Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Discrete Fourier Transform
In the following Figure 6.2, the difference between the DFT and discrete version of FT are compared. The upper left is a
non-periodic signal with N samples in 0 ≤ n ≤ N −1
in which zeros are given to all outside the N records. The upper
right is its discrete version of Fourier transform which is a continuous function. The lower left is the signal in which the
N samples are regarded as one period and the record has been extended to the whole axis − ∞ < n < ∞ . Therefore, like
Fourier series, its periodic discrete spectrum is shown in the lower right figure.
x([n]
x([n]
1
0.5
0
-0.5
-1
1
0.5
0
-0.5
-1
-5 0 5 10 15
n
one period
-5 0 5 10 15
n
|X(Ω)|
|X[k]|
5
4
3
2
1
one period
FT
0
0 5 10 15
0 ≤ Ω ≤ 6π
DFT
5
4
3
2
1
one period
0
0 5 10 15 20
k
Figure 6.2 The discrete version of FT for a non-periodic signal and DFT for a periodic signal.
In essence, applying the DFT is to decompose a periodic signal to a series of cosine and sine functions represented by
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