1 Chapter 9 Detection of Spread-Spectrum Signals

2 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.

3 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 N 0 /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.

4 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 –f c is the known carrier frequency –ψis the carrier phase assumed to be constant over the observation interval 0 ≦ t ≦ T.

5 The spreading waveform p(t) which subsumes the random data modulation, is given by (2-2) with the {p i } 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 H 1 that the signal is present and the hypothesis H 0 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 N 0 /2.

6 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 H 1 and the value of θ. – is the conditional density function of r given hypothesis H 0 – is the expectation over the random vector θ.

7 (9.4) (9.5) where s i the are the coefficients of the signal. Substituting these equations into (9-3) yields (9.6) (9.7)

8 If N is the number of chips, each of duration received in the observation interval, then there are 2 N 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)

9 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)

10 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)

11 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.

12 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 H 0 only noise is present, and the one-sided power spectral density over the signal band is N 0 –Under H 1 both signal and noise are present, and the power spectral density is N 1 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)

13 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.

14 Figure 9.1: Radiometers: (a) passband, (b) baseband with integration, and (c) baseband with sampling at rate 1/W and summation.

15 The filter has center frequency f c bandwidth W, and produces the output (9.20) –n(t) is bandlimited white Gaussian noise with a two-sided power spectral density equal to N 0 /2. Squaring and integrating y(t) taking the expected value, and observing that n(t) is a zero-mean process, we obtain (9.21)

16 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)

17 The sampling theorems for deterministic and stochastic processes provide expansions of (9.25) (9.26) (9.27) (9.28)

18 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)

19 The associated autocorrelation functions are (9.31) Therefore, (7-29) becomes (9.32) where the {A i } and {B i } the are statistically independent Gaussian random variables with unit variances and means (9.33)

20 Thus, 2V/N 0 has a noncentral chi-squared (χ 2 ) distribution with 2γdegrees of freedom and a noncentral parameter (9.35) The probability density function of Z=2V/N 0 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)

21 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)

22 Integrating (7-40) by γ-1 parts times yields the series (9.42) Since correct detection occurs if V >V t 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)

23 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, N 0 can be estimated by averaging sampled radiometer outputs when it is known that no signal is present.

24 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)

25 Figure 9.2: Probability of detection versus for wideband radiometer with and various values of TW. Solid curves are the dashed curve is for

26 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)

27 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 P D =0.99 and various values of P F and h.

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 d n and is the phase associated with the ith hop. The parameters and the components of d n are modeled as random variables.

29 Dividing the integration interval in (9-7) into N h 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.

30 Figure 9.4: General structure of optimum detector for frequency- hopping signal with hops and M frequency channels.

31 Each of the N d 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.

32 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.

33 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.

34 A major contributor to the huge computational complexity of the optimum detectors is the fact that with N s data symbols per hop and an alphabet size q there may be data sequences per hop. Consequently, the computational burden grows exponentially with N s. The preceding theory may be adapted to the detection of fast frequency hopping signals with MFSK as the data modulation.

35 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.

36 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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