Post on 25-Jun-2018
Detection of Frequency Hopping Signals Using
a Sweeping Channelized Radiometer
Janne J. Lehtomaki∗, Markku Juntti
Mailing address: (J. J. Lehtomaki)
Janne Lehtomaki
Centre for Wireless Communications (CWC)
P.O. Box 4500
90014 University of Oulu
FINLAND
Mailing address: (M. Juntti)
Markku Juntti
Centre for Wireless Communications (CWC)
P.O. Box 4500
90014 University of Oulu
FINLAND
Tel.: (J. J. Lehtomaki): +358 8 553 2862
Tel.: (M. Juntti): +358 8 553 2834
Email: (J. J. Lehtomaki) janne.lehtomaki@ee.oulu.fi
Email: (M. Juntti) markku.juntti@ee.oulu.fi
Fax.: +358 8 553 2845
Preprint submitted to Elsevier Science 3 February 2005
Unusual symbols used in the article: � (page 22)
The number of pages: 40
The number of tables: 0
The number of figures: 6
Key words: channelized radiometer, frequency sweeping, intercept receiver, signal
detection.
List of Symbols
C Set of local decisions in all hops and channels
R Set of normalized radiometer outputs
η Hard-decision threshold for individual radiometers
γH Instantaneous hop SNR
γH Average hop SNR
Γ Gamma function
κi ith cumulant of the per hop decision variable
Λ Optimal detection statistic
λ Non-centrality parameter
Λij Local likelihood ratio
Q Generalized Marcum’s Q function
� Term in the efficient representation of the Bessel function, see equation (17)
c Constant term related to the local likelihood ratio, see equation (15)
2
Cij Local decision based on the radiometer output
d1 Term in the approximation of the logarithm of the likelihood ratio, see
equation (19)
d2 Term in the approximation of the logarithm of the likelihood ratio
EH Received signal energy per hop
Eij Energy of the frequency hopping signal in the time–frequency area
corresponding to the radiometer output j in hop i
fj Signal is present in hop channel j
fWi Discrete density function of the per hop decision variable
fW Discrete density function of the final decision variable
H0 Noise-only hypothesis
H1 Signal–and–noise hypothesis
Iv Modified Bessel function of the first kind
K Number of detection phases within each hop
kM Final detection threshold
M Number of hops observed per decision
N Number of radiometers
N0 One-sided power spectral of additive white Gaussian noise process
NH Number of non-overlapping hop channels
3
Neff Total number of radiometer outputs within a hop
p0 Probability of false alarm per hop
p1 Probability of detection per hop
pI Probability of intercept per hop
PD Final probability of detection
PFA Final probability of false alarm
QD Individual radiometer’s probability of detection
QFA Individual radiometer’s probability of false alarm
Rk0 kth raw moment of the per hop decision variable Wi in the noise–only case
Rk1 kth raw moment of Wi in the signal–and–noise case assuming interception
Rij Normalized (scaled with 2/N0) radiometer output
t Term in the efficient representation of the Bessel function,
see equation (17)
TH Hop duration
TR Individual radiometer’s integration time
uk Term in the efficient representation of the Bessel function,
see equation (17)
Vij Measured energy
W Final decision variable
4
WH Hop bandwidth
Wi Per hop decision variable
WR Individual radiometer’s bandwidth
z Term in the approximation of the logarithm of the likelihood ratio, see equation (19)
Zk1 kth raw moment of Wi in the signal–and–noise case
5
Abstract
The paper presents a novel performance analysis of a frequency sweeping channel-
ized radiometer when the signal to be detected is a frequency hopping signal. Often
technology limits the instantaneous bandwidth that can be used. When frequency
sweeping (search strategy) is used, the center (carrier) frequency of the receiver is
changed rapidly to increase the probability of intercept (POI). Conventional log-
ical OR, sum and maximum based methods are used to combine the channelized
radiometer outputs. Exact results are derived for the sum based channelized ra-
diometer. A novel accurate approximation of the performance of the sweeping max-
imum based channelized radiometer is presented. An efficient method is presented
for accurately calculating the likelihood ratio used in optimal detection. The effects
of fading are analyzed. Numerical results show that although sweeping increases
POI, the final probability of detection is not increased if the number of hops ob-
served is large. When the number of hops observed is small, sweeping can increase
performance in the case of fading channel.
1 Introduction
In unknown signal detection, one important task is to decide whether only
noise or noise and signal is present. Detection is a key function in electronic
support (ES) receivers. These receivers are used to search, locate and identify
sources of electromagnetic radiation [23, p. 6]. The Neyman–Pearson detection
criterion [10, p. 62-63] is typically used. The best detector in the Neyman–
Pearson sense is the one that gives the best probability of detection for a
given probability of false alarm. Interception occurs when at least part of the
signal energy can be used for detection. This means that antenna pointing and
6
frequency band of the signal and the ES receiver coincide at a certain time
instant. Detection is possible when the signal is intercepted.
A radiometer is an energy (or power) measurement device. It integrates energy
in a frequency band and can be used for detection by alerting when the en-
ergy integrated exceeds a threshold [28]. In the integrate-and-dump case, the
radiometer output is sampled periodically [8, p. 13]. A channelized radiometer
integrates energy in many bands simultaneously by using multiple radiome-
ters. It can be used for detection of a frequency hopping signal by combining
different radiometer outputs and by summing the outputs corresponding to
different time-intervals. This summing is sometimes called binary integration.
In block-by-block detection, summing is done over non-overlapping intervals.
Final decision is made by comparing the sum value against another threshold.
In binary moving-window (BMW) detection, the summing is done in overlap-
ping blocks and alarm is made when the sum crosses a threshold in upwards
direction [8, p. 20-23]. Nemsick [19] has studied cell-averaging type algorithms
in the context of the channelized radiometer (see also [13]). Usually in de-
tection studies, non-fading additive white Gaussian noise (AWGN) channel is
assumed. The simple AWGN channel can be used, for example, for modelling
air-to-air channels. However, in many practical situations fading occurs. In
[17], effects of movement between the intercept receiver and the signal source
on a total power radiometer detecting frequency hopped signals have been
studied. Therein, it was assumed that the channel consists of a direct compo-
nent and a reflected component (multipath fading).
