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Nyquist's Theorem - Concept

The sampling frequency determines the limit of audio frequencies that can be reproduced digitally. One of the most important rules of sampling is called the Nyquist Theorem, which states that the highest frequency which can be accurately represented is one-half of the sampling rate. So, if we want a full 20 kHz audio bandwidth, we must sample at least twice that fast, i.e. over 40 kHz.

For a practical example of the application of his theorem we can examine the technologies used for digital voice transmission. Voice data for telephony purposes is limited to frequencies less than 4,000 Hz (cycles per second). Because of the requirement for guardbands, the actual frequency that can be utilized over voice grade lines is limited to 3,000 Hz. This limitation has nothing to do with Nyquist's Theorem, but instead is based upon the assumption made by Bell Labs that this frequency rate was good enough to make the voice intelligible (agreed it's not CD quality :).

According to Nyquist, it would take 8,000 samples (2 times 4,000) to capture a 4,000 Hz signal perfectly. That means 8,000 data points must be recorded concerning the amplitude of the analog signal. Now, this does not dictate how precise the information is concerning each sample. But, for the purposes of digital representation (based, of course, on binary arithmetic) one of a 256 range of values can be recorded in each 8 bit sample.

Therefore, if we take a 4K Hz voice grade signal and quantitize it in 256 levels and code each sample using one byte, then it requires 64,000 bits per second to communicate this digitally encoded voice signal in real time (8 bits * 8,000 samples per second = 64K bps) over a circuit.

It is not surprising, therefore, that T-carrier circuits were designed around this requirement, since they are primarily designed to carry voice signals that have been converted from analog to digital signals. The 4K Hz signal is sampled and then transmitted as a time-division multiplexed [TDM] synchronous baseband signal.

For example, look at the DS-1 signal which passes over a T-1 circuit. For DS-1 transmissions, each frame contains 8 bits per channel and there are 24 channels. Also, 1 "framing bit" is required for each of the 24 channel frames.

     24 channels * 8 bits per channel + 1 framing bit = 193 bits per frame. 
     193 bits per frame * 8,000 "Nyquist" samples = 1,544,000 bits per second.
And it just so happens that the T-1 circuit is 1.544 Mbps.--not a coincidence. Each of the 24 channels in a T-1 circuit carries 64Kbps. There are many other places in data communication where Nyquist's theorems can be applied.