Transcript Document

Chapter 9
Detection of
Spread-Spectrum Signals
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• This chapter presents a statistical analysis of the unauthorized
detection of spread-spectrum signals.
• The basic assumption is that the spreading sequence or the
frequency-hopping pattern is unknown and cannot be accurately
estimated by the detector.
• Thus, the detector cannot mimic the intended receiver.
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7.1 Detection of Direct-Sequence Signals
• A spectrum analyzer usually cannot detect a signal with a power
spectral density below that of the background noise, which has
spectral density N0/2.
• Thus, a received
is an approximate necessary, but not
sufficient, condition for a spectrum analyzer to detect a directsequence signal.
• If
detection may still be probable by other means. If
not, the direct-sequence signal is said to have a low probability of
interception.
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Ideal Detection
• We make the idealized assumptions that the chip timing of the
spreading waveform is known and that whenever the signal is
present, it is present during the entire observation interval.
• The spreading sequence is modeled as a random binary sequence,
which implies that a time shift of the sequence by a chip duration
corresponds to the same stochastic process.
• Consider the detection of a direct-sequence signal with PSK
modulation:
(9.1)
– S is the average signal power
– fc is the known carrier frequency
– ψis the carrier phase assumed to be constant over the
observation interval 0≦t ≦T.
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• The spreading waveform p(t) which subsumes the random data
modulation, is given by (2-2) with the {pi} modeled as a random
binary sequence.
• To determine whether a signal is present based on the observation
of the received signal, classical detection theory requires that one
choose between the hypothesis H1 that the signal is present and
the hypothesis H0 that the signal is absent.
• Over the observation interval, the received signal under the two
hypotheses is
(9.2)
where n(t) is zero-mean, white Gaussian noise with two-sided
power spectral density N0/2.
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• The coefficients in the expansion of the observed waveform in
terms of νorthonormal basis functions constitute the received
vector
• Letθdenote the vector of parameter values that characterize the
signal to be detected.
• The average likelihood ratio [1], which is compared with a
threshold for a detection decision, is
(9.3)
–
is the conditional density function of r given
hypothesis H1 and the value of θ.
–
is the conditional density function of r given
hypothesis H0
–
is the expectation over the random vector θ.
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(9.4)
(9.5)
where si the are the coefficients of the signal.
• Substituting these equations into (9-3) yields
(9.6)
(9.7)
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• If N is the number of chips, each of duration received in the
observation interval, then there are 2N equally likely patterns of
the spreading sequence.
• For coherent detection, we set in (9-1), substitute it into (9-7),
and then evaluate the expectation to obtain
(9.8)
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• For the more realistic noncoherent detection of a direct-sequence
signal, the received carrier phase is assumed to be uniformly
distributed over [0, 2π)
• Substituting (9-1) into (9-7), using a trigonometric expansion,
dropping the irrelevant factor that can be merged with the
threshold level, and then evaluating the expectation over the
random spreading sequence, we obtain
(9.10)
where
is chip i of pattern j
(9.11)
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• The modified Bessel function of the first kind and order zero is
given by
(9-13)
• Replace cosμwith cos(μ+ψ) for any in (9-13).
(9-14)
• The average likelihood ratio of (7-10) becomes
(9.15)
(9.16)
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• These equations define the optimum noncoherent detector for a
direct-sequence signal.
• The presence of the desired signal is declared if (9-15) exceeds a
threshold level.
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Radiometer
• Suppose that the signal to be detected is approximated by a zeromean, white Gaussian process.
• Consider two hypotheses that both assume the presence of a zeromean, bandlimited white Gaussian process over an observation
interval 0≦t ≦T.
– Under H0 only noise is present, and the one-sided power
spectral density over the signal band is N0
– Under H1 both signal and noise are present, and the power
spectral density is N1 over this band.
• Using ν orthonormal basis functions as in the derivation of (9-4)
and (9-5) and ignoring the effects of the bandlimiting, we find
that the conditional densities are approximated by
(9.17)
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• Calculating the likelihood ratio, taking the logarithm, and
merging constants with the threshold, we find that the decision
rule is to compare
(9.18)
to a threshold.
• If we let
and use the properties of orthonormal basis
functions, then we find that the test statistic is
(9.19)
• A device that implements this test statistic is called an energy
detector or radiometer shown in Figure 9.1.
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Figure 9.1: Radiometers: (a) passband, (b) baseband with integration, and (c)
baseband with sampling at rate 1/W and summation.
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• The filter has center frequency fc bandwidth W, and produces the
output
(9.20)
– n(t) is bandlimited white Gaussian noise with a two-sided
power spectral density equal to N0/2.
