Upload
hongyuan
View
213
Download
0
Embed Size (px)
Citation preview
A New Time-frequency Representation
Based on Ambiguity Function Analysis
and Its Application in Parameter
Estimation of FH Signals
Guo Jiantao1,2
, Wang Hongyuan1
1.Department of Electronics and Information Engineering, Huazhong University of
Science and Technology Wuhan,Hubei Province, China,430074
2.College of Physics and Electronic Engineering, Xinyang Normal University,
Henan, China, 464000
Abstract—This paper introduces a new kernel to
compute time-frequency representation (TFR) of
Frequency Hopping signal based on the ambiguity
function (AF) analysis. By control two parameters
of the kernel in time lag and frequency lag
directions, the new representation makes a good
improvement in time-frequency resolution and
suppresses the influence of cross terms. The
application in parameter estimation of FH signal is also show to validate the proposed representation.
Key words-frequency hopping; kernel function;
parameter estimation
I. INTRODUCTION
Frequency hopping (FH) has become one of the most widely used and effective technologies in military anti-jamming and anti-interception applications because of its low probability of interception, good capability against interference, and good ability against fading channel [1]. Estimating all the signal parameters, i.e. hop frequencies, hop duration and time offset in the noise environment is hard and hot in signal analysis at present.
Classical Fourier techniques can not reveals the time-varying spectra characteristics of FH signals as it is non-stationary signals, while joint time-frequency representations (TFR) are transformations that describe the energy density of the signal simultaneously in time and frequency domain, so time frequency analysis is a powerful and effective tools, such as short-time Fourier transform (STFT) [2], wavelet transform (WT) [3] and Wigner-Ville distribution (WVD) [4]. However, the STFT assumes that the signal is qusi-stationary and analyzes the signal by taking the FT of the windowed signal. But windowing signals leads to a tradeoff in time resolution versus frequency resolution. Wavelet
analysis has to choose mother wavelet and identify proper scale for hop frequency extraction. Although WVD has various interesting properties, it introduces cross terms or interference terms which make the transform difficult to interpret. To cope with these problems, a new TFR should be designed for special FH signal applications.
The purpose of this paper is twofold: (i) to propose a signal-dependent TFR which achieves a high degree of both cross-component suppression and auto-component concentration, and (ii) to demonstrate by simulation studies that it is useful in parameter estimation for FH signal in random noise and has higher performance than smoothed pseudo WVD.
This paper is organized as follows. After this introduction, we describe the assumed signal model and derive signal ambiguity function (AF) formulation. In Section 3 we introduce a proper time-frequency representation matched to the ambiguity function auto-term characteristics and design a methodology for selecting the parameter of the proposed TFR. We consider the parameter estimation performance of FH signal by contract with the smoothed pseudo WVD (SPWVD) in Section 4. A discussion and conclusion are offered in the final section.
II. SIGNAL MODEL AND ITS AMBIGUITY
FUNCTION ANALYSIS
A. Signal Model
The FH signal x (t) is modeled as time-frequency shift result of a single tone x0(t), i.e.
Ttettxtxk
tfj
kk ≤<−= 0)()( 2
0
π (1)
978-1-4244-2108-4/08/$25.00 © 2008 IEEE 1
where
ftj
HHT eTTtrecttxH
πα 2
0 )()( −−=,
)(trecthT is equal to one for
],[22hh TT
t −∈and zeros elsewhere. fk is the generic hop frequency, Th is the hop duration, Th is the hop timing.
Consider FH signal having the following parameters in this paper: hop frequency, belonging to a given finite alphabet {5, 45, 25,20 15, 35, 40, 10, 30}Hz; the duration of each hop is 32; the hop timing is 16; number of samples used to estimate the spectrum is 256 and the sample frequency is 100Hz.
B. Ambiguity Function Analysis
The ambiguity function A ( , v) of the signal
x (t) is defined as ∞
∞−−+= dtetxtxvA
tjωτττ )()(),(2
*
2 (2)
In conjunction with the signal model in (1),
the AF may be written as
Let tTnTt hh′=−−+ ατ
2, we obtain
where Arect ( , v) is the AF of rectangular
function )(trecthT , which is given by
>
<⋅=
−
h
hv
TvvTj
rect
T
TevA
hh
ττ
τ πτππ
0),(
)(sin2
(4)
Based on the results in (3), we can consider
the AF of random FH signal consisting of the
weighted sum of N2 time-frequency shifted
versions of Arect ( , v) which construct the zonal
structure paralleling to axis. Center interval in
the time lag direction corresponds to the
difference in the time shift which is same as hop
duration Th, while in the frequency lag
corresponds to the distance in the frequency shift
which is random for the frequency hopping.
