Z-transform Question & Answers July 11, 2023 By Wat Electrical This article lists 100 Z-transform MCQs for engineering students. All the Z-transform Questions & Answers given below include a hint and a link wherever possible to the relevant topic. This is helpful for users who are preparing for their exams, interviews, or professionals who would like to brush up on the fundamentals of Z-transform. The Z-transform is a mathematical transformation commonly used in digital signal processing and control systems theory. It is an extension of the discrete-time Fourier transform and provides a powerful tool for analyzing and manipulating discrete-time signals. The Z-transform converts a discrete-time signal, which is a sequence of values sampled at equally spaced time intervals, into a complex function of a complex variable, denoted as Z. The Z-transform has several important properties, such as linearity, time shifting, scaling, and convolution. It also provides a way to determine the system’s transfer function, which relates the input and output signals in a linear time-invariant (LTI) system. The Z-transform is widely used in various applications, including digital filter design, system analysis, and control system design. It provides a powerful mathematical framework for analyzing and designing discrete-time systems, making it an essential tool in the field of digital signal processing. 1). The Z-transform of a unit impulse function is __________________? 1 z 1/z z^n Hint 2). The Z-transform of a delayed version of a sequence x[n] by M samples is ___________________? z^M * X(z) z^-M * X(z) X(z) / z^M X(z) * z^-M Hint 3). If the ROC of a Z-transform includes the unit circle (|z| = 1), the signal x[n] is ____________________? Right-sided Left-sided Two-sided Causal Hint 4). The poles of a rational Z-transform determine __________________? The time-domain representation of the signal The stability of the system The frequency response of the system The ROC of the Z-transform Hint 5). Who introduced the Z-transform? Karl Ziegler Max Zorn Peter Zorn Laplace Hint 6). In which year was the Z-transform first introduced? 1850 1900 1945 1950 Hint 7). What is the Z-transform primarily used for? Image processing Signal processing Quantum mechanics Fluid dynamics Hint 8). Which of the following statements about the Z-transform is true? It transforms a time-domain signal into a frequency-domain signal. It is used to calculate the derivative of a signal. It only works for discrete-time signals. It is equivalent to the Laplace transform. Hint 9). The Z-transform is an extension of which other transform? Fourier transform Laplace transform Hilbert transform Haar transform Hint 10). The inverse Z-transform is used to __________________? Convert a frequency-domain signal into a time-domain signal. Calculate the integral of a signal. Determine the poles and zeros of a system. Perform convolution in the frequency domain. Hint 11). Which of the following is a commonly used method to find the inverse Z-transform? Partial fraction decomposition Fourier series expansion Convolution theorem Fast Fourier Transform (FFT) Hint 12). Which of the following statements is true about the Z-transform? It is defined for both continuous-time and discrete-time signals. It is only defined for continuous-time signals. It is only defined for discrete-time signals. It is not related to signal processing. Hint 13). What is the inverse Z-transform of X(z) = (2z)/(z – 0.5)? 