Torrieri D. Principles of Spread-Spectrum Communication Systems

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CHAPTER 7. DETECTION OF SPREAD-SPECTRUM SIGNALS

termining whether one or more than one signal has been detected, narrowband

interference can be rejected by the methods of Section 2.7.

7.2

Detection of Frequency-Hopping Signals

An interception receiver intended for the detection of frequency-hopping signals

may be designed according to the principles of classical detection theory or

according to more intuitive ideas. The former approach is useful in setting

limits on what is possible, but the latter approach is more practical and flexible

and less dependent on knowledge of the characteristics of the frequency-hopping

signals.

Ideal Detection

To enable a tractable analysis, the idealized assumptions are made that the

hopset is known and that the hop epoch timing, which includes the hop-

transition times is known. Consider slow frequency-hopping signals with CPM

(FH/CPM), which includes continuous-phase MFSK. The signal over the

hop interval is

where S is the average signal power, is the carrier frequency associated with

the hop, is the CPM component that depends on the data sequence

and is the phase associated with the hop. The parameters

and the components of are modeled as random variables. The derivation

of the average likelihood ratio (7-7) is still valid, but the vector has different

parameters as components.

The M carrier frequencies in the hopset are assumed to be equally likely

over a given hop and statistically independent from hop to hop for hops.

Dividing the integration interval in (7-7) into parts, averaging over the M

frequencies, and dropping the irrelevant factor 1/

, we obtain

where the condition in the argument of indicates that the carrier frequency

over the hop is the expectation is over the remaining random parameters

and and is the energy per hop. The decomposition in (7-56) indicates

that the general structure of the detector has the form illustrated in Figure 7.4.

The average likelihood ratio of (7-56) is compared with a threshold to determine

whether a signal is present. The threshold may be set to ensure the tolerable

false-alarm probability when the signal is absent. Assuming that is

7.2.

DETECTION OF FREQUENCY-HOPPING SIGNALS

399

Figure 7.4: General structure of optimum detector for frequency-hopping signal

with hops and M frequency channels.

the same for every hop and carrier frequency, we may drop the irrelevant factor

in (7-57), which only affects the threshold level.

Each of the data sequences that can occur during a hop is assumed to be

equally likely. For coherent detection of FH/CPM [5], we set in (7-55),

substitute it into (7-57), and then evaluate the expectation to obtain

where irrelevant factors have been dropped. This equation indicates how

in Figure 7.4 is to be calculated for each hop and each frequency channel

corresponding to carrier frequency Equations (7-56) and (7-58) define the

optimum coherent detector for any slow frequency-hopping signal with CPM.

For noncoherent detection of FH/CPM [4], the received carrier phase is

assumed to be uniformly distributed over during a given hop and statis-

tically independent from hop to hop. Substituting (7-55) into (7-57), averaging

over the random phase in addition to the sequence statistics, and dropping

irrelevant factors yields

where

and

Equations (7-56), (7-59), (7-60), and (7-61) define the optimum noncoherent

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CHAPTER 7. DETECTION OF SPREAD-SPECTRUM SIGNALS

Figure 7.5: Optimum noncoherent detector for slow frequency hopping with

CPM: (a) basic structure of frequency channel for hop with parallel cells for

candidate data sequences, and (b) cell for data sequence

detector for any slow frequency-hopping signal with CPM. The means of pro-

ducing (7-59) is diagrammed in Figure 7.5.

A major contributor to the huge computational complexity of the optimum

detectors is the fact that with data symbols per hop and an alphabet size

there may be data sequences per hop. Consequently, the com-

putational burden grows exponentially with However, if it is known that

the data modulation is CPFSK with a modulation index where is a

positive integer, the computational burden has a linear dependence on

Even then, the optimum detectors are extremely complex when the number of

frequency channels is large.

The preceding theory may be adapted to the detection of fast frequency-

hopping signals with MFSK as the data modulation. Since there is one hop per

MFSK channel symbol, the information is embedded in the sequence of carrier

frequencies. Thus, we may set and in (7-58) and (7-59).

For coherent detection, (7-58) reduces to

Equations (7-56) and (7-62) define the optimum coherent detector for a fast

frequency-hopping signal with MFSK. For noncoherent detection, (7-59), (7-60),

7.2.

DETECTION OF FREQUENCY-HOPPING SIGNALS

401

and (7-61) reduce to

Equations (7-56), (7-63), and (7-64) define the optimum noncoherent detector

for a fast frequency-hopping signal with MFSK. Performance analyses for the

detectors of fast frequency-hopping signals are given in [5].

Instead of basing detector design on the average likelihood ratio, one might

apply a composite hypothesis test in which the presence of the signal is detected

while simultaneously one or more of the unknown parameters under hypothe-

sis are estimated. To simultaneously detect the signal and determine the

frequency-hopping pattern, (7-56) is replaced by the generalized likelihood ratio:

where the equations and subsystems for remain the same. Equation

(7-65) indicates that a maximum-likelihood estimate of is made for each

hop. Thus, an optimum test to determine the frequency channel occupied

by the frequency-hopping signal is conducted during each hop. Although the

detection performance is suboptimal when the generalized likelihood ratio is

used to design a detector, this detector provides an important signal feature

and is slightly easier to implement and analyze [4], [5].

