2011 7th International Workshop on Systems, Signal Processing and their Applications (WOSSPA)
AN AUTOMATIC TIME-FREQUENCY PROCEDURE FOR INTERFERENCE SUPPRESSION BY EXPLOITING THEIR GEOMETRICAL FEATURES Nicoletta Saulig, Victor Sucic∗
Boualem Boashash
Faculty of Engineering, University of Rijeka Vukovarska 58, HR-51000 Rijeka, Croatia
[email protected];
[email protected]
College of Engineering, Qatar University PO Box: 2713, Doha, Qatar
[email protected]
ABSTRACT
2. INTERFERENCE GEOMETRY IN THE WVD
This paper presents an adaptive method for interference suppression in the Wigner-Ville distribution. The structure of the artifacts in the Wigner-Ville distribution has been analyzed to optimally mask the signal Wigner-Ville distribution by the Pseudo Wigner-Ville distribution. The resulting time-frequency distribution outperforms in terms of representation quality the Wigner-Ville distribution, the Pseudo Wigner-Ville distribution with fixed smoothing windows, as well as the Modified B distribution. 1. INTRODUCTION Time-frequency representations provide many advantages in the analysis of non-stationary signals. A signal representation with respect to both time and frequency offers a complete information of the number of signal components, their respective bandwidths and instantaneous frequency (IF) laws, as well as the components relative amplitudes [1]. The Wigner-Ville distribution preserves good energy concentration around the IF of linear frequency modulated (FM) signals, but suffers from pronounced interferences in case of non-linear FM or multicomponents [1]. One approach in the interference reduction is the masking of the WVD by another time-frequency distribution (TFD) [1]. The masking process requires a TFD with a reduced level of interference, inevitably having lower energy concentration around the IF. This paper introduces an automatic approach in the interference reduction, based on an adaptive masking algorithm which generates, at each time moment, the optimal Pseudo WVD (PWVD) that ensures the minimal interference in the resulting distribution, while preserving signal energy concentration around the components IFs. Note, however, that unlike in other existing high-resolution time-frequency methods (e.g. [2]), the TFD obtained with the method proposed in this paper not only reduces the interference terms, but also preserves the morphology of the signal WVD.
The WVD, defined as [1]: Wz (t, f ) =
∞
−∞
z(t +
τ ∗ τ ) z (t − ) e−j2πf τ dτ 2 2
(1)
represents a monocomponent signal as a knife-edge ridge whose crest is the IF law in the (t, f ) plane [1]. The mathematical model of the WVD implies a good energy concentration around the instantaneous frequency but also the appearance of the cross-terms midway between any two points belonging to the IF trajectory [1]. In fact, the rule of interference construction in the WVD can be summarized as follows. Two points belonging to the signal will interfere to create a third point which will be located on their geometrical midpoint. The amplitude of the interference will be proportional to the double product of the amplitudes of the interfering points. In addition, the interferences oscillate perpendicularly to the line joining the two signal points, with the frequency proportional to the distance between these two points [1]. In general, the mechanism of interference generation can be analyzed by considering the signal three neighbouring points as shown in Figure 1.
Fig. 1. Schematic representation of interference location The three points P−1 , P0 and P1 are respectively defined by the coordinates (x−1 , y−1 ), (x0 , y0 ) and (x1 , y1 ). Let x0 = (x1 + x−1 )/2 (P0 is the midpoint on the abscise). By considering the general interference rule and the similarity of the triangles in Figure 1, it follows that:
∗ This
work is a part of the research project ”Optimization and Design of Time-Frequency Distributions”, No. 069-0362214-1575, which is financially supported by the Ministry of Science, Education and Sports of the Republic of Croatia.
978-1-4577-0690-5/11/$26.00 ©2011 IEEE
x1 − x−1 (y1 − y0 ) + (y0 − y−1 ) = =2 d + y0 − y−1 x0 − x−1
311
(2)
and consequently d=
y1 + y−1 − 2y0 2
(3)
where d is the distance between the points P0 and M (the Δy between midpoint between P−1 and P1 ). The increase Δx the neighbouring points P−1 and P0 is: y0 − y−1 Δy = . Δx x0 − x−1
(4)
The expression (4) corresponds to the numerical derivatives of the point P−1/2 (the midpoint between P−1 and P0 ). Thus, the numerical derivatives of the points P−1/2 and P1/2 (the midpoints between P1 and P0 ) are: P−1/2
y0 − y−1 y1 − y0 , P1/2 = . = Δx Δx
case of nonlinear FMs. The inner artifacts can be significantly attenuated by performing the smoothing of the WVD in the frequency direction [3]. The obtained distribution is the Pseudo WVD [1]: ∞ τ τ P Wz (t, f ) = h(τ ) z(t + )z ∗ (t − ) e−j2πf τ dτ (10) 2 2 −∞
(5)
Therefore
P1/2 − P−1/2
=
Δx
y1 −y0 Δx
−y1 − y0Δx , Δx
(6)
so the numerical second derivative of the point P0 is:
P1/2 − P−1/2 y1 + y−1 − 2y0 Δ2 P0 = . = Δx2 Δx Δ2 x
(7)
Now, for Δx = 1,
P0 = y1 + y−1 − 2y0 .
