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

e_jiantao@163.com

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

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),(

)()(

)()(),(

1

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)(21

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)(2)(2

)(21

0

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n

N

m

TnTvjmTnTfj

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dtenTmTtrecttrectee

tdeenTmTtrectetrectvA

hhhhm

nmhhhhm

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+−+−⋅=

+−−⋅=

′+−−′′=

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=

−++−

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∞−

+−−

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=

−+−−−

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+−−′−

=

′

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 )()(

)()(),(

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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

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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.

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