2(0.5)^n 2(2)^n (1/2)(0.5)^n (1/2)(2)^n Hint 14). What is the region of convergence (ROC) of a causal sequence? Inside the unit circle Outside the unit circle On the unit circle Any region Hint 15). Which property of the Z-transform states that a time shift in the time domain corresponds to a multiplication by a complex exponential in the frequency domain? Linearity property Time shifting property Convolution property Time reversal property Hint 16). Which property of the Z-transform states that a constant scaling in the time domain corresponds to a scaling by the same constant in the frequency domain? Linearity property Time shifting property Convolution property Scaling property Hint 17). Which property of the Z-transform states that convolution in the time domain corresponds to multiplication in the frequency domain? Linearity property Time shifting property Convolution property Scaling property Hint 18). Which property of the Z-transform states that a complex conjugate in the time domain corresponds to a complex conjugate in the frequency domain? Linearity property Time shifting property Convolution property Complex conjugate property Hint 19). Which property of the Z-transform states that if a sequence is right-sided, its ROC (Region of Convergence) includes the unit circle? Linearity property Time shifting property ROC property Stability property Hint 20). Which of the following Z-transform pairs represents the discrete-time exponential function x[n] = (a^n)/n!, where a is a constant? X(z) = e^(-a)*z^(-1) X(z) = e^(-a)*z^(-1)/(1 – z^(-1)) X(z) = e^(-a)*z^(-1)/(1 – az^(-1)) X(z) = e^(-a)*z^(-1)/(1 – e^(-a)*z^(-1)) Hint 21). What is the Z-transform of the discrete-time convolution of x(n) = [1, 2, 1] and y(n) = [2, 1, 2]? X(z) = z^2 + 3z + 2, Y(z) = z^2 + 3z + 2 X(z) = z^2 + 4z + 2, Y(z) = z^2 + 2z + 4 X(z) = z^2 + 3z + 2, Y(z) = z^2 + 2z + 4 X(z) = z^2 + 4z + 2, Y(z) = z^2 + 3z + 2 Hint 22). The Z-transform of the discrete-time convolution of x(n) = [1, 2, 1] and y(n) = [2, 1, 2] has how many poles? 0 1 2 3 Hint 23). What is the Z-transform of the sequence x(n) = [1, 2, 1]? X(z) = z^2 + 2z + 1 X(z) = z^2 + z + 1 X(z) = z^2 + 3z + 2 X(z) = z^2 + 4z + 2 Hint 24). What is the Z-transform of the sequence y(n) = [2, 1, 2]? Y(z) = 2z^2 + z + 2 Y(z) = 2z^2 + z + 2z Y(z) = 2z^2 + 3z + 2 Y(z) = 2z^2 + 4z + 2 Hint 25). What is the Z-transform of the discrete-time convolution of x(n) = [1, 2, 1] and y(n) = [2, 1, 2] evaluated at z = 1? Z[x(n) * y(n)]|z=1 = 6 Z[x(n) * y(n)]|z=1 = 7 Z[x(n) * y(n)]|z=1 = 8 Z[x(n) * y(n)]|z=1 = 9 Hint Z-transform MCQ for Quiz 26). What is the convolution of x(n) = [1, 2, 1] and y(n) = [2, 1, 2]? [1, 2, 1] [2, 5, 6, 5, 2] [2, 3, 6, 3, 2] [2, 4, 4, 2] Hint 27). What is the Z-transform of the convolution of x(n) = [1, 2, 1] and y(n) = [2, 1, 2]? X(z) = 2z^2 + 5z + 2, Y(z) = 2z^2 + 5z + 2 X(z) = z^2 + 3z + 2, Y(z) = z^2 + 3z + 2 X(z) = 2z^2 + 3z + 2, Y(z) = 2z^2 + 3z + 2 X(z) = z^2 + 4z + 2, Y(z) = z^2 + 4z + 2 Hint 28). What is the inverse Z-transform of X(z) = 2z^2 + 5z + 2? [2, 5, 2] [1, 2, 1] [1, 3, 2] [1, 2, 1, 0, 0] Hint 29). Which transformation is used to scale an object along the Z-axis? Translation Rotation Scaling Shearing Hint 30). Which transformation is used to rotate an object around the Z-axis? Translation Rotation Scaling Shearing Hint 31). Which transformation is used to move an object along the Z-axis? Translation Rotation Scaling Shearing Hint 32). Which transformation is used to skew an object along the Z-axis? Translation Rotation Scaling Shearing Hint 33). Which transformation is responsible for changing the perspective of an object along the Z-axis? Translation Rotation Scaling Projection Hint 34). What does the scaling factor of 1 do to an object during a Z-axis scaling transformation? Enlarges the object Shrinks the object Flips the object No change to the object Hint 35). During a Z-axis rotation, the X and Y coordinates of an object __________________? Remain unchanged Are rotated around the Z-axis Are scaled uniformly Are translated along the Z-axis Hint 36). What does a positive Z-axis shearing transformation do to an object? Squeezes the object along the X-axis Stretches the object along the Y-axis Skews the object along the Z-axis Reflects the object across the Z-axis Hint 37). Which transformation flips an object over the XY-plane? Translation Reflection Scaling Rotation Hint 38). What is the effect of shearing along the Z-axis in a 3D scene? Changing the object's position Changing the object's size Changing the object's shape Changing the object's orientation Hint 39). The initial value theorem of the Z-transform states that _________________? The initial value of the Z-transform is equal to the sum of all the initial values of the individual terms in the time-domain sequence. The initial value of the Z-transform is equal to the initial value of the time-domain sequence The initial value of the Z-transform is equal to the final value of the time-domain sequence The initial value of the Z-transform is always zero Hint 40). If the Z-transform of a sequence is given by X(z) = (1 – z^(-1))/(1 – 2z^(-1)), what is the initial value of the corresponding time-domain sequence? 