Perfomance of Binary Receivers in AWGN

In preceding sections, we have derived optimal receiver structures for detecting which of two alternative signals is received in the presence of additive Gaussian noise.

In this section, we will examine the performance of the receivers.

Since we already know that the receivers are optimal, it may seem strange to be interested in their performance.

After all, we know we cannot build anything better under the assumed conditions.

On the other hand, it is useful to know the performance for at least two reasons:

1. The performance will give us great insight into signal design.

2. If, for cost reasons, we desire to build a suboptimal receiver, it is useful to know how much performance degradation we are suffering.

We limit ourselves here to receiver structures that are optimal under noise conditions that are Gaussian and white.

In addition, the structures are assumed to be phase coherent, implying that a perfect estimate of the received phase is available.

Furthermore, we address the performance of the continuous-time signal receivers.

Matched Filter and Decision Statistics

If hypothesis \( H_0 \) is true, then the measured signal, assuming no variation in the received signal amplitude, is represented by

\[ y(t) = u_0(t) + z(t), \quad 0 \leq t \leq T \]

For simplicity, the received signal amplitude introduced by a constant channel is assumed to be unity.

It follows that the decision variable \( U_0 \) is

\[ U_0 = \Re \left\{ \int_0^T y(t) u_0^*(t) \, dt \right\} - \mathcal{E}_0 \]

and \( U_1 \) is

\[ U_1 = \Re \left\{ \int_0^T y(t) u_1^*(t) \, dt \right\} - \mathcal{E}_1 \]

Since the noise \( z(t) \) is a zero-mean white Gaussian process, it follows that \( U_0 \) and \( U_1 \) are both Gaussian random variables. Hence, we can characterize them completely by determining their means, variances, and covariance.

Energy of Complex Baseband Signal

\[ \boxed{ \int_0^T u_0(t) u_0^*(t) \, dt = \int_0^T |u_0(t)|^2 \, dt = 2 \mathcal{E}_0 } \]

\[ \boxed{ \int_0^T u_1(t) u_1^*(t) \, dt = \int_0^T |u_1(t)|^2 \, dt = 2 \mathcal{E}_1 } \]

White Noise Power Spectral Density

\[ \boxed{ E \left\{ z^*(t) z(t + \tau) \right\} = 2 N_0 \delta(\tau) } \]

Mean and Variance of \(U_0\) given \(H_0\)

The mean of \( U_0 \), given that \( H_0 \) is correct, is

\[\begin{split} \begin{align*} E\{U_0 | H_0\} &= E \left\{ \Re \left[ \int_0^T (u_0(t) + z(t)) u_0^*(t) \, dt \right] \right\} - \mathcal{E}_0 \\ &= E \left\{ \frac{1}{2} \int_0^T \left[ u_0(t) u_0^*(t) + z(t) u_0^*(t) \right] dt + \frac{1}{2} \int_0^T \left[ u_0^*(t) + z^*(t) \right] u_0(t) dt \right\} - \mathcal{E}_0 \\ &= \frac{1}{2} \int_0^T u_0(t) u_0^*(t) \, dt + \frac{1}{2} \int_0^T u_0^*(t) u_0(t) \, dt - \mathcal{E}_0 \\ &\boxed{= \mathcal{E}_0} \end{align*} \end{split}\]

The conditional complex variance is

\[\begin{split} \begin{align*} \text{Var}(U_0 | H_0) &= E \left\{ \left[ \frac{1}{2} \int_0^T z(t_1) u_0^*(t_1) \, dt_1 + \frac{1}{2} \int_0^T z^*(t_2) u_0(t_2) \, dt_2 \right] \right. \\ &\quad\left. \times \left[ \frac{1}{2} \int_0^T z^*(t_3) u_0(t_3) \, dt_3 + \frac{1}{2} \int_0^T z(t_4) u_0^*(t_4) \, dt_4 \right] \right\} \end{align*} \end{split}\]

Since \( E\{z(t_i) z(t_j)\} = E\{z^*(t_i) z^*(t_j)\} = 0 \) for all arguments and \( E\{z(t_i) z^*(t_j)\} = 2N_0 \delta(t_i - t_j) \), it follows that

\[\begin{split} \begin{align*} \text{Var}(U_0 | H_0) &= \frac{1}{4} \int_0^T \int_0^T E\{z(t_1) z^*(t_3)\} u_0^*(t_1) u_0(t_3) \, dt_3 \, dt_1 \\ &\quad+ \frac{1}{4} \int_0^T \int_0^T E\{z(t_4) z^*(t_2)\} u_0^*(t_4) u_0(t_2) \, dt_4 \, dt_2 \\ &= \frac{N_0}{2} \int_0^T |u_0(t_1)|^2 \, dt_1 + \frac{N_0}{2} \int_0^T |u_0(t_2)|^2 \, dt_2 \\ &\boxed{= 2 \mathcal{E}_0 N_0} \end{align*} \end{split}\]

Cross-Correlation Coefficient

We define

\[ \boxed{ \rho = \rho_r + j \rho_i = \frac{1}{2 \sqrt{\mathcal{E}_0 \mathcal{E}_1}} \int_0^T u_0(t) u_1^*(t) \, dt } \]

as a complex-valued cross-correlation coefficient with real and imaginary parts \( \rho_r \) and \( \rho_i \), respectively.

