Binary Data Transmission

  
Figure: Basic Binary Digital Communication System

The basic system is shown in Figure . The source produces a sequence of binary digi ts or bits every T seconds which are represented as a logical 0 or a logical 1. The modulator produces an analog waveform which is defined as

The modulator ``maps'' logical levels 0 and 1 to analog waveforms which have a duration T seconds. The transmission channel has two effects on the transmitted signal: is attenuated and white Gaussian noise is added. These channel effects are modeled by an attenuation by followed by the addition of which is a zero mean Gaussian random process with power spectral density

Thus the received signal is

Unlike demodulators for carriers modulated by continuous analog signals such as AM of FM, the demodulator for digitally modulated carriers do not have to reconstruct the modulating message signal. The goal of the demodulator is to determine whether a logical 0 or a logical 1 was originally transmitted. Clearly, any demodulator should do this making as few mistakes as possible. When the demodulator makes a mistake, the event is termed a bit error. The probability that the demodulator makes a mistake is termed the probability of bit error which is denoted . The optimal demodulator is the demodulator which minimizes the probability of bit error. The structure of such a demodulator is shown in Figure . The optimal demodulator consists of a demodulator filter, a sampler, and a threshold comparitor. The output of the demodulator filter is given by

The demodulator filter output is sampled every T seconds by the sampler. The sample times are synchronized with the bit intervals. (Note that this requires that the beginning and ending time of each pulse be known at the receiver. This is accomplished using a symbol synchronizer.) The sample outputs (for ) are compared to a threshold k. The demodulator makes its decision based on this comparison. The demodulator output is

Since is a Gaussian random process, is a Gaussian random variable with mean and variance

Thus the sampler output

(for i=0,1) is also a Gaussian random variable with mean and variance

The transfer function of the demodulator filter and the value of the threshold k are selected to minimize the probability of bit error. These choices constitute the optimum receiver.

Optimum Receiver

The probability of bit error is determined using Baye's Theorem:

Without loss of generality, assume

H3>Determine k

The threshold value k is determined by taking the derivative of the above equation, setting it equal to zero, and solving for k. (Consult your favorite calculus text).

For the extremely important special case where

the threshold value k which minimizes the probability of error is

For this special case

so that

Determine

Since the function erfc(x) decreases as x increases, the probability of error is minimized by maximizing the argument of the erfc() function. Thus the filter should be chosen to maximize

or what is equivalent, to maximize

The numerator may be expressed in terms of using the inverse Fouier Transform:

while the denominator is given by

Thus, maximizing () is equivalent to maximizing

Equation () may be bounded using the Schwarz inequality (see page 464 of the text):

where equality is achieved only when

Thus () defines the optimum filter . The impulse response of this filter is

The filter is ``matched'' to the input signals and and is called a matched filter. The receiver structure is shown in Figure 7.9 on page 466 of the text. Using the matched filter, the probability of error for equally likely symbols is

where is the correlation coefficient defined as (see equation 7.55 on page 467 of the text):

Correlation Receiver

The optimum receiver consists of a matched filter whose output sampled at the end of each symbol interval. Consider the output of one branch of the match filter due to an arbitrary signal :

The last line represents an alternate realization of the matched filter reciever called a correlation receiver since () is the correlation between signals and . The correlation receiver is shown in Figure .