Answer:
To determine the Nyquist rate of the given signals, we need to consider the highest frequency component present in each signal. The Nyquist rate is defined as twice the highest frequency component of the signal.
(a) For signal x1(t) = 4sinc^3(4200πt):
The function sinc^3(4200πt) has a bandwidth of 1/2π, which means its highest frequency component is 1/2π. Therefore, the Nyquist rate for x1(t) is 2 × 1/2π = 1/π.
(b) For signal x2(t) = 5sinc^6(4200πt):
The function sinc^6(4200πt) has a bandwidth of 1/2π, which means its highest frequency component is 1/2π. Therefore, the Nyquist rate for x2(t) is 2 × 1/2π = 1/π.
(c) For signal x3(t) = x1(t) + 2x2(t):
Since x1(t) and x2(t) both have the same Nyquist rate of 1/π, their sum, x3(t), will also have the same Nyquist rate of 1/π.
(d) For signal x4(t) = x1(t)⋅x2(t):
The Nyquist rate of the multiplication of two signals is equal to the sum of their individual Nyquist rates. Therefore, the Nyquist rate for x4(t) is 1/π + 1/π = 2/π.
In summary:
(a) Nyquist rate for x1(t) = 1/π.
(b) Nyquist rate for x2(t) = 1/π.
(c) Nyquist rate for x3(t) = 1/π.
(d) Nyquist rate for x4(t) = 2/π.