Digital Signal Processing Reference
In-Depth Information
Finally, the steady-state response is identified as
Hðe
j
U
Þ
sin
ðn
U
þ
:
Hðe
j
U
ÞÞuðnÞ
y
ss
ðnÞ¼K
For this particular filter, the transient response exists for only the first sample in the system response.
By substituting
n ¼
0 into
yðnÞ
and after simplifying algebra, we achieve the response for the first
output sample:
yð
0
Þ¼y
tr
ð
0
Þþy
ss
ð
0
Þ¼
0
:
5
K
sin
ð
U
Þ
0
:
5
K
sin
ð
U
Þ¼
0
Note that the first output sample of the transient response cancels the first output sample of the steady-
state response, so the combined first output sample has a value of zero for this particular filter. The
system response reaches the steady-state response after the first output sample. At this point, we can
conclude that
Peak amplitude of steady state response at
U
Peak amplitude of sinusoidal input at
U
Steady-state magnitude frequency response
¼
Hðe
j
U
Þ
K
Hðe
j
U
Þ
¼
¼
K
Steady-state phase frequency response
¼
Phase difference
¼
:
Hðe
j
U
Þ
U
¼
0
:
75
p
, respectively.
Next, we examine the properties of the filter frequency response
Hðe
j
U
Þ
. From Euler's identity and
the trigonometric identity, we know that
e
jð
U
þk
2
pÞ
¼
cos
ð
U
þ k
2
pÞþj
sin
ð
U
þ k
2
pÞ
¼
cos
U
þ j
sin
U
¼ e
j
U
where
k
is an integer taking values of
k ¼
0
;
1
;
2
;/
. Then the frequency response has the
following property (assuming all input sequences are real):
1.
Periodicity
a.
Frequency response:
Hðe
j
U
Þ¼Hðe
jð
U
þk
2
pÞ
Þ
¼jHðe
jð
U
þk
2
pÞ
Þj
c.
Phase response:
:
Hðe
j
U
Þ¼
:
Hðe
j
U
þk
2
p
Þ
The second property is given without proof (see proof in Appendix D):
b.
Magnitude frequency response:
Hðe
j
U
Þ
2.
Symmetry
¼jHðe
j
U
Þj
b.
Phase response:
:
Hðe
j
U
Þ¼
:
Hðe
j
U
Þ
Since the maximum frequency in a DSP system is the folding frequency,
f
s
=
2, where
f
s
¼
1
=T
,
and T designates the sampling period, the corresponding maximum normalized frequency of the
system frequency can be calculated as
a.
Magnitude frequency response:
Hðe
j
U
Þ
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