9.8 Laboratory Experiment on Signal Processing
Purpose: By performing this experiment, the student will receive a better understanding
of the use and power of the FFT algorithm in evaluating the corresponding discrete (time)
Fourier transforms, continuous-time Fourier transform, and discrete-time convolution. As
the 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 find
the DFT of some discrete-time signals. In addition, we demonstrate the use of the IFFT
in recovering original discrete-time signals.
(a) Consider the discrete-time signal
! #"%$&
and find analytically its DTFT.
(b) Use the MATLAB function X=fft(x,N) to find the DFT of the preceding
signal for
'
)(
!*++,
(
. Use the MATLAB function x=ifft(X,N) to recover
the original discrete-time signal. Plot the DFTs and IDFTs, and comment on the results
obtained.
(c) Consider the signal
- ./
0
+1%121%+
&3& #"%$&
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<DI+=@?KJ#B
, can be used for an approximate
evaluation of the continuous-time Fourier transform. In this formula,
E
G
is the sampling
interval used for sampling the continuous-time signal
=MLNB
into
=
EGBPO
-
, and
<I=@?KJ#B
is the corresponding DFT.
(a) Consider the continuous-time signals presented in Figures 3.22 and 3.23. Sample
these signals with
EHG
12
and find DFTs of the obtained discrete-time signals. Calculate
and plot the corresponding magnitude spectra for the approximate Fourier transforms and
compare them to the results obtained analytically.
(b) Repeat part (a) with
EG
1
.
Part 3. Discrete-time signal convolution can be efficiently evaluated via the
DTFT and its convolution property. The relation
Q
R6- TSVUW
implies
XY=M?JZB
<P=8?KJ#B![P=M?JZB
. Hence, discrete-time convolution via DFT can be evaluated as
Q
\
]&^V_-`Vab^V_-`
=
Wc
B
^V_-`
=
U-
Bd
. Note that such an obtained signal
Q
c
is, in general,
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