{"id":3731,"date":"2021-08-03T10:14:01","date_gmt":"2021-08-03T04:44:01","guid":{"rendered":"https:\/\/www.goseeko.com\/blog\/?p=3731"},"modified":"2025-12-06T15:18:28","modified_gmt":"2025-12-06T09:48:28","slug":"what-is-the-sampling-theorem","status":"publish","type":"post","link":"https:\/\/www.goseeko.com\/blog\/what-is-the-sampling-theorem\/","title":{"rendered":"What is the Sampling Theorem?"},"content":{"rendered":"\n

Before understanding sampling theorem we first need to understand sampling. Sampling usually refers to conversion of continuous signal into short duration of pulses each pulse followed by a skip period when no signal is available. The uniformly sampled signal is below.<\/p>\n\n\n\n

Following are two popular sampling operations:<\/p>\n\n\n\n

 1. Single rate or periodic sampling<\/p>\n\n\n\n

 2. Multi-rate sampling<\/p>\n\n\n\n

Figure below shows the structure and operation of a finite pulse width sampler, where (a) represents the basic block diagram and (b) illustrates the function of the same. T is the sampling period and p is the sample duration.<\/p>\n\n\n\n

<\/strong><\/p>\n\n\n\n

<\/strong><\/p>\n\n\n\n

Figure (a): Basic block diagram<\/p>\n\n\n\n

<\/p>\n\n\n\n

 Figure (b): Sampler output<\/p>\n\n\n\n

Finite pulse width sampler converts a continuous time signal into a pulse modulated or discrete signal. The most common type of modulation in the sampling and hold operation is the pulse amplitude modulation.<\/p>\n\n\n\n

The block diagram of sampler is below, having a pulse train of p seconds and sampling period of T seconds.<\/p>\n\n\n\n

p(t)= unit pulse train with period T<\/p>\n\n\n\n

<\/p>\n\n\n\n

Us<\/sub>(t)=unit step function<\/p>\n\n\n\n

<\/p>\n\n\n\n

In frequency domain we write p(t) as<\/p>\n\n\n\n

<\/p>\n\n\n\n

p(t)=1 for 0\u2264 t \u2264p<\/p>\n\n\n\n

The output of the ideal sampler is written as<\/p>\n\n\n\n

<\/p>\n\n\n\n

The approximation of the output sampler will now be <\/p>\n\n\n\n

<\/p>\n\n\n\n

Shannon\u2019s Sampling theorem<\/strong><\/h2>\n\n\n\n

In sampling the signal m(t) is multiplied with periodic pulse trains. Let M(\u03c9) the spectrum of the input signal be band limited with the maximum frequency of fm<\/sub> as shown in figure.<\/p>\n\n\n\n

<\/strong><\/p>\n\n\n\n

Input Signal m(t)<\/p>\n\n\n\n

<\/p>\n\n\n\n

Figure 2((fs<\/sub>>2 fm<\/sub>)<\/p>\n\n\n\n

<\/strong><\/p>\n\n\n\n

Figure: 3 (fs<\/sub><2 fm<\/sub>)<\/p>\n\n\n\n

The frequency spectrum of this signal when we sample the impulse is in figure 2 (for fs<\/sub>>2 fm<\/sub>). In figure 3 for (fs<\/sub><2 fm<\/sub>). From figure 2 and figure 3 we can see that as long as fs<\/sub>\u22652fm<\/sub> the original signal is present in the sampled signal and can be extracted from it by the low pass filter. This is known as Shannon\u2019s Sampling theorem. This theorem states that the information present in a signal is fully preserved in the sampled form as long as the sampling frequency is at least twice the maximum frequency contained in the signal.<\/p>\n\n\n\n

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