Work involves blind detection of Frequency Hopping Spread Spectrum Signal in operating in HF band.

1. Performance Analysis of Wavelet Based Blind
Detection and Hop Time Estimation Algorithm for
Frequency Hopping Signals in HF Band
Saira Anwar, Dr. Adnan Ahmed Khan, Dr. Imran Touqeer, Dr. Mohammed Imran
College of Signals, National University of Sciences and Technology,
Islamabad, Pakistan.
Abstract—The paper is based on the detection and
characterization of frequency hopping signals (FH), operating in
HF band by using Discrete Haar Wavelet Transform. Our
proposed work is focused on the blind detection and hop time
estimation of FH. It does not require any prior information about
the hop frequencies, hop pattern or modulation type. For
reception of FH signal, the first step is to detect the presence of
the signal. The detection technique used in this work is wavelet
based transient detection. For the hop time estimation of FH
signal we first extract the phase information from the temporal
correlation function (TCF) of received signal. Then, by applying
certain de-noising techniques, discrete wavelet transform (DWT)
is used for extracting the time of hopping. Simulation results in
the form of detection statistics are presented in Additive White
Gaussian Channel (AWGN).
Keywords—Frequency Hopping Spread Spectrum (FHSS), Temporal
Correlation Function (TCF), Discrete Wavelet Transform (DWT)
I. INTRODUCTION
Frequency hopping spread spectrum (FHSS) modulations
are well-known for their low probability of interception,
privacy and anti-jamming capability. FHSS is therefore
preferably used in military/commercial radios. FHSS uses a
narrow band data signal and that signal is usually FSK
modulated. This modulated signal “hops” in random and
hopping is carried out in pseudo-random (predictable)
sequence in a regular time from one frequency to another in
spread bandwidth [1].
On account of the underlying advantages of FHSS, various
techniques have been used in literature for the detection and
parameter characterization of FH Signals. Parameter estimation
of FH Signal incorporates determination of its time of hop
(hopping instant), hop period, hop frequencies, hop sequence
and hop rate. The methodologies used for the detection of FH
signal and extraction of its features depend on the bandwidth of
FH signal and also its frequency band. Detection techniques
include energy detection, maximum likelihood approach, auto-
correlation based detection [2], polyphase filter banks and
wavelet detection. A comparison of few techniques is given in
[3]. Energy detection is not robust to noise and its detection
performance degrades at low SNR. Maximum Likelihood
detector has very good performance but it requires prior
knowledge of the FH signal so it cannot be used for non-
cooperative detection. The use of polyphase filter banks using
Fast Fourier Transform (FFT) was presented in [4]. Time-
frequency distributions have also been used for estimating the
features of FH signals [5]. In [6], reassigned smoothed pseudo
Wigner-Ville Distribution (SPWVD) is employed to determine
hop frequencies, frequency hop pattern and hop rate of FH
signal. Compressive sensing is proposed [7 – 10] for the
detection of wideband FH signals in the Industrial, Scientific
and Medical (ISM) band. In [11 – 13], wavelet analysis is used
for detection and for extracting the time of hop. Wavelet re-
arrangement algorithm is presented in [14] for estimating FH
hop rate and hop frequencies.
Our work is based on the Detection and Estimation of the
hopping instant of FH Signal in HF band, by using DWT.
Wavelet Analysis is a time-frequency analysis technique that
allows Multi-Resolution Analysis (MRA). It has the ability to
zoom in (so as to have a detailed view) or zoom out (to have a
global view) of the signal. In an FH Signal, when the frequency
hops from one value to another, a switching transient is
generated. Discrete Wavelet Transform (DWT), using Haar
Wavelet, is applied in our work. It is very efficient in detecting
discontinuities and breakdown points in a signal. In our
proposed algorithm, DWT is applied to TCF of the FH Signal.
