Final answer:
To construct the piecewise signal x(t) with a sampling frequency of 20 Hz and a duration of 30 seconds, the continuous signal must be sampled at a rate of 20 samples per second and evaluated according to its definition in the specified intervals.
Step-by-step explanation:
To construct the signal x(t) sampled at a sampling frequency of fs=20 Hz with a duration of x(t) is 30 s, we will have to define x(t) in its respective intervals and then sample the continuous signal at the specified rate.
The signal x(t) is piecewise defined as follows:
5t - 25, for 0 ≤ t < 10
10 cos(4t), for 10 ≤ t < 15
10e−t/5, for 15 ≤ t ≤ 30
Sampling the signal means measuring its value at discrete time intervals. Since the sampling frequency is 20 Hz, we take a sample every 1/20th of a second, or every 0.05 seconds. To sample this signal, we evaluate x(t) at t = 0, 0.05, 0.10, ..., up to t = 30 seconds, according to its definition in each interval.
For a practical implementation, one would use a programming language or software with numerical capabilities, such as MATLAB or Python, to generate the discrete samples based on the piecewise function. In MATLAB, for example, one could use the 'linspace' function to create a time vector and then define x(t) using element-wise operations.