Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Discrete Fourier Transform
It can be proved that the FFT algorithm has saved huge computing time by reducing from
2
N complex multiplications
to N log 2 N , i.e. saved N
8 8
times of computation. For example, if N=8, = = 2. 67 ; if N=1024,
log 2
N
log2
8 3
1024 1024
= = 102.4 . i.e. saved 102 times of multiplications. The longer the data length N, the more time can
log 1024 10
2
be saved relative to the direct calculation of the DFT.
Problems
Q6.1 What are the features of the DFT coefficients X[k] of an N-sample signal which is
a) Real,
b) Real and even,
c) Real and odd, and
d) Complex
Q6.2 For the digital sequence
a) x[n] = [1 -1 ],
b) x[n]= [3 -2],
c) x[n]=[1 -1 0 0],
d) x[n] = [1 0 0 1],
e) x[n]=[1 2 1 3].
Calculate the Discrete Fourier Transform (DFT) .
Q6.3 Explain how the Fast Fourier Transform (FFT) algorithm can be faster than direct calculation of the Discrete Fourier
Transform (DFT).
Q6.4 Answer the following questions:
--
With reference to the Fast Fourier Transform (FFT), why is the length, N, normally chosen as an integer power
of 2
--
In brief, what is the reason that the FFT algorithm can be faster than direct calculation of the Discrete Fourier
Transform (DFT)
--
If the length of a sequence is not yet an integer power of 2, how is it possible to take advantage of the FFT
algorithm
99
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