2.2 Binary data transmission (deterministic signals)

where no channel coding is used, or one symbol of binary code in a system with errorcorrecting code and hard decisions, etc. Numbering the messages 0 and 1 and assuming that signals s0(t) and s1(t) (again deterministic!) are used for their transmission, we can represent the minimum distance decision rule (2.3) as:

where placement of the decision symbols points directly to when one or the other of two decisions is made. The same can be rewritten in correlation-based form following from rule (2.8):

^

H1

with correlations zk, k ¼ 0, 1 of each signal and observation y(t) defined by equation (2.7) and Ek ¼ kskk2, k ¼ 0, 1, being the kth signal energy given by (2.4). Optimal rules (2.9) and (2.10) of distinguishing between two signals can be explained geometrically in a very clear way. Two signal vectors s0 and s1 always lie in a signal plane SP. Observation vector y does not necessarily fall onto this plane but the closeness of it to one or the other signal is determined by the closeness to them of the projection y0 of y onto SP (see Figure 2.4a). Therefore, we can divide SP into two half-planes by the straightline bound passing strictly perpendicular to the straight line connecting the signal vectors,

^ ^ 0

and base decisions H0, H1 on y hitting the corresponding half-plane (Figure 2.4b). It is also seen from Figure 2.4b that the probability of mistaking one signal for the other (error probability) depends on the distance between vectors s0 and s1 in comparison with the range of random fluctuations of y0 caused by channel noise. According to (2.10), the actually received signal s0(t) will be erroneously taken for the wrong one s1(t) if and only if the correlation difference is lower than the threshold (E0 E1)/2. Therefore, the probability p01 of such an error is found as:

Figure 2.4 Signal plane and decision half-planes

where Pr (AjB) stands for the conditional probability of event A given that event B occurred, and W(zjs0(t)) is the conditional probability density function (PDF) of correlation difference z in (2.10), given that the signal s0(t) is actually received. One of the remarkable features of the Gaussian process is that any linear transform of it again produces a Gaussian process. Therefore z, as a result of a linear transform of the Gaussian observation y(t) (see (2.7) and (2.10)), has the Gaussian PDF:

integration of which according to (2.11) results in:

where

is the complementary error function.

The expectation of z conditioned in the received signal (the bar above will be used from now on to symbolize expectation) and variance 2 ¼ varfzg of z can be found directly from equations (2.7) and (2.10). When the signal s0(t) is assumed true, i.e.

is called the correlation coefficient of the signals sk(t), sl(t), Ek, El being their energies. As is seen, geometrically 01 is simply the cosine of the angle between the signals s0(t), s1(t) (or the signal vectors s0, s1), and hence characterizes closeness or resemblance of the signals.

To find 2 ¼ varfzg we rely on the fact that it is not affected by a deterministic component of the observation y(t), i.e. in the situation in question the signal s0(t), since the noise is additive. Therefore we can virtually remove the signal from y(t), putting

After this let us calculate the variance of correlation (2.7) of y(t) and some arbitrary signal s(t):

:;

where the squared integral is presented as a double integral with separable variables, order of integration and averaging is changed (expectation of sum is sum of expectations!) and, finally, averaging is applied to the only random cofactor in the integrand.

Recall now, that due to uniformity of the spectrum of white noise over the entire frequency range, its autocorrelation function (statistical average of product of samples at two time moments) is the Dirac delta function: n(t)n(t0) ¼ (N0/2) (t t0). In other words, any two samples of white noise, notwithstanding how close in time, are uncorrelated. Using this result in the integral above along with the sifting property of the delta function:

ZT

sðt0Þ ðt0 tÞ dt0 ¼ sðtÞ

0

leads to:

where E is the energy of the signal s(t).

In the case under consideration, as (2.10) and (2.7) show, substitution s(t) ¼ s0(t) s1(t) should be made in (2.15), i.e. E is energy Ed of the signal difference s0(t) s1(t). Deriving it gives:

Taking into account the geometrical content of the correlation coefficient and energy, this is just the cosine theorem from ‘school’ mathematics.

Now, substitute (2.13), (2.15) and (2.16) into (2.12), arriving at:

error probability Pe does not depend on a priori probability w of sending the signal s0(t) and is expressed by the equation:

It is quite obvious now that the only way to achieve high data transmission fidelity is to make the distance between the two signals d(s0, s1) large enough. Certainly, d(s0, s1) can be increased at the cost of large signal energies or vector lengths, as is seen from (2.16). But what is the optimal signal pair when recourse to this ‘brute force’ approach is limited, i.e. signal energies are fixed beforehand? Consider first the very typical case of signals of equal energies: E0 ¼ E1 ¼ E, meaning that signal intensity is not utilized as a message indicator. Decision rule (2.10) is now reduced to just comparison between z0 and z1 or, equivalently, to testing the polarity of their difference:

^

H0

z ¼ z0 z1 ><0

^

H1

Of course, to maximize the distance between two vectors of fixed lengths, one should

make them antipodal, as shown in Figure 2.5a. Then the angle between s0, s1 ¼ , p

cos ¼ 01 ¼ 1 and d(s0, s1) ¼ 2 E, which turns (2.18) into the following:

representing the minimal achievable error probability in binary data transmission, signal energy E being fixed. The correlator and matched filter, which will often be

referred to in this book, are devices typically used to physically calculate correlation z, p

and parameter q ¼ 2E/N0 is nothing but signal-to-noise ratio (SNR) at the correlator or matched filter output.

Figure 2.5 Signal pairs in various binary transmission modes