Our simulation results show the detection and hop time
estimation performance in different SNRs. Paper is organized
in sections. Section II gives the concept of DWT, Section III
describes system model, Section III contains the detection &
estimation algorithm and Section IV provides the simulation
results in Additive White Gaussian Noise (AWGN) and finally
the conclusions are given in Section V.
II. DISCRETE WAVELET TRANSFORM
Temporal Correlation Function (TCF) provides time
correlation of non-stationary signals. TCF of any signal x(t)
can be written as [7]:
��� , � = + � ∗
− � (1)
Here,‘t’ is the signal’s time, ‘�′ is the lag time and * is the
complex conjugate.
After applying pre-processing techniques to the TCF phase
plot (explained in the following section), we have used DWT
to detect frequency transition points.

2. In DWT, filters of different cutoff frequencies are used to
examine the signal at different frequencies (scales). The signal
is passed through a number of high pass filters and low pass
filters to study the high frequency and low frequency
components of the signal, respectively. The filtering
operations change the resolution of the signal and the scale is
changed by upsampling and downsampling operations.
A scaling function provides a series of approximations of a
signal, such that each differs by a factor of 2 in scale (or
resolution) from its neighboring approximations. Wavelets
provide the difference in information between adjacent
approximations. Scaling functions together with wavelets are
used to analyze the signal. The scaling function actually solves
the issue of the infinite number of wavelets required to cover
the spectrum down to zero and lays down a lower limit for
wavelets. A wavelet has a band-pass spectrum and a scaling
function has a low-pass spectrum, so a series of dilated
wavelets along with a scaling function can be considered as a
filter bank.
We can say that any function, x(t) can be expanded as:
x(t) = xa(t) + xd(t) (2)
where, xa(t) is an approximation of the function x(t), using
scaling functions, and xd(t) is the difference between the
function x(t) and its approximation expressed as a sum of
wavelets. The low frequencies of x(t) are given by xa(t) ( it
gives the average value of x(t) in every integer interval). On
the other hand, xd(t) gives the detail. When described in terms
of the iterative filter bank, the low pass filter provides the
approximation and the high pass filter gives the detail
coefficients.
III. SYSTEM MODEL
A two-hop FH signal is considered. The frequency range is 2 –
30 MHz, sampling frequency is 60 MHz and minimum
hopping time, Thop_min is chosen to be 256 sample points
(4.27µs). Frequency differential (minimum difference between
hop frequencies) is 2 kHz. In this set up, 500 trial experiments
are conducted. In each experiment, the signal jumps from a
hop frequency to another, at a random hop time. Both
frequencies belong to a finite alphabet in the range 2 – 30
MHz. The Detection & Estimation Algorithm, presented in
our paper, detects the presence of hop in each frame and
estimates the time of hop. To completely test the performance
of algorithm, 10% of the experiments are no hop frames.
Mathematically, a FH signal can be written as [5]:
= � ∑ � �ℎ − �ℎ − � 2��� �− �ℎ−�
+
(2)
Here 0 < t < T, is Additive white Gaussian Noise, fk is a
hop frequency in the spread bandwidth.
� �ℎ (t) = {
∈ −
�ℎ
2
,
�ℎ
2
ℎ
(3)
IV. DETECTION AND ESTIMATION ALGORITHM
Algorithm steps are elaborated as under:
1) The FH Signal is generated with the following
specifications:
a) Frequency Bandwidth used for spreading: 2 – 30
MHz
b) Sampling Frequency, fs = 60 MHz
c) Signal is segmented into frames, with each frame
having a length of Thop-min
d) Thop-min is chosen to be 256 sample points. This
assumption ensures that there will be at most one
hop in each frame.
e) Minimum frequency differential ∆f = 2kHz
f) No. of hop frequencies in the spread bandwidth,
Nf = (fmax – fmin)/ ∆f = 14,000
g) With the above mentioned specifications, the FH
signal thus generated either has
i) no hop in a frame or
ii) it has a hop from any frequency f1, to any
frequency f2, such that
2 MHz ≤ f1, f2 ≥ 30 MHz, and |f1 – f2|≥ 2 kHz
2) Simulation involves five hundred trial experiments,
which are conducted with six different SNR values, ranging
from 15dB to -3dB. The algorithm finds out
a) Presence or absence of a frequency hop within a
frame
b) The hop time estimate, in case of a frequency hop
within a frame
3) The real signal is transformed into an analytical
signal. An Analytical Signal has no negative frequency
components. It facilitates many mathematical calculations.
