Nyquist's Signal Sampling Theory

In the late 1920s, the only technology to preserve musical recordings was to copy sound waves in wax. Harry Nyquist, an AT&T scientist, thought there was a better way. He wrote a landmark paper (Nyquist, Harry, "Certain topics in Telegraph Transmission Theory," published in 1928) describing the criteria for what we know today as sampled data systems. Nyquist taught us that for periodic functions, if you sampled at a rate that was at least twice as fast as the signal of interest, then no information (data) would be lost upon reconstruction. And since Fourier had already shown that all alternating signals are made up of nothing more than a sum of harmonically related sine and cosine waves2, then audio signals are periodic functions and can be sampled without lost of information following Nyquist's instructions. This became known as the Nyquist frequency, which is the highest frequency that may be accurately sampled, and is one-half of the sampling frequency.

Harry Nyquist (1920's) showed that to distinguish unambiguously between all signal frequency components we must sample at least twice the frequency of the highest frequency component, Figure 1.

Figure 1: In the diagram, the high frequency signal is sampled twice every cycle. If we draw a smooth connecting line between the samples, the resulting curve looks like the original signal. This avoids aliasing3. The highest signal frequency allowed for a given sample rate is called the Nyquist frequency.

Harry Nyquist thought of a way to take an analog signal (such as voice) and code it (just like with the Morse code) using ones (1) and zeros (0). For this, he invented something called a "CODEC" or coder-decoder. This thing that today is the size of a fingernail (a microchip) measures the input analog signal, codes the result of the measurement and sends this code down the telephone lines and trunks. It does so often enough so its peer at the other end of the line can reconstruct the voice signal almost as good as it was at the calling side. N. Erd calls the measuring of the signal "sampling." Good old Harry Nyquist also recommended that the number of samples per second for a good representation of the signal has to be twice as big as the number of Hertz of the fastest sine wave contained in the analog signal. Since the telephone only allows 4 kHz through the phone line, sampling for voice is done 8000 times per second.

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