Using this definition of \( \rho \), and in a manner similar to the preceding calculation, we can determine the conditional mean and complex variance of \( U_1 \), given \( H_0 \). As shown in the exercises, these are

\[\begin{split} \begin{align*} E\{U_1 | H_0\} &= E \left\{ \Re \left[ \int_0^T \left( u_0(t) + z(t) \right) u_1^*(t) \, dt \right] - \mathcal{E}_1 \right\} \\ &\boxed{= 2 \rho_r \sqrt{\mathcal{E}_0 \mathcal{E}_1} - \mathcal{E}_1} \end{align*} \end{split}\]

and

\[ \boxed{ \text{Var}(U_1 | H_0) = 2 \mathcal{E}_1 N_0 } \]

The conditional complex covariance of \( U_0 \) and \( U_1 \) is derived as

\[\begin{split} \begin{align*} \text{Cov}(U_0 U_1 | H_0) &= E \left\{ \left[ \frac{1}{2} \int_0^T z(t_1) u_0^*(t_1) \, dt_1 + \frac{1}{2} \int_0^T z^*(t_2) u_0(t_2) \, dt_2 \right] \right. \\ &\quad \left. \times \left[ \frac{1}{2} \int_0^T z^*(t_3) u_1(t_3) \, dt_3 + \frac{1}{2} \int_0^T z(t_4) u_1^*(t_4) \, dt_4 \right] \right\} \\ &= \frac{1}{4} \int_0^T \int_0^T E\{z(t_1) z^*(t_3)\} u_0^*(t_1) u_1(t_3) \, dt_3 \, dt_1 \\ &\quad+ \frac{1}{4} \int_0^T \int_0^T E\{z^*(t_2) z(t_4)\} u_0(t_2) u_1^*(t_4) \, dt_4 \, dt_2 \\ &= \frac{N_0}{2} \int_0^T u_0^*(t_1) u_1(t_1) \, dt_1 + \frac{N_0}{2} \int_0^T u_0(t_2) u_1^*(t_2) \, dt_2 \\ &\boxed{= 2 \rho_r \sqrt{\mathcal{E}_0 \mathcal{E}_1} N_0} \end{align*} \end{split}\]

Error Probability

In communications problems, we are often concerned with the probability of error.

When \( H_0 \) is correct, this is \( P_{10} \).

We know an error occurs if \( U_1 > U_0 \) or \( U_1 - U_0 > 0 \).

Consequently, we define a new random variable

\[\boxed{ \mathcal{V} = U_1 - U_0 } \]

Assuming \( H_0 \) is correct, the mean and variance of \( \mathcal{V} \) are

\[\boxed{ \mu_{\mathcal{V}} = E\{\mathcal{V} | H_0\} = 2 \rho_r \sqrt{\mathcal{E}_0 \mathcal{E}_1} - \mathcal{E}_0 - \mathcal{E}_1 } \]

and

\[ \boxed{ \sigma_{\mathcal{V}}^2 = V\{\mathcal{V} | H_0\} = 2 N_0 \left( \mathcal{E}_0 + \mathcal{E}_1 - 2 \rho_r \sqrt{\mathcal{E}_0 \mathcal{E}_1} \right) } \]

The probability of error is

\[\begin{split} \begin{align*} P_{10} &= P(\mathcal{V} > 0 | H_0) \\ &= \frac{1}{\sqrt{2 \pi \sigma_{\mathcal{V}}^2}} \int_0^{\infty} \exp \left( -\frac{(v - \mu_{\mathcal{V}})^2}{2 \sigma_{\mathcal{V}}^2} \right) dv \\ &= \frac{1}{2} \, \text{erfc} \left( - \frac{\mu_{\mathcal{V}}}{\sqrt{2 \sigma^2_{\mathcal{V}}}} \right) \\ &\boxed{= \frac{1}{2} \, \text{erfc} \left( \sqrt{\frac{\mathcal{E}_0 + \mathcal{E}_1 - 2 \rho_r \sqrt{\mathcal{E}_0 \mathcal{E}_1}}{4 N_0}} \right) } \end{align*} \end{split}\]

Based on the symmetry of the derivation, we have

\[ P_{01} = P_{10} \]

If both hypotheses are equally likely, it follows that the average binary error \( P_E \) is also this quantity.

In many communications examples, the two signals are chosen to have equal energy, i.e.,

\[ \mathcal{E} = \mathcal{E}_0 = \mathcal{E}_1 \]

In this case, \( P_{10} \) simplifies to

\[ \boxed{ P_E = \frac{1}{2} \, \text{erfc} \left( \sqrt{\frac{\mathcal{E} (1 - \rho_r)}{2 N_0}} \right) } \]

Figure 5.9 shows a plot of \(P_E\) as a function of \( \mathcal{E} / N_0 \) for two different values of \( \rho_r \).

The best value that \( \rho_r \) can attain under these conditions is \( \rho_r = -1 \). In this case, \( u_1(t) = -u_0(t) \). This condition is known as antipodal signaling.

Another common value is \( \rho_r = 0 \), which is known as orthogonal signaling.

These two values of \( \rho_r \) are used in Figure 5.9.

As the figure and \(P_E\) indicate, there are three parameters that a communications system designer can influence which will improve the binary error-rate performance:

1. The signals should be chosen so that the real part of the correlation coefficient, \( \rho_r \), is as close to \(-1\) as possible.

2. All else being equal, better performance is attained if the transmitted power (and hence the received energy \( \mathcal{E} \)) can be increased.

3. If \( N_0 \) can be reduced, performance will improve. Note that in radio systems, \( N_0 \) is controlled, in large part, by the noise temperature and noise figure of the receiver’s first preamplifier. Hence, some control is possible.

The use of error-correction coding will also improve performance by essentially expanding the alphabet size from binary to \( M \)-ary.

MATLAB Tutorials

• Analyze Performance with Bit Error Rate Analysis App