Detection of

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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
  direct-sequence 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
• ?is 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 ?orthonormal basis functions
  constitute the received vector
• Let?denote 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 ?.
• is the conditional density function
  of r given hypothesis H0
• is the expectation over the random vector ?.

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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, 2p)
• 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?) 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 zero-mean, white Gaussian
  process.
• Consider two hypotheses that both assume the
  presence of a zero-mean, 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 ? 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 (?2)
  distribution with 2?degrees of freedom and a
  noncentral parameter
• (9.35)
• The probability density function of Z2V/N0 is
• (9.36)
• Using the series expansion in of the Bessel
  function and then setting ? 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 (9-32) 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 ?-1 parts times yields the
  series
• (9.42)
• Since correct detection occurs if V gtVt 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 gt 1.
• Figure 9.3 shows versus TW for PD0.99 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 frequency-hopping 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, 2p) 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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