As per Nyquist theorem, to reproduce the original analog signal one necessary condition is that the sampling rate be at least _______ the hightest frequency present in the original signal.

Question

Accepted Answer

The correct option is (1):
The **Nyquist-Shannon Sampling Theorem** is a foundational principle in digital signal processing that defines the minimum sampling rate required to convert an analog signal into a discrete signal such that the original continuous signal can be perfectly reconstructed. This critical minimum rate is known as the Nyquist rate. The theorem explicitly states that the sampling rate, $f\_s$, must be at least **twice** the highest frequency component, $f\_{max}$, present in the original analog signal ($f\_s \geq 2 f\_{max}$). By sampling at this rate, you ensure that the frequency components of the sampled signal do not overlap, which prevents an irreversible distortion known as aliasing, thereby guaranteeing the ability to accurately reproduce the original waveform.

The incorrect option is (2):
The option **equal to** ($f\_s = f\_{max}$) is insufficient because it does not provide the minimum margin required by the theorem. If the sampling rate were merely equal to the highest frequency, the system would only capture a single sample per cycle of the highest frequency component. More importantly, in the frequency domain, the spectrum of the sampled signal would be touching the copies of the original signal's spectrum, which means that perfect separation and reconstruction would only be theoretically possible using an unrealizable ideal "brick-wall" anti-aliasing filter. Therefore, for practical, distortion-free reconstruction, the sampling rate must be *greater than* twice the highest frequency, or at least twice.

The incorrect option is (3):
Choosing **half of** the highest frequency is a clear case of *undersampling*. If the sampling rate is $f\_s = 0.5 f\_{max}$, it means the system is taking fewer than two samples per cycle of the highest frequency component. Sampling below the Nyquist rate violates the theorem's necessary condition and inevitably leads to severe aliasing. In this scenario, the high-frequency components of the original signal are misrepresented as lower-frequency components in the digital signal, permanently corrupting the information and making it impossible to reconstruct the original analog signal accurately.

The incorrect option is (4):
While a sampling rate of **four times** the highest frequency ($f\_s = 4 f\_{max}$) would certainly be a sufficient condition for reconstruction, it is not the *minimum necessary condition* that the question asks for ("at least"). Engineers often use a sampling rate much higher than the Nyquist rate in practical applications to allow for the use of real-world, non-ideal anti-aliasing filters and to provide a safety margin. However, the theoretical lower bound established by the Nyquist theorem for perfect reconstruction is simply **twice** the highest frequency.