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Nyquist Shannon Theorem | Signal to Noise Ratio - Calculation & Formula

In 1924, working at Bell Labs, Henry Nyquist, realized that the transmission capacity of a perfect channel is also finite. Nyquist discovered that it was not necessary to capture the entire analog signal to find the original signal, samples of the signal could be taken at various points to recover the original signal.

Sampling is the process of converting a signal (for example, a function of continuous time or space) into a numeric sequence. He derived an equation expressing the maximum data rate for a finite bandwidth noiseless channel.

Nyquist theorem

States that if an arbitrary signal is passed through a low-pass filter (which allows only low frequencies to pass) of bandwidth B, the filtered signal can be completely reconstructed by making only 2B (exact) samples per second. If the signal consists of L discrete levels, Nyquist's theorem states:

maximum data rate = 2 x H (log2) x L

where the max data rate is represented in bits/sec.

Example on Nyquist Signal Noise Ratio:

A noiseless 3-kHz channel cannot transmit binary (i.e. L=2) signals at a rate exceeding 6000 bps. Note that the signal is passed through the low-pass filter in order to avoid Aliasing. In 1948, Shannon carried Nyquist's work further and wanted to derive an equation for the case of a noisy channel and he was successful. The amount of noise in the channel /medium is measured by a ratio know as signal to noise ratio in which signal's power and noise present is taken into consideration. The Signal's power is represented by "S" whereas noise power is represented by "N". So, the ratio is represented by S/N. If we represent the ratio by SNR, then the value of signal to noise ratio is taken in decibels i.e SNR= 10(log10)S/N. For example SNR with S/N of 100 is 20db.

Shannon theorem

States that the maximum data rate of a noisy channel with bandwidth 'B' Hz and having Signal to Noise ratio SNR can be found from the following equation:

Calculating signal to noise ratio formula

maximum data rate = H x (log2) x (1 + S/N)

Example:

A channel of 3000-Hz bandwidth with a signal to noise ratio of 30 dB (typical parameters of the analog part of the telephone system) can never transmit much more than 30,000 bps. Shannon's result applies to any channel subject to thermal noise.

Nyquist Shannon Theorem, Nyquist-Shannon Sampling theorem

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