Final answer:
The Nyquist rate for the given signal x(t) is calculated by identifying the highest frequency component, which is 150 Hz. According to the Nyquist theorem, the sampling rate must be at least twice this frequency, giving a Nyquist rate of 300 Hz.
Step-by-step explanation:
To calculate the Nyquist rate for sampling a continuous-time signal, you must first identify the highest frequency component in the signal. The given signal is:
x(t) = 5cos(100πt) + 10cos(200πt) - 15cos(300πt)
To find the highest frequency, look at the coefficients of t inside the cosine functions. The given frequencies are 50 Hz, 100 Hz, and 150 Hz, corresponding to 100π, 200π, and 300π, respectively (since π rad/s is equivalent to 50 Hz). The 150 Hz frequency is the highest.
According to the Nyquist theorem, the sampling rate must be at least twice the highest frequency to accurately capture the signal without aliasing. Therefore, the Nyquist rate for the given signal is:
Nyquist rate = 2 × (highest frequency) = 2 × 150 Hz = 300 Hz