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. Which of the following method is used to find the inverse z-transform of a signal?

Counter integrationAll of the mentionedPartial fraction expansionExpansion into a series of terms

2. What is the steady-state response of the system described by the difference equation y(n)=0.5y(n-1)+x(n) when the input signal is x(n)= 10cos(πn/4)u(n) and the system is initially at rest?

13.56cos(πn/4 -28.7o)13.56cos(πn/4 +28.7o)u(n)None of these13.56cos(πn/4 -28.7o)u(n)

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

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

4. The ROC of z-transform of any signal cannot contain poles.

MaybeFalseTrueDon't Know

5. 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)

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

\(\frac{z}{4(z+1)} + \frac{3z}{4(z-1)} + \frac{z}{2(z+1)^2}\) \(\frac{z}{4(z+1)} + \frac{3z}{4(z-1)} + \frac{z}{2(z-1)^2}\) \(\frac{z}{4(z+1)} + \frac{3z}{4(z-1)} – \frac{z}{2(z+1)^2}\) \(\frac{z}{4(z+1)} + \frac{z}{4(z-1)} + \frac{z}{2(z+1)^2}\)

7. If all the poles of H(z) are inside the unit circle, then the system is said to be ____________

Only causalNone of theseOnly BIBO stableBIBO stable and causal

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

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

9. Which of the following signals have a pole-zero plot as shown below?

anu(n)u(an)a.u(n)none of the mentioned

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

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

11. What is the partial fraction expansion of the proper function X(z)=\(\frac{1}{1-1.5z^{-1}+0.5z^{-2}}\)?

\(\frac{2z}{z-1}-\frac{z}{z+0.5}\) \(\frac{2z}{z-1}+\frac{z}{z-0.5}\) \(\frac{2z}{z-1}+\frac{z}{z+0.5}\) \(\frac{2z}{z-1}-\frac{z}{z-0.5}\)

12. Sampling rate conversion by the rational factor I/D is accomplished by what connection of interpolator and decimator?

ParallelNone of theseConvolutionCascade

13. 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 _____________

Rational z-transformUnilateral z-transformBilateral z-transformNone of these

14. Which of the following is true?

\(W_N^*=\frac{1}{N} W_{N^{-1}}\) \(W_N-1=W_{N^*}\) \(W_N-1=\frac{1}{N} W_{N^*}\) None of these

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

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

16. If the ROC of the system function is the exterior of a circle of radius r < ∞, including the point z = ∞, then the system is said to be ___________

None of theseCausalStableAnti causal

17. Which of the following has to be performed in sampling rate conversion by rational factor?

InterpolationEither interpolation or decimationDecimationNone of these

18. What is the one sided z-transform of x(n)=δ(n-k)?

0z-k1zk

19. What is the system function of the system described by the difference equation y(n)=0.5y(n-1)+2x(n)?

\(\frac{0.5}{1-2z^{-1}}\) \(\frac{2}{1-0.5z^{-1}}\) \(\frac{2}{1+0.5z^{-1}}\) \(\frac{0.5}{1+2z^{-1}}\)

20. If X(z) has M finite zeros and N finite poles, then which of the following condition is true?

|N+M| zeros at origin(if N < M)|N-M| poles at origin(if N < M)|N-M| zeros at origin(if N < M)|N+M| poles at origin(if N < M)

21. What is the ROC of a causal infinite length sequence?

None of these|z|>r1|z|<r1r2<|z|<r1

22. What is the proper fraction and polynomial form of the improper rational transform
X(z)=\(\frac{1+3z^{-1}+\frac{11}{6} z^{-2}+\frac{1}{3} z^{-3}}{1+\frac{5}{6} z^{-1}+\frac{1}{6} z^{-2}}\)?

1+2z-1+\(\frac{\frac{1}{3} z^{-1}}{1+\frac{5}{6} z^{-1}+\frac{1}{6} z^{-2}}\) 1-2z-1+\(\frac{\frac{1}{6} z^{-1}}{1+\frac{5}{6} z^{-1}+\frac{1}{6} z^{-2}}\) 1+2z-1–\(\frac{\frac{1}{6} z^{-1}}{1+\frac{5}{6} z^{-1}+\frac{1}{6} z^{-2}}\) 1+2z-1+\(\frac{\frac{1}{6}z^{-1}}{1+\frac{5}{6} z^{-1}+\frac{1}{6} z^{-2}}\)

23. If x(n)=aⁿ, then what is one sided z-transform of x(n-2)?

\(\frac{z^{-2}}{1-az^{-1}} + a^{-1}z^{-1} + a^{-2}\) \(\frac{z^{-2}}{1-az^{-1}} – a^{-1}z^{-1} + a^{-2}\) \(\frac{z^{-2}}{1-az^{-1}} + a^{-1}z^{-1} – a^{-2}\) \(\frac{z^{-2}}{1+az^{-1}} + a^{-1}z^{-1} + a^{-2}\)

24. Which of the following is true regarding the number of computations requires to compute an N-point DFT?

N2 complex additions and N(N-1) complex multiplicationsN2 complex additions and N(N+1) complex multiplicationsN2 complex multiplications and N(N-1) complex additionsN2 complex multiplications and N(N+1) complex additions

25. 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.

TrueFalseDon't KnowMaybe

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