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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then
Ys XsHs,
6:1:7
where Y(s), H(s), and X(s) are the Laplace transforms of y(t), h(t), and x(t), respectively.
Thus convolution in the time domain is equivalent to multiplication in the Laplace (or
frequency) domain.
In (6.1.7), H(s) is the transfer function of the system defined as
Hs
Ys
Xs
I
0
hte
Àst
dt,
6:1:8
where h(t) is the impulse response of the system. The general form of a transfer function
is expressed as
Hs
b
0
b
1
s Á Á Á b
LÀ1
s
LÀ1
a
0
a
1
s Á Á Á a
M
s
M
Ns
Ds
:
6:1:9
The roots of N(s) are the zeros of the transfer function H(s), while the roots of D(s) are
the poles.
Example 6.2: The input signal xt e
À2t
ut is applied to an LTI system, and the
output of the system is given as
yt e
Àt
e
À2t
À e
À3t
ut:
Find the system's transfer function H(s) and the impulse response h(t).
From Table 6.1, we have
Xs
1
s 2
and Ys
1
s 1
1
s 2
À
1
s 3
:
From (6.1.8), we obtain
Hs
Ys
Xs
1
s 2
s 1
À
s 2
s 3
:
This transfer function can be written as
Hs
s
2
6s 7
s 1s 3
1
1
s 1
1
s 3
:
From Table 6.1, we have
ht dt e
Àt
e
À3t
ut:
244
DESIGN AND IMPLEMENTATION OF IIR FILTERS
Summary :
From Table 6.1, we have Xs 1 s 2 and Ys 1 s 1 1 s 2 À 1 s 3 : From (6.1.8), we obtain Hs Ys Xs 1 s 2 s 1 À s 2 s 3 : This transfer function can be written as Hs s 2 6s 7 s 1s 3 1 1 s 1 1 s 3 : From Table 6.1, we have ht dt e Àt e À3t ut: 244 DESIGN AND IMPLEMENTATION OF IIR FILTERS
Tags :
transfer,function,system,à2t,617,impulse,hae,618,là1,à3t,roots,table,laplace