(8)
By comparing Eqs. (8) and (3), it follows that the position of the interference point M can be expressed as a function of the second derivative of the midpoint P0 :
d=
P0 , 2
M (x0 , y0 + d) = M (x0 , y0 +
P0 ) . 2
(9)
By applying this reasoning to a group of three points (atoms) in the immediate proximity of the points P−1 , P0 and P1 , which are lying on the ideal IF trajectory of the signal, the resulting interference is shown in Figure 2. The interference between the adjacent atoms will lie on the IF trajectory, and due to the close position of the atoms will not oscillate, practically bending the atoms together. The external atoms P−1 and P1 create an interference centered at M that oscillate perpendicularly to the line connecting them (the darker areas represent the first two negative sidelobes, and the lighter area is the first positive sidelobe). Thus, what is perceived as the signal component is, in fact, the superposition of the component itself and the interference of the adjacent signal atoms. The interference contribution on the IF trajectory assures a larger amplitude of the signal compared to the amplitude of the interference between non-adjacent atoms. 3. THE ALGORITHM FOR OPTIMAL MASKING OF THE WVD 3.1. The monocomponent case As shown in Section 2, the WVD presents potentially misleading inner interference terms along the IF trajectory in
312
The duration of the lag window h(τ ) is inversely proportional to the grade of the frequency smoothing. Since the window choice in the PWVD allows a gradual frequency smoothing, the aim of the optimal window selection is to find a window by which the product of the WVD and the PWVD will result in a distribution with the lowest interference level, preserving the component bandwidth of the WVD. The lag window that satisfies these requirements has been shown to be the one by which the PWVD will be stretched along the frequency axis so that the zero-crossing point of the mainlobe of the PWVD overlaps with the minimum of the first sidelobe of the WVD. Figure 3 shows time slices of the WVD, the PWVD with the optimal masking time window, and the distribution resulting from their product, of the test signal from Figure 4. The distribution obtained by the optimal mask-
WVD Opt. masked WVD PWVD (Opt. win.)
1
Amplitude
Fig. 2. Simplified representation of interference between three Gaussian atoms in the time-frequency plane
0.5
0
−0.5 0.32
0.34
0.36
0.38
0.4
Frequency [Hz]
Fig. 3. Time slices of the WVD, optimal masked WVD and PWVD at the time instant t = 90 s of the signal in Figure 4 ing results in the interference decrement without affecting the signal instantaneous bandwidth. Since the choice of the optimal window depends on the distance of the first sidelobe from the signal, the manual search of the optimal window for each time moment would be extremely inefficient. Section 2 shows that the distance of the first sidelobe from the signal is one half of the numerical second derivative of the centers of the first pair of interfering
atoms. In practice, in the case of a signal with continuous IF, the character of the second derivative of the discrete IF (defined by the centers of the atoms) will be preserved in the second derivative of the continuous IF. In order to obtain a simple method for the optimal window selection, based on our extensive simulations showing an exponential dependency of the optimal Hamming window on the second IF derivative, the optimal window length can be defined as: w(t0 ) = k |at0 |l + m , at0
(11)
Δ2 fi (t) = = fi (t0 + 1) + fi (t0 − 1) − 2fi (t0 ) (12) Δt2 t0
(a)
with the coefficients k = 0.4161, l = −0.4379, m = 19.33, for at0 ≤ 8 · 10−5 , and k = 0.1021, l = −0.6148, m = 10.77, otherwise. 3.2. The multicomponent case
In the case of multicomponent signals, the WVD will present pronounced outer interference terms (Ax (t)) appearing in between the signal components [1]. Outer interference terms, unlike the inner interferences, correspond to the dominant peaks of the WVD [1]. In order to provide the optimally masked WVD of a multicomponent signal, an estimate of the components separation is first required. For this purpose the algorithm described in [4], [5] will be used, and the extracted components will be represented using the modified B distribution [1] (with the time and lag windows of length N/10). Next, we proceed by masking each component individually as in the monocomponent signal case. In the algorithm last step, the components are simply summed up. 4. EXAMPLES To illustrate the performance of the optimal masked WVD in terms of interference suppression and energy concentration when compared to the WDV and PWVD with the fixed window, the sidelobe amplitude AS (t) relative to the mainlobe √ amplitude, and the instantaneous bandwidth Bi (t) (at 2/2 of the normalized amplitude), have been considered [1]. When the analytic fi (t) of the signal is not available, it is estimated from the maxima of the WVD of the signal by the least square polynomial approximation. The WVD of a bat chirp signal [6] (Figure 4(a)) shows pronounced inner-artifacts along the signal component. Figure 4(b) shows that the optimal masking has led to a significant interference suppression, with no influence on the original bandwidth, which would result from the classical frequency smoothing (as in the PWVD). Table 1 shows the quality parameters referring to the time slices of the WVD, optimally masked WVD, PWVD with the Hamming window of length 91 s and a reduced interference distribution, the Modified B distribution (MBD) [7] with the Hamming window of length 91 s and the parameter β = 0.1, of the bat chirp signal [6], at the times t1 = 45 s and t2 = 90 s. The masked WVD outperforms
313
(b)
Fig. 4. (a) WVD of a real test signal. (b) Optimally masked WVD of a real test signal the WVD in terms of the parameter As up to 10 times, and halves the instantaneous bandwidth Bi (t) of the PWVD. When compared to the MBD, the proposed distribution shows 30−40% reduction in the sidelobe amplitude, while the instantaneous bandwidth is narrower by up to 35%. Figure 5(a) shows the WVD of a two component signal with hyperbolic and parabolic FMs, respectively. The components separation and optimal masking brought the complete suppression of the outer interferences (Ax (t) = 0), while significantly reducing inner interferences (Figure 5(b)). The representation quality indicators of each component (denoted as C1 and C2 , respectively) are shown in Table 2. As in the monocomponent case, the optimal masking has resulted in considerable improvements in the quality indicators. 5. CONCLUSION An adaptive algorithm for the optimal masking of the WVD by the PWVD is proposed. The structure of the innerartifacts of the WVD determines the lag smoothing window duration of the PWVD. In the case of multicomponent signals, the proposed masking method reduces the interference between the components as well as the innerartifacts. The simulation results show that the distribution
Table 2. REPRESENTATION QUALITY PARAMETERS FOR A TWO-COMPONENT SIGNAL fi |As | Bi [×10−3 ] t C1 C2 C1 C2 C1 C2 WVD 100 0.427 0.111 0.936 0.644 4.2 4.9 300 0.385 0.039 0.761 0.474 3.7 2.3 PWVD (Hamming, 91 s) 100 0.426 0.109 0.041 0.074 7.2 7.2 300 0.387 0.038 0.042 0.017 7.2 7.1 Masked WVD 100 0.427 0.110 0.081 0.090 3.3 4.2 300 0.385 0.039 0.089 0.041 3.4 1.3 MBD (β = 0.1, Hamming, 91 s) 100 0.427 0.109 0.09 0.073 6 6.2 300 0.386 0.038 0.23 0.169 5.5 5.4
(a)
6. ACKNOWLEDGEMENT The author wishes to thank Curtis Condon, Ken White, and Al Feng of the Beckman Institute of the University of Illinois for the bat data and for permission to use it in this paper. 7. REFERENCES [1] B. Boashash, Time frequency signal analysis and processing: A comprehensive reference, Elsevier, Oxford, UK, 2003.
(b)
Fig. 5. (a) WVD of a two-component test signal. (b) Optimal masked WVD of a two-component test signal
[2] P. Flandrin, F. Auger and E. Chassande-Mottin, Timefrequency reassignment - From principles to algorithms, in : Applications in time-frequency signal processing, A. Papandreou-Suppappola ed., CRC Press, chap. 5, pp. 179-203, 2003.
Table 1. REPRESENTATION QUALITY PARAMETERS FOR THE BAT SIGNAL t fi |As | Bi [×10−3 ] WVD 45 0.408 0.67 10.5 90 0.347 0.56 8.2 PWVD (Hamming, 91 s) 45 0.403 0.189 15 90 0.345 0.051 14.7 Masked WVD 45 0.406 0.073 9 90 0.347 0.067 7.3 MBD (β = 0.1, Hamming, 91 s) 45 0.392 0.102 13.9 90 0.334 0.117 11.4
[3] F. Hlawatsch and P. Flandrin, Wigner Distribution Theory and applications in Signal Processing, Elsevier, Amsterdam, The Neverthelands,1997. [4] B. Barkat and K. Abed-Meraim, ”Algorithms for blind components separation and extraction from the time-frequency distribution of their mixture”, EURASIP J. Appl. Signal Process., vol. 2004, pp. 2025-2033, 2004. [5] J. Lerga, V. Sucic and B. Boashash, ”An efficient algorithm for instantaneous frequency estimation of nonstationary multicomponent signals in low SNR”, EURASIP J. Adv. Signal Process., vol. 2011, 2011. [6] http://dsp.rice.edu/software/bat-echolocation-chirp
obtained by the adaptive masking has better quality representation when compared to either WVD, PWVD, or the MBD, while still preserving the signal WVD morphology. Future work will include a comparison with other masking TFDs including the spectrogram, as well as a review of the loss of properties caused by the proposed method and the relevance to specific applications.
[7] Z. M. Hussain and B. Boashash, ”Adaptive instantaneous frequency estimation of multicomponent FM signals using quadratic time-frequency distributions”, Signal Processing, IEEE Transactions on, vol. 50, pp. 1866-1876, 2002.
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