Question

(10pt) An analog voltage signal is given by V(t) = 3 + cos(2πt) + 4cos(4πt) + 2cos(6πt). Suppose the given analog signal is recorded in a digital form through a digital data acquisition system. The system samples the signal discretely at a sampling frequency of 10Hz. Is the sampling frequency appropriate to avoid aliasing issue? Explain why the selected sampling frequency will (or will not) result in aliasing problem.

Answer 1

The sampling frequency of 10 Hz is appropriate for the given analog signal, as it exceeds twice the highest frequency component (3 Hz) present in the signal, thus avoiding the aliasing issue according to the Nyquist theorem.

Step-by-step explanation:

To determine if the sampling frequency is appropriate and to avoid the aliasing issue, we must refer to the Nyquist theorem. According to the Nyquist theorem, the sampling frequency must be at least twice the highest frequency present in the analog signal to prevent aliasing. The analog voltage signal V(t) = 3 + cos(2πt) + 4cos(4πt) + 2cos(6πt) consists of three cosines with the highest frequency being 3 Hz (from the term 2cos(6πt)). Therefore, the minimum sampling frequency to avoid aliasing should be at least 2 × 3 Hz, which is 6 Hz. The provided sampling frequency of 10 Hz is higher than this required minimum, so it will prevent aliasing, ensuring the digital representation accurately reflects the analog signal.