Question

The sequence x[n]=(−1)ⁿ is obtained by sampling the continuous-time sinusoidal signalx(t)=cosω₀t at 1-ms intervals, i.e., cos(ω₀nT)=(−1),T=10⁻³ s Determine three distinct possible values of ω₀

Answer 1

The question is about finding three distinct possible angular frequencies (ω0) that would result in a sampled sequence alternating between 1 and -1 for a cosine wave sample at 1-ms intervals. The values of ω0 are determined using the formula ω0 = πm/T, where m is an odd integer and T is the sampling interval.

Step-by-step explanation:

The student is asking about determining possible values of the angular frequency ω0 for a cosine function representing a sinusoidal wave sample at specific time intervals. The sequence x[n]=(−1)n suggests that the cosine function evaluated at discrete times is alternating between 1 and -1, which are the peak values for a cosine function. Therefore, we are looking for the values of ω0 that will make the cosine function alternate between these values at the given sampling rate of 1 ms (T=10−3 s).

Let's consider a single period of the cosine wave. We know when cos(ω0t) is −1 or 1, the angle ω0t must be an integral multiple of π for it to reach the peak values. Because the sequence is sampled every 1 ms, we need to find the frequencies at which the sample points correspond to these integral multiples of π at the 1-ms intervals. The general form of the frequency at these intervals can be given by ω0 = πm/T:

1. ω0 = π×1/(10−3) rad/s
2. ω0 = π×2/(10−3) rad/s
3. ω0 = π×3/(10−3) rad/s

Note that m must be an odd integer for the sample points to alternate between 1 and -1. Thus, the values mentioned above are the first three distinct odd multiples of π divided by the sampling interval, resulting in three distinct possible values of ω0.