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

{1,2/3,4/7,8/15,16/31,….}{1,3/2,7/4,15/8,31/16,….}None of these{1/2,3/4,7/8,15/16,31/32,….}

2. The Z-Transform X(z) of a discrete time signal x(n) is defined as ____________

\(\sum_{n=-\infty}^{\infty}x(n)z^n\) \(\sum_{n=-\infty}^{\infty}x(n)z^{-n}\) None of these\(\sum_{n=0}^{\infty}x(n)z^n\)

3. For what kind of signals one sided z-transform is unique?

All signalsCausal signalAnti-causal signalNone of these

4. What is the z-transform of the signal x(n) = -αⁿu(-n-1)?

\(-\frac{1}{1-\alpha z^{-1}}\);ROC |z|>|α| \(-\frac{1}{1+\alpha z^{-1}}\);ROC |z|\(-\frac{1}{1-\alpha z^{-1}}\);ROC |z|\(\frac{1}{1-\alpha z^{-1}}\);ROC |z|

5. A linear time invariant system is said to be BIBO stable if and only if the ROC of the system function _____________

Includes unit circleIs an unit circleNone of theseExcludes unit circle

6. If X(z) is the z-transform of the signal x(n) then what is the z-transform of aⁿx(n)?

X(az-1)X(az)X(a-1z)None of these

7. What is the inverse z-transform of X(z)=log(1+az^-1) |z|>|a|?

x(n)=(-1)n+1 \(\frac{a^{-n}}{n}\), n≥1; x(n)=0, n≤0x(n)=(-1)n-1 \(\frac{a^{-n}}{n}\), n≥1; x(n)=0, n≤0None of these

8. Is the discrete time LTI system with impulse response h(n)=aⁿ(n) (|a| < 1) BIBO stable?

Don't KnowFalseTrueMaybe

9. 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-1)-2(3)nu(-n-1)(0.5)nu(n)+2(3)nu(n)(0.5)nu(n)-2(3)nu(-n-1)-[(0.5)n+2(3)n]u(-n-1)

10. If one or more poles are located near the unit circle, then the rate of decay of signal is _________

None of theseRapidSlowConstant

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

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

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

(2n-0.5n)u(n)(2+0.5n)u(n)(2-0.5n)u(n)None of these

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

None of theseConstant signalDecaying signalRising signal

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

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

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

CascadeConvolutionNone of theseParallel

16. What is the transient 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?

(0.5)nu(n)6.3(0.5)n0.5(6.3)nu(n)6.3(0.5)nu(n)

17. 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)

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

5+7z-1+z-35+7z+z3z2+2z+5+7z-1+z-3z-2+2z-1+5+7z+z3

19. What is the set of all values of z for which X(z) attains a finite value?

Feasible solutionNone of theseRadius of convergenceRadius of divergence

20. If W₄¹⁰⁰=W_x²⁰⁰, then what is the value of x?

81642

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

01zkz-k

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

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

23. The impulse response of a relaxed LTI system is h(n)=aⁿu(n), |a|<1. What is the value of the step response of the system as n→∞?

\(\frac{a}{1-a}\) \(\frac{1}{1-a}\) \(\frac{1}{1+a}\) \(\frac{a}{1+a}\)

24. What is the z-transform of the signal x(n)=naⁿu(n)?

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

25. 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)=NckX(k)=ck/NNone of theseX(k)=N/ck