A Radon Transform Based Method to Determine the Numerical Spread as a Measure of Non-Stationarity

Transcription

1 A Radon Transform Based Method to Determine the Numerical Spread as a Measure of Non-Stationarity ROBERT A. HEDGES and BRUCE W. SUTER Air Force Research Laboratory IFGC 525 Brooks Rd. Rome, NY U.S.A. Abstract: Establishing measures for local stationarity is an open problem in the field of time-frequency analysis. One promising theoretical measure, known as the spread, provides a means for quantizing potential correlation between signal elements. In this paper we undertake the issue of generalizing techniques developed by the Authors to better estimate the spread of a signal. Existing techniques estimate the spread as the rectangular region of support of the associated expected ambiguity function oriented parallel to the axes. By applying Radon Transform techniques we can produce a parameterized model which describes the orientation of the region of support providing tighter estimates of the signal spread. Examples are provided that illustrate the enhancement of the new method. Keywords: Non-Stationarity, Radon Transform 1 Introduction In a recent publication, Donoho, Mallat, and von Sachs [1] pose an open problem of devising a measure of local stationarity by measuring the departure from pure stationarity. They provide a list of ideal properties for a measure of local stationarity as follows: Quantitative The measure of local stationarity should reduce to a nonnegative dimensionless constant, where zero means no departure from stationarity and large values mean large departures from stationarity. Measurable The measure should be directly computable from the covariance, and should not involve complex implicit functionals of the covariance, such as the eigenfunctions of the best onestep-ahead predictor. Robust The measure should work well for processes which are not homogeneously stationary, i.e., which on a small subset of their domain are changing rapidly. Analytically Powerful The measure should have direct implications for many derived quantities about which one would like to have information. They then note that the notion of spread, as introduced by W. Kozek [2, 3] comes close to satisfying these requirements ::: we think spread is a major conceptual advance, which points the way towards important future developments. However, they go on to point out, the spread is not robust. [1] In [4] the investigation the issues of implementation arising from finite, discrete, data was addressed through direct implementation of existing theory. This work will demonstrate shortcomings in this theory and address them using Radon transform techniques. 2 Theoretical Spread To establish a context for our work we first summarize the theoretical framework for spread of a random process x(t) defined for continuous t [2,3,5]. The (generalized) Ambiguity Function (AF) [5]ofa deterministic signal x(t) is given by A () x = (;) x(t +( 1 2,))x (t,( 1 2 +))e,{2t dt:

2 The expected (generalized) ambiguity function EAF EA () x (t; ) of a nonstationary random process x(t) is the expectation of the AF EA () (;) x = E = x(t +( 1 2,))x (t,( 1 2 +))e,{2t dt R x (t +( 1 2,);t, (1 2 + ))e,{2t dt: Let R () x (t; ) R x (t +( 2 1,);t, (1 2 +)),then EA () x (;) = R () x (t; )e,{2t dt: (1) If the EAF is zero about a given TF lag point (12;12), then any two TF points (t1;f1)and (t2;f2) with t1, t2 = 12 and f1, f2 = 12 are uncorrelated. The EAF indicates the potential correlation between Time-Frequency points separated by the time lag and the frequency lag [3]. Let [, ; ][, ; ]be the smallest rectangle (centered at the origin of the (;) plane) which contains the effective support of the EAF, i.e., jea () x (;)j for jj > or jj >. Define the spread of x as the area of this rectangle: =4 : (2) From these definitions we can view the spread as a product of temporal correlation,, and spectral correlation. In [2] the classification of a signal as underspread if 1 and overspread otherwise. 3 Numerical Spread In [4] we addressed the issues of implementing the above measures of discrete signals. To define numerical spread we can simply rewrite equation (1) in terms of discrete variables. NEA () x [ m; k ]= and N,1 X tn= r () x [t n; m ]e,{2tn k=n r () x [t n; m ]=r x [t n +(, 1 2 ) m;t n,(+ 1 2 ) m] (3) In order to implement such a scheme on discrete data we first must determine a value for. Itwasshownthat the EAFs obtained for different choices of are equal up to a phase factor [3], EA ( 1) x (;) =e {2( 1, 2 ) EA ( 1) x (;) jea ( 1) x (;)j = jea ( 1) x (;)j: Thus, with respect to the calculation of spread for continuous variables, the factor is arbitrary. For our work, we chose = 1=2 since it fits the above criterion and has the added benefit that the EAF becomes the Fast Fourier transform along the diagonals of the correlation matrix. We can now define the Numerical Expected Ambiguity Function, NEAF, as N,1 X NEA x [ m ; k ]= r (1=2) x [t n ; m ]e,{2tn k=n tn= N,1 X = ~r x [t n ;t n, m ]e,{2tn k=n tn= (4) where the ~r x is the extended autocorrelation as defined in [4]. In order to facilitate the calculation of the region of support we project the signal onto the and axes and determine the support of these projections. Define = j^j : = j^j : E(;)d E(;)d <;for > ^ <;for >^ ; (5) : (6) where delta is a pre-determined threshold. Thenumerical spread is defined as the effective support of the NEAF =4 : 4 Radon Transform Approach One shortcoming of the previous approaches is that of orientation. The spread (theoretical or numerical) is calculated from and as estimates of support in the direction and direction. Thus the rectangle of support is always oriented parallel to the axis, this may lead to over-approximation of the spread and ignores any information concerning correlation between and. Matzet al. refine the definition of underspread to strictly underspread if the minimal rectangle of support is oriented parallel to the and axis [5]. However, there is no mention as to how to determine this

