Sampling theorem and proof  Statement:  A continuous time signal can be represented in its samples and can be recovered back when sampling frequency f s  is greater than or equal to the twice the highest frequency component of message signal. i. e. f s ≥ 2 f m . f s ≥ 2 f m . Proof:  Consider a continuous time signal x(t). The spectrum of x(t) is a band limited to f m  Hz i.e. the spectrum of x(t) is zero for |ω|>ω m . Sampling of input signal x(t) can be obtained by multiplying x(t) with an impulse train δ(t) of period T s . The output of multiplier is a discrete signal called sampled signal which is represented with y(t) in the following diagrams: Here, you can observe that the sampled signal takes the period of impulse. The process of sampling can be explained by the following mathematical expression:        Sampled signal y ( t ) = x ( t ) . δ ( t ) . . . . . . ( 1 ) The trigonometric Fourier series representation of  δ δ (t) is give