Modified Gabor Wigner Transform For Crisp Time Frequency Representation

Nabeel Ali Khan, Dr. M Noman Jaffri Department of Electrical Engineering

Federal Urdu University Islamabad, Pakistan

nabeel_alikhan@yahoo.co.uk, mnjafri@yahoo.com

Dr. Syed Ismail Shah Department of Telecom Engineering

Iqra University Islamabad, Pakistan

syedismailshah@gmail.com

Abstract—Gabor Wigner transform (GWT) avoids the cross terms and gives high clarity in time frequency (t-f) domain. The transform is however unable to provide a clear picture in situations when auto and cross components of Wigner transform of the multi component signal overlap. This paper presents a new simple hybrid approach that overcomes the limitations of GWT in an efficient manner by making use of fractional Fourier transform and image processing technique. Experimental results demonstrate superiority of proposed scheme.

Keywords- Gabor Transform, Wigner Transform, Fractional Fourier Transform, Gabor Wigner Transform

I. INTRODUCTION Analysis of non-stationary signals is a challenging task,

which involves monitoring of frequency changes of the signal over time. The introduction of t-f signal processing has led to new tools to represent and characterize the time-varying contents of non-stationary signals using time frequency distributions (TFD) [1, 6]. By distributing the signal energy over the t-f plane, TFD provide the analyst with information unavailable from the signal time-domain representation or its frequency-domain representation. This includes the number of components present in the signal, the time durations and frequency bands over which these components are defined, their' relative amplitudes, phase information, and the instantaneous frequency (IF) [7,8] laws that components follow in the t-f plane. A signal-processing engineer is mostly confronted with the task of processing frequencies of spectral peaks that require unambiguous and accurate information about IF present in mono/multi component signals. This has made the IF, a parameter of practical importance in situations such as seismic, radar, sonar, communications, and biomedical application [16-17].

It has been shown that the bilinear distributions result in a blurred version of the true TFD [1, 10]. Spectrogram suffers from the window effect that governs its resulting t-f resolution. Combination of Spectrograms or Spectrogram with adaptive window selection may just reduce blurring effect [4]. Wigner-Ville distribution (WVD) is the prototype of distributions that are qualitatively different from Spectrogram, produces ideal concentration along the IF for linear frequency modulated signals, but presence of cross terms reduces their practical effectiveness. Moreover, for IF

variations of higher order WVD cannot produce the ideal concentration.

The impediments that plague TFD are dealt in various ways and many techniques are developed to partially or completely remove them. [2-6, 11-14].

Recently S. Pei and J. Ding have demonstrated that the GWT can combine the advantages of the Gabor transform (GT) and the WVD by avoiding the cross term problem while maintaining the clarity as good as that of the WVD and will be a powerful tool for analyzing the character of a signal in the t-f domain [2]. However this approach fails to provide useful results once the cross terms are superimposed on the auto components of WVD in a multi component signal.

In this paper, the problem is analyzed mathematically and a simple hybrid approach, based on fractional fourier transform and image processing techniques, is proposed as a solution. The proposed method makes use of image processing techniques to segment the GT image of a signal. GT being a linear operator avoids the cross terms problem and is found suitable for filter design, multiplexing and signal compression [2]. These segments are identified as signal components and are filtered out using fractional filters from original signal as illustrated in [2]. GWT of all the mono-component filtered signals is then computed and results are summed up to get a crisp TFD.

Rest of the paper is organized as follows. Section 2 presents the mathematical description and analysis of the problem. The proposed hybrid approach to obtain crisp TFD is illustrated in Section 3. Discussion over experimental results and conclusions are given in section 4 and 5 respectively.

II. MATHEMATICAL ANALYSIS OF GABOR WIGNER TRANSFORM

In this section mathematical analysis of GWT is presented. Moreover its ability to eliminate cross terms and reduce blurring as well as its limitations is also discussed.

WVD distribution can be described as Fourier transform of instantaneous auto correlation function. Mathematically,

Where )(ts is a multi-component signal given below

τττω ωτ detststWD j−+−= ∫ )2/()2/(),( * (1)

∑=

=N

ii tsts

1

)()( (2)

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Where )(tsi is each signal component. WVD due to its quadratic nature will produce cross terms

Where

GT is just the Short time Fourier transforms with Gaussian window function.