In a practical implementation, instantaneous bandwidth may be smaller than
the signal bandwidth. In such a system, the simplest possibility is to use the
same center (carrier) frequency at all times. Miller et al. [18] have analyzed a
7
system for detecting slow frequency hopping signals where the bandwidth of
each radiometer is increased so that the intercept probability is increased. This
is a useful approach mainly in systems where the number of radiometers is
limited. Another possibility is to change the center frequency of the receiver,
i.e, to perform frequency sweeping [8, p. 32],[15]. Sweeping faster than hop
dwell time has been proposed in [15], where the performance of sweeping sum
based envelope receiver implemented with the Fast Fourier Transform (FFT)
was analyzed. Dillard and Dillard [8, p. 32–34, p. 112] have analyzed a system
using one radiometer that is continuously swept and the center frequency
is changed with a constant slope. Approximate results are given there for a
system using BMW detection to combine the radiometer outputs.
Beaulieu et al. [2] have presented the optimum detector for fast frequency hop-
ping signals. It is closely related (in the noncoherent case) to the Woodring-
Edell detector [14]. The difference is that envelope detectors are used instead
of bank of radiometers. Therein, a simplified detector called multiple-hop max-
imum likelihood (MML) is also introduced. It corresponds to a channelized
radiometer using logical–OR function. A compressive receiver is an alternative
to channelized receivers. Its use for detecting fast frequency hopped signals has
been studied in [26]. In the compressive receiver, the signal is mixed with a
chirp-type signal and is filtered with a pulse compression filter. Typically the
decision whether signal is present or not is made based on a fixed number of
samples. In sequential detection, the number of samples can vary depending
on the specific values of the samples. Snelling [25] has studied sequential de-
tection of fast frequency hopped signals. Therein, it was shown that optimal
sequential test requires (on average) less samples than a fixed test.
In this paper, we study interception and detection of slow frequency hopping
8
signals using a channelized radiometer. Analysis of the effects of frequency
sweeping on a channelized radiometer is presented. Different methods to com-
bine the channelized radiometer outputs are analyzed. These methods are
logical OR–sum, sum–sum and max–sum. The performance of a logical–OR
based channelized radiometer is calculated by applying the results derived in
[18]. In addition to these practical methods, optimum detection is also ana-
lyzed. Optimal detection statistics conditioned on the normalized radiometer
outputs are presented.
The contributions of this paper can be summarized as follows:
• We derive exact results for the sum–sum channelized radiometer by calcu-
lating the relevant discrete density functions.
• The performance of a maximum–sum based channelized radiometer is found
with a novel application of the shifted log–normal approximation.
• An efficient method for calculating the numerically demanding likelihood
ratio used in the optimal detection is proposed and its accuracy is studied.
• The effects of the sweeping speed on the detectors mentioned above are
analyzed. Numerical results are presented for two cases. Case (a) has a large
number of observed hops and case (b) has a small number of observed hops.
• The effects of fading on the logical OR–sum, sum–sum and maximum–sum
based detectors are analyzed.
The paper is organized as follows. Statistical assumptions and the system
model are described in Section 2. Analysis of the logical OR–sum, sum–sum
and the max–sum methods is presented in Section 3 for the non-fading channel.
9
In Section 4, the optimal detector and the likelihood ratio are discussed. In
Section 5, the effects of fading are analyzed. In Section 6, we find the required
signal-to-noise ratio (SNR) as a function of the sweeping speed assuming the
signal to be detected uses similar parameters as the SINCGARS combat radio.
Finally, the conclusions are drawn in Section 7.
2 System Model
The detection problem is to decide between hypotheses H0 and H1 based on
the received signal r (t)
H0 : r (t) = n (t)
H1 : r (t) = s (t) + n (t) ,
where H0 denotes the noise–only hypothesis and H1 denotes the signal–and–
noise hypothesis. The noise n (t) is assumed to be a white Gaussian process
with one-sided power spectral density N0 and the signal to be detected s (t)
is assumed to be a frequency hopping signal with NH non-overlapping hop
channels. The hop bandwidth is WH and the hop duration is TH . All the
signal energy is assumed to be contained within the hop bandwidth and the
signal energy is assumed to be approximately evenly distributed over the hop
duration [8].
10
2.1 Basic Channelized Radiometer
Fig. 1 shows an intercept receiver that has N radiometers with adjacent fre-
quency ranges, which form the channelized radiometer, each with bandwidth
WR. The total bandwidth of the intercept receiver is NWR. The channelized
radiometer measures the received signal energy in adjacent channels with in-
tegration time TR. In the case of hard-decision processing, these energies are
compared to a threshold η. After hard-decision, the outputs from different
channels are combined with either a logical–OR [7],[8, p. 26],[18] operation
or a sum [12, 15]. The radiometer outputs from different channels can also
be directly combined by taking the largest output (maximum). Let M denote
the total number of observed time-intervals per decision. In all cases, the com-
bined outputs corresponding to different time-intervals are summed to form
the final decision variable W . The sum W is compared to a threshold kM . If
the sum is larger than or equal to the threshold it is decided that a signal was
present in addition to just noise, i.e., hypothesis H1 is accepted. Otherwise it
is decided that only noise was present in the received signal, i.e., hypothesis
H0 is selected. For simplicity, we assume synchronization with the hop timing
and that the frequency ranges of the individual radiometers match that of the
hop channels [14, 15]. Thus, TR = TH , M is the number of hops per decision,
and WR = WH . When the synchronization assumption is not valid there is
random splitting of the signal energy in time and/or frequency. This can be
approximately taken into account by adding energy loss to the required SNR
[8, p. 72-77]. According to Dillard [8], the effective energy loss resulting from
a random splitting of the signal energy in time into two cells of integration
is typically 0.5-2 dB. Asynchronous operation does not always result in SNR
11
loss: if the detection probability is limited by the intercept probability (no
matter how high SNR), asynchronous operation helps because it increases the
probability of intercept [18].
