Bandwidth-limited signals
The transmission of the ASCII character `b' is encoded in an 8-bit byte. The bit pattern that is to be transmitted is 01100010. Figure 2.3 shows the voltage output of the transmitting computer.

The Fourier analysis of this signal yields the coefficients:

and
The root-mean-square amplitudes,or the first few terms are shown on
the right-hand side of Figure 2.3. These values are of interest because their squares are proportional to the energy transmitted at the corresponding frequency.
No transmission facility can transmit signals without losing some power in the process. If all the Fourier components were equally diminished, the resulting signal would be reduced in amplitude but not distorted.

Now let us consider how the signal of Figure 2.3 would look if the bandwidth were ~ low that only the lowest frequencies were transmitted. Figure 2.4 shows the signal that results from a channel that allows only the first harmonic to pass through.

The time T required to transmit the character depends on both the encoding method and the signaling speed (the number of times per second that the signal changes its value, say, its voltage).
The number of changes per second is measured in BAUD. In the example, Os and is are being used as signal levels, so the bit rate is equal to the baud rate. The bit rate is used to describe a medium's capacity and is measured in bits per second (bps).

The maximum data rate of a channel

As early as 1924, H. Nyquist derived an equation expressing the maximum data rate for a finite bandwidth noiseless channel.
Nyquist proved that if an arbitrary signal has been seen through a low-pass filter of bandwidth H, the filter signal can be completely reconstructed by making only 2H (exact) samples per second. Sampling the line faster than 2H times per second is pointless because the higher frequency components that such sampling could recover have already been filtered out. If the signal consists of V discrete levels, Nyquist's theorem states:

For example, a noiseless 3-kHz channel cannot transmit binary (i.e. two-level) signals at the rate exceeding 6,000 bps.
So far we have considered only noiseless channels. If random noise is present, the situation deteriorates rapidly. The amount of thermal noise present is measured by the ratio of the signal power to the noise power and is called the signal to noise ratio.
If we denote the signal power by S and the noise power by N, the signal to noise ratio is SIN. Usually, the ratio itself is not quoted; instead, the quantity 10 log10 SIN is given. The units used are called decibels (dB). Thus, an SIN ratio of 10 corresponds to 10 dB, 100 corresponds to 20 dB, 1,000 corresponds to 30 dB, and so on.
Shannon's major result is that the maximum data rate of noisy channel whose bandwidth is Hz, and whose signal-to-noise ratio is SIN is given by,
Figure 2.4 Signal for first harmonic.

For example, a channel of 3,000 Hz bandwidth, and a signal to thermal noise ratio of 30 dB (typical parameters of the analog part of the telephone systems) can never transmit much more than 30,000 bps, no matter how many or few signal levels are used; no matter how often or not samples are taken.

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