The analytic form of the signal is used as a substitute for its
real expression when it is required to suppress cross
interference. While performing time-frequency analysis, real
signals result in more cross interference as compared to the
analytic form of signals. Moreover, on removing the negative
frequency components, there is no loss of data as well. Before
time frequency analysis a real data signal is converted to its
respective analytic form.
Mathematically, the analytic expression of FH signal is given
as [7]:
xa(t)= 2�� �
[ − ( − �ℎ )] + 2�� �
[ − �ℎ +
− − � ]
for 0 ≤ t ≤ T, (4)
Thop is the time of hop of the signal from one frequency f1 to
f2. u(t) is a unit step function defined as:
= {
,
,

3. 4) TCF of the analytic form of FH signal can be
obtained by substituting (4) in (1) as follows:
��� , � = 2� 2� �
[ − � − ( + � − �ℎ )][ −
� − ( − � − �ℎ )] + 2� 2� �
[ ( + � − �ℎ + ) −
+ � − � ][ ( − � − �ℎ + ) − − � − � ]
+ 2�[ � −� �+ � +� �]
[ + � − ( + � − �ℎ )][ ( −
� − �ℎ + ) − − � − � ] + [ − � − ( − � −
�ℎ )][ ( + � − �ℎ + ) − + � − � ]
(5)
In simplified notation, (5) can also be written as:
TCF(t, τ) =TCF1(t, τ)+TCF2(t, τ)+TCF12(t, τ) (6)
The three terms on the right hand side of (6) appear as non-
overlapping triangular shaped regions which constitute the
complete TCF expression. The unit step functions are used to
model the boundary lines between the three triangular shaped
TCF1(t, τ), TCF2(t, τ) and TCF12(t, τ) phase regions. These
triangular regions have an orientation of 45˚. TCF1(t, τ) and
TCF2(t, τ) may be called the auto-term triangle and TCF12(t, τ)
as cross-term triangular region.
Fig. 1 shows the TCF phase plot for the signal xa(t), given by
equation (3), for f1=25.89 MHz, f2=24.2 MHz and Thop= 139
samples. It is clear from figure that Thop is the time at which
the triangular region covered by TCF1(t, τ) ends and the
triangular region covered by TCF2(t, τ) starts.
Fig. 1. TCF phase plot
5) Phase of TCF is determined from the angle of TCF.
The phase of TCF1(t, τ) and TCF2(t, τ) are 2πf1τ and 2πf2τ
respectively and are constant with respect to the variable ‘t’.
TCF1(t, τ) depends on f1 and τ. Similarly, TCF2(t, τ) depends
on f2 and τ . It means that for a given value of frequency and
lag, TCF1(t, τ) shows a constant phase. Same is the case for
TCF2(t, τ). On the other hand, the phase of TCF12(t, τ) is a
function of f1, f2, time (t) and lag (τ). Thus for a fixed value of
τ, TCF phase expressed as a function of ‘t’ undergoes
changes in its slope value when going from one region to
another one. Moreover, the phase behavior in the cross-term
region is linear.
After obtaining the TCF phase plot, some pre-processing
techniques are necessary for de-noising. These include phase
unwrapping, median filtering and differentiation. Steps 6 - 9
explain these techniques.
6) Across TCF12(t,τ), the phase behavior is periodic
over 2π producing discontinuities at regular intervals. The
phase of TCF is unwrapped along the time axis. Phase
unwrapping eliminates the discontinuities due to periodicity in
phase and reduces the spikes due to noise. It transforms jumps
greater than π between successive points to their 2π
complement.