2.2 Sweeping Channelized Radiometer
Typically N < NH . In such a case, the detector can step the received fre-
quency band K times within each hop (synchronization is still assumed) to
increase the probability of intercept [15]. This means the integration time
per radiometer output is reduced, TR = TH/K. The radiometer outputs are
sampled before the center frequency is changed. This means that the total
number of radiometer outputs within a hop is Neff = KN . These outputs are
the measured energies. We index these outputs so that in the first phase the
outputs have indices 1, 2, . . . ,N ; in the second phase the outputs have indices
N +1, N +2, . . . , 2N , and so on. Due to the synchronization assumption, this
detection structure is analytically equivalent to Neff radiometers with band-
width WH and integration time TH/K [15]. Time–frequency product of each
individual radiometer is TRWR = (TH/K)WH . In the following, it is assumed
that TRWR is an integer or that it is rounded to an integer. Fig. 2 shows the
detection structure for the case of K = 3 detection phases per hop duration.
Sweeping makes the analytically equivalent instantaneous bandwidth large,
but the integration time is reduced. The probability of intercept per hop is
pI = Neff/NH ≤ 1, because all hop channels are assumed to be equally likely
(random frequency hopping). Note that in some systems it is possible to use
only a part of the channels. However, this is not considered here. In Fig. 2,
channels 1–12 are searched during one hop and there are 15 possible hop chan-
12
nels. Therefore, pI = 80%. The probability of intercept is the same if channels
2–13, 3–14 or 4–15 are searched instead. It is possible to combine sweeping
faster than hop dwell time and this ”hop level” sweeping (which does not af-
fect the results). For example, let us assume that NH = 16, N = 4 and K = 2.
Now channels 1–8 can be searched in the first hop and channels 9–16 in the
second hop (and 1–8 in the third hop and so on). If K = 1, the search pattern
could be 1–4, 5–8, 9–12, 13–16. If K = 4, all the channels are searched during
a single hop.
The energy of the frequency hopping signal in the time–frequency area of the
radiometer that intercepts the signal is assumed to be Eij = EH/K, where
EH is the received energy per hop, i is the hop index, i ∈ {1, 2, · · · , M},
and j is the radiometer output index, j ∈ {1, 2, · · · , Neff}. In practice, this
assumption is an approximation, and the signal may leak some energy also
to adjacent radiometers. Let Rij = 2Vij/N0, where Vij is the measured en-
ergy, denote the normalized (scaled with 2/N0) radiometer output. The local
decision Cij based on the (normalized) radiometer output is
Cij =
1, Rij > η
0, otherwise
The final decision variable is
W =M∑i=1
Wi, (1)
where Wi =Neff∑j=1
Cij assuming a sum is used to combine the local decisions.
A logical–OR operation can alternatively be used to combine all Neff local
decisions within a hop. In this case, the per hop decision variable Wi = 1 if at
13
least one Cij = 1, j ∈ {1, 2, · · · , Neff}; otherwise it is zero. The maximum–
based intercept receiver sums per hop maxima of the radiometer outputs [5,
11, 20], so that Wi = maxj {Rij}. In [5] the envelope detector outputs were
combined using the maximum and the performance was analyzed using the
normal approximation. Therein, the proposed detector was called the sum–of–
largest–envelopes (SLE) receiver. In [11] the channelized radiometer outputs
were combined with the maximum. Therein, numerical convolution and the
normal approximation were used. This can be viewed to be an extension of the
sum–of–largest–envelopes–squared (SLES) receiver [20] to a situation where
TRWR ≥ 1. In [20] it was found that the SLES receiver has slightly better
performance than the SLE receiver.
The statistical properties of the local decisions and the resulting decision rules
are analyzed in the next section. The optimal channelized radiometer output
combining method, based on the average-likelihood ratio, is considered in Sec-
tion 4.
3 Performance analysis
The statistical properties of the local decisions are discussed in Section 3.1.
Analysis of the logical OR–sum detector is presented in Section 3.2, followed
by the sum–sum detector analysis and the max–sum detector analysis in Sec-
tions 3.3 and 3.4, respectively. The analysis of the logical OR–sum detector in
Section 3.2 is based on [18]. It is presented to enable comparisons.
14
3.1 Instantaneous Radiometer Outputs
In the noise–only case, the distribution function of Rij can be approximated
by the chi-square distribution with 2TRWR degrees of freedom. In the deter-
ministic signal–and–noise case, the distribution can be approximated by the
non-central chi-square distribution with 2TRWR degrees of freedom and non-
centrality parameter λ = 2Eij/N0 [28]. The above results are usually called
exact although they are approximations in the case of the conventional analog
implementation [8].
The probability of false alarm is the probability that the radiometer output ex-
ceeds a threshold when only noise is present. The individual radiometers have
the probability of false alarmQFA = P (Rij > η|Eij = 0) = P (Cij = 1|Eij = 0).
It is given by [8, eq. 3.2]
QFA =
∞∫η
xTRWR−1e−x/2
2TRWRΓ (TRWR)dx, (2)
where η is the threshold and Γ is the gamma function [1, 6.1.1]. Based on (2)
we can calculate the threshold for the required probability of false alarm. The
probability of detection QD is the probability that the threshold is exceeded
when signal and noise are present, i.e., it is QD = P (Rij > η|Eij > 0) =
P (Cij = 1|Eij > 0). Assuming a deterministic signal [8, p. 57],[28]
QD = QTRWR
(√2Eij/N0,
√η), (3)
where QTRWR() is the generalized Marcum’s Q function [21, eq. (2-1-122)] with
parameter TRWR. The energy required for a given probability of detection QD
15
can be approximated with [18, Eq. (13a)]
Eij/N0 ≈ 1
2
([√η − (2TRWR − 1) /2 − Q−1 (QD)
]2
− (2TRWR − 1) /2
), (4)
where Q−1 is the inverse of the tail integral of the normal distribution. This
result with the exact threshold, i.e., inverse of the function (2), is a very
accurate approximation, especially when QD is relatively high.
