Torrieri D. Principles of Spread-Spectrum Communication Systems

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408

CHAPTER 7. DETECTION OF SPREAD-SPECTRUM SIGNALS

vation interval be if it is required that 3 standard deviations of the

estimate do not exceed

The receiver operating characteristic (ROC) is a traditional plot depicting

versus for various values of TW or The ROC may be calcu-

lated from (7-49) and (7-46). Plot the ROC for the wideband radiometer

with and no noise-measurement error. Let TW = 104 and

105.

Derive (7-50) using the method described in the text.

Find conditions under which (7-51) indicates that a negative energy is

required. What is the physical implication of this result?

Show that the first two terms in (7-54) give the intercepted power neces-

sary to achieve and the specified value of either or F.

Consider a channelized radiometer that is to detect a frequency-hopping

signal with N = 1 hop. (a) Find in terms of (b) If

and is small, derive the required for

specified values of and

7.4

References

M. D. Srinath, P. K. Rajasekaran, and R. Viswanathan, Introduction

to Statistical Signal Processing with Applications. Englewood Cliffs, NJ:

Prentice Hall, 1996.

A. Polydoros and C. L. Weber, “Detection Performance Considerations

for Direct-Sequence and Time-Hopping LPI Waveforms,” IEEE J. Select.

Areas Commun., vol. 3, pp. 727–744, September 1985.

D. A. Hill and E. B. Felstead, “Laboratory Performance of Spread Spec-

trum Detectors,” IEE Proc.-Commun., vol. 142, pp. 243–249, August

1995.

B. K. Levitt et al., “Optimum Detection of Slow Frequency-Hopped Sig-

nals,” IEEE Trans. Commun., vol. 42, pp. 1990–2000, Feb./March/April

1994.

N. C. Beaulieu, W. L. Hopkins, and P. J. McLane, “Interception of

Frequency-Hopped Spread-Spectrum Signals,” IEEE. J. Select. Areas

Commun., vol. 8, pp. 853–870, June 1990.

L. E. Miller, J. S. Lee, and D. J. Torrieri, “Frequency-Hopping Signal

Detection Using Partial Band Coverage,” IEEE Trans. Aerosp. Electron.

Syst., vol. 29, pp. 540–553, April 1993.

D. Torrieri, Principles of Secure Communication Systems, 2nd ed. Boston:

Artech House, 1992.

Appendix

Inequalities

A.1

Jensen’s Inequality

A function defined on an open interval I is convex if

for in I and Suppose that has a continuous, nondecreasing

derivative on I. The inequality is valid if or 1

If and

Simplifying this result, we obtain (A-1). If a similar analysis again

yields (A-1). Thus, if has a continuous, nondecreasing derivative on I, it

is convex.

Lemma. If is a convex function on the open interval

, then

for all in I, where is the left derivative of

Proof: If then substituting into (A-1)

gives

which yields

410

APPENDIX A. INEQUALITIES

If and then (A-1) implies that

which yields

Inequality (A-3) indicates that the ratio decreases m

onoton

ically as from above and (A-4) implies that this ratio has a lower bound.

Therefore, the right derivative exists on

. If then (A-1)

with implies that

which yields

This inequality indicates that the ratio increases monotoni-

cally as from below and (A-4) implies that this ratio has an upper bound.

Therefore, the left derivative exists on I, and (A-4) yields

Taking the limits as and in (A-3) and (A-5), respectively, and

then using (A-6), we find that (A-2) is valid for all

Jensen’s inequality. If X is a random variable with a finite expected

value

[

], and is a convex function on an open interval containing the

range of X, then

Proof: Set and in (A-2), which gives

Taking the expected values of the random variables on

both sides of this inequality gives Jensen’s inequality.

A.2

Chebyshev’s Inequality

Consider a random variable X with distribution Let denote

the expected value of X and P[A] denote the probability of event A. From

elementary probability it follows that

A.2. CHEBYSHEV’S INEQUALITY

411

Therefore,

Let denote the variance of X. If then (A-8) becomes

Chebyshev’s inequality:

As an application, let Then Chebyshev’s inequality indicates that

Therefore, the probability that a random variable is

within three standard deviations of its mean value is at least 8/9.

