This set of Tough Digital Signal Processing Questions focuses on “Properties of Z Transform-2”.

1. What is the signal x(n) whose z-transform X(z)=log(1+az^{-1});|z|>|a|?

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2. If Z{x1(n)}=X1(z) and Z{x2(n)}=X2(z) then Z{x1(n)*x2(n)}=?

a) X1(z).X2(z)

b) X1(z)+X2(z)

c) X1(z)*X2(z)

d) None of the mentioned

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Explanation: According to the convolution property of z-transform, the z-transform of convolution of two sequences is the product of their respective z-transforms.

3. What is the convolution x(n) of the signals x1(n)={1,-2,1} and x2(n)={1,1,1,1,1,1}?

a) {1,1,0,0,0,0,1,1}

b) {-1,-1,0,0,0,0,-1,-1}

c) {-1,1,0,0,0,0,1,-1}

d) {1,-1,0,0,0,0,-1,1}

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4. If Z{x1(n)}=X1(z) and Z{x2(n)}=X2(z) then what is the z-transform of correlation between the two signals?

a) X1(z).X2(z^{-1})

b) X1(z).X2(z^{-1})

c) X1(z).X2(z)

d) X1(z).X2(-z)

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Explanation: We know that rx1x2(l)=x1(l)*x2(-l)

Now Rx1x2(z)=Z{x1(l)}.Z{x2(-l)}=X1(z).X2(z

^{-1})

5. If x(n) is causal, then lim┬(z→∞)X(z)=?

a) x(-1)

b) x(1)

c) x(0)

d) Cannot be determined

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Explanation: According to the initial value theorem, X(z)=x(0)+x(1)z -1+x(2)z

^{-2}+….

When z→∞, z -n tends to 0 because n>0.

So lim┬(n→∞)〖X(z)〗=x(0).

6. If Z{x(n)}=X(z) and the poles of X(z) are all inside the unit circle, then the final value of x(n) as n→∞ is given by i.e., lim┬(n→∞)x(n)=?

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7. What is the z-transform of the signal x(n)=δ(n-n0)?

a) z^{n0}

b) z^{-n0}

c) z^{n-n0}

d) z^{n+n0}

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8. If X(z) is the z-transform of the signal x(n), then what is the z-transform of x*(n)?

a) X(z*)

b) X*(z)

c) X*(-z)

d) X*(z*)

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Explanation: According to the conjugation property of z-transform, we have

Z{x*(n)}= X*(z*)

9. If x(n) is an imaginary sequence, then the z-transform of the real part of the sequence is:

a) 1/2[X(z)+X*(z*)]
b) 1/2[X(z)-X*(z*)]
c) 1/2[X(-z)-X*(z*)]
d) 1/2[X(-z)+X*(z*)]
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Explanation: If x(N) is an imaginary sequence, then the real part of x(n) is given as

Real{x(n)}= 1/2[x(n)+x*(n)] According to linearity property of z-transform, we get

Z{ Real{x(n)}}= 1/2[X(z)+X*(z*)]

10. What is the signal whose z-transform is given as

a) x1(n)*x2(n)

b) x1(n)*x2(-n)

c) x1(n).x2(n)

d) x1(n)*x2*(n)

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11. What is the z-transform of the signal x(n)= x1(n).x2*(n)?

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Explanation:

We know that Z{x*(n)}=X*(z*)

Now from the multiplication property in time domain we get,

12. If x1(n)={1,2,3} and x2(n)={1,1,1}, then what is the convolution sequence of the given two signals?

a) {1,2,3,1,1}

b) {1,2,3,4,5}

c) {1,3,5,6,2}

d) {1,2,6,5,3}

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Explanation: Given x1(n)={ 1,2,3}=>X1(z)=1+2z -1+3z -2

x2(n){1,1,1}=>X2(z)=1+z -1+z -2

Now from the convolution in time domain property of z-transform, we have

Z{ x1(n)* x2(n)}= X1(z). X2(z)

=> X(z)=1+2z

^{-1}+6z

^{-2}+5z

^{-3}+3z

^{-4}=>x(n)={1,2,6,5,3}

13. What is the z-transform of the signal x(n)=cos(jω0n)u(n)?

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14. What is the z-transform of the signal defined as x(n)=u(n)-u(n+N)?

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15. What is the z-transform of the signal x(n)=[5(3n)-9(7n)]u(n)?

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**Sanfoundry Global Education & Learning Series – Digital Signal Processing.**

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