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Thus, we would expect that a single die would yield a particular outcome, on the average, of 1 time out of 6. Similarly, with many dice we would expect that 1/6 of the dice tossed would yield a particular outcome.

Throughout this text we shall assume that the random processes with which we shall have occasion to deal are ergodic.

Hence, if we should select some sample function, a knowledge of the value of n(t)at time t would be of no assistance in improving our ability to predict the valoe attained by that same sample function at time t+τ.

Such noise is referred to as"white"noise in analogy with the consideration

that white light is a combination of all colors, that is, colors of all

frequencies.

The pulses are of the same form but have random amplitudes and

statistically independent random times of occurrence.

Shannon's theorem says that it is possible, in principle,to devise a means whereby a communications system will transmit information an arbitrarily small probability of error provided that the information rate R is less than or equal to a rate C called the channel capacity.

This negative theorem states that if the information rate R exceeds a specified value C, the error probability will increase toward unity as M increases, and that also, generally, in this case where RC,increasing the complexity of the coding results in an increase in the probability of error.

The levels have been separated by an interval λσ, where λ is a number presumed large enough to allow recognition of individual levels with an acceptable probability of error,

Assuming an even number of levels, the levels are located at voltages ±λσ/2,±3λσ/2, etc,

Thus for a fixed signal power and in the presence of white gaussian noise the channel capacity approaches an upper limit with increasing bandwidth.

If the SNR is increased to S/N=15 and B decreased to 3 kHz, the channel capacity remains the same. With a 3 kHz bandwidth the noise power will be 3/4 as large as with 4 kHz. Thus the signal power will have to be increased by the factor （3/4×15/7）=1.6.

Quantization amounts to selecting a set of discrete voltage levels (say eight

in the example, 1 volt apart) and rounding off the exact samples to the nearest discrete or quantum level.

Clearly, information is lost in the process of rounding off exact sample

values to the nearest quantum level

Perhaps the easiest to visualize is a nonlinear network acting on the message prior to quantization by a quantizer with uniformly sepatated quantum levels.

However,in practice the equivalent effect is often achieved by direct

quantization with a quantizer having nonuniformly separated quantum levels.

DPCM works well with data that is reasonably continuous and exhibits

small gradual changes such as photographs with smooth tone transitions.

The introduction of redundant bits increases the raw data rate used in the link, hence increases the bandwidth requirement for a fixed source data rate

It is well-known that the use of orthogonal signaling allows the probability of error to become arbitrarily small by expanding the signal set, i. e., by making the number of waveforms M→∞, provided that the SNR per bit exceeds the Shannon limit of SNR≥-1.6 dB.

A channel coder operates on digital message (or source) data by encoding the source information into a code sequence for transmission through the channel

Block codes can be used to improve the performance of a communications system when other means of improvement (such as increasing transmitter power or using a more sophisticated demodulator) are impractical

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