The Nyquist rate for the signal x(t) = 2 cos (2000 πt) cos (5000 πt), is:

Concept:

Nyquist rate is the minimum rate at which a signal can be sampled without introducing errors, which is twice the highest frequency present in the signal.

Nyquist rate = 2 × highest frequency present in the signal

fs = 2fm

Calculation:

\(x\left( t \right) = 2~cos \left( {2000\;\pi t} \right)\cos \left( {5000\;\pi t} \right)\)

\( = \left[ {\cos \left( {3000\;\pi t} \right)+\cos \left( {7000\;\pi t} \right)} \right]\)

ω1 = 3000 π ⇒ f1 = 1500 Hz

ω2 = 7000 π ⇒ f2 = 3500 Hz

fm = max (f1, f2) = 3500 Hz

Nyquist rate, fs = 2fm = 2 × 3500 = 7 kHz