Z Transform and its Application – Analysis of the LTI Systems topics include: Z transforms and its properties, types of Z transforms which include rational, inverse and one sided Z transform and their analysis. The Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex valued frequency-domain (the z-domain or z-plane) representation. It can be considered a discrete-time equivalent of the Laplace transform (the s-domain or s-plane). Z transform is used for linear filtering. z transform is also used for finding Linear convolution,... Show more

Z Transform and its Application – Analysis of the LTI Systems topics include: Z transforms and its properties, types of Z transforms which include rational, inverse and one sided Z transform and their analysis.

The Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex valued frequency-domain (the z-domain or z-plane) representation. It can be considered a discrete-time equivalent of the Laplace transform (the s-domain or s-plane).

Z transform is used for linear filtering. z transform is also used for finding Linear convolution, cross-correlation and auto-correlations of sequences. 4. In z transform user can characterize LTI system (stable/unstable, causal/anti-causal) and its response to various signals by placements of pole and zero plot.

LTI systems are used to predict long-term behavior in a system. So, they are often used to model systems like power plants. Another important application of LTI systems is electrical circuits. These circuits, made up of inductors, transistors, and resistors, are the basis upon which modern technology is built.

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Digital Signal Processing Practice Test: Z Transform and its Application - Analysis of the LTI Systems

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25 Questions

1. What is the nature of the signal whose pole-zero plot is as shown?

None of theseRising signalConstant signalDecaying signal

2. Which of the following justifies the linearity property of z-transform?[x(n)↔X(z)].

x(n)+y(n) ↔ X(z)+Y(z)x(n)y(n) ↔ X(z)Y(z)x(n)+y(n) ↔ X(z)Y(z)x(n)y(n) ↔ X(z)+Y(z)

3. What is the z-transform of the signal x(n)=aⁿ(sinω₀n)u(n)?

\(\frac{az^{-1} sin\omega_0}{1-2 az^{-1} cos\omega_0- a^2 z^{-2}}\)\(\frac{az^{-1} sin\omega_0}{1+2 az^{-1} cos\omega_0+a^2 z^{-2}}\)\(\frac{az^{-1} sin\omega_0}{1-2 az^{-1} cos\omega_0+a^2 z^{-2}}\) \(\frac{(az)^{-1} cos\omega_0}{1-2 az^{-1} cos\omega_0+a^2 z^{-2}}\)

4. If Y(z) is the z-transform of the output function, X(z) is the z-transform of the input function and H(z) is the z-transform of system function of the LTI system, then H(z)=?

\(\frac{Y(z)}{X(z)}\) \(\frac{X(z)}{Y(z)}\) None of theseY(z).X(z)

5. What is the partial fraction expansion of X(z)=\(\frac{1+z^{-1}}{1-z^{-1}+0.5z^{-2}}\)?

\(\frac{z(0.5-1.5j)}{z-0.5-0.5j} – \frac{z(0.5+1.5j)}{z-0.5+0.5j}\) \(\frac{z(0.5-1.5j)}{z-0.5-0.5j} + \frac{z(0.5+1.5j)}{z-0.5+0.5j}\) \(\frac{z(0.5+1.5j)}{z-0.5-0.5j} – \frac{z(0.5-1.5j)}{z-0.5+0.5j}\) \(\frac{z(0.5+1.5j)}{z-0.5-0.5j} + \frac{z(0.5-1.5j)}{z-0.5+0.5j}\)

6. What are the values of z for which the value of X(z)=0?

PolesNone of theseZerosSolutions

7. If X⁺(z) is the one sided z-transform of the signal x(n), then \(\lim_{n \rightarrow \infty} x(n)=\lim_{z\rightarrow 1}(z-1) X^+(z)\) is called Final value theorem.

