Proving DSP Sampling Theorem [duplicate]

1. How do you prove the sampling theorem?
2. What is sampling theorem in DSP?
3. How do you reconstruct a signal from its samples?
4. How do you prevent aliasing in sampling?

How do you prove the sampling theorem?

Proof of Sampling Theorem. To prove the sampling theorem, we need to show that a signal whose spectrum is band-limited to f_m Hz, can be reconstructed exactly without any error from its samples taken uniformly at a rate f_s > 2 f_m Hz. Let us consider a continuous time signal x(t) whose spectrum is band-limited to f_m Hz.

What is sampling theorem in DSP?

The sampling theorem specifies the minimum-sampling rate at which a continuous-time signal needs to be uniformly sampled so that the original signal can be completely recovered or reconstructed by these samples alone.

How do you reconstruct a signal from its samples?

The reconstruction process consists of replacing each sample by a sinc function, centered at the time of the sample and scaled by the sample value x(nT) times 2f_c/ f_s and adding all the functions so created. Suppose the signal is sampled at exactly Nyquist rate f_s= 2f_m, Then f_m= f_s/2 = f_s- f_m and F_m= 1/2 = 1- F_m.

How do you prevent aliasing in sampling?

The solution to prevent aliasing is to band limit the input signals—limiting all input signal components below one half of the analog to digital converter's (ADC's) sampling frequency. Band limiting is accomplished by using analog low-pass filters that are called anti-aliasing filters.