Where )(τh is a Gaussian window. Short time Fourier Transform is a linear transformation and does not suffer from cross terms problem. However, it blurs t-f plane. Mathematically,

Where ),( ωtGTi is GT of signal component )(tsi . Mathematically, GWT is defined as multiplication of WVD and GT. Two different signals say )(tsi and )(ts j have non-overlapping time frequency signature (crossing chirps and very closely placed chirps are ignored in this analysis) therefore the product of their time frequency representations will be zero. This reduces equation (9) into If auto-terms of WVD of a signal does not overlap with its cross-terms then GWT becomes a linear transformation. Mathematically,

GT transform is cross-term free blurred image of WVD distributions. Multiplying GT and WVD results in an image which is de-blurred and free of cross terms only if auto-terms do not overlap cross terms

III. THE PROPOSED APPROACH In this section a modification over GWT is proposed.

Proposed approach isolates signal components and analyzes isolated signal component separately using GWT. Blurred time frequency image is obtained by computing GT image of multi-component signal. It is assumed that all signal components forms a non overlapping segments in t-f plane. This assumption makes task of image segmentation easier. GT image is segmented using an 8-bit connectivity criterion. Each segment iS in t-f plane corresponds to individual signal

components )(tsi . Segments are separated by drawing lines between them. Lines are drawn using support vector machines which gives a maximum margin classifier. These lines give order of fractional filters [2] which are then used to isolate signal components. Isolated signal components can be analyzed by computing their GWT or WVD. GWT is chosen in this paper as it gives crisp cross term free representation of both linearly and nonlinearly frequency modulated signals [2].

A. Algorithm Steps of algorithm are

1. Find out blurred Time frequency Image of multi-

component signal given in (2) using GT. 2. Image is segmented using 8-bit connectivity

criteria into N-segments. Each segment

iS corresponds to a signal component )(tsi . 3. For each segment iS

• Find out N lines (l1L to lNL) that separates segment iS from the regions that are on top of iS (For linearly separable regions only one line would be enough).

• Find out N lines llH to lNH to separate iS from the regions that are on bottom.

• Apply high pass fractional filtering using lines l1H to lNH and low pass fractional filtering using l1L to lNH respectively to isolate component signal )(tsi .

• Compute GWT of filtered signal

4. Finally add all the GWTs.

IV. RESULTS Two examples of synthetic multi-component signals are

being considered to evaluate the objectives, which include signals containing three parallel linear chirps and three parabolic chirps.

kj

tWDtWDtWDN

j

N

kjk

N

iii

≠

+= ∑∑∑= == 1 11

),(),(),( ωωω (3)

τττω ωτ detststWD jiiii

−∫ +−= )2/()2/(),( * (4)

τττω ωτ detststWD jjiij

−∫ +−= )2/()2/(),( *(5)

∫ −−= τττω ωτ desthtGT j)()(),(

∑=

=N

ii tGTtGT

1

),(),( ωω

(6)

ωτττω jii ethstGT −∫ −= )()(),(

(7)

),(),(),( ωωω tGTtWDtGWT =

(8)

kj

tWDtGT

tWDtGTtGWT

jk

N

i

N

j

N

ki

N

iiii

≠

+=

∑∑∑

∑

= = =

=

),(),(

),(),(),(

1 1 1

1

ωω

ωωω (10)

),(),(1

ωω ∑=

=M

ii tGWTtGWT (11)

(9)

120

A. Three linear chirps First example is a synthetic signal composed of three

parallel chirps. It is synthesized in such a manner that the cross terms of two chirps lies exactly on the middle chirp. Mathematically,

Sampling frequency was chosen to be 200 Hz and time

duration was taken from -1 to 1. Results are shown in figure 1. It is clear that in GWT middle chirp is corrupted by cross-terms of other two chirps. However, modified GWT does not suffer from cross-term interference problem

B. Three Quadratic chirps Second example is a synthetic signal composed of three parallel quadratic chirps. It is synthesized in such a manner that the cross terms of two chirps lies exactly on the middle chirp. Mathematically, Sampling frequency was chosen to be 200 Hz and time duration was taken from -1.5s to 1.5s. It is clear that GWT of upper two chirps are corrupted by cross-terms. However, modified GWT does not suffer from cross-term interference problem.

