Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Spectral Analysis by DFT
7.4 Windowing
In the DFT, only a component at an exact harmonic frequency gives rise to a single and well-defined spectral line. In
fact, practical digital signals normally contain a majority of fractional frequencies and few of them exact harmonics. This
means that the spectral leakage is generally present, and it may lead to inaccuracy in analysis and interpretation. It is
therefore common practice to taper two ends of the original signal before applying the DFT, reducing or removing any
discontinuities at its two ends. This can be achieved by multiplying the signal with a suitable window function. For a
signal x [ n],
n = 0,1,2,... N −1
, applying an equal length window function w [ n],
n = 0,1,2,... N −1, a windowed
signal is given by
x w
[ n]
= x[
n]
w[
n]
(7.5)
--
Rectangular window (No Window)
w [ n]
= 1
0 ≤ n < N
(7.6)
The windowed signal x w
[n]
is not tapered by this rectangular window. The spectral leakage is fully present.
-Triangular window
| 2n
− N + 1|
w [ n]
= 1−
0 ≤ n < N (7.7)
N
The windowed signal x w
[n]
will be tapered by the straight slopes of the triangle.
-Hamming window
( 2n
− N 1)
+ π
w [ n]
= 0.54 + 0.46cos
0 ≤ n < N (7.8)
N
The windowed signal x w
[n]
will be tapered by the cosine function.
9.8 Performance of windows
In the time domain, applying a window is to multiply by the window function w [n]
:
x w
[ n]
= x[
n]
w[
n]
In the frequency domain, according to the DFT modulation property in Eq.(6.8), two spectra are in convolution:
X w
[ k]
= X [ k]
∗W[
k]
(7.9)
104
Download free ebooks at bookboon.com