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Ans: The sampling theorem is a fundamental bridge between continuous-time signals "analog signals" and discrete-time signals "digital signals". It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth.

Sampling is a process of converting a signal (for example, a function of continuous time and/or space) into a numeric sequence (a function of discrete time and/or space).

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In the field of digital signal processing, the sampling theorem is a fundamental bridge between continuous-time signals (often called "analog signals") and discrete-time signals (often called "digital signals").

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to get back the original signal after transmission of signal from carrier signal.its frequency should be greater than or equal totwice of the particular signal frequency

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The sampling theorem states that for a limited bandwidth (band-limited) signal with maximum frequency fmax, the equally spaced sampling frequency fs must be greater than twice of the maximum frequency fmax, i.e.,

fs > 2·fmax

in order to have the signal be uniquely reconstructed without aliasing.

The frequency 2·fmax is called the Nyquist sampling rate. Half of this value, fmax, is sometimes called the Nyquist frequency.

   The sampling theorem is considered to have been articulated by Nyquist in 1928 and mathematically proven by Shannon in 1949. Some books use the term "Nyquist Sampling Theorem", and others use "Shannon Sampling Theorem". They are in fact the same sampling theorem.

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Converting the continues signal into discrete samples without loosing the data or modifications in it.

This process is called sampling theorem.

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A bandlimited signal can be reconstructed exactly if it is sampled at a rate atleast twice the maximum frequency component in it.”

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The sampled signal should have frequency twice that of the original signal.Otherwise signal cannot be recovered

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The Sampling theorem states that " If the sampling rate in any pulse modulation system exceeds twice the maximum signal frequency, the original can be reconstructed in the receiver with minimal distortion."

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Sampling Theorem says “A bandlimited signal can be reconstructed exactly if it is sampled at a rate atleast twice the maximum frequency component in it.” The sampling theorem is a fundamental bridge between continuous-time signals (often called "analog signals") and discrete-time signals (often called "digital signals"). It establishes a sufficient condition for a sample rate that permits a discrete sequence of samples to capture all the information from a continuous-time signal of finite bandwidth.

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The signals we use in the real world, such as our voices, are called "analog" signals.  To process these signals in computers, we need to convert the signals to "digital" form.  While an analog signal is continuous in both time and amplitude, a digital signal is discrete in both time and amplitude.  To convert a signal from continuous time to discrete time, a process called sampling is used.  The value of the signal is measured at certain intervals in time. Each measurement is referred to as a sample.  (The analog signal is also quantized in amplitude, but that process is ignored in this demonstration.  See the Analog to Digital Conversion page for more on that.)  When the continuous analog signal is sampled at a frequency F, the resulting discrete signal has more frequency components than did the analog signal.  To be precise, the frequency components of the analog signal are repeated at the sample rate.  That is, in the discrete frequency response they are seen at their original position, and are also seen centered around +/- F, and around +/- 2F, etc.  How many samples are necessary to ensure we are preserving the information contained in the signal?  If the signal contains high frequency components, we will need to sample at a higher rate to avoid losing information that is in the signal.  In general, to preserve the full information in the signal, it is necessary to sample at twice the maximum frequency of the signal.  This is known as the Nyquist rate.  The Sampling Theorem states that a signal can be exactly reproduced if it is sampled at a frequency F, where F is greater than twice the maximum frequency in the signal. 

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Sampling Theorem: “A bandlimited signal can be reconstructed exactly if it is sampled at a rate atleast twice the maximum frequency component in it.”

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