frequency domain signal processing using matlab

8/3/2019 Frequency Domain Signal Processing Using MATLAB

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Frequency Domain SignalFrequency Domain SignalProcessing Using MATLABProcessing Using MATLAB

Mohammad Sadgh TalebiSharif University of Technology


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Fourier Transform inFourier Transform in MatlabMatlab

Y = fft(X,n)

Computes n-point Discrete FourierTransform (DFT) of each column of X

with a FFT algorithm

If length(x) < n => zero-padding

If length(x) > n => truncate x


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Y = ifft(X,n)

Similarly, IFFT computes n-point InverseDiscrete Fourier Transform (IDFT) of

each column of X with a IFFT algorithm

If length(x) < n => zero-padding

If length(x) > n => truncate x


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Signal Analysis >Signal Analysis >

ConvolutionConvolution

fftshift(X) swaps the left and right halves

of X. For matrices, fftshift(X) swaps thefirst quadrant with the third and the

second quadrant with the fourth.

Similarly, ifftshift(X) neutralizes the

results of fftshift.


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ConvolutionConvolutionx(1:20)=0;

x(21:30)=1;

x(31:50)=0;

figure(1);

plot(x);

X=fft(x);

figure(2);

subplot(211);

plot(abs(X));subplot(212);

plot(angle(X));

figure(3);

subplot(211);

plot(abs(fftshift(X)));

subplot(212);

plot(angle(fftshift(X)));


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Step ResponseStep Response


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Step ResponseStep Response


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Bode DiagramsBode Diagrams Filtering

Frequency Domain Time Domain


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Frequency Domain Filtering

Take FFT from input signal and just multiply itby frequency response of filter. Finally take

inverse FFT from result.

Alternatively, with the knowledge of Pole-Zeroplot or Transfer Function, you can filter any

signal using filter command.


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Y = FILTER(B,A,X)

Filter data with an infinite impulseresponse (IIR) or finite impulse response

(FIR) filter

y(n) = b(1)*x(n) + b(2)*x(n-1) + ... +

b(nb+1)*x(n-nb)- a(2)*y(n-1) - ... -

a(na+1)*y(n-na)


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Time Domain Filtering

Frequency response of desired filter yields theimpulse response of filter, thus filtering can be

carried out using:

Conv Fftfilt

However, the main problem, i.e. computing the

impulse response is still left.


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Filter DesignFilter Design

Several approaches for computing

impulse response of filters, with a desiredcharacteristics

Parks-McClellan , (firpm and remez

commands)

Traditional approximations for filters, such

as Butterworth, Chebychev, Elliptic, etc.

Filter Design & Analysis Tool


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Analog filter frequency responseAnalog filter frequency response

H = FREQS(B,A,W) returns the complex

frequency response of the analog filterspecified by coefficient vectors b and a.

[H,W] = FREQS(B,A,F) picks f number offrequencies on which to compute thefrequency response h.


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Digital filter frequency responseDigital filter frequency response

[H,W] = FREQZ(B,A,N) returns the N-point complexfrequency response vector H and the N-point frequency

vector W in radians/sample of the filter.

frequency response is evaluated at N points equally

spaced around the upper half of the unit circle. If Nisn't specified, it defaults to 512.

[H,W] = FREQZ(B,A,N,'whole') uses N points around

the whole unit circle.


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Example:Example:

Frequency Response Using FREQZFrequency Response Using FREQZa=[1 3 2 1];

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

semilogy(w,abs(H));


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Example:Example:

Frequency Response Using FREQSFrequency Response Using FREQS Evaluate frequency response of

a = [1 0.4 1]; b = [0.2 0.3 1];

w = logspace(-1,1);

h=freqs(b,a,w)

mag = abs(h); phase = angle(h);

subplot(2,1,1), loglog(w,mag)

subplot(2,1,2), semilogx(w,phase)


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Example:Example:

Frequency Response Using FREQSFrequency Response Using FREQS


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Butterworth FilterButterworth Filter

The frequency response of the

Butterworth filter is maximally flat (hasno ripples) in the passband, and rolls off

towards zero in the stopband.

A Butterworth LP prototype has the

following characteristic function:


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Butterworth Filter (ContButterworth Filter (Contd)d)


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Chebyshev FilterChebyshev Filter

having a steeper roll-off and morepassband/stopband ripple than Butterworthfilters

Chebyshev filters have the property that theyminimize the error between the idealized filtercharacteristic and the actual over the range ofthe filter, but with ripples in the passband or

stopband.


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Chebyshev Filter (ContChebyshev Filter (Contd)d)

Chebyshev Type I

It has no ripple in the stopband, but hasripple in the passband. The transfer function

is:

where Tn() is a chebyshev polynomial.


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Chebyshev Type II

It has no ripple in the passband, but hasripple in the stopband. The transfer function

is:

where Tn() is a chebyshev polynomial


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Elliptic FilterElliptic Filter

is a filter with equiripple behavior in

both the passband and the stopband.