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Appendix B

Adaptive Filters

The input and weight vectors of an adaptive filter are

where T denotes the transpose and the components of the vectors may be real

or complex. The filter output is the scalar

The derivation of the optimal filter weights depends on the specification of a

performance criterion or estimation procedure. A number of different estimators

of the desired signal can be implemented by linear filters that produce (B-2).

Unconstrained estimators that depend only on the second-order moments of x

can be derived by using performance criteria based on the mean square error

or the signal-to-noise ratio of the filter output. Similar estimators result from

using the maximum-a-posteriori or the maximum-likelihood criteria, but the

standard application of these criteria includes the restrictive assumption that

any interference in x has a Gaussian distribution.

The difference between the desired response and the filter output is the

error signal:

The most widely used method of estimating the desired signal is based on the

minimization of the expected value of the squared error magnitude, which is

proportional to the mean power in the error signal. Let H denote the conjugate

transpose and an asterisk denote the conjugate. We obtain

where

414

APPENDIX B. ADAPTIVE FILTERS

is the N × N Hermitian correlation matrix of x and

is the N × 1 cross-correlation vector. If we assume that when

then must be positive definite.

In terms of its real part and its imaginary part a complex weight

vector is defined as

The gradient of with respect to the real-valued vector

defined as the column vector with components Let

and denote the N × 1 gradient vectors with respect to and

respectively. The complex gradient with respect to W is defined as

Let and denote the components of and

respectively. Letr denote a real-valued function of W and W*.

Regarding W and W* as independent variables, we assume that is an analytic

function of each when W* is held constant and an analytic function of each

when W is held constant. We define as the gradient with respect to

W*. Since

The chain rule of calculus then implies that

Thus,

This result allows a major simplification in calculations.

Since and (B-4) yield

Since and imply that a necessary condition

for the optimal weight is obtained by setting Thus, if is

positive definite and hence nonsingular, the necessary condition provides the

Wiener-Hopf equation for the optimal weight vector:

To prove the optimality, we substitute into (B-4) to obtain the mean

square error

415

Equations (B-4), (B-13), and (B-14) imply that

Since is positive definite, this equation shows that the Wiener-Hopf equa-

tion provides a unique optimal weight vector and that (B-14) gives the minimum

mean square error.

Since the computational difficulty of inverting the correlation matrix is con-

siderable when the number of weights is large, and insofar as time-varying signal

statistics may require frequent computations, adaptive algorithms not entailing

matrix inversion have been developed. Suppose that a performance measure,

(W), is defined so that it has a minimum value when the weight vector has its

optimal value. In the method of steepest descent, the weight vector is changed

along the direction of the negative gradient of the performance measure. This

direction gives the largest decrease in P(W). If the signals and weights are

complex, separate steepest-descent equations can be written for the real and

imaginary parts of the weight vector. Combining these equations, we obtain

where the adaptation constant controls the rate of convergence and the sta-

bility. For complex signals and weights, a suitable performance measure is

The application of (B-12) and (B-16) leads to the steepest-

descent algorithm:

This ideal algorithm produces a deterministic sequence of weights and does

not require a matrix inversion, but it requires the knowledge of and

However, the possible presence of interference means that is unknown. In

the absence of information about the direction of the desired signal, is also

unknown.

The least-mean-square (LMS) algorithm is obtained when is estimated

by is estimated by and (B-3) is applied in (B-17).

The LMS algorithm is

For a fixed value of the product is an unbiased estimate of

According to this algorithm, the next weight vector is obtained by

adding to the present weight vector the input vector scaled by the amount of

error. It can be shown that, for an appropriate value of the mean of the weight

vector converges to the optimal value given by the Wiener-Hopf equation.

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Appendix C

Signal Characteristics

C.1

Bandpass Signals

The Hilbert transform provides the basis for signal representations that facilitate

the analysis of bandpass signals and systems. The Hilbert transform of a real-

valued function is

Since its integrand has a singularity, the integral is defined as its Cauchy prin-

cipal value:

provided that the limit exists. Since (C-1) has the form of the convolution of

with results from passing through a linear filter with an

impulse response equal to The transfer function of the filter is given by

the Fourier transform

where This integral can be rigorously evaluated by using contour

integration. Alternatively, we observe that since is an odd function,

where is the signum function defined by