Explore core principles of digital control systems with a focus on sampling theory and Z-Transform applications. This quiz challenges your understanding of discrete-time signals, aliasing, sampling rates, and digital domain analysis techniques.
A continuous-time signal contains frequency components up to 4 kHz. According to the Nyquist-Shannon theorem, what is the minimum sampling rate required to avoid aliasing?
Explanation: The Nyquist-Shannon sampling theorem states that the sampling rate must be at least twice the highest frequency present in the signal, so 2 × 4 kHz equals 8 kHz. Options like 2 kHz and 4 kHz are under the minimum and would lead to aliasing. While 12 kHz is above the minimum and also prevents aliasing, it is not the minimum required rate. Thus, 8 kHz is the correct choice.
Which property of the Z-Transform allows the conversion of time shifts in a sequence into algebraic multiplication in the Z-domain?
Explanation: The time-shifting property of the Z-Transform links a time shift in the discrete sequence to a multiplication by a power of z inverse (z^-k) in the Z-domain. The linearity property addresses how sums of signals are transformed but not shifts. The convolution property relates to multiplication in the Z-domain corresponding to convolution in time. The initial value property is used to find the initial value of a sequence from the Z-transform. Only the time-shifting property directly relates to this conversion.
If a digital control system samples a 3.5 kHz sine wave at a rate of 5 kHz, which phenomenon is most likely to distort the sampled data?
Explanation: Sampling at 5 kHz for a 3.5 kHz signal violates the Nyquist criterion, leading to aliasing where the true frequency components overlap and produce misleading lower frequencies. Quantization noise relates to amplitude, not frequency, distortion. Zero-order hold refers to signal reconstruction, not aliasing. Time delay is a separate issue affecting phase or system response, not frequency distortion. Aliasing is the correct answer in this context.
What is obtained by taking the Z-Transform of a discrete-time system’s impulse response?
Explanation: The Z-Transform of the discrete-time impulse response yields the system's transfer function in the Z-domain, which is essential for digital system analysis. Fourier series coefficients represent periodic signals, not system dynamics. The Laplace transform applies to continuous-time systems rather than discrete ones. The sampled time waveform references time-domain data and not its Z-domain representation. Therefore, the Z-domain transfer function is the correct outcome.
When the sampling frequency of a digital system is increased while keeping all else constant, what is the immediate effect on the reconstructed analog output’s quality?
Explanation: Higher sampling frequency moves the Nyquist limit upwards, reducing aliasing and allowing a more accurate reconstruction of the original analog signal. Quantization errors are related to analog-to-digital bit depth, not sampling rate. Phase distortion may occur due to filter characteristics, not directly from increased sampling. High-frequency noise is not automatically amplified unless additional system properties introduce it. Thus, improved reconstruction and reduced aliasing is the correct effect.