It minimizes the maximum error in both

bands, as opposed to a Chebyshev filterwhich exhibits equiripple behavior in thepassband, or the inverse Chebyshev filter

which has ripples in the stopband.


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Elliptic Filter (ContElliptic Filter (Contd)d)

The magnitude of the frequency response of

a lowpass elliptic filter is given by:

where Rn() is a chebyshev rational function.


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Elliptic Filter (ContElliptic Filter (Contd)d)


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Filters : ComparisonFilters : Comparison


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Butterworth FilterButterworth Filter

butter designs lowpass, bandpass, highpass,

and bandstop digital and analog Butterworthfilters.

[B,A] = BUTTER(N,Wn) designs an Nth order

lowpass digital Butterworth filter and returnsthe filter coefficients in length N+1 vectors B

(numerator) and A (denominator). The cutoff

frequency Wn must be 0


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Butterworth Filter (ContButterworth Filter (Contd)d)

If Wn is a two-element vector, Wn = [W1 W2],BUTTER returns an order 2N bandpass filterwith passband W1 < W < W2.

[B,A] = BUTTER(N,Wn,'high') designs ahighpass filter.

[B,A] = BUTTER(N,Wn,'low') designs a lowpassfilter.

[B,A] = BUTTER(N,Wn,'stop') is a bandstop

filter if Wn = [W1 W2].


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Chebyshev FilterChebyshev Filter

[b,a] = cheby1(n,R,Wp) designs an order nChebyshev lowpass digital Chebyshev filterwith normalized passband edge frequency Wpand R dB of peak-to-peak ripple in thepassband.

Normalized passband edge frequency is thefrequency at which the magnitude response of

the filter is equal to -R dB.


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Butterworth ExampleButterworth Example

[b,a]=butter(7,.32);

[H,w]=freqz(b,a,128);

plot(w,abs(H));

Filt D i ithFilt D i ith


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Filter Design withFilter Design with

ParksParks--McClellanMcClellan FRIPM command designs a linear-phase FIR

filter using the Parks-McClellan algorithm.

Chebyshev approximation is used to fit theoptimal fit between desired and actualresponse.

b = firpm(n,f,a): returns n+1 coefficients ofthe order n FIR filter whose frequency-amplitude characteristics match those given

by vectors f and a.

Filt D i ithFilt D i ith


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ParksParks--McClellan (ContMcClellan (Contd)d) f is a vector of pairs of normalized frequency

points, specified in the range between 0 and1 in increasing order.

a is a vector containing the desiredamplitudes at the points specified in f.

f and a must be the same length. The lengthmust be an even number.

Filt D i ithFilter Design ith


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ParksParks--McClellan (ContMcClellan (Contd)d) The desired amplitude at frequencies

between pairs of points (f(k), f(k+1)) for k oddis the line segment connecting the points(f(k), a(k)) and (f(k+1), a(k+1)).

The desired amplitude at frequenciesbetween pairs of points (f(k), f(k+1)) for keven is unspecified. The areas between such

points are transition or "don't care" regions.

E ample ofExample of


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Example ofExample of

FIRPM Filter DesignFIRPM Filter Designf = [0 0.3 0.4 0.6 0.7 1];

a = [0 0 1 1 0 0];

b = firpm(17,f,a);

[h,w] = freqz(b,1,512);


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How to read a soundHow to read a sound

[y,Fs,bits] = wavread('filename',[N1 N2])

Read Microsoft WAVE (.wav) sound file

specified in the filename and load its data to

vector y ( samples N1 through N2).

Fs returns the sampling frequency and bits

returns the number of bits used for encoding.


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How to write a signal as a soundHow to write a signal as a sound

wavwrite(y,Fs,N,'filename')

writes the data stored in the variable y to a

WAVE file called filename. Fs is sample

rate and N is the number of bits used forencoding. Amplitude values outside the

range [-1,+1] are clipped prior to writing.

N = 8, 16, 24, or 32


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How to make a sound noisy!How to make a sound noisy!

By using wavread command, first we

convert a sound into a vector, then weadd noise term to it.

Common forms of noises are Gaussian

and Uniform.


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Gaussian NoiseGaussian Noise

Gaussian Noise

Y = randn(m,n) generates normallydistributed random numbers (noises)

In order to generate a Gaussian noisesequence with arbitrary statistics, multiply

the output of randn by the standard

deviation and then add the desired mean


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UniformUniform NoiseNoise

Uniform Noise

Y = rand (m,n) generates unifromlydistributed random numbers (noises) over[0,1] interval.

In order to generate a uniform distribution ofrandom numbers on a specified interval[a,b], multiply the output of rand by (b-a),

then add a.


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QuestionsQuestions

surely, in the creation of the heavensand the earth, there are signs for theowners of wisdom

The Holy Quran

Thanks for your attendance.

frequency domain signal processing using matlab

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