A continuous signal is sampled at regular intervals, every 10 ms. What is the maximum frequency that can be captured by this sampling process without incurring any distortion of the original signal? If the original signal contains noise at a frequency of 95 Hz, would that noise be accurately represented by the sampling process? If not, at what frequency would that noise be aliased?

Question

A continuous signal is sampled at regular intervals, every 10 ms. What is the maximum frequency that can be captured by this sampling process without incurring any distortion of the original signal? If the original signal contains noise at a frequency of 95 Hz, would that noise be accurately represented by the sampling process? If not, at what frequency would that noise be aliased?

Answer 1

To determine the maximum frequency that can be accurately captured by the given sampling process, we can use the Nyquist-Shannon sampling theorem. According to this theorem, the sampling rate must be at least twice the highest frequency component of the original continuous signal in order to avoid distortion during the sampling process.

In this case, the sampling interval is 10 ms, which is equivalent to a sampling rate of 1/10 ms = 100 samples per second (or Hz). Therefore, based on the Nyquist-Shannon sampling theorem, the maximum frequency that can be captured without distortion is half of the sampling rate, which is 100 Hz / 2 = 50 Hz.

If the original signal contains noise at a frequency of 95 Hz, it would not be accurately represented by the sampling process. Since the maximum frequency that can be captured without distortion is 50 Hz, the noise at 95 Hz would be aliased. Aliasing occurs when a signal at a frequency higher than the Nyquist frequency is incorrectly reconstructed as a lower frequency during the sampling process.

To determine the frequency at which the noise would be aliased, we can use the concept of folding or aliasing. When a signal is sampled, any frequency component above the Nyquist frequency gets aliased or folded back into the frequency range below the Nyquist frequency.

In this case, the Nyquist frequency is 50 Hz. So, to determine the aliased frequency of the noise at 95 Hz, we need to find the equivalent frequency within the range of 0 to 50 Hz. Since the Nyquist frequency represents the folding point, we can subtract the original frequency from the Nyquist frequency: 50 Hz - 95 Hz = -45 Hz.

Thus, the noise at 95 Hz would be aliased as -45 Hz, or more accurately, as the positive difference between -45 Hz and the Nyquist frequency, which is 50 Hz. Consequently, the noise at 95 Hz would appear in the sampled signal as a 50 Hz noise component, leading to distortion.