Explanation:
To determine the Nyquist sampling rate for the given signal x(t), we need to consider the highest frequency component present in the signal. In this case, the highest frequency component is determined by the sinc(150πt) term.

Sinc function:
The sinc function is defined as sinc(x) = sin(x)/x. It has a main lobe centered at x = 0 and side lobes that extend infinitely on both sides. The width of the main lobe is inversely proportional to the frequency.

Frequency of sinc(150πt):
To determine the frequency of sinc(150πt), we need to find the period of the function. The period of a sinc function is given by T = 2π/B, where B is the width of the main lobe. In this case, B = 150π. Therefore, the period of sinc(150πt) is T = 2π/(150π) = 1/75.

Highest frequency component:
The highest frequency component in the signal x(t) is the reciprocal of the period, which is 75.

Nyquist sampling rate:
According to the Nyquist-Shannon sampling theorem, the sampling rate should be at least twice the highest frequency component in the signal to avoid aliasing. Therefore, the Nyquist sampling rate for the given signal x(t) is 2 * 75 = 150 Hz.

However, it is always recommended to choose a sampling rate higher than the Nyquist rate to ensure accurate reconstruction of the signal. A commonly used rule of thumb is to choose a sampling rate that is at least 2-3 times the highest frequency component. In this case, choosing a sampling rate of 2-3 times 75 Hz would give a range of 150-225 Hz.

In the given answer, the Nyquist sampling rate is rounded up to the nearest integer, which is 300 Hz. This ensures that the sampling rate is higher than the minimum required rate of 150 Hz and provides a sufficient margin for accurate signal reconstruction.

X(t) = sinc (100 πt)+ 2 sinc²(150 πt)