Wideband Radiometer

Among the many alternatives to the optimum detector, two of the most useful

are the wideband radiometer and the channelized radiometer. The wideband

radiometer is notable in that it requires virtually no detailed information about

the parameters of the frequency-hopping signals to be detected other than their

rough spectral location. The price paid for this robustness is much worse per-

formance than more sophisticated detectors that exploit additional information

about the signal [4]. The channelized radiometer is designed to explicitly ex-

ploi

the spectral characteristics of frequency-hopping signals. In its optimal

form, the channelized radiometer gives a performance nearly as good as that of

ideal detector. In its suboptimal form, the channelized radiometer trades

performance for practicality and the easing of the required a priori information

about the signal to be detected.

Channelized Radiometer

A channelized radiometer comprises K parallel radiometers, each of which has

the form of Figure 7.1 and monitors a disjoint portion of the hopping band of a

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CHAPTER 7. DETECTION OF SPREAD-SPECTRUM SIGNALS

Figure 7.6: Channelized radiometer.

frequency-hopping signal, as depicted in Figure 7.6. The largest of the sampled

radiometer outputs is compared to a threshold stored in a comparator. If

the threshold is exceeded, the comparator sends a 1 to the summer; otherwise

it sends a 0. If the hop dwell epochs are at least approximately known, the

channelized radiometer may improve its detection reliability by adding the 1’s

produced by N consecutive comparator outputs corresponding to multiple fre-

quency hops of the signal to be detected. A signal is declared to be present if the

sum V equals or exceeds the integer which serves as a second threshold. The

two thresholds are are jointly optimized for the best system performance.

Ideally, K = M, the number of frequency channels in a hopset, but many

fewer radiometers may be a practical or economic necessity; if so, each ra-

diometer may monitor frequency channels, where Because

of insertion losses and the degradation caused by a power divider, it is unlikely

that many more than 30 parallel radiometers are practical. An advantage of

each radiometer covering many frequency channels is the reduced sensitivity to

imprecise knowledge of the spectral boundaries of frequency channels. Since it

is highly desirable to implement the parallel radiometers with similar circuitry,

their bandwidths are assumed to be identical henceforth.

To prevent steady interference in a single radiometer from causing false

alarms, the channelized radiometer must be able to recognize when one of its

constituent radiometers produces an output above the threshold for too many

consecutive samples. The channelized system may then delete that constituent

radiometer’s output from the detection algorithm or it may reassign the ra-

diometer to another spectral location.

In the subsequent analysis of the channelized radiometer of Figure 7.6, the

observation interval of the parallel radiometers, which is equal to the sampling

interval, is assumed to equal the hop duration The effective observation

time of the channelized radiometer, should be less than the mini-

mum expected message duration to avoid processing extraneous noise. Let

denote the probability that a particular radiometer output at the sampling time

exceeds the comparator threshold when no signal is present. This probability

is given by the right-hand side of (7-42). Therefore, a derivation similar to that

of (7-47) indicates that if the sampling times are aligned with the frequency

7.2.

DETECTION OF FREQUENCY-HOPPING SIGNALS

403

transitions, then the threshold necessary to achieve a specified is

where B is the bandwidth of each of the frequency channels encompassed

by a radiometer passband. The probability that at least one of the K parallel

radiometer outputs exceeds is

assuming that the channel noises are statistically independent because the ra-

diometer passbands are disjoint. The probability of a false alarm of the chan-

nelized radiometer is the probability that the output V equals or exceeds a

threshold

To solve this equation for in terms of we observe that the incomplete

beta function is defined as

where is the beta function and In terms of this function,

(7-68) becomes

The inverse of the incomplete beta function, which we denote by may

be easily computed by Newton’s method or approximations [6]. Therefore, if

(7-66), (7-67), and (7-70) may be combined to determine the

approximate threshold necessary to achieve a specified

where it is assumed that does not vary across the hopping band and, hence,

there is one and one for all the parallel radiometers.

The number of sampling intervals during which the signal is present is

where is the intercepted signal duration. For simplicity, it is assumed

that is an integer. Furthermore, we assume that at most a single radiometer

receives significant signal energy during each sampling interval. Let denote

the probability that a particular radiometer output exceeds the threshold when

a signal is present in that radiometer. Derivations similar to those of (7-45)

and (7-49) 38.88 imply that

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CHAPTER 7.

