9.8 Laboratory Experiment on Signal ProcessingPurpose: By performing this experiment, the student will receive a better understandingof the use and power of the FFT algorithm in evaluating the corresponding discrete (time)Fourier transforms, continuous-time Fourier transform, and discrete-time convolution. Asthe computational tool, we will use the MATLAB functions fft and ifft.

Part 1. In this part we use the FFT algorithm, as implemented in MATLAB, to findthe DFT of some discrete-time signals. In addition, we demonstrate the use of the IFFTin recovering original discrete-time signals.

(a) Consider the discrete-time signal

! #"%$&

and find analytically its DTFT.(b) Use the MATLAB function X=fft(x,N) to find the DFT of the preceding

signal for ' )(!*++,

(

. Use the MATLAB function x=ifft(X,N) to recoverthe original discrete-time signal. Plot the DFTs and IDFTs, and comment on the resultsobtained.

(c) Consider the signal

- ./

0

+

1%121%+

&3& #"%$&

and repeat parts (a) and (b).(d) Consider the signal whose nonzero values are between 45 and 4

*

,

respectively defined by /6

(

7

(

8+

#9

;:

, and repeat parts (a) and (b).Comment on the results obtained.

Part 2. Formula (9.71), =@?A#BDCFEHG

the wrapped signal, so that the corresponding convolution is called mod- e circularconvolution [1].

Use the formula to find the convolution of the discrete-time signals defined inProblems 6.15 and 6.16. Do the results obtained represent unwrapped or wrapped signals?

SUPPLEMENT:

t2-1

0-1 2

0-1

1

t

1

x x(t) (t)1 2

FIGURE 3.22: Two Fourier transformable signals

1(t)x 2(t)x

t

t

20-2

1-1

2

(a) (b)

1

-10

FIGURE 3.23: Time domain signals

Discrete-time signals in Problem 6.15 are defined as

fgh ijkmln

o

np

q

iYk0r

s

q

iYk

q

q

iYk0t

r uv&wx+y&z#{%|&x~}

fh ijk.~

iYkr

s

tiYkt

r uv&wxy&z#{%|&x

Discrete-time signals in Problem 6.16 are defined as

f

g

hcijk

l

n

n

n

n

n

o

n

n

n

n

n

p

s

t ik

s

q

t ikr

q

ik

q

s

q

ikt

ik

r uv&wxy!z#{%|&x

}

f

hcijk

l

n

n

n

o

n

n

n

p

q

ikr

tik

q

ikt

tik

ruv!wxy!z#{2|!x