Fig. 2. TCF Phase plotted as a function of time at SNR=9dB and τ=25, after
unwrapping
TCF phase p(t) of a signal x(t) can be unwrapped as [7]:
Unwrap (p (t)) = {
, � | − − | �
+ � � − − < −�
− � � − − > �
(7)
7) A median filter is used for the suppression of impulse
errors or short-term discontinuities superimposed on a signal.
A Median Filter of length 5 is applied to the unwrapped phase
of TCF. Median filter is used for de-noising.
8) The unwrapped phase of TCF is differentiated along
the time axis. This changes the ramp function into a pseudo
pulse.
Fig. 3. TCF phase plot after Differentiation
0 20 40 60 80 100 120 140
0
50
100
150
200
250
300
Lag(Samples)
Time(Samples)
-3
-2
-1
0
1
2
3
Thop
=139

4. 9) Another Median Filter of length 25 is applied to the
differentiated phase.
10) Wavelets are best suited for detecting discontinuities
in a signal or any of its high order derivatives. For this
purpose, the chosen wavelet should be able to represent the
highest order derivative present in the signal function. Any
wavelet with, at least, p vanishing moments can be used to
detect a discontinuity in the p - 1 derivative [11]. Haar wavelet
is the shortest of all wavelets and is, therefore, best suited for
detecting discontinuities in a signal. It has only one vanishing
moment and hence can only detect discontinuity in the p – 1 =
0 derivative (i.e. the signal itself).
Discrete Wavelet Transform is calculated for the processed
TCF phase, along the time axis using Haar Wavelet. It
determines the discontinuity at the edges of the cross-terms
region of TCF phase plot.
11) Wavelet detail coefficients of first two scales, d1 and
d2 of DWT are summed up.
12) 45˚/135˚ summation is performed across all values of
lag. This results in a detection vector which has a time index.
13) The data of the detection vector is checked against a
threshold.
14) The time index of a peak in the thresholded detection
vector, gives an estimate of hop time.
Fig.4 Detection Vector
V. RESULTS
Simulations are carried out in such a way that 500 experiments
are conducted for each of the six SNRs, ranging from 15dB to
-3dB. Random hop times, Thop and random hop frequencies,
f1and f2 are generated, such that 2 MHz ≤ f1, f2 ≥ 30 MHz, and
|f1 – f2| ≥ 2 kHz. In each experiment, the Detection and Hop
Time Estimation Algorithm, described in section III is applied
to a frame of the two hop FH signal. Length of a frame is
taken as 256 sample points (4.27 µs). Each frame has a
random hop time and a random set of hop frequencies. The
algorithm detects the presence or absence of hop in each
frame.
Table 1 provides the results of detection as probability of
detection, probability of false alarm and the percentage of
error. The column labeled ‘k’ refers to the threshold
(determined from the ROC Curve) corresponding to each
SNR.
Detection results given in Table 5.2 show that for an SNR of 3
dB, if 8.6% misclassification and a probability of false alarm
of 0.5 is acceptable then, 96% of the hops in a FH signal can
be detected.
TABLE I. DETECTION STATISTICS OF 500 SIMULATIONS
CORRESPONDING TO EACH SNR
SNR
(dB)
k PD PFA %age
error
15 140 0.998 0.04 0.6
10 30 0.991 0.3 3.8
6 15 0.989 0.4 5.0
3 11 0.960 0.5 8.6
0 1 0.933 0.4 10.0
-3 3 0.524 0.26 45.4
We have analyzed the performance of our detection algorithm
by changing the frequency differential and the spread
bandwidth also. We have applied our algorithm to two other
setups of FH parameters. First we changed the frequency
differential to 5 kHz and took the same spread bandwidth i.e.