3.2 Logical OR–Sum Decision Rule
The probability of false alarm per hop when using logical–OR is [18]
p0 = P (Wi = 1|H0) = 1 − (1 −QFA)Neff , (5)
because a false alarm occurs if at least one radiometer output exceeds the
threshold. The probability of detection per hop is [18]
p1 = P (Wi = 1|H1) =
pI
(1 − (1 −QD) (1 −QFA)Neff−1
)+ (1 − pI) p0,
(6)
because at most one time–frequency cell (radiometer output) can have signal
energy. The probability of this occurring is the probability of intercept, pI .
If the signal is not intercepted, the probability of detection is the false alarm
probability. The total probability theorem can be applied to combine these
two mutually exclusive events to get the result in (6). Assuming that the hop
positions are independent, the final probability of false alarm over M observed
16
hops is [7, 18]
PFA = P (W ≥ kM |H0) =M∑
i=kM
M
i
pi
0 (1 − p0)M−i, (7)
where kM is the final threshold that is used after summing the last M logical–
OR outputs. The final probability of detection is similarly [7, 18]
PD = P (W ≥ kM |H1) =M∑
i=kM
M
i
pi
1 (1 − p1)M−i. (8)
3.3 Sum–Sum Decision Rule
In [15], hard decision envelope detector has been studied. Therein, each lo-
cal decision is based on initial decisions which are based on magnitudes (en-
velopes) of the corresponding FFT outputs. These local decisions from all
channels and hops are summed and compared to a threshold, i.e., there are
three different thresholds. The detector performance was evaluated with the
normal approximation. In the sum–sum channelized radiometer considered
here the local decisions Cij are based on the radiometer outputs. There are
only two thresholds to be optimized. For the case studied in [15], a high proba-
bility of detection and a large number of hops, the normal approximation gives
reasonably good results. However, in most cases, the normal approximation is
not very accurate. In [12] exact results have been used. However, therein the
studied scenario was different and the probability of intercept (POI) was not
taken into account.
17
Here we will present an exact method to calculate the probability of detection
and false alarm for the sum–sum channelized radiometer. When only noise
is present, W follows the binomial distribution. Therefore, the probability of
false alarm is
PFA =MNeff∑i=kM
MNeff
i
Qi
FA (1 −QFA)MNeff−i . (9)
In the signal–and–noise case, the discrete density function of Wi is
fWi(x) = P (Wi = x|H1) = pIfWi
(x|H1,inter.) + (1 − pI) fWi(x|H0) , (10)
where fWi(x|H1,inter.) is convolution of a binomial density with parameters
Neff −1 and QFA and a binomial density with parameters 1 and QD. Function
fWi(x|H0) is a binomial density with parameters Neff and QFA. The density
function of the decision variable W is
fW (x) = fWi(x) ∗ fWi
(x) ∗ · · · ∗ fWi(x)︸ ︷︷ ︸
M
, (11)
which can be efficiently calculated with the FFT, and the final probability of
detection is found with
PD = 1 −kM−1∑x=0
fW (x). (12)
3.4 Max–sum Decision Rule
Because the conditional probability density functions in the signal–and–noise
case and the noise–only case are known [5, 11], we can calculate the corre-
18
sponding raw moments of the per hop decision variable Wi. Let Rk0 = E
{W k
i
}denote the kth raw moment in the noise–only case and Rk
1 = E{W k
i
}denote
the kth raw moment in the signal–and–noise case assuming interception. In
the noise–only case, the mean κ1, variance κ2 and third central moment κ3 of
the per hop maximum are
κ1 = R10
κ2 = R20 − (R1
0)2
κ3 = 2 (R10)
3 − 3R10R
20 + R3
0
The raw moments of the per hop maximum when taking the POI per hop
(pI) into account are Zk1 = pIR
k1 + (1 − pI)R
k0 . The mean, variance and third
central moment can be calculated in similar way as in the noise–only case.
Now, because the central moments of a sum of independent random variables
add up to the order of 3 we know the mean, variance and the third central
moment of decision variable W in both the noise–only case and the signal–
and–noise case. Instead of the traditional normal approximation that we used
in [11], we propose the use of more accurate shifted log–normal approximation
that is matched to the first three central moments of the sum [22]. The shifted
log–normal approximation is first used to obtain the threshold and then to
obtain the probability of detection. The shifted log–normal approximation is
especially useful when the number of observed hops is relative small. Fig.
3 shows comparison, corresponding to [11, Fig. 4], between the shifted log–
normal and normal approximations. It can be seen that the shifted log–normal
approximation is more accurate.
19
The method presented above requires knowledge of the raw moments of the
per hop maximum. In [11], the first two moments, to be used with the normal
approximation, were calculated with numerical integration. It is possible to
find the raw moments analytically by writing out the density function of the
maximum and performing symbolic integration using similar techniques as in
[5, 16]. As an example, in the special case TRWR = 1, the following simple
result is obtained
Rv0 = Neff2vΓ (v + 1)
×
Neff−1∑i=0
Neff − 1
i
(−1)Neff−1−i (Neff − i)−(v+1)
. (13)
However, this process is excessively tedious for practical values of the param-
eters (large TRWR and Neff ). Therefore, numerical integration will be used.
4 Optimal detection
4.1 Thresholded Outputs
A optimal detector conditioned on the observable C = {Cij}, i ∈ {1, 2, · · · , M},
j ∈ {1, 2, · · · , Neff}, i.e., the set of local decisions in all hops and channels,
has been derived in [15]. There it was derived in the context of the enve-
lope detector, but the result can also be directly applied in the case of the
channelized radiometer.
In [15], it was shown that at low signal-to-noise-ratios the sum–sum statistic is
20
asymptotically optimal. This does not mean that a detector based on the sum–
sum performs better than a detector based on the logical–OR. Actually, when
only one signal is present, the logical–OR based detector performs slightly
better.