DETECTION OF SPREAD-SPECTRUM SIGNALS

where and is the energy per hop dwell time. Let denote

the probability that the threshold is exceeded by the sampled maximum of

the parallel radiometer outputs. It is assumed that when a signal is present it

occupies any one of M frequency channels with equal probability and that all

radiometer passbands are within the hopping band. Consequently, the signal

has probability of being in the passband of a particular radiometer and

probability of being in the passband of some radiometer. Since a

detection may be declared in response to a radiometer that does not receive the

signal,

where the second term vanishes if the radiometer passbands cover the hopping

band so that The probability of detection associated with the

observation interval when the signal is actually present during of the

hop intervals is

If at least the minimum duration of a frequency-hopping signal is known, the

overestimation of N might be avoided so that The detection proba-

bility then becomes

A suitable, but not optimal, choice for the second threshold is when

the full hopping band is monitored by the channelized radiometer. In general,

numerical results indicate that

is a good choice for partial-band monitoring.

If detection decisions are made in terms of fixed observation intervals of

duration and successive intervals do not overlap except possibly at

end points, then the false alarm rate defined in (7-48) is an appropriate design

parameter. This type of detection is called block detection, and the false-alarm

rate is

To prevent the risk of major misalignment of the observation interval with the

time the signal is being transmitted, either block detection must be supple-

mented with hardware for arrival-time estimation or the duration of successive

7.2.

DETECTION OF FREQUENCY-HOPPING SIGNALS

405

observation intervals should be less than roughly half the anticipated signal

duration.

A different approach to mitigating the effect of a misalignment, called binary

moving-window detection, is for the observation interval to be constructed by

dropping the first sampling interval of the preceding observation interval and

adding a new sampling interval. A false alarm is considered to be a detection

declaration at the end of the new interval when no signal is actually present.

Thus, a false alarm occurs only if the comparator input for an added sampling

interval exceeds the threshold, the comparator input for the discarded sampling

interval did not, and the count for the preceding observation interval was

Therefore, the probability of a false alarm is

where

It follows that the false-alarm rate is

Since the right-hand side of this equation is proportional to the first term of

the series in (7-68),

This inequality indicates that the false alarm rate is nearly times as large for

moving-window detection as it is for block detection. Thus, moving-window

detection usually requires a higher comparator threshold for the same false-

alarm rate and, hence, more signal power to detect a frequency-hopping signal.

However, moving-window detection with inherently limits the

misalignment between the occurrence of the intercepted signal and some obser-

vation interval. If the signal occurs during two successive obsevation intervals,

then for one of the observation intervals, the misalignment is not more than

As an example, it is assumed that there are M = 2400 frequency channels,

the signal duration is known, and there is no misalignment so that

Block detection is used so that (7-77) is applicable,

and Figure 7.7 plots versus for the channelized radiometer

with full hopping-band coverage so that and several values of

K and N. The figure also shows the results for a wideband radiometer with

and N = 150 or 750. It is observed that

the channelized radiometer with K = 30 is much better than the wideband

radiometer when N = 150, but loses its advantage for when N =

750. The substantial advantage of the channelized radiometer with K = M and

is apparent. As N increases, the channelized radiometer can retain its

advantage over the wideband radiometer by increasing K accordingly.

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CHAPTER 7. DETECTION OF SPREAD-SPECTRUM SIGNALS

Figure 7.7: Probability of detection versus for channelized and wideband

radiometers with full coverage,

and

Figure 7.8: Probability of detection for channelized radiometer with different

percentages of coverage,

7.3. PROBLEMS

407

In Figure 7.8, and K = 30, but and are variable.

The fraction of the hopping band monitored by the channelized radiometer is

denoted by the monitored fraction It is observed that when

there is only a small performance loss for the channelized

radiometer despite the fact that The relative insensitivity of

the channelized radiometer to small errors in is a major advantage over the

wideband radiometer. The figure illustrates the following trade-off when K and

M are fixed: as decreases, fewer frequency channels are monitored, but less

noise enters a radiometer. The net result is beneficial when is reduced to 20

percent. However, the figure indicates that for percent or 5 percent, the

hopping-band coverage becomes inadequate to enable a greater than 0.995

and 0.96, respectively, regardless of Thus, there is a minimum fraction

of the hopping band that must be monitored to ensure a specified

As (7-72) indicates that Therefore, (7-73) implies

that Suppose that is raised to a sufficiently high

level that and, hence, If detection is to be accomplished for

the minimum monitored fraction, then is the best choice for the second

threshold. For and yields

Since (7-82) implies that even if the realization of a

specified requires the minimum monitored fraction

Thus, if and then Many other aspects

of the channelized radiometer, including the effects of timing misalignments,

are examined in [6].

7.3

Problems

If the form and the parameters of a signal to be detected are known,

optimum detection in white Gaussian noise can be accomplished by

an ideal matched filter or correlator. Let denote the received sig-

nal and T denote the observation interval. (a) Write an expression for

the sampled output of an ideal matched filter. (b) This output is com-

pared to a threshold to determine whether the target signal is present.

Suppose that it is present and coincides with an observation interval. As-

suming that is zero-mean and has two-sided power spectral density

and that the signal energy is what are the mean and the vari-

ance of (c) What is the probability of detection What is the

probability of false alarm Express in terms of (d) What is

the value of necessary to ensure specified values of and

An unbiased estimate of may be obtained from the wideband radiome-

ter output when the target signal is absent. How long must the obser-