1 2 -1 -2 Hint 41). The initial value theorem of the Z-transform can be used to calculate the initial value of a time-domain sequence when ___________________? The Z-transform of the sequence is known The final value of the sequence is known Both the Z-transform and the final value of the sequence are known. None of the above. Hint 42). If the Z-transform of a sequence is X(z) = 1/(1 – z^(-1)), what is the initial value of the corresponding time-domain sequence? 1 0 Infinity Cannot be determined Hint 43). Which of the following statements best describes the initial value theorem? It relates the initial condition of a system to the initial value of its z-transform. It relates the final condition of a system to the final value of its z-transform. It relates the initial condition of a system to the final value of its z-transform It relates the final condition of a system to the initial value of its z-transform Hint 44). The initial value theorem states that if the z-transform of a discrete-time signal X(z) is rational and has no poles at z = 0, then _____________________? X(z) must have a zero at z = 0 X(z) must have a zero at z = 0 X(z) is not defined at z = 0 X(z) must be a constant value Hint 45). According to the initial value theorem of the z-transform, what does the value of a discrete-time signal at time n = 0 represent? The value of the signal at the first sample The average value of the signal over all time The sum of all previous samples of the signal The maximum value of the signal Hint 46). The initial value theorem is applicable for ________________? Causal signals only Anti-causal signals only Both causal and anti-causal signals Non-causal signals Hint 47). If the z-transform of a signal X(z) has a pole at z = 1, then according to the initial value theorem __________________? The initial value of x[n] is zero The initial value of x[n] is infinity The initial value of x[n] cannot be determined None of the above Hint 48). The initial value theorem is used to determine the initial condition of a system in the _______________? Frequency domain Time domain Laplace domain None of the above Hint 49). If a sequence x[n] has a z-transform X(z), and another sequence y[n] has a z-transform Y(z), what is the z-transform of the sequence obtained by multiplying x[n] by a constant scalar a? X(a * z) a * X(z) X(z) / a X(z) + a Hint 50). The Final Value Theorem of the Z-transform states that the final value of a discrete-time signal can be determined by evaluating the ________________ of its Z-transform? Pole Zero Residue Limit Hint Z-transform MCQ for Exams 51). The Final Value Theorem is applicable only when the region of convergence includes _________________? The unit circle The left half-plane The right half-plane The entire z-plane Hint 52). Which condition must be satisfied for the Final Value Theorem to be used? The signal must be causal The signal must be non-causal The signal must be periodic The signal must be anti-causal Hint 53). Consider a rational Z-transform representation H(z) = (1 - z^(-1))/(1 - 0.5z^(-1)). What is the final value of the corresponding time-domain sequence? 