4.2 Direct Outputs
The optimal detection statistic conditioned on the observable R = {Rij}, i.e.,
the set of normalized radiometer outputs, is [9]
Λ (R) =M∏i=1
1
NH
NH∑j=1
Λij (Ri| fj), (14)
where Ri denotes the set of normalized radiometer outputs in the hop i and
fj denotes the hypothesis that the signal is present in channel j. It is assumed
that the signal is equally likely to be in any channel and these channels are
independent from hop to hop. For the sweeping system, the likelihood ratio
Λij is 1 if Neff < j ≤ NH and if j ≤ Neff it is ([8, Eq. (A.12)])
Λij (Ri| fj) = c ·R−TRWR−1
2ij ITRWR−1
(√2EHRij
KN0
), (15)
where ITRWR−1 is the modified Bessel function of the first kind [1, 9.6] with
order TRWR − 1 and
c = 2TRWR−1
2
(EH
KN0
)−TRWR−1
2
Γ (TRWR) e− EH
KN0 . (16)
If no sweeping is used, i.e., K=1, and if also Neff = NH (14) and (15) form
the classical Woodring-Edell (WE) detector [14]. In this case, the constant
term can (16) can be ignored.
21
It can be observed that the optimal detection statistic depends on the signal-
to-noise ratio. The signal-to-noise ratio is usually unknown (it may be esti-
mated after detection) so the optimal detector is not realizable in practice.
However, it is important to study the optimal detector so that the upper
bound on detection performance is found. Note that because R contains more
information than C, the performance of a detector using the statistic (14) is
better than that of the optimal detector conditioned on thresholded radiome-
ter outputs C. The max–sum channelized radiometer also uses the observable
R. This allows it to have in most cases clearly better performance than the
logical–OR or sum–sum detectors. If details of the signal modulation would
be known (for example that the signal is a SFH/CPM signal) more specialized
optimal detection methods could be used [14]. It would be possible to evaluate
the performance of the optimal detector with approximations similar to those
used in [3, 4, 14]. However, in this paper we use simulations.
4.3 Calculating the likelihood ratio
For large values of TRWR, there exists an efficient representation of the Bessel
function, namely [1, 9.7.7]
Iv (z) =1√2πv
ev�(1 + (z/v)2
)1/4
{1 +
∞∑k=1
uk (t)
vk
}, (17)
where ( =(1 + (z/v)2
)1/2+ln (z/v)−ln
(1 +
√1 + (z/v)2
), t = 1
/√1 + (z/v)2
and u1 (t) = 1/8t− 5/24t3. For values of uk(t) with k > 1 refer to [1, 9.3.9].
22
Let us use the following approximation [6]
d2 = ln{
1 +∞∑
k=1
uk(t)vk
}≈ 1
8vt− 5
24vt3 + 1
16v2 t2 − 3
8v2 t4
+ 516v2 t
6 + 25384v3 t
3 − 531640v3 t
5 + 221128v3 t
7 − 11051152v3 t
9
. (18)
By taking logarithm of the likelihood ratio (15) and using (17)–(18) we find
ln Λij (Ri| fj) ≈ d1 + (v2 + z2)1/2
−v ln(
1 +√
1 + (z/v)2)− 1
4ln(1 + (z/v)2
)+ d2,
(19)
where v = TRWR − 1, z =√
2EHRij
KN0,
d1 = v ln 2 + ln Γ (v + 1) − EH
KN0
− v ln v − 1
2ln (2πv) (20)
and d2 is given by (18). If the term d2 is ignored, the approximation (19) is still
rather accurate. The likelihood ratio can be found with the exponential of (19).
Fig. 4 shows the relative error of the individual likelihood ratio Λij (Ri| fj)
calculated with this approximation, i.e., the absolute error divided by the cor-
rect value. The parameters used are EH/KN0 = 30 and TRWR = 250. It is
observed that the approximation (19) with the term d2 is very accurate. There-
fore, when evaluating the performance of the optimal detector with simulations
we will use exponential of (19) with the term d2. This method of evaluating
the likelihood ratio works with a wide range of input values, because first the
logarithm of the likelihood ratio is evaluated. Actually, the likelihood ratio is
almost linear in a logarithmic scale. This means that potentially even simpler
yet still rather accurate expressions could be developed. One way to achieve
this would be to use suitable approximations for the terms in (19).
23
Numerical effects can also affect the calculation of the final decision variable
(14), which is a product of the sums of the individual likelihood ratios. It is
possible to take a logarithm of the final decision variable, which results in
a sum of logarithms of sums of the individual likelihood ratios. However, in
this case it is still necessary to sum exponentials (logarithm cannot be moved
inside a sum).
5 Performance Under Fading
We assume a frequency-nonselective (in the used hop channels) fading, con-
stant over each hop and independent from hop to hop. The frequency-nonselective
fading results in multiplicative distortion of the signal [21, p. 772-773]. Let us
denote the multiplicative scaling factor α. It is Rayleigh-distributed. The in-
stantaneous SNR is now γH = α2EH/N0(phase shift does not affect energy).
The instantaneous SNR (in each hop) is a random variable that follows the
chi-square distribution with two degrees of freedom [21, p. 773] with average
value γH , i.e., the density function is P (γH) = 1/γHe−γH/γH . The results are
given as a function of the average hop SNR, i.e, the average SNR is specified
and the instantaneous SNRs in each hop are independently fluctuating around
the the specified value.
Performance analysis of the logical OR–sum and the sum–sum based receivers
requires finding each individual radiometer’s probability of detection QD (as-
suming that signal is present in the time-frequency area of the radiometer).
