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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. Discrete-time signals
B. Analog signals only
C. Continuous-time signals
D. Periodic 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. Discrete-Time Fourier Transform (DTFT)
D. Continuous Fourier Transform

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

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

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 summation converges
B. The system is unstable
C. The poles lie on the unit circle
D. The signal is causal

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

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

8 The ROC of a z-transform can contain:

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

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

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

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

Properties of the z-transform Easy
A. Quadratic
B. Linear
C. Logarithmic
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. Addition
C. Multiplication
D. Convolution

14 The property is known as:

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

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

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

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

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

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

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

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. Transfer function
B. Impulse spectrum
C. Step response
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. Outside the unit circle
B. Inside the unit circle
C. On the unit circle
D. At the origin only

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 circle ()
B. A cross ()
C. A square
D. A triangle

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. The interior of a circle:
C. An annular ring:
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 includes the origin
B. The ROC excludes all poles and the unit circle
C. The ROC includes the unit circle
D. The ROC extends to

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. Gaussian elimination
B. Laplace transform inversion
C. Partial fraction expansion
D. Fourier series expansion

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. Stable only for negative inputs
C. Marginally stable, since the pole lies on the unit circle
D. Stable, since the pole lies inside the unit circle

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. Compute the impulse response of the system
B. Perform the inverse z-transform numerically
C. Plot the poles and zeros of a system in the z-plane
D. Plot the magnitude frequency response only

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. Stable
B. Marginally stable
C. Unstable
D. Non-causal

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. filter
B. tf2zp
C. freqz
D. conv

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 minimum-phase since the pole is inside the unit circle
B. The system is stable and causal but not minimum-phase (zero outside unit circle)
C. The system is unstable because it has a zero at
D. The system is both stable and minimum-phase with all-pass characteristics

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. only
B. excluding the unit circle
C. Entire -plane except possibly and
D. Entire -plane including and

51 Find the inverse z-transform of , .

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

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. 3 poles; 2 zeros
B. 2 poles at ; zeros at and
C. 2 poles at ; one zero at
D. poles at and ; zeros at and

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,k] = residuez(b,a)
B. [z,p,k] = zp2tf(b,a)
C. [z,p,k] = tf2zp(b,a)
D. [z,p] = roots(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. Instability because complex poles always cause growth
D. A notch (null) at because the zero dominates

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
D. and for all

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

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 filter automatically becomes an ideal integrator
B. Very slowly decaying, long impulse response sensitive to coefficient quantization
C. The output is identically zero due to pole-zero cancellation
D. The impulse response becomes exactly periodic with no decay
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