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Sampling Theorem and signal reconstruction

Answer 1

Given that

Sampling Theorem:

A continuous signal or an analog signal can be represented in the digital version in the form of samples. Here, these samples are also called as discrete points. In sampling theorem, the input signal is in an analog form of signal and the second input signal is a sampling signal, which is a pulse train signal and each pulse is equidistance with a period of “Ts”.

The sampling theorem can be defined as the conversion of an analog signal into a discrete form by taking the sampling frequency as twice the input analog signal frequency. Input signal frequency denoted by Fm and sampling signal frequency denoted by Fs.

  Sampling theorem states that “continues form of a time-variant signal can be represented in the discrete form of a signal with help of samples and the sampled (discrete) signal can be recovered to original form when the sampling signal frequency Fs having the greater frequency value than or equal to the input signal frequency Fm.

Signal reconstruction:

  The sampling process produces a discrete time signal from a continuous time signal by examining the value of the continuous time signal at equally spaced points in time. Reconstruction, also known as interpolation, attempts to perform an opposite process that produces a continuous time signal coinciding with the points of the discrete time signal.

  Let F be any sampling method, i.e. a linear map from the Hilbert space of square-integrable functions L^2 to complex space C^n.

In our example, the vector space of sampled signals C^n is n-dimensional complex space. Any proposed inverse R of F (reconstruction formula, in the lingo) would have to map C^n to some subset of L^2. We could choose this subset arbitrarily, but if we're going to want a reconstruction formula R that is also a linear map, then we have to choose an n-dimensional linear subspace L^2.