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Unit 6: The Z-transform - Practice Quiz

ECE220 — Signal And Systems 60 Questions
0 Correct 0 Wrong 60 Left
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1 The z-transform of a discrete-time signal is defined as:

the z-transform Easy
A.
B.
C.
D.

2 The z-transform is primarily used for the analysis of which type of signals?

the z-transform Easy
A. Periodic signals only
B. Continuous-time signals
C. Discrete-time signals
D. Analog signals only

3 The z-transform of the unit impulse is:

the z-transform Easy
A.
B.
C.
D.

4 The z-transform can be viewed as a generalization of which transform?

Introduction Easy
A. Laplace Transform
B. Hilbert Transform
C. Continuous Fourier Transform
D. Discrete-Time Fourier Transform (DTFT)

5 Evaluating the z-transform on the unit circle yields the:

the z-transform Easy
A. DTFT of the signal
B. Laplace transform
C. Continuous Fourier transform
D. Autocorrelation of the signal

6 The Region of Convergence (ROC) of a z-transform is the set of values of for which:

The region of convergence for the z transform Easy
A. The signal is causal
B. The summation converges
C. The system is unstable
D. The poles lie on the unit circle

7 In the z-plane, the ROC is typically bounded by:

The region of convergence for the z transform Easy
A. Straight horizontal lines
B. Parabolas through the origin
C. Circles centered at the origin
D. Straight vertical lines

8 The ROC of a z-transform can contain:

The region of convergence for the z transform Easy
A. No poles
B. Exactly one pole
C. All the poles
D. Only the zeros

9 For a right-sided (causal) sequence, the ROC is:

The region of convergence for the z transform Easy
A. The interior of a circle
B. The entire z-plane
C. A ring between two circles
D. The exterior of a circle

10 For a left-sided sequence, the ROC is:

The region of convergence for the z transform Easy
A. A ring between two circles
B. The exterior of a circle
C. The unit circle only
D. The interior of a circle

11 The z-transform is a __ operation with respect to its input signals.

Properties of the z-transform Easy
A. Logarithmic
B. Linear
C. Quadratic
D. Nonlinear

12 A time shift of corresponds to multiplication of by:

Properties of the z-transform Easy
A.
B.
C.
D.

13 Convolution of two signals in the time domain corresponds to which operation in the z-domain?

Properties of the z-transform Easy
A. Division
B. Convolution
C. Addition
D. Multiplication

14 The property is known as:

Properties of the z-transform Easy
A. Time reversal
B. Differentiation
C. Conjugation
D. Scaling in the z-domain

15 A common method to compute the inverse z-transform of a rational function is:

The inverse z transform Easy
A. Gaussian elimination
B. Partial fraction expansion
C. Numerical integration only
D. Fourier series expansion

16 Which of the following is NOT a standard method for finding the inverse z-transform?

The inverse z transform Easy
A. Partial fraction expansion
B. Contour integration
C. Power series (long division)
D. Laplace inversion

17 To uniquely determine the inverse z-transform from , one must also know the:

The inverse z transform Easy
A. Phase spectrum
B. Signal power
C. ROC
D. Sampling rate

18 For a discrete-time LTI system, the z-transform of the impulse response is called the:

Analysis and characterisation of LTI systems using z-transforms Easy
A. Impulse spectrum
B. Step response
C. Transfer function
D. Frequency deviation

19 A causal and stable discrete-time LTI system has all its poles located:

Analysis and characterisation of LTI systems using z-transforms Easy
A. Inside the unit circle
B. At the origin only
C. Outside the unit circle
D. On the unit circle

20 In a pole-zero plot of the z-plane, zeros are usually marked with the symbol:

Software simulation of system representation and pole zero analysis Easy
A. A cross ()
B. A square
C. A triangle
D. A circle ()

21 The z-transform of the discrete-time signal is:

the z-transform Medium
A.
B.
C.
D.

22 The z-transform of the unit impulse is:

the z-transform Medium
A.
B.
C.
D.

23 For a right-sided sequence, the region of convergence (ROC) of the z-transform is:

The region of convergence for the z transform Medium
A. The entire z-plane except infinity
B. An annular ring:
C. The interior of a circle:
D. The exterior of a circle:

24 The z-transform corresponds to a causal signal. Its ROC is:

The region of convergence for the z transform Medium
A.
B.
C.
D.

25 Which condition on the ROC guarantees that an LTI system is stable?

The region of convergence for the z transform Medium
A. The ROC extends to
B. The ROC includes the origin
C. The ROC includes the unit circle
D. The ROC excludes all poles and the unit circle

26 If , then the z-transform of is:

Properties of the z-transform Medium
A.
B.
C.
D.

27 Using the convolution property, if , then equals:

Properties of the z-transform Medium
A.
B.
C.
D.

28 If , the z-transform of (scaling in the z-domain) is:

Properties of the z-transform Medium
A.
B.
C.
D.

29 The z-transform differentiation property gives the transform of as:

Properties of the z-transform Medium
A.
B.
C.
D.

30 The inverse z-transform of , ROC , is:

The inverse z transform Medium
A.
B.
C.
D.

31 For with ROC , the inverse z-transform is:

The inverse z transform Medium
A.
B.
C.
D.

32 Which technique is most commonly used to invert a rational z-transform with distinct poles?

The inverse z transform Medium
A. Partial fraction expansion
B. Laplace transform inversion
C. Fourier series expansion
D. Gaussian elimination

33 Using long division (power series method) on yields a sequence corresponding to:

The inverse z transform Medium
A.
B.
C.
D.

