barcode lib ssrs For the DTFT that is given, we see that in Software

Make Code 128 in Software For the DTFT that is given, we see that

Therefore,

23 .7

For the sequence x ( n ) plotted in the figure below,

23 .8

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

+ 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 ) .

+ 0.5e-iw + 0 . 7 5 e - z J " ] ~ ( e i " )= [l - 0 5 - j w + e - 3 ~ " ] ~ ( e J " )

y(n)

+x(n - 3 )

Find a difference equation to realize a linear shift-invariant system that has a frequency response

To find a difference equation for H ( e j W ) .we must first express tan w in terms of complex exponentials:

jlej" + e - j " ] ~ ( e j " ) = [ei"

-e-j"]~(ej")

Inverse transforming, we obtain the following difference equation:

+ 1 ) -x(n - 1)

By introducing a delay and dividing by j, this difference equation may be written in the more standard form

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,

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:

) ) Substituting this expression for V ( e J W into the first equation and solving for Y ( e J W gives

( ~ 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 " ) + )