The probability of detection for a fixed SNR is given by (3). The probability
24
of detection when taking into account the fluctuating energy is
QD =
∞∫0
QTRWR
(√2γH/K,
√η)
1
γH
e−γH/γHdγH . (21)
By using results of Swerling [27, p. 277], (21) can be written as (see also [24,
Eq. (25)])
QD = QTRWR−1
(0,√η)
+[(
1 + 1γH/K
)TRWR−1
×(
1 −QTRWR−1
(0,√
η1+ 1
γH/K
))e− η/2
1+γH/K
].
(22)
In contrast to the situation with no fading, the shifted log-normal approxi-
mation was found to have rather poor accuracy in the case of fading channel
and it was not used. Instead, the performance of the maximum based receiver
was evaluated with numerical convolutions similar to those used when ana-
lyzing the sum based receiver (see also [11]). This requires more numerical
computations than the shifted log-normal approximation, but gives the exact
performance assuming numerical convolutions have sufficient accuracy (the
distributions are not discrete as they were with the sum based receiver).
6 Numerical results
We assume that the signal to be detected shares some parameters with the
SINGCARS radio [18]: WH = 25 kHz, TH = 0.01 sec and NH = 2320. Other
parameters used here are PFA = 10−3 and N = 464 channels in the receiver.
The probability of intercept for these parameters is pI = KN/NH = 0.2K.
For example, when K = 1, so that no sweeping is performed, pI = 0.2. The
other possible values are 0.4, 0.6, 0.8 and 1. We will study two cases, (a) the
25
number of hops observed per decision M = 300 and (b) M = 16.
The required energy for the sum–sum receiver, for a given threshold, was
calculated by first solving (9) for QFA as a function of kM and the desired false
alarm probability. Then the inverse of the chi-square cumulative distribution
function was used to find the threshold η for individual radiometers. Now QD
can be found by using (3) (no fading) or (22) (fading). The final probability of
detection was found with (12). The required SNR per cell corresponding to the
required probability of detection, for a given threshold kM , was found with a
search. Searching was also used to find the optimal threshold kM . The required
SNR for a logical OR–sum based channelized radiometer was calculated using
procedures similar to those in [18]. Optimal thresholds found with a search
were used. The required SNR for a maximum–based receiver was discovered
by using the shifted log–normal approximation (no fading) or with numerical
convolutions (fading). The optimal threshold of the maximum–based receiver
does not depend on SNR.
The required SNR for the optimal detector was found with simulation. Due
to a very large number of random variables to be generated, simulations in
the case (a) are excessively time consuming. Therefore, the performance of the
optimal detector was evaluated only in the case (b).
In the case (a) we additionally studied hard decision envelope detector pro-
posed in [15]. Its performance was evaluated via normal approximations similar
to those in [15]. We assumed that each envelope detector output containing
signal has equal signal component. In practice, the strength of signal compo-
nent varies and the results can be interpreted to be an approximation allowing
comparisons with other structures. Results we obtained were the same as those
26
in [15]. It would be possible to apply the exact results presented in this paper
for the sum–sum detector also for the envelope detector. Since we concentrate
on the channelized radiometer this was not pursued here further.
Fig. 5 shows the required energy per hop in the case (a) to achieve PD = 0.999
with the envelope detector [15], the sum–sum based channelized radiometer,
the logical OR–sum based channelized radiometer and the maximum based
channelized radiometer. It can be observed from Fig. 5 that the sum–sum re-
ceiver has practically the same performance as the logical OR–sum receiver.
It is anticipated that when multiple signals are present, the sum–sum based
receiver is better than the receiver using the logical OR–sum. Envelope detec-
tion is about 1 dB worse than the radiometer based solutions. When there is
no fading, the maximum based intercept receiver is the best of the receivers
discussed here, except when POI is 20%. When channel is fading, the logical
OR–sum and sum–sum receivers have better performance than the maximum
based intercept receiver. It is seen that when the number of hops observed is
large sweeping does not increase the probability of detection. It is more impor-
tant to have large detection SNR than to have large probability of intercept.
This is in line with the result in [15], obtained for the envelope detector based
system. When channel is fading the detection SNR is sometimes much higher
than the average value (it is a random variable). This explains why fading
actually improves performance. If the frequency band of the transmitter is
unknown, some type of frequency sweeping (at least in the hop level) should
be used. Otherwise the probability of intercept can be very low, even zero.
Fig. 6 shows the required energy per hop in the case (b) to achieve PD = 0.99
with the sum–sum based channelized radiometer, the logical OR–sum based
channelized radiometer, the maximum based channelized radiometer and the
27
optimum detector using detection statistic (14). It was not possible to get the
required PD without sweeping. This is because without sweeping, the prob-
ability that at least one radiometer intercepts the signal in any of the hops
is 1 − (1 − 0.2)16, which is 0.97185. Therefore, the maximum probability of
detection with any detector is 0.97185 · 1 + (1 − 0.97185) · 0.001, which is
smaller than 0.99. It is possible to get the required performance if K ≥ 2, i.e,
probability of intercept is greater than or equal to 40 %. It can be observed
from Fig. 6, that if the probability of intercept is greater than 40 % and there
is no fading, the maximum based detector is the best of the practical detec-
tors. When the probability of intercept is 100 %, the maximum based detector
has performance very close to that of the optimal detector. When channel is
fading, the logical OR–sum and sum–sum receivers have better performance
than the maximum based intercept receiver. It can be observed that also in
this case the sum–sum receiver and the logical OR–sum based receiver have
almost equal performance. In simulations, it was noticed that the thresholds
given by the shifted log–normal approximation are accurate. However, when
K = 2 there was a small difference between the required SNR given by simula-
tions and the approximation (0.08 dB). When K > 2 the difference was much
smaller (0.01–0.02 dB). This is because the shifted log–normal approximation
is not so good fit to the distribution of the decision variable when K = 2 as
when K > 2. In the case of fading, numerical convolutions were used instead
of the shifted log–normal approximation. In case (b) fading degrades perfor-
mance. This is because the number of intercepted hops can be small. When the
detection SNR is fading, quite often only a few of the intercepted hops have
”enough” detection SNR. When the number of intercepted hops increases, the
performance in the case of fading increases. The best performance in the case
of fading is achieved when K = 5 (POI is 100%).
