Final answer:
To generate a sampled signal, the original signal needs to be sampled at a given frequency. The period of the sampled signal in samples is 200/π ≈ 63.66 samples and in seconds is approximately 1.273 seconds.
Step-by-step explanation:
To generate a sampled signal from a periodic signal, we need to sample it with a certain frequency. In this case, the sampling frequency is given as 50 Hz. To generate the sampled signal x₁[n], we can sample the original signal x₁(t) at a rate of 50 samples per second. We can use the formula x₁[n] = x₁(t) at intervals of 1/fₛ or every 1/50s. Since the period of the original signal is 1/f, we can find the period of the sampled signal by counting the number of samples in one period.
For the given signal x₁(t) = 3cos(4πt), the period of the original signal can be found as T = 1/f, where f is the frequency. In this case, f = 4π. Therefore, the period T is 1/(4π). To find the period of the sampled signal in samples, we divide the period T by the sampling interval or time step size, which is 1/50s. So the period of the sampled signal in samples is (1/(4π))/(1/50) = 200/π ≈ 63.66 samples. In seconds, the period of the sampled signal is 63.66/50 ≈ 1.273 seconds.
To plot the sampled signal for a time duration of 1.2 seconds, we can generate enough samples for the duration and then plot the samples against time. Since the sampling frequency is 50 Hz, we need 50*1.2 = 60 samples to cover a duration of 1.2 seconds. We can then plot the samples against time.