Question

Determine the minimum sampling frequency necessary for the signal x(t)=4.0+2.0 cos 150πt+1.0 cos(450πt+π/2) to be unambiguously represented by the samples. Determine the formula for the discrete-time signal x[n] obtained by sampling x(t) at four times the minimum sampling frequency .

Answer 1

The minimum sampling frequency necessary for the signal x(t) to be unambiguously represented by samples is twice the highest frequency component in the signal. The formula for the discrete-time signal x[n] obtained by sampling x(t) at four times the minimum sampling frequency is 4.0 + 2.0 cos(6nπ/4) + 1.0 cos(18nπ/4 + π/2).

Step-by-step explanation:

The minimum sampling frequency necessary to unambiguously represent the signal x(t) is twice the highest frequency component in the signal. In this case, the highest frequency component is 450π, so the minimum sampling frequency is 900π.

To obtain the discrete-time signal x[n] by sampling x(t) at four times the minimum sampling frequency, we multiply the continuous-time signal by the sampling interval, which is equal to 1/4 times the minimum sampling frequency. Therefore, the formula for the discrete-time signal x[n] is 4.0 + 2.0 cos(6nπ/4) + 1.0 cos(18nπ/4 + π/2), where n is an integer representing the sample number.