barcode lib ssrs it follows that the frequency response of the overall system is H (d('"-"') as we found before. in Software

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it follows that the frequency response of the overall system is H (d('"-"') as we found before.
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If h ( n ) is the unit sample response of an ideal low-pass filter with a cutoff frequency w, = n/4, find the frequency response of the filter that has a unit sample response g ( n ) = h(2n).
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To find the frequency response of this system. we may work the problem in one of two ways. The first is to note that because the unit sample response of an ideal low-pass filter with a cutoff frequency w,. = n / 4 is
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then which is the unit sample response of a low-pass filter with a magnitude of and a cutoff frequency w = n / 2 . The second way to work this problem is to find the frequency response of the system that has a unit sample response g(n) = h(2n), given that H(el'") is the frequency response of a system with a unit sample response h(n). Although more difficult than the first approach. this will give a general expression for the frequency response G(eJw)in terms of H(ejw)that may be applied to any system. To find the frequency response, we must evaluate the sum
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we may write the frequency response as
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CHAP. 21
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FOURIER ANALYSIS
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In terms of H ( d w ) ,the first term may be written as
whereas the second term is
With ~ ( e jthe frequency response of a low-pass filter with a cutoff frequency w, = n/4, this gives the same result ~ ) as before.
21 .0
Consider the high-pass filter that has a cutoff frequency w, = 3n/4as shown in the following figure:
(a) Find the unit sample response, h(n). (h) A new system is defined so that its unit sample response is h l ( n ) = h(2n). Sketch the frequency ( response, H Ie j " ) , of this system.
(a) The unit sample response may be found two different ways. The first is to use the inverse DTFT formula and
perform the integration. The second approach is to use the modulation property and note that if I
H ( e J u ) may be written as H ( @ ) = HIp(eJ("-"' 1
H I p ( e J W= )
for lo[ 5 4 otherwise
Therefore, it follows from the modulation property that
h ( n ) = eJn"hlp(n) ( - l ) " h l p ( n ) =
With
we have
(6) The frequency response of the system that has a unit sample response h I ( n ) = h ( 2 n ) may be found by evaluating the discrete-time Fourier transform sum directly:
n=-00
n=-m
n even
FOURIER ANALYSIS However, an easier approach is to note that
[CHAP.2
is which is a low-pass filter with a cutoff frequency of n / 2 and a gain of f . A plot of H , ( ~ J " )shown in the following figure:
Interconnection of Systems
The ideal filters that have frequency responses as shown in the figure below are connected in cascade.
For an arbitrary input x(n), find the range of frequencies that can be present in the output y(n). Repeat for the case in which the two systems are connected in parallel.
If these two filters are connected in cascade, the frequency response of the cascade is
Therefore, any frequencies in the output, y(n), must be passed by both filters. Because the passband for Hl(ejW) is Iwl > n/3, and the passband for H2(eJU) n / 4 < Iwl < 3x14, the passband for the cascade (the frequencies for is and which both I H I(eJ1')1 I Hz(ejU)lare equal to I ) is
With a parallel connection, the overall frequency response is
Therefore, the frequencies that are contained in the output are those that are passed by either filter, or
Consider the following interconnection of linear shift-invariant systems:
CHAP. 21
FOURIER ANALYSIS
Find the frequency response and the unit sample response of this system.
To find the unit sample response, let x ( n ) = S(n). The output of the adder is then
Because w ( n ) is input to an LSI system with a unit sample response h 2 ( n ) ,
where
hz(n) =
- -~ ~ 2n ,
' I"
sin(nn/2) ( ~ j f ~ = ~ j l j ~eJnwdw - ) d ~ = nn 2 n -,,Z
Therefore, the unit sample response of the overall system is
To find the frequency response, note that
Therefore,
~ ( e j " = w ( ~ ~ " ) H : ( ~ J = )[I - e - j " ] ~ ) "'
Consider the interconnection of LSI systems shown in the following figure:
x(n)
hzb)
hl(n)
hdn)
hdn)
(a) Express the frequency response of the overall system in terms of H I (ej"), H2(ejw), FZ3(ejo), and H4 (ejw ). (b) Find the frequency response if
(a) Because h 2 ( n )is in parallel with the cascade of h 3 ( n )and h 4 ( n ) ,the frequency response of the parallel network is
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