Final answer:
The largest sample rate to avoid aliasing is at least twice the maximum frequency, which is 26 radians per second. To cause aliasing such that both frequencies appear the same, the sample rate must be below 8 radians per second, as twice the lower frequency is 8 radians per second.
Step-by-step explanation:
The student's question pertains to the sampling of two sinusoidal signals and finding a sample rate that prevents aliasing while also finding one that causes the two signals to appear the same when sampled. In order to solve this problem, the concept of the Nyquist rate must be employed, which states that the sampling rate must be at least twice the highest frequency present in the signal to avoid aliasing. The student has provided two frequency values, 4 and 13, which represent the frequencies of the sinusoidal signals. The highest frequency 13 radians per second dictates that the Nyquist rate would need to be at least 26 radians per second to avoid aliasing. However, if the sample rate is chosen to be less than this, specifically at a rate that is a common multiple of both frequencies, the two signals could appear identical when sampled due to aliasing. For instance, if we want to find a sampling rate that causes both frequencies to alias to the same frequency, we must look for a rate that when divided into the original frequencies results in the same integer quotient or a harmonic. The largest sampling rate that causes aliasing to occur could be found by finding the greatest common divisor (GCD) of the two frequencies, but since the question is about a non-zero sample rate greater than 0 that achieves aliasing, we need a sample rate lower than twice the lowest frequency provided, thus the sample rate could be any value below 8 radians per second (since twice the lowest frequency is 8 radians per second).