As with analog linear systems, we need to find the frequency
response of discrete-time systems. We used impedances to derive
directly from the circuit's structure the frequency
response. The only structure we have so far for a discrete-time
system is the difference equation. We proceed as when we used
impedances: let the input be a complex exponential signal. When
we have a linear, shift-invariant system, the output should also
be a complex exponential of the same frequency, changed in
amplitude and phase. These amplitude and phase changes comprise
the frequency response we seek. The complex exponential input
signal is
xn=Xⅇⅈ2πfn
x
n
X
2
f
n
. Note that this input occurs for
all values of
nn. No need to worry about initial
conditions here. Assume the output has a similar form:
yn=Yⅇⅈ2πfn
y
n
Y
2
f
n
. Plugging these signals into the
fundamental difference equation, we have
Yⅇⅈ2πfn=a1Yⅇⅈ2πfn-1+…+apYⅇⅈ2πfn-p+b0Xⅇⅈ2πfn+b1Xⅇⅈ2πfn-1+…+bqXⅇⅈ2πfn-q
Y
2
f
n
a1
Y
2
f
n
1
…
ap
Y
2
f
n
p
b0
X
2
f
n
b1
X
2
f
n
1
…
bq
X
2
f
n
q
(1)
The assumed output does indeed satisfy the difference equation
if the output complex amplitude is related to the input
amplitude by
Y=b0+b1ⅇ-ⅈ2πf+…+bqⅇ-ⅈ2πqf1-a1ⅇ-ⅈ2πf-…-apⅇ-ⅈ2πpfX
Y
b0
b1
2
f
…
bq
2
q
f
1
a1
2
f
…
ap
2
p
f
X
This relationship corresponds to the system's frequency response
or, by another name, its transfer function. We find that any
discrete-time system defined by a difference equation has a transfer
function given by
Hⅇⅈ2πf=b0+b1ⅇ-ⅈ2πf+…+bqⅇ-ⅈ2πqf1-a1ⅇ-ⅈ2πf-…-apⅇ-ⅈ2πpf
H
2
f
b0
b1
2
f
…
bq
2
q
f
1
a1
2
f
…
ap
2
p
f
(2)
Furthermore, because
any discrete-time signal can
be expressed as a superposition of complex exponential signals and
because linear discrete-time systems obey the Superposition Principle,
the transfer function relates the discrete-time Fourier transform of
the system's output to the input's Fourier transform.
Yⅇⅈ2πf=Xⅇⅈ2πfHⅇⅈ2πf
Y
2
f
X
2
f
H
2
f
(3)
Example 1
The frequency response of the simple IIR system (difference
equation given in
a previous example)
is given by
Hⅇⅈ2πf=b1-aⅇ-ⅈ2πf
H
2
f
b
1
a
2
f
(4)
This Fourier transform occurred in a previous example; the
exponential signal spectrum portrays the
magnitude and phase of this transfer function. When the
filter coefficient
aa is
positive, we have a lowpass filter; negative
aa results in a highpass filter.
The larger the coefficient in magnitude, the more pronounced
the lowpass or highpass filtering.
Example 2
The length-
qq boxcar filter
(difference equation found in
a previous example) has
the frequency response
Hⅇⅈ2πf=1q∑m=0q-1ⅇ-ⅈ2πfm
H
2
f
1
q
m
0
q1
2
f
m
(5)
This expression amounts to the Fourier transform of the
boxcar signal. There we found that this frequency response
has a magnitude equal to the absolute value of
dsincπf
dsinc
f
; see the
length-10 filter's frequency response. We see that boxcar
filters--length-
qq signal
averagers--have a lowpass behavior, having a cutoff
frequency of
1q
1
q
.
Problem 1
Suppose we multiply the boxcar filter's coefficients by a sinusoid:
bm=1qcos2π
f
0
m
bm
1q
2
f
0
m
Use Fourier transform properties to determine the transfer
function. How would you characterize this system: Does it
act like a filter? If so, what kind of filter and how do you
control its characteristics with the filter's coefficients?
[
Click for Solution 1 ]
Solution 1
It now acts like a bandpass filter with a center frequency of
f0f0
and a bandwidth equal to twice of the original
lowpass filter.
[
Hide Solution 1 ]
These examples illustrate the point that systems described (and
implemented) by difference equations serve as filters for
discrete-time signals. The filter's order
is given by the number
pp of denominator coefficients in
the transfer function (if the system is IIR) or by the number
qq of numerator coefficients if the
filter is FIR. When a system's transfer function has both terms,
the system is usually IIR, and its order equals
pp regardless of
qq. By selecting the coefficients
and filter type, filters having virtually any frequency response
desired can be designed. This design flexibility can't be found
in analog systems. In the next section, we detail how analog
signals can be filtered by computers, offering a much greater
range of filtering possibilities than is possible with circuits.
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