• Squaring and integrating y(t) taking the expected value, and
observing that n(t) is a zero-mean process, we obtain
(9.21)
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• A bandlimited deterministic signal can be represented as
(9.22)
• The Gaussian noise emerging from the bandpass filter can be
represented in terms of quadrature components as
(9.23)
• Substituting (9-23) and (9-22) into (9-20), squaring and
integrating y(t) and assuming that
we obtain
(9.24)
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• The sampling theorems for deterministic and stochastic processes
provide expansions of
(9.25)
(9.26)
(9.27)
(9.28)
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• We define
• Substituting expansions similar to (7-25) into (7-24) and then
using the preceding approxi-mations, we obtain
(9.29)
where it is always assumed that TW ≧1
• A test statistic proportional to (9-29) can be derived for the
baseband radiometer of Figure 9.1(c) and the sampling rate 1/W.
• The power spectral densities of
are
(9.30)
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• The associated autocorrelation functions are
(9.31)
• Therefore, (7-29) becomes
(9.32)
• where the {Ai} and {Bi} the are statistically independent
Gaussian random variables with unit variances and means
(9.33)
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• Thus, 2V/N0 has a noncentral chi-squared (χ2) distribution with
2γdegrees of freedom and a noncentral parameter
(9.35)
• The probability density function of Z=2V/N0 is
(9.36)
• Using the series expansion in of the Bessel function and then
setting λ= 0 in (9-36), we obtain the probability density function
for Z in the absence of the signal:
(9.37)
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• The direct application of the statistics of Gaussian variables to (932) yields
(9.38)
(9.39)
• Equation (9-38) approaches the exact result of (9-21) as TW
increases.
• A false alarm occurs if when the signal is absent. Application of
(9-37) yields the probability of a false alarm:
(9.40)
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• Integrating (7-40) by γ-1 parts times yields the series
(9.42)
• Since correct detection occurs if V >Vt when the signal is present,
(9-36) indicates that the probability of detection is
(9.43)
• The generalized Marcum Q-function is defined as
• A change of variables in (9-43) and the substitution of (9-35)
yield
(9.45)
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• Using (9-38) and (9-39) with and the Gaussian distribution, we
obtain
(9.46)
(9.47)
• In the absence of a signal, (9-21) indicates that
• Thus, N0 can be estimated by averaging sampled radiometer
outputs when it is known that no signal is present.
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• The false alarm rate is
(9.48)
• If V is approximated by a Gaussian random variable, then (9-38)
and (9-39) imply that
(9.49)
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Figure 9.2: Probability of detection versus
for wideband radiometer
with
and various values of TW. Solid curves are the
dashed curve is for
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• Inverting (9-49). Assuming that
we obtain the necessary value:
(9.50)
• The substitution of (9-47) into (9-50) and a rearrangement of
terms yields
(9.51)
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• As TW increases, the significance of the third term in (9-51)
decreases, while that of the second term increases if h > 1.
• Figure 9.3 shows versus TW for PD=0.99 and various values of
PF and h.
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7.2 Detection of Frequency-Hopping Signals
Ideal Detection
• The idealized assumptions:
– The hopset is known
– 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
(9.55)
–
is the CPM component that depends on the data
sequence dn and
is the phase associated with the ith hop.
• The parameters
and the components of dn are modeled as
random variables.
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• Dividing the integration interval in (9-7) into Nh parts, averaging
over the M frequencies, and dropping the irrelevant factor 1/M,
we obtain
(9.56)
(9.57)
• The average likelihood ratio of (9-56) 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.
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• Figure 9.4: General structure of optimum detector for frequencyhopping signal with hops and M frequency channels.
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• Each of the Nd data sequences that can occur during a hop is
assumed to be equally likely.
• For coherent detection of FH/CPM, we set
in (9-55),
substitute it into (9-57), and then evaluate the expectation to
obtain
(9.58)
• Equations (9-56) and (9-58) define the optimum coherent detector
for any slow frequency-hopping signal with CPM.
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• For noncoherent detection of FH/CPM, the received carrier phase
is assumed to be uniformly distributed over [0, 2π) during a
given hop and statistically independent from hop to hop.
• Substituting (9-55) into (9-57), averaging over the random phase
in addition to the sequence statistics, and dropping irrelevant
factors yields
(9.59)
(9.60)
(9.61)
• Equations (9-56), (9-59), (9-60), and (9-61) define the optimum
noncoherent detector for any slow frequency-hopping signal with
CPM.
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• Figure 9.5: 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 n.
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• A major contributor to the huge computational complexity of the
optimum detectors is the fact that with Ns data symbols per hop
and an alphabet size q there may be
data sequences per
hop.
• Consequently, the computational burden grows exponentially
with Ns .
• The preceding theory may be adapted to the detection of fast
frequency hopping signals with MFSK as the data modulation.
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• we may set
in (9-58) and (9-59).
• For coherent detection, (9-58) reduces to
(9.62)
• Equations (7-56) and (7-62) define the optimum coherent detector
for a fast frequency-hopping signal with MFSK.
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• For noncoherent detection, (7-59), (7-60), and (7-61) reduce to
(9.63)
(9.64)
• Equations (9-56), (9-63), and (9-64) define the optimum
noncoherent detector for a fast frequency-hopping signal with
MFSK.
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