By putting the terms for m=n together, the
auto ambiguity component is now
),(),(1
0
)(22
,2 vAeevA rect
N
m
TmTvjfj
autoxhhm ττ
ταπτπ ⋅=−
=
−+ (5)
In the above equation, Arect ( , v) is
multiplied by an N term sum. In conjunction
with (5), the auto AF in the frequency lag
direction is still a sample function although its
main lobe width is changing with the time lag
absolute value which minimum occurs at =0,
and broadens gradually as | | increase. The
central or main lobe is narrow in frequency and broad along the time lag axis while the side lobe
oscillates in sign and flares out in frequency
direction, with a characteristic dumb-bell shape.
An example of an auto AF is given in Figure 1.
Time lag
Fig. 1 auto component of the AF of FH signal
3
1. Kernel function design
The general formula for Cohen’s Class of
Quadratic TFRs for a time signal x (t) is given
by [5] ∞
∞−
∞
∞−
−−Φ= dvdevvAft jvtfj
x τττ τ),(),(),(P (6)
where ( , v) is the Doppler-lag kernel of the
distribution Px (t, f).The properties of a quadratic
TFR are completely determined by its kernel.
Although there are great variations in the shape
of the kernel ( , v), it is still difficult to
analyze multi-component signals or
mono-component nonlinear frequency
modulated signals using fixed kernel for the
serious cross terms interference. On the other
hand, adaptive method using different windows
at each time instance to achieve a good TFR is so computationally expensive that it can not
process real-time signals [6, 7].
We proposed a new signal-dependent TFR
based on the discussion detailed in the previous
No
rmali
zed
fre
qu
ency
lag
),(
)()(
)()(),(
1
0
1
0
)(2)(2
)(21
0
1
0
)(2)(2
)(21
0
)(21
0
)(2
2
2
2
nmhhrect
N
n
N
m
TnTvjmTnTfj
ffvtj
hh
N
n
N
m
TnTvjmTnTfj
TnTtvjN
m
nTmTtfj
hh
N
n
tfj
x
ffvmTnTAee
dtenTmTtrecttrectee
tdeenTmTtrectetrectvA
hhhhm
nmhhhhm
hhhhmn
+−+−⋅=
+−−⋅=
′+−−′′=
−
=
−
=
−++−
∞
∞−
+−−
=
−
=
−+−−−
∞
∞−
++−′−
=
+−−′−
=
′
τ
τ
ττ
τ
τ
υ
απτπ
παπτπ
απτππ
∞
∞−
−
=
−−−−
=
−−+
∞
∞−
−−−−−+=
−+=
dteeTmTtrecteTnTtrect
dtetxtxvA
tvjN
m
TmTtfj
hh
N
n
TnTtfj
hh
tvj
x
hhmhhn παπταπτ
πττ
ττ
αα
τ
21
0
)(2
2
1
0
)(2
2
2
2
*
2
22 )()(
)()(),(
978-1-4244-2108-4/08/$25.00 © 2008 IEEE 2
section. The kernel function ( , v) of the
proposed TFR is chosen in the ambiguity
domain as a 2-D function around the origin
( ) )(sin),(, ατβττ +=Φ vchv (7)
where h (·) is the weighting function in the time
lag direction which we will employ Kaiser
Window and α and are the real parameters
that control the kernel main lobe width at the time lag and frequency lag directions. From this
equation, it is not hard to see that the new kernel
is a low-pass function. By controlling the kernel
parameters, we can obtain a high-resolution in
time-frequency domain in addition to cross
terms reduction for FH signals.
According to the document [8], the
proposed TFR satisfies many of the important
properties listing below:
1 It is real since
( ) ( )vv −−Φ=Φ ,, * ττ .
2 It is time-shift invariant since the
Doppler-lag kernel ( )v,τΦ is not a function of
time t.
3 It is frequency-shift invariant since
the Doppler-lag kernel ( )v,τΦ is not a
function of frequency f.