V. CONCLUSION A new hybrid approach to get crisp TFD by compensating distortions caused due to inherent shortcoming of GWT, for a particular class of signals is presented. The paper presented a mathematical description of the problem and evaluated the performance of the approach to obtain interference free TFD which are not possible by simple GWT. Experimental results demonstrate the effectiveness of the approach. In our future work we will extend our technique to handle more complicated signals like sinusoidal frequency modulated signals and crossing chirps etc.

REFERENCES

[1] G Cohen, L.,“Time Frequency Analysis”, , Prentice-Hall, NJ, 1995 [2] Soo-Chang Pei and Jian-Jium Ding, “Relations between Gabor

Transforms and Fractional Fourier Transforms and their applications for signal processing”. IEEE Transactions on Signal Processing,vol. 55 no.10 pp. 4839-4850, October 2007.

[3] J.Jeong and J.Williams, “Kernel design for reduced interference distributions” IEEE Transaction on signal processing., vol. 40 no.2 pp. 402-412, Feb 1992.

[4] R. G. Baraniuk and D. L. Jones, “A signal dependent time-frequency representation: Optimal kernel design,” IEEE Trans. Signal Process., vol. 41, no. 4, pp. 1589–1601, Apr. 1993.

[5] B. Ristic and B. Boashash, “Kernel design for time-frequency signal analysis using the radon transform,” IEEE Trans. Signal Process., vol. 41, no. 5, pp, .May 1993,

[6] S. Qian and J. M. Morris,“Wigner distribution decomposition and cross-terms deleted representation,” Signal Process., vol. 27, pp. 125–14, 1992.

[7] B. Boashash, "Estimating and interpreting the instantaneous frequency of a signal-Part 1: Fundamentals", Proc. IEEE, Vol. 80, pp. 519-538, Apr. 1992.

[8] B. Boashash, "Estimating and interpreting the instantaneous frequency of a signal. II. Algorithms and applications," Proc.IEEE, vol. 80, no. 4, pp. 540-568, Apr. 1992.

[9] S. Qian, D. Chen, "Joint time-frequency analysis," IEEE Signal Processing Magazine, vol. 16, no. 2, pp. 52-67 Mar. 1999.,

[10] L. Cohen, "Time-frequency distributions-A review," Proc. IEEE, vol. 77 pp. 941-981, July 1989.

[11] I. Shafi, J. Ahmad, S. I. Shah, and F. M. Kashif, “Evolutionary time–frequency distributions using Bayesian regularised neural network model”, IET Trans. on Signal Processing, Vol. 1, No. 2, pp. 97-106, June 2007.

[12] L. Stankovic, "Highly Concentrated Time-Frequency Distributions: Pseudo Quantum Signal Representation", IEEE Trans. on Signal Processing, Vol. 45, No. 3, pp. 543-551, March 1997.

[13] Sridhar Krishnan, "A New Approach for Estimation of Instantaneous Mean Frequency of a Time-Varying Signal," EURASIP Journal on Applied Signal Processing 2005:17, pp. 2848—2855

[14] .L. B. Almeida, “The fractional Fourier transform and time-fre-quency representations,” IEEE Trans. Signal Processing, vol. 42, pp. 3084–3091. Nov. 1994,

[15] V. Chen and H. Ling, “Time-Frequency Transforms for Radar Imaging and Signal Analysis”, Artech House, Boston. 2002.

[16] Gramatikov B, Brinker J, Yi-chun S, Thakor, “NV. Wavelet analysis and time-frequency distributions of the body surface ECG before and after angioplasty.” Comp Meth Prog Biomed: 62:87-98, 2000

tjtjttjtjtj eeets ππππππ 408808608 222

)( +++ ++=

tjtjttjtjtj eeets ππππππ 208508808 333

)( +++ ++=

(12)

(13)

121

Figure 1. a) Modified GWT for three quadratic chirps b) GWT for three quadratic chirps c) Modified GWT for three linear chirps d) GWT for l three linear

chirps.

122