Question

When the cosine waveform x(t) = 3 cos(500π) is sarmpled at a rate fs, we obtain a discrete-time signal x[n] given by x[n] = x (n/f,) = 3 cos(800π n/f.) for-oo < n < oo. In the following parts, assume that f, = 3600 Hz.

Determine how many samples are taken in one period of the cosine wave x(t). This answer is the average number of samples per period, which is an integer in this case.

Answer 1

The number of samples taken in one period of the cosine wave x(t) = 3 cos(500πt) with a sampling rate of 3600 Hz is 14 samples, as the period of the wave is 1/250 seconds and the sampling rate multiplies to give 14.4, which is rounded to the nearest integer.

Step-by-step explanation:

The student is asking how to determine the number of samples taken in one period of a cosine wave when sampled at a certain rate. Given the continuous-time cosine waveform x(t) = 3 cos(500πt), and the discrete-time signal resulting from sampling this waveform at a rate fs = 3600 Hz, x[n] = 3 cos(≈800πn/fs), the goal is to find the average number of samples per period, which will be an integer value.

The period of the continuous-time signal is the inverse of the frequency, given by T = 1/f, where f is the frequency of the cosine wave in hertz (Hz). The frequency of the given waveform x(t) is 500π radians per second, which equates to 250 Hz (since 500π rad/s is 500π/2π = 250 cycles per second).

The period T is therefore 1/250 seconds. As the sampling rate is 3600 samples per second, the number of samples taken in one period of the cosine wave is fs × T, which is 3600 × 1/250 = 14.4. Since we need an integer number of samples, we'll round to 14 samples per period.