2 MHz ≤ f1, f2 ≥ 30 MHz. In the other case, f1 and f2 are
generated such that 1 MHz ≤ f1, f2 ≥ 24 MHz, and
|f1 – f2| ≥ 2 kHz. This set up has been used in [10]. Figure 5
shows plots of Detection Probability PD versus SNR, for all
three sets of FH parameters. Detection performance is reliable
at SNRs above 3dB. By applying our algorithm to three sets of
specifications, we conclude that the proposed algorithm works
reliably for blind detection of any FH signal in HF band.
Detection probability is comparable at SNR levels above 0dB
but at SNR below 0dB, detection probability is better for FH
signals having a large frequency differential. Since our
algorithm is based on the phase information of TCF, detection
performance improves when frequencies in the spread
spectrum hop set are widely spaced.
The results of hop time estimation are expressed as percentage
of the difference from the actual hop time of the FH signal.
Table 2 shows all the hop time estimation results.

5. Fig.5 Comparison of Detection Performance with varying frequency
differentials
In table 2, column labeled 1% indicates the probability of the
detected hops having an estimated hop time within 1% of
Thop_min (256 sample points). 1% of Thop_min means within 2
samples of the true hop time. Sampling is done at a frequency
of 60 MHz, so one sample corresponds to 16.67 ns. For an
SNR of 6 dB, the column labeled 1% indicates that estimated
hop time of 47.4% of the detected hops is within 2 samples of
the actual hop time. Similarly the column labeled 5% indicates
that the estimated hop time of 81.2% of the detected hops is
within 12 samples of the true hop time.
TABLE II. HOP TIME ESTIMATION RESULTS GIVEN AS
PROBABILITIES OF ESTIMATED HOPS HAVING A GIVEN DISTANCE,
(REPRESENTED AS PERCENTAGE OF THOP_MIN) FROM TRUE HOP
TIME
SNR
(dB)
1% 5% 10% 15% 20% 30% 40% 50%
15 0.796 0.966 0.980 0.984 0.984 1 1 1
10 0.676 0.906 0.926 0.932 0.938 0.944 0.946 0.946
6 0.474 0.812 0.848 0.868 0.884 0.898 0.898 0.904
3 0.35 0.71 0.77 0.816 0.836 0.874 0.9 0.904
0 0.122 0.34 0.48 0.594 0.682 0.78 0.832 0.872
-3 0.086 0.154 0.252 0.33 0.398 0.456 0.498 0.526
The accuracy of Hop Time Estimation in terms of probability
of estimation (Pe) is shown in Fig. 6.
Figure shows that with an increase in SNR, the probability of
hop time estimation within a given number of samples time
also increases. The lowest plot in the figure corresponds to the
probability of detected hops having estimated hop time within
1% of Thop_min (2 samples time i.e.33.34 ns).
Fig.6 Probability of Estimation vs SNR
The middle plot corresponds to the probability of detected
hops having estimated hop time within 5% of Thop_min (12
samples time i.e.0.2 µs) and the uppermost plot corresponds to
the probability of detected hops having estimated hop time
within 10% of Thop_min (25 samples time i.e. 0.417 µs).
VI. CONCLUSIONS
The Detection and Hop Time estimation Algorithm presented
in our paper is based on the application of Discrete Haar
Wavelet Transform on TCF phase of FH Signal in HF band.
Haar wavelet used in this work gives good time localization of
frequency hop. HF frequency band is used by Military for
security purposes.
Results indicated by Detection and Hop Time Estimation
statistics show that detection performance is acceptable at
SNR levels above 3 dB. At 3 dB probability of detection is
0.96 with 8.6% misclassifications (determined by false alarms
and misses). Hop time estimation accuracy is also reliable at
SNR levels above 3 dB. Performance analysis of the proposed
algorithm with varying spread bandwidths and frequency
differentials of hop set frequencies, shows the flexibility of
Detection and Hop Time Estimation Algorithm.
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0.4
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