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23 .8
A linear shift-invariant system has a frequency response
H ( e j w ) = elo
1 1.1 + c o s w
Find an LCCDE that relates the input to the output.
CHAP. 21
FOURIER ANALYSIS
To convert H ( e J w )into a difference equation, we must first express H ( e J W )in terms of complex exponentials. Expanding the cosine into a sum of two complex exponentials, we have
Multiplying numerator and denominator by 2e-Jw gives
Cross-multiplying, we have
+ 2.2e-i'u + e - 2 i " ] ~ ( e J " )= 2 X ( e l w )
+ 2 . 2 y ( n - I) + y ( n - 2 ) = 2 x ( n )
which leads to the following difference equation when we take the inverse DTFT of each term:
y(n)
Find the frequency responseof a linear shift-invariant system whose input and output satisfy the difference equation y ( n ) - 0.5y(n - 1 ) = x(n) 2x(n - I) x ( n - 2)
To find the frequency response, we begin by finding the DTFT of each term in the difference equation
- OSe-j")Y(eJ")
+ 2e-1" + e - ~ ' " ) x ( e J ~ )
Because ~ ( e j " = ~ ( e J " ) / X ( e j " ) we have ) .
Write a difference equation to implement a system with a frequency response
With after cross-multiplying, we have [I
+ 0.5e-iw + 0 . 7 5 e - z J " ] ~ ( e i " )= [l - 0 5 - j w + e - 3 ~ " ] ~ ( e J " )
+ 0 . 5 y ( n - 1 ) + 0.75y(n - 2 ) = x ( n ) - 0 . 5 x ( n
Taking the inverse DTFT of each term gives the desired difference equation
y(n)
+x(n - 3 )
Find a difference equation to realize a linear shift-invariant system that has a frequency response
H ( e J W )= tan w
To find a difference equation for H ( e j W ) .we must first express tan w in terms of complex exponentials:
I ej" - e-I" sin w t a n w = -- j elw e-Jw cos w
With H ( e j w ) = Y ( e j w ) / X ( e j " ) we have, after cross-multiplying.
jlej" + e - j " ] ~ ( e j " ) = [ei"
-e-j"]~(ej")
Inverse transforming, we obtain the following difference equation:
jy(n
+ 1) + j y ( n
1) = x(n
+ 1 ) -x(n - 1)
FOURIER ANALYSIS
[CHAP. 2
By introducing a delay and dividing by j, this difference equation may be written in the more standard form
Find a difference equation to implement a filter that has a unit sample response
To find a difference equation for this system, we must first find the frequency response H (elw). Expressing h(n) in terms of complex exponentials,
it follows that the frequency response is
Therefore, the difference equation for this system is
A system with input x ( n ) and output y ( n ) is described by the following set of coupled linear constant
coefficient difference equations:
Find a single linear constant coefficient difference equation that describes this system, and find the frequency response H ( e l W ) .
To find the frequency response for this system of difference equations, we first express each equation in the frequency domain:
) , Using the last two equations to express V ( e J W in terms of X ( e J W ) we have
) ) Substituting this expression for V ( e J W into the first equation and solving for Y ( e J W gives
CHAP. 21 Therefore, the frequency response is
FOURIER ANALYSIS
Cross-multiplying, we have
~ ( ~ i w -[ ie-iw + i e - j 2 w ] = x ) l
( ~ I ~ ) [ ~ - I+ "
qe-2i" + ze-3iw]
and taking the inverse DTFT of each term gives the difference equation for the system:
y(n) - i y ( n - I )
+ i y ( n - 2 ) = x(n - I ) + qx(n - 2) + 2x(n - 3 )
A linear shift-invariant system with input x ( n ) and output v ( n ) is described by the difference equation
This system is cascaded with another system with input v ( n ) and output y ( n ) that is described by the difference equation y ( n ) = l,Y( n - 1) + 0 ) What value of cr will guarantee that y ( n ) = x ( n )
Substituting the first equation into the second, we obtain a single difference equation that describes the overall system. that is, y(n) = y(n - 1) x ( n ) a x ( n - 1) Taking the DTFT of both sides of the equation, we have
y ( e j w ) = j e - j " ~ ( e ~ w )x ( ~ J "+ a e - ~ w ~ ( e j " ) + )
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