28
The same methods can be used also with signals that have other parameters
(higher hop rate, larger bandwidth). Actually, the results depend only on
the time-bandwidth product, the number of channels in the signal and in
the receiver, the number of observed hops and the required PD and PFA per
decision. For example, if the hop rate is 10 000 hops/s and the bandwidth
of the channels is 2.5 MHz then the time-bandwidth product is 250. If the
other parameters do not change, the results are equal to those presented in
the paper.
7 Conclusions
The logical OR–sum channelized radiometer, the sum–sum channelized ra-
diometer, the max–sum channelized radiometer and the optimal detector us-
ing frequency sweeping have been analyzed. When sweeping is performed there
are multiple detection phases within each hop, i.e, sweeping is faster than the
hop dwell time. The numerical results presented here, for a slow frequency
hopping signal having parameters similar to those of the SINGCARS combat
radio, support the following conclusions. If the number of hops observed per
decision is large, frequency sweeping degrades the performance compared to
a system that does not apply frequency sweeping (with or without fading).
If the number of hops observed is small, sweeping is often necessary to get
the desired performance. When the channel is fading best performance is ob-
tained by using fast sweeping. Using a sum is only slightly worse than using
a logical–OR. If there is no fading, the maximum based intercept receiver has
the best performance of the practical receivers discussed here unless the prob-
ability of intercept (POI) is small. When POI is large, the performance of the
29
maximum based receiver is close to that of the optimal receiver. In the case of
fading, logical OR–sum and sum–sum receivers have better performance than
the maximum based receiver.
8 Acknowledgments
This work was supported by Finnish Defence Forces Technical Research Cen-
tre. The work of J. J. Lehtomaki was supported by Nokia Foundation and the
Graduate School in Electronics, Telecommunications and Automation, GETA.
The authors wish to thank the reviewers for their suggestions to improve the
manuscript. We also thank Ari Pouttu, Johanna Vartiainen, Harri Saarnisaari
and Keijo Ruotsalainen for their help with the article.
References
[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions
with Formulas, Graphs, and Mathematical Table. National Bureau of
Standards, 1964.
[2] N. C. Beaulieu, W. L. Hopkins, and P. J. McLane. Interception of
frequency-hopped spread-spectrum signals. IEEE Journal on Selected
Areas in Communications, Vol. 8, No. 5, June 1990, pp. 853–870.
[3] U. Cheng, M. K. Simon, A. Polydoros, and B. K. Levitt. Statistical
models for evaluating the performance of coherent slow frequency-hopped
M-FSK intercept receivers. IEEE Trans. Commun., Vol. 42, No. 234,
February/March/April 1994, pp. 689–699.
[4] U. Cheng, M. K. Simon, A. Polydoros, and B. K. Levitt. Statistical
30
models for evaluating the performance of noncoherent slow frequency-
hopped M-FSK intercept receivers. IEEE Trans. Commun., Vol. 43, No.
234, February/March/April 1995, pp. 1703–1712.
[5] C. D. Chung. Generalised likehood-ratio detection of multiple-hop
frequency-hopping signals. IEE Proc.–Commun., Vol. 141, No. 2, April
1994, pp. 70–78.
[6] E. R. B. de Mello, V. B. Bezerra, and N. R. Khusnutdinov. Ground
state energy of massive scalar field inside a spherical region in the global
monopole background. Journal of Mathematical Physics, Vol. 42, No. 2,
February 2001, pp. 562–581.
[7] R. A. Dillard. Detectability of spread-spectrum signals. IEEE Trans.
Aerosp. Electron. Syst., Vol. 15, No. 4, July 1979, pp. 526–537.
[8] R. A. Dillard and G. M. Dillard. Detectability of Spread-Spectrum Signals.
Artech House, Norwood, Massachusetts, 1989.
[9] R. A. Dillard and G. M. Dillard. Likelihood-ratio detection of frequency-
hopped signals. IEEE Trans. Aerosp. Electron. Syst., Vol. 32, No. 2, April
1996, pp. 543–553.
[10] S. M. Kay. Fundamentals of Statistical Signal Processing: Detection The-
ory. Prentice Hall, Upper Saddle River, New Jersey, 1998.
[11] J. J. Lehtomaki. Maximum based detection of slow frequency hopping
signals. IEEE Commun. Lett., Vol. 7, No. 5, May 2003, pp. 201–203.
[12] J. J. Lehtomaki. Performance comparison of multichannel energy detec-
tor output processing methods. Proc. Seventh Int. Sympos. on Signal
Processing and Its Applications, Paris, France, July 2003, pp. 261–264.
[13] J. J. Lehtomaki, M. Juntti, and H. Saarnisaari. CFAR strategies for
channelized radiometer. IEEE Signal Processing Letters, Vol. 12, No. 1,
January 2005, pp. 13–16.
31
[14] B. K. Levitt, U. Cheng, A. Polydoros, and M. K. Simon. Optimum
detection of slow frequency-hopped signals. IEEE Trans. Commun., Vol.
42, No. 234, February/March/April 1994, pp. 1990–2000.
[15] B. K. Levitt, M. K. Simon, A. Polydoros, and U. Cheng. Partial-band de-
tection of frequency-hopped signals. Proc. IEEE Globecom’93, Houston,
USA, November/December 1993, pp. 70–76.
[16] C. H. Lim and H. S. Lee. Performance of order-statistics CFAR detector
with noncoherent integration in homogenous situations. IEE Proc. Radar
and Signal Processing, Vol. 140, No. 5, October 1993, pp. 291–296.
[17] S. J. MacMullan. The effect of multipath fading on the radiometric detec-
tion of frequency hopped signals. Proc. IEEE Third International Sympo-
sium on Spread Spectrum Techniques and Applications, Oulu, Finland,
July 1994, pp. 243–247.
[18] 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, No. 2, April 1993, pp. 540–553.
[19] L. W. Nemsick and E. Geraniotis. Adaptive multichannel detection of
frequency-hopping signals. IEEE Transactions on Communications, Vol.
