Sampling Theorem Quiz: Nyquist, Aliasing, and Reconstruction Quiz

Explore core principles of the sampling theorem, including the Nyquist rate, effects of aliasing, and methods for reconstructing signals from samples. This quiz is designed to enhance understanding of digital signal processing fundamentals and real-world sampling scenarios.

1. Nyquist Rate Definition

   What is the Nyquist rate for a continuous-time signal with a highest frequency component of 7 kHz?

   1. 21 kHz
   2. 7 kHz
   3. 14 kHz
   4. 3.5 kHz

   Explanation: The Nyquist rate is twice the highest frequency present in the signal, so for a 7 kHz component it is 14 kHz. Sampling at 3.5 kHz or 7 kHz would result in aliasing because the sampling rate would be insufficient. The value 21 kHz exceeds the requirement but isn't the minimum needed. Only 14 kHz precisely satisfies the Nyquist theorem for this signal.

2. Consequences of Undersampling

   If a signal is sampled below its Nyquist rate, which phenomenon is most likely to occur?

   1. Aliasing
   2. Overshoot
   3. Quantization error
   4. Clipping

   Explanation: Aliasing occurs when a signal is sampled below its Nyquist rate, resulting in distortion as higher frequencies appear as lower frequencies in the reconstructed signal. Overshoot generally relates to system response, not sampling rate. Quantization error is associated with signal amplitude resolution rather than frequency. Clipping refers to amplitude being limited, not sampling frequency.

3. Identifying Aliased Signals

   A sinusoidal signal of 8 kHz is sampled at 10 kHz. Which frequency will appear in the reconstructed signal due to aliasing?

   1. 8 kHz
   2. 10 kHz
   3. 2 kHz
   4. 18 kHz

   Explanation: Sampling below the Nyquist rate causes higher frequencies to map to lower ones; 8 kHz sampled at 10 kHz aliases to |10-8| = 2 kHz. 8 kHz is the original, not the aliased frequency. 10 kHz is the sampling rate and does not manifest as the signal frequency. 18 kHz is unrelated, as aliasing cannot create a frequency higher than the original or the sampling rate.

4. Reconstruction Filter Purpose

   What is the main purpose of the low-pass reconstruction filter in the process of digital-to-analog conversion?

   1. To amplify noise in the signal
   2. To increase the sampling rate
   3. To introduce phase distortion
   4. To remove high-frequency artifacts resulting from the sampling process

   Explanation: The main role of a reconstruction (low-pass) filter is to eliminate high-frequency artifacts, often called 'images,' that arise during sampling. Amplifying noise is not its function, and doing so would degrade signal quality. Increasing the sampling rate is handled by resampling or interpolation, not by filtering. Introducing phase distortion is undesirable and not an objective of reconstruction filtering.

5. Reconstructing a Bandlimited Signal

   Given ideal sampling at exactly twice the highest frequency of a bandlimited signal, what is true about perfect reconstruction?

   1. Perfect reconstruction is theoretically possible if the sampling theorem conditions are met.
   2. Perfect reconstruction cannot occur regardless of sampling conditions.
   3. Perfect reconstruction only happens for signals with no frequency components.
   4. Perfect reconstruction requires sampling below the Nyquist rate.

   Explanation: According to the sampling theorem, perfect reconstruction of a bandlimited signal is possible if the signal is sampled at or above the Nyquist rate. Suggesting that it cannot occur conflicts with the theorem's foundation. Sampling below the Nyquist rate causes aliasing and loss of information. The idea that only constant (zero-frequency) signals can be perfectly reconstructed is incorrect.