TrueDon't KnowMaybeFalse

8. What is the ROC of z-transform of finite duration anti-causal sequence?

z=0z=∞Entire z-plane, except at z=∞Entire z-plane, except at z=0

9. If all the poles of H(z) are outside the unit circle, then the system is said to be _____________

BIBO stable and causalOnly BIBO stableOnly causalNone of these

10. What is the step response of the system y(n)=ay(n-1)+x(n) -1\(\frac{1}{1+a}\)(1-an+2)u(n)\(\frac{1}{1-a}\)(1+an+2)u(n)\(\frac{1}{1-a}\)(1-an+2)u(n)\(\frac{1}{1+a}\)(1+an+2)u(n)

11. Which of the following series has an ROC as mentioned below?

αnu(n)α-nu(n)α-nu(-n)

12. What is the causal signal x(n) having the z-transform X(z)=\(\frac{1}{(1+z^{-1})(1-z^{-1})^2}\)?

[1/4(-1)n+3/4-n/2]u(-n-1)[1/4(-1)n+3/4-n/2]u(n)[1/4+3/4(-1)n-n/2]u(n)[1/4(-1)n+3/4+n/2]u(n)

13. What is the ROC of the signal x(n)=δ(n-k), k>0?

Entire z-plane, except at z=0z=∞z=0Entire z-plane, except at z=∞

14. A linear time invariant system is characterized by the system function H(z)=\(\frac{1}{1-0.5z^{-1}}+\frac{2}{1-3z^{-1}}\). What is the h(n) if the system is anti causal?

(0.5)nu(n)+2(3)nu(n)-[(0.5)n+2(3)n]u(-n-1)(0.5)nu(n)-2(3)nu(-n-1)(0.5)nu(-n-1)-2(3)nu(-n-1)

15. What is the inverse z-transform of X(z)=\(\frac{1}{1-1.5z^{-1}+0.5z^{-2}}\) if ROC is |z| < 0.5?

{…..62,30,14,6,2,0,0}{….62,30,14,6,2}{2,6,14,30,62…..}{0,0,2,6,14,30,62…..}

16. What is the one sided z-transform X⁺(z) of the signal x(n)={1,2,5,7,0,1}?

5+7z-1+z-3z2+2z+5+7z-1+z-3z-2+2z-1+5+7z+z35+7z+z3

17. What is the inverse z-transform of X(z)=\(\frac{1}{1-1.5z^{-1}+0.5z^{-2}}\) if ROC is |z|<0.5?

[-2+0.5n]u(n)[-2-0.5n]u(-n-1)[-2+0.5n]u(-n-1)[-2-0.5n]u(n)

18. The z-transform of a signal x(n) whose definition is given by \(X(z)=\sum_{n=0}^{\infty} x(n)z^{-n}\) is known as _____________

Unilateral z-transformRational z-transformNone of theseBilateral z-transform

19. What is the z-transform of the signal x(n)=(0.5)ⁿu(n)?

\(\frac{1}{1-0.5z^{-1}};ROC |z|>0.5\) \(\frac{1}{1+0.5z^{-1}};ROC |z|>0.5\) \(\frac{1}{1+0.5z^{-1}};ROC |z|<0.5\) \(\frac{1}{1-0.5z^{-1}};ROC |z|<0.5\)

20. A linear time invariant system is characterized by the system function H(z)=\(\frac{1}{1-0.5z^{-1}}+\frac{2}{1-3z^{-1}}\). What is the ROC of H(z) if the system is causal?

|z|<3|z|<0.5|z|>0.5|z|>3

21. If X(k) is the N point DFT of a sequence whose Fourier series coefficients is given by c_k, then which of the following is true?

X(k)=ck/NX(k)=NckNone of theseX(k)=N/ck

22. What is the z-transform of the following finite duration signal?

2z2 + 4z + 5 +7z-1 + z-32 + 4z + 5z2 + 7z3 + z52 + 4z-1 + 5z-2 + 7z-3 + z-52 + 4z + 5z2 + 7z3 + z4

23. What is the ROC of z-transform of an two sided infinite sequence?

|z|>r1|z|<r1r2<|z|<r1None of these

24. The z-transform X(z) of the signal x(n)=aⁿu(n) has:

One pole at z=a and one zero at z=0One pole at z=0 and one zero at z=0One pole at z=0 and one zero at z=aOne pole at z=a and one zero at z=a

25. What are the values of z for which the value of X(z)=0?

SolutionsNone of thesePolesZeros

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