0 0.5 1 Undefined Hint 54). If the Z-transform H(z) has poles outside the unit circle and satisfies the conditions for the Final Value Theorem, what can be said about the final value of the corresponding time-domain sequence? It is zero It is non-zero It is undefined It cannot be determined Hint 55). The Final Value Theorem can be applied to which type of signals? Finite-duration signals Infinite-duration signals Only periodic signals Only random signals Hint 56). If the Z-transform H(z) has a pole at z = 1, what can be said about the final value of the corresponding time-domain sequence? It is zero It is non-zero It is infinite It cannot be determined Hint 57). Which property of the Z-transform is crucial for applying the Final Value Theorem? Linearity Time-reversal Time-shifting Stability Hint 58). The Final Value Theorem provides information about the ________________ of a signal. Initial value Steady-state value Maximum value Minimum value Hint 59). For a stable system, the Final Value Theorem is useful in finding the _______________ of the output. Transient response Steady-state response Frequency response Hint 60). If the Z-transform H(z) has a pole at z = 0, what can be said about the final value of the corresponding time-domain sequence? It is zero It is non-zero It is infinite It cannot be determined Hint 61). The Final Value Theorem is analogous to which theorem in the Laplace transform domain? Initial Value Theorem Final Value Theorem Convolution Theorem Parseval's Theorem Hint 62). If the ROC of the Z-transform includes the unit circle and the final value of the corresponding time-domain sequence is finite, what can be said about the poles of the Z-transform? They are all inside the unit circle They are all on the unit circle They are all outside the unit circle They can be anywhere in the z-plane Hint 63). Consider a rational Z-transform representation H(z) = (1 - 2z^(-1))/(1 - 0.5z^(-1)). If the signal x[n] has a finite final value, what is the value of x[∞]? 1 2 3 Undefined Hint 64). Which theorem is used to find the initial value of a discrete-time signal in the Z-domain? IVT FVT Both IVT and FVT None of the above Hint 65). Which theorem is used to find the final value of a discrete-time signal in the Z-domain? IVT FVT Both IVT and FVT None of the above Hint 66). In the Z-domain, the Final Value Theorem is mathematically expressed as ___________________? lim(n->∞) z^n = 0 lim(n->∞) (1 - z^-1) = 0 lim(n->∞) z^n / (z - 1) = 0 lim(n->∞) z^-n = 0 Hint 67). Which condition should be satisfied by the pole locations for the Final Value Theorem to be valid? Poles inside the unit circle Poles outside the unit circle Poles on the unit circle No condition on pole locations Hint 68). The z-transform of a discrete-time sinusoidal function x[n] = sin(ωn) is given by ________________________? X(z) = z/(z^2 - 2zcos(ω) + 1) X(z) = z/(z^2 + 2zcos(ω) + 1) X(z) = z/(z^2 - 2zsin(ω) + 1) X(z) = z/(z^2 + 2zsin(ω) + 1) Hint 69). The Z-transform of the sequence x[n] = {1, 2, 3, 4} is given by ____________________? X(z) = 1 + 2z^(-1) + 3z^(-2) + 4z^(-3) X(z) = 1 + 2z + 3z^2 + 4z^3 X(z) = z + 2z^2 + 3z^3 + 4z^4 X(z) = 1 + 2z^2 + 3z^4 + 4z^6 Hint 70). The Z-transform of the sequence x[n] = {1, -1, 1, -1, ...} is given by ____________________? X(z) = 1 - z^(-2) X(z) = 1 - z^(-1) X(z) = 1 + z^(-1) X(z) = 1 + z^(-2) Hint 71). The Z-transform of the sequence x[n] = {0.5^n} for n ≥ 0 is given by ____________________? X(z) = 1/(1 - 0.5z^(-1)) X(z) = 1/(1 + 0.5z^(-1)) X(z) = 0.5/(1 - z^(-1)) X(z) = 0.5/(1 + z^(-1)) Hint 72). The Z-transform of the sequence x[n] = {2^n} for n ≥ 0 is given by ____________________? X(z) = 1/(1 - 2z^(-1)) X(z) = 2/(1 - 2z^(-1)) X(z) = 1/(1 + 2z^(-1)) X(z) = 2/(1 + 2z^(-1)) Hint 73). The Z-transform of the sequence x[n] = {-1, -2, -3, -4} is given by ____________________? X(z) = -1 - 2z^(-1) - 3z^(-2) - 4z^(-3) X(z) = -1 - 2z^(-1) - 3z^(-2) - 4z^(-3) X(z) = -z - 2z^2 - 3z^3 - 4z^4 X(z) = -1 - 2z^2 - 3z^4 - 4z^6 Hint 74). The Z-transform of the sequence x[n] = {(-0.5)^n} for n ≥ 0 is given by ____________________? X(z) = 1/(1 + 0.5z^(-1)) X(z) = 1/(1 - 0.5z^(-1)) X(z) = -0.5/(1 - z^(-1)) X(z) = -0.5/(1 + z^(-1)) Hint 75). Find the inverse Z-transform of X(z) = (z - 0.5)/(z^2 - z - 0.5) is ____________________? x[n] = (-0.5)^n x[n] = 2^n x[n] = n(0.5)^n x[n] = n(-0.5)^n Hint 76). The Z-transform allows for the analysis and representation of discrete-time signals in the __________ domain? Time Frequency Amplitude Phase Hint 77). The Z-transform provides a powerful mathematical tool for analyzing __________ systems? Continuous-time Analog Discrete-time Digital Hint 78). The Z-transform can simplify complex difference equations into _______________ equations? Algebraic Differential Integral Exponential Hint 79). The Z-transform can be used to determine the stability of a discrete-time system by examining the _______________? Impulse response Step response Transfer function Frequency response Hint 80). The Z-transform provides a convenient way to represent and manipulate ________________ signals? Deterministic Stochastic Continuous-time Periodic Hint 81). The Z-transform allows for the application of various mathematical operations such as differentiation, integration, and _________________? Convolution Differentiation Fourier analysis Laplace transform Hint 82). The Z-transform is particularly useful in the analysis and design of ________________ systems? Analog Continuous-time Feedback Control Hint 83). What is one of the disadvantages of the Z-transform? It cannot be used for non-causal signals. It is computationally complex. It only applies to continuous-time signals. It is not applicable to linear systems. Hint 84). The Z-transform assumes which of the following? Causality Time-invariance Linearity All of the above Hint 85). The Z-transform is not suitable for which type of signals? Discrete-time signals Continuous-time signals Nonlinear signals Causal signals Hint 86). The Z-transform may yield a complex-valued result when ___________________? The input signal is real-valued The input signal is complex-valued. The system is linear. The system is time-invariant Hint 87). One limitation of the Z-transform is that it requires the signal to be ___________________? Infinite in duration Finite in duration. Symmetric Periodic Hint 88). The Z-transform is less suitable for analyzing systems with ___________________? High-frequency components Low-frequency components Nonlinear behavior Stable behavior Hint 89). The Z-transform may lose information about the input signal's ____________________? Magnitude Phase Frequency content Causal nature Hint 90). The Z-transform assumes which type of system behavior? Time-varying Time-invariant Causal Nonlinear Hint 91). The Z-transform is limited in its ability to handle signals with ______________________? Rapid changes Slow changes Steady-state behavior Non-causal behavior Hint 92). The Z-transform can become unstable when __________________? The input signal is causal The input signal is non-causal The system is stable The system is non-linear Hint 93). The Z-transform is an essential tool in which branch of mathematics? Calculus Probability theory Complex analysis Linear algebra Hint 94). The Z-transform can be used to solve which type of convolution operation for discrete-time signals? Circular convolution Linear convolution Time-domain convolution Frequency-domain convolution Hint 95). The Z-transform is widely employed in which field to analyze and process digital signals? Medicine Communications Finance Automotive engineering Hint 96). The Z-transform is particularly useful for solving which type of difference equations? Linear difference equations Nonlinear difference equations Partial difference equations Stochastic difference equations Hint 97). The Z-transform provides a concise representation of discrete-time signals in which form? Differential equation State-space model Transfer function Pole-zero plot Hint 98). The region of convergence (ROC) in the Z-transform represents the values of z for which the transform is ________________? Convergent Divergent Undefined Equal to zero Hint 99). What is the Z-transform of the sum of two sequences x1[n] and x2[n]? X(z) = X1(z) + X2(z) X(z) = X1(z) + X2(z) X(z) = X1(z) / X2(z) X(z) = X1(z) - X2(z) Hint 100). What is the Z-transform of the difference between two sequences x1[n] and x2[n]? X(z) = X1(z) + X2(z) X(z) = X1(z) * X2(z) X(z) = X1(z) / X2(z) X(z) = X1(z) - X2(z) Hint Time's up