Real-time digital signal processing: implementations, ... changes in the input signal is limited by its internal clock rate, so that it may be slow to
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aligned below it. This flip-and-slide form of linear convolution can be illustrated in
Figure 5.5. Note that shifting xÀl to the right is equivalent to shift b
l
to the left one
unit at each sampling period.
As shown in Figure 5.5, the input sequence is extended by padding L À 1 zeros to its
right. At time n 0, the only non-zero product comes from b
0
and x(0) which are time
aligned. It takes the filter L À 1 iterations before it is completely overlapped with the
input sequence. The first L À 1 outputs correspond to the transient behavior of the FIR
filter. For n ! L À 1, the filter aligns over the non-zero portion of the input sequence.
That is, the signal buffer of FIR filter is full and the filter is in the steady state. If the
input is a finite-length sequence of M samples, there are L M À 1 output samples and
the last L À 1 samples also correspond to transients.
x(n
-1) x(n-2)x(n-3)
b
0
x(0)
b
0
x(1)
b
1
x(1)
b
1
x(n
-1)
b
0
x(2)
b
0
x(n)
b
1
x(0)
b
2
x(0)
b
2
x(n
-2)
b
3
x(n
-3)
b
0
b
1
b
2
b
3
n = 0:
n = 1:
n = 2:
n
3:
x(0)
x(0)
x(0)
x(1)
x(1)
x(2)
x(n)
Figure 5.4 Graphical interpretation of linear convolution, L 4
b
0
b
0
b
1
b
1
b
2
b
2
b
3
b
3
x(n)
x(1)
y(n)
y(0)
x(n
-1) x(n-2) x(n-3)
0
0
0
Figure 5.5 Flip-and-slide process of linear convolution
FIR FILTERING
191
Summary :
x(n -1) x(n-2)x(n-3) b 0 x(0) b 0 x(1) b 1 x(1) b 1 x(n -1) b 0 x(2) b 0 x(n) b 1 x(0) b 2 x(0) b 2 x(n -2) b 3 x(n -3) b 0 b 1 b 2 b 3 n = 0: n = 1: n = 2: n 3: x(0) x(0) x(0) x(1) x(1) x(2) x(n) Figure 5.4 Graphical interpretation of linear convolution, L 4 b 0 b 0 b 1 b 1 b 2 b 2 b 3 b 3 x(n) x(1) y(n) y(0) x(n -1) x(n-2) x(n-3) 0 0 0 Figure 5.5 Flip-and-slide process of linear convolution FIR FILTERING 191
Tags :
filter,figure,sequence,input,linear,conolution,samples,fir,right,nonzero,flipandslide,aligned,time