filter required

B = NaN;

A = NaN;

return;

  

elseif fc(1)==0

% Lower bound is 0: design lowpass filter

[B,A] = butter(order, fc(2), 'low');

  

elseif isinf(fc(2)) || fc(2)==1

% Upper bound is Inf or Nyquist freq: design highpass filter

[B,A] = butter(order, fc(1), 'high');

  

else

[B,A] = butter(order, fc);

  

end

  

% Use the filter ----------------------------------------------------------

switch direction

case 'forward'

x = FilterM(B, A, x);

case 'backward'

x = FilterM(B, A, x, [], 'reverse');

case 'both'

if ispc

x = filter(B, A, x);

x(end:-1:1,:) = filter(B, A, x(end:-1:1,:));

else

x = FiltFiltM(B, A, x);

end

otherwise

error('Unrecognised filter direction: %s',direction);

end

end

I hope u can understand this. ..