40, No. 9, September 1992, pp. 1502–1511.
[20] W. Ng and N. C. Beaulieu. Noncoherent interception receivers for fast
frequency-hopped spread spectrum signals. Proc. Canadian Conference
on Electrical and Computer Engineering’94, Halifax, Canada, September
1994, pp. 348–351.
[21] J. G. Proakis. Digital Communications, 3rd edition. McGraw-Hill, 1995.
[22] K. L. Q. Read. A lognormal approximation for the collector’s problem.
The American Statistician, Vol. 52, No. 2, May 1998, pp. 175–180.
[23] D. C. Schleher. Introduction to Electronic Warfare. Artech House, Nor-
32
wood, Massachusetts, 1986.
[24] D. A. Shnidman. Radar detection probabilities and their calculation.
IEEE Transaction on Aerospace and Electronic Systems, Vol. 31, No. 3,
July 1995, pp. 928–950.
[25] W. E. Snelling and E. Geraniotis. Sequential detection of unknown
frequency-hopped waveforms. IEEE Journal on Selected Areas in Com-
munications, Vol. 7, No. 4, May 1989, pp. 602–617.
[26] W. E. Snelling and E. Geraniotis. Analysis of compressive receivers for the
optimal interception of frequency-hopped waveforms. IEEE Transactions
on Communications, Vol. 42, No. 1, January 1994, pp. 127–138.
[27] P. Swerling. Probability of detection for fluctuating targets (originally
published in 1954 as RAND research memo RM-1217). IRE Transactions
on Information Theory, Vol. 6, No. 2, April 1960, pp. 269–308.
[28] H. Urkowitz. Energy detection of unknown deterministic signals. Pro-
ceedings of the IEEE, Vol. 55, No. 4, April 1967, pp. 523–531.
33
Figure captions:
Fig. 1 A channelized radiometer using binary integration and various output
processing methods.
Fig. 2 The detection structure in the synchronous case, 15 possible hop chan-
nels, 4 radiometers, 3 detection phases, signal intercepted in the first phase by
the radiometer 3, Neff = 12.
Fig. 3 Theoretical (normal and shifted log–normal approximations) and sim-
ulated miss probabilities for the maximum based channelized radiometer, 100
hops observed, 464 radiometers in the receiver, signal has 2320 FH-channels,
K=1 (POI 20%), PFA = 10−3 and the time–frequency product of the radiome-
ters TRWR = 250.
Fig. 4 Relative error of the approximation to the individual likelihood ra-
tio Λij (Ri| fj), EH/KN0 = 30, TRWR = 250 and j ≤ Neff .
Fig. 5 Required hop SNR, WH = 25 kHz, TH = 0.01, NH = 2320, M = 300,
PD = 0.999 and PFA = 10−3.
Fig. 6 Required hop SNR, WH = 25 kHz, TH = 0.01, NH = 2320, M = 16,
PD = 0.99 and PFA = 10−3.
34
Optional harddecision DFA QQ ,
Binaryintegration
BPF ( RW ) &Radiometer 1
η≥
η≥
η≥
Processingof the harddecisionoutputsfrom theradiometers(LogicalOR, Sum,…)
Processingof the directoutputsfrom theradiometers(Maximum,…)
�=
M
i 1
Mk≥
Finalthreshold
DFA PP ,BPF ( RW ) &Radiometer 2
BPF ( RW ) &Radiometer N
Fig. 1. A channelized radiometer using binary integration and various output pro-
cessing methods.
35
4=N
HT
KTH /
3=K
15=HN
Radiom. 4, Output 4
Radiom. 1, Output 1
Radiom. 4, Output 12
Radiom. 4, Output 8Radiom. 1, Output 9
Radiom. 1, Output 5
Radiom. 2, Output 2
Radiom. 3, Output 7Radiom. 2, Output 6
Radiom. 3, Output 11Radiom. 2, Output 10
Signal withenergy EH
Energy EH / K
Fig. 2. The detection structure in the synchronous case, 15 possible hop channels,
4 radiometers, 3 detection phases, signal intercepted in the first phase by the ra-
diometer 3, Neff = 12.
36
18 18.5 19 19.510
−4
10−3
10−2
10−1
100
SNR per hop [dB]
mis
s pr
obab
ility
Maximum based (simulated)Shifted log normal approximationNormal approximation
Fig. 3. Theoretical (normal and shifted log–normal approximations) and simulated
miss probabilities for the maximum based channelized radiometer, 100 hops ob-
served, 464 radiometers in the receiver, signal has 2320 FH-channels, K=1 (POI
20%), PFA = 10−3 and the time–frequency product of the radiometers TRWR = 250.
37
100 200 300 400 500 600 700 800 900 100010
−14
10−12
10−10
10−8
10−6
10−4
10−2
Rij
Rel
ativ
e er
ror
term d2 ignored
with term d2
Fig. 4. Relative error of the approximation to the individual likelihood ratio
Λij (Ri| fj), EH/KN0 = 30, TRWR = 250 and j ≤ Neff .
38
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 116
17
18
19
20
21
22
Probability of Intercept
Req
uire
d ho
p S
NR
[dB
]
Fading channel
No fading
Envelope detector based systemSum−sum channelized radiometerOR−sum channelized radiometerMax−sum channelized radiometer
Fig. 5. Required hop SNR, WH = 25 kHz, TH = 0.01, NH = 2320, M = 300,
PD = 0.999 and PFA = 10−3.
39
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 121
21.5
22
22.5
23
23.5
24
24.5
25
25.5
26
Probability of Intercept
Req
uire
d ho
p S
NR
[dB
]
Fading channel
No fading
Not possible to get the required performance
Sum−sum channelized radiometerOR−sum channelized radiometerMax−sum channelized radiometerOptimum detector
Fig. 6. Required hop SNR, WH = 25 kHz, TH = 0.01, NH = 2320, M = 16,
PD = 0.99 and PFA = 10−3.
40