34 An LTI system has transfer function . The system is causal and:

Analysis and characterisation of LTI systems using z-transforms Medium
A. Unstable, since the pole lies outside the unit circle
B. Marginally stable, since the pole lies on the unit circle
C. Stable, since the pole lies inside the unit circle
D. Stable only for negative inputs

35 A causal LTI system is described by . Its transfer function is:

Analysis and characterisation of LTI systems using z-transforms Medium
A.
B.
C.
D.

36 For a causal LTI system with , the poles are located at:

Analysis and characterisation of LTI systems using z-transforms Medium
A. and
B. and
C. and
D. and

37 The frequency response of a stable LTI system is obtained from by evaluating it at:

Analysis and characterisation of LTI systems using z-transforms Medium
A.
B.
C.
D.

38 In MATLAB, the command zplane(b, a) where b and a are numerator and denominator coefficients is used to:

Software simulation of system representation and pole zero analysis Medium
A. Plot the magnitude frequency response only
B. Plot the poles and zeros of a system in the z-plane
C. Perform the inverse z-transform numerically
D. Compute the impulse response of the system

39 In a pole-zero plot, poles are conventionally marked with and zeros with . If all poles lie inside the unit circle for a causal system, the system is:

Software simulation of system representation and pole zero analysis Medium
A. Marginally stable
B. Stable
C. Non-causal
D. Unstable

40 Given transfer function coefficients, which MATLAB function converts them into pole-zero-gain form?

Software simulation of system representation and pole zero analysis Medium
A. freqz
B. conv
C. filter
D. tf2zp

41 Consider the two-sided signal with . What is its bilateral z-transform and ROC?

the z-transform Hard
A. , ROC:
B. , ROC:
C. , ROC:
D. , ROC:

42 A signal has z-transform with poles at and . Which ROC corresponds to a stable but non-causal system?

The region of convergence for the z transform Hard
A.
B.
C.
D.

43 Given with ROC , find .

The inverse z transform Hard
A.
B.
C.
D.

44 If , what is the z-transform of ?

Properties of the z-transform Hard
A.
B.
C.
D.

45 An LTI system has . Which statement about stability and minimum-phase is correct?

Analysis and characterisation of LTI systems using z-transforms Hard
A. The system is unstable because it has a zero at
B. The system is both stable and minimum-phase with all-pass characteristics
C. The system is stable and causal but not minimum-phase (zero outside unit circle)
D. The system is minimum-phase since the pole is inside the unit circle

46 Using the initial value theorem, find for the causal signal with .

Properties of the z-transform Hard
A.
B.
C.
D.

47 For with , the inverse z-transform for is:

The inverse z transform Hard
A.
B.
C.
D.

48 A causal LTI system is described by . What are the pole locations?

Analysis and characterisation of LTI systems using z-transforms Hard
A. and
B. and
C. and
D. and

49 If with ROC and with ROC , the ROC of the convolution is at least:

Properties of the z-transform Hard
A.
B.
C.
D.

50 A finite-length signal is nonzero only for . What is the ROC of ?

The region of convergence for the z transform Hard
A. Entire -plane including and
B. excluding the unit circle
C. Entire -plane except possibly and
D. only

51 Find the inverse z-transform of , .

The inverse z transform Hard
A.
B. for even , zero for odd
C. for all
D.

52 A system function is . How many poles and zeros are in the finite -plane (excluding origin/infinity trivial ones counted properly)?

Analysis and characterisation of LTI systems using z-transforms Hard
A. poles at and ; zeros at and
B. 2 poles at ; zeros at and
C. 2 poles at ; one zero at
D. 3 poles; 2 zeros

53 Applying the final value theorem, find for , ROC .

Properties of the z-transform Hard
A.
B. The theorem does not apply here
C.
D.

54 The z-transform of is:

the z-transform Hard
A. ,
B. ,
C. ,
D. ,

55 In MATLAB, given b = [1 -0.5] and a = [1 -1.5 0.56], which command correctly returns poles, zeros, and gain of the transfer function ?

Software simulation of system representation and pole zero analysis Hard
A. [z,p] = roots(b,a)
B. [z,p,k] = residuez(b,a)
C. [z,p,k] = tf2zp(b,a)
D. [z,p,k] = zp2tf(b,a)

56 A pole-zero plot generated by zplane shows poles at and a zero at . What can you infer about the system's magnitude response?

Software simulation of system representation and pole zero analysis Hard
A. A resonant peak near normalized frequency due to poles near the unit circle
B. A flat all-pass response since gain is unity
C. A notch (null) at because the zero dominates
D. Instability because complex poles always cause growth

57 An all-pass system has . For this to be stable and all-pass, which condition on holds, and what is ?

Analysis and characterisation of LTI systems using z-transforms Hard
A. real and
B. and
C. and for all
D. and

58 For with ROC , the inverse z-transform is:

The inverse z transform Hard
A.
B.
C.
D.

59 Which statement best captures why the z-transform generalizes the discrete-time Fourier transform (DTFT)?

Introduction Hard
A. The z-transform evaluates the signal on , and reduces to the DTFT on the unit circle when the ROC includes it
B. The DTFT is the z-transform evaluated at
C. The z-transform only exists for signals whose DTFT diverges everywhere
D. The z-transform is identical to the DTFT but uses instead of

60 You simulate a filter with poles very close to the unit circle (radius ). What numerical/practical issue is most likely to appear in the impulse response computed via filter in floating point?

Software simulation of system representation and pole zero analysis Hard
A. The impulse response becomes exactly periodic with no decay
B. The output is identically zero due to pole-zero cancellation
C. Very slowly decaying, long impulse response sensitive to coefficient quantization
D. The filter automatically becomes an ideal integrator
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