2. Performance Analysis and
Comparison
In this section, we evaluate the performance of
the proposed TFR by comparing it with the
SPWVD. The comparison here is two-fold: one
is based on the time-frequency resolution, and
the second is based on the ability to estimate FH
signal parameters accurately. We give equal-energy contours
ambiguity-domain kernel functions of the
proposed TFR and the SPWVD in Figs.2 top and
bottom. Comparing them with the auto
component of AF of FH signal in Fig. 1, we find
that the shape of the former is more suitable than
the latter. The kernel AF of the proposed TFR is
much broader at time lag direction by adjusting
parameter than at the frequency lag direction to obtain higher frequency resolution in the
time-frequency domain. In addition, it is
narrower around the origin of the ambiguity
plane along the frequency lag axis through adjusting the parameter to suppress the cross
component of the AF of the FH signal where the
distortion is more troublesome. However, the
SPWVD is simply similar to a 2-D Gaussian
function regardless of the detailed TF structure
of the FH signal.
Time lag
(Top)
Time lag
(Bottom)
Fig. 2 Ambiguity-domain weighting function ( , v) of
(top) the proposed TFR and (bottom) the SPWVD
Secondly, slices of the two distributions at
time sample instant t=132 are plotted in Fig.3.
From this figure, we can observe that the
proposed TFR is better distribution in terms of
narrower lobe around the signal instantaneous
frequency (35Hz) in the analysis of the
noiseless FH signal under consideration than the SPWVD.
Frequency (Hz)
Fig. 3 Slices of the proposed TFR taken at time instant
t=132.
Finally, a statistical performance
comparison of the distributions is considered in
an additive white Gaussian noisy environment.
The signal-to-noise ratio (SNR), which is
defined as the power of the signal over the
power of the noise, is varied from -10dB to 15
dB. For each SNR value, the number of
realizations is 500 in our simulations.
Parameter estimation results which the algorithm
is same to [4] are displayed in Fig.4. The figure shows that the proposed distribution surpasses
No
rmali
zed
fre
qu
ency
lag
N
orm
ali
zed
am
pli
tud
e N
orm
aliz
ed f
req
uen
cy l
ag
978-1-4244-2108-4/08/$25.00 © 2008 IEEE 3
the SPWVD in robustness, especially for hop
duration which the threshold outperforms by
1dB and for time offset which improves at high
SNRs.
3. Conclusion
In this paper, we presented a new type of
kernel for the TFR to analyze FH signal. The
proposed distribution outperforms the SPWVD
in terms of time-frequency resolution and cross
term suppression. We have also shown that the
proposed distribution is more accurate in
parameter estimation of FH signal in additive
white Gaussian noise.
4. References [1] N.Beaulieu, W.Hopkins and P.Mclane, “Interception of
frequency-hopped spread-spectrum signals,” IEEE J.
Select.Areas Commu., vol.8, no. 5, pp. 853-870, 1990.
[2] Xu Mankun, Ping Xijian, and Li Tianyun, et al, “A new
time-frequency spectrogram analysis of FH signals by
image enhancement and mathematical morphology,”
Fourth International Conference on Image and Graphics,
pp. 610-615, 2007.
[3] Fargues Monique, P.Overdyk, and Howard F,
“Wavelet-based detection of frequency hopping signals,”
Conference Record of the 31st Asilomar Conference on
Signals,Systems & Computers,vol.1,pp.515-518,1997.
[4] Barbarossa S, “Parameter estimation of spread spectrum
frequency hopping signals using time-frequency
distributions,” First IEEE Signal Processing Workshop
on Signal Processing Advances in Wirelss
Communications, pp.213-216, 1997.
[5] L.Cohen, “Time-frequency distribution-a review,” Proc.
IEEE, vol.77, no.7, pp.941-981, 1989.
[6] Henry K. Kwork, D.L. Jones, “Improved instantaneous
frequency estimation using an adaptive short-time
Fourier transforms,” IEEE transactions on signal
processing, vol.48, no.10, pp.2964-2972, 2000.
[7] L. Stankovic, V. Katkovnik, “Algorithm for the
instantaneous frequency estimation using time-frequency
distributions with adaptive window width,” IEEE Signal
Process. Lett., vol.5, no.9, pp.207-223, 1998.
[8] J.Jeong and W.J.Williams, “Kernel design for reduced
interference distributions,” IEEE Trans. Signal
Processing, vol.40, pp.402-412, 1992.
SNR (dB) SNR (dB) SNR (dB)
Fig. 4 Performance comparison of proposed TFR (solid asterisk line) and the SPWVD (solid line) for FH signal. (Left) hop timing, (center) hop during, and (right) hop frequency estimation variance with different values of SNR.
Var
ian
ce
Var
ian
ce
Var
ian
ce
978-1-4244-2108-4/08/$25.00 © 2008 IEEE 4