lab material on convolution in time and frequency domain

7/29/2019 lab material on convolution in time and frequency domain

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0.1 AIM::

1.To observe the spectra of the given discrete time signals.

2. To illustrate the convolution theorem.

3. To undersample,sample at nyquist rate and to over sample a discrete timesignal and to observe their spectra.

0.2 Background reading::

0.2.1 FOURIER SERIES::Any non sinusoidal periodic signal can be represented as the sum of infinitesinusoidal signals of different frequencies (or) any periodic signal can be rep-resented as a linear combination of harmonically related complex exponentials.This series is called as FOURIER SERIES and is defined as follows

Trigonometric Fourier series(T.F.S)::

x(t) = a0 +

i=1

ancos0t + bnsin0t (1)

Where 0 Fundamental frequency and a0, an, bn are T.F.S coefficients

a0 =1

T

T0

x(t)dt an =2

T

T0

x(t)cosn0tdt bn =2

T

T0

x(t)sinn0tdt

Exponential /Complex Fourier Series::

x(t) =

i=

cnejn0t (2)

Where

cn =

1

T

T

0 x(t)ejn0t

dt (3)

But in this Fourier series representation, irrespective of the number of harmonicswe are adding ,always at the point of discontinuity we are observing 9 percentovershoot !!!( i.e Gibbs Phenomenon). Hence for convergence of Fourier Series,we have the following Properties to be satisfied.

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Convergence Of Fourier Series::(Dirchlet conditions)

1. x(t) must be absolutely integrable over a period.

T0

| x(t) | dt < (4)2. x(t) must have finite number of discontinuities ,maxima and a minima withina period.

0.2.2 FOURIER TRANSFORM::

Fourier Transform is the extension of Fourier Series for Non Periodic signals andit provides a frequency domain representation of time domain signals. Trans-formation is the process in which one domain is converted to another domainsuch that the signal analysis becomes easy.fourier transform of a signal x(t) isgiven as,Analysis Equation,

X() =

x(t)ejtdt (5)

Synthesis Equation,

X() =1

2

X()ejtdt (6)

Convergence of Fourier Transform::

1. F.T is defined for stable and energy signals

2. F.T is not defined for signals of type neither energy nor power signals

0.2.3 Fourier Series for discrete time signals::

Consider a discrete periodic signal x[n] with period N, where the fundamental

period is 0 =2

N. The fourier series expression is given by,

X[n] =N1k=0

akejk0n ak =

1

N

N1k=0

x[n]ejk0n (7)

0.2.4 FUNDAMENTAL PERIOD::

Consider a periodic signal with period T, f(t) = f(t + T) The fundamental pe-riod of the signal is the smallest real positive number T0 for which the periodicfunction f(t) = f(t + T) holds true.

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0.3 Implementation::

1. The spectra of the given signals x1(n) = cos

2n and x2(n) = cosn

3are

generated and plotted and these are compared with the spectra obtained byusinf matlab functions.

2.The fourier transform of the given unit step signal is plotted and observed.

3.By considering two sinusoidal signals,Convolution theorem is verified by look-ing at the convoluted output in both frequency and time domain.

4.The effect of undersampling,nyquist sampling and oversampling are observedby respectively sampling the discrete time signal with corresponding frequen-

cies.

0.4 Results::

1.Fourier transform of the given two signals,x1(n) = cos

2n and x2(n) =

cosn

3

Now the spectra of the signals will show us two impulse functions at the corre-sponding signal frequencies.

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2.Fourier transform of unit impulse signal.

plot of a unit step function and its fourier transform.unit step function is dis-continuous at t = 0

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3.Convolution of two discrete time signals and its spectra

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it shows the equivalence of convolution in time domain and multiplication in fre-quency domain.Using the FFT, convolution by multiplication in the frequencydomain can be hundreds of times faster than conventional convolution in timedomain.4.Sampling of discrete time signal,

a sinusoidal discrete signal was taken and it is sampled at different frequen-cies.fig 2 shows how aliasing effects the signal when undersampled.when criti-cally sampled ,some negligible interferences occurs as shown in fig 1.when it isoversampled there is no interference which shows that reconstruction is possible

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without any distortion.

0.5 Disussion ::

1.Here the given two signal are plotted in time domain and the spectra of thetwo signals are plotted.The spectra generated by matlab functions are also gen-erated and plotted.Here we observe that as the frequencies of the two discretetime signals are different,they are shifted to two different frequencies in thefrequency domain i.e two impulse functions at two frequencies f and f of therespective signal.

2.The given unit step signal and its spectra which is an impulse function areplotted here.

3.Here we considered two discre time signals and the convoluted signal ob-tained in time domain is plotted.The spectra of the convoluted signal is alsoplotted.Now to illustrate the convolution theorem, we plotted the spectra offirst and second signals.Next we multiplied the two frequency domain spectraand obtained the spectra of the convoluted signal.

4.A sinusoidal signal is considered and it is sampled at three different frequenciesundersampling,oversampling and nyquist sampling.By observing the spectra ofthe three sampled signals,we can say that in under sampling, the information islost as we observed the loss of a frequency component in undersampling.Henceunder sampling is not preferred in sampling a signal.

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lab material on convolution in time and frequency domain

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