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barcode lib ssrs For the DTFT that is given, we see that in Software
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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 2eJw gives
Crossmultiplying, we have
+ 2.2ei'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 shiftinvariant 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
 OSej")Y(eJ") + 2e1" + 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 crossmultiplying, we have [I
+ 0.5eiw + 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 shiftinvariant 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"  eI" sin w t a n w =  j elw eJw cos w
With H ( e j w ) = Y ( e j w ) / X ( e j " ) we have, after crossmultiplying.
jlej" + e  j " ] ~ ( e j " ) = [ei" ej"]~(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
Crossmultiplying, we have
~ ( ~ i w [ ieiw + i e  j 2 w ] = x ) l
( ~ I ~ ) [ ~  I+ " qe2i" + ze3iw] 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 shiftinvariant 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 " ) + )