3 contour plot of E(,) contour plot of E(,) contour plot of E(,) Projection of E(,) onto Axis Projection of E(,) onto Axis 8 Projection of E(,) onto Axis 8 Projection of E(,) onto Axis Figure 1: Three Expected Ambiguity Functions, from left to right, corresponding to r = ;r =,:6 and r = :99 with 1 = 2;2 = 3 in equation (7). Each function has identical and projections, thus existing methods would calculate the spread to be equal for all three cases, which is obviously incorrect. orientation. In Figure 1 we see three examples of expected ambiguity functions of the form inspired by an example in [6]: p 1 E(;) = 212(1, r 2 ) 1 2 exp, 2(1, r 2 ) 2 1, 2r : (7) The values of spread components and from equation (5) and (6) are independent of the parameter r, thus the resulting spread is the same for each example. This result is unsatisfactory since upon examination we can see that the area of support is not a constant. What we are observing is that the orientation of the rectangle we use to estimate the support is important. We can extend the idea of projection onto any arbitrary axis and estimate the support from this projection using the Radon transform. Definition 4..1 Assume E(;) satisfies the following properties: 1. E(;) is defined on R 2 2. E(;) is absolutely convergent i.e., je(;)jdd < 1; then the Radon Transform [7] of E, is defined as E(; ) = ; E(;)( cos + sin, )dd: (8) Observe that the delta function is one dimensional and thus the integral over the plane reduces to an integral along the line given by = cos + sin. Note that E(; ) is not defined on a (circular) polar coordinate system. This is best demonstrated by the fact that E(;) is not constant and thus 1 6= 2 may imply that E(;1) 6= E(;2) making the origin in circular polar coordinates multiply defined. The appropriate space for E(; ) is the surface of the unit cylinder R S 1. The cylinder need only be a half cylinder since E(; ) = E(,; + ): For purposes of illustration, the cylinder will be opened up with on one axis and on the other. We can utilize the Radon transform of the ambiguity function to determine the coordinates of the maximum intensity in Radon space (;). This point in turn corresponds to the line of maximum intensity in ambiguity space ( = cos + sin ). Utilizing this correspondence between ambiguity space and Radon space, we will assume the features in ambiguity space are concentrated about this line. We then align one of

4 (; ) Ambiguity Space Radon Space Figure 2: Correspondence between the line o = x cos + y sin in ambiguity space and the point (;)in Radon space. Figure 3: Orientation of spread components. the axis with this line and denote as. The other axis, denoted, is orthogonal to with their intersection a distance from the origin as shown in Figure 2. We can now define spread with respect to our new axis. Let [, ; ][, ; ]be the smallest rectangle (centered at the origin of the ( ; ) plane) which contains the effective support of the EAF, i.e., jea () x ( ; )j for j j > or j j > : Define the spread of x as the area of this rectangle: =4 : (9) We can now write a general equation for spread with respect to the angles of orientation o and? o. Thus we calculate and as follows: = j^j : E(; ) n = j^j : E(; o? ) <;for >^ <;for > ^ : (1) : (11) This is consistent with equations (5) and (6) with = and? = =2. See Figure 3 for an illustration comparing the two regions of support. 5 Application of New Techniques to Examples. In Figure 4 we have applied the Radon transform technique to the same data sets in Figure 1. By properly orienting our support region we may differentiate between cases. The first case (row one) was oriented so that = and thus the projections remain unchanged. In example two (row two) we see that be reorienting the axis along the angles = 1:8 radians and? = 1:8 + =2 radians, the new projections, and thus the signal spread had changed. Example three (row three) demonstrates a dramatic change in the spread by reorienting the axes by = 2:1642 radians and? = 2: =2 radians and calculating and. 6 Summary and Conclusions We have introduced the use of Radon Transform techniques to the calculation of signal spread. Through these techniques we are able to produce a two parameter model of the and orientation of the support of an expected ambiguity function. The use of the Radon Transform as a tool to investigate signal spread shows great promise beyond its use here. An immediate extension would be to construct a four parameter model to describe orientation and location of the support. Another area of intrest is the issue of robustness, which known to be a problem [1], we believe that the Radon transform in conjunction with multiresolution techniques can be used to determine regions of local stationarity and increase robustness. The combination of Radon and multiresolution techniques is similar to Candès and Donoho s work on ridgelets [8]. Further research utilizing these techniques to aid in the calculation of signal representation on local regions of support will provide both an enhanced framework for our future work in the area of adaptive signal representations, including the adaptation of novel multirate and wavelet signal processing techniques [9, 1].

5 1 contour plot of E(,) Contour plot of Radon Transform of E(,) 1 φ=φ φ=φ 8 Projections onto and axes φ= φ=pi/ φ contour plot of E(,) Contour plot of Radon Transform of E(,) 15 1 φ=φ φ=φ Projections onto and axes φ= φ=pi/ φ contour plot of E(,) Contour plot of Radon Transform of E(,) 1 φ=φ φ=φ 8 Projections onto and axes φ= φ=pi/ φ 2 2 Figure 4: The three expected ambiguity functions from Figure 1 are plotted in the first column along with the contour plots of there Radon transforms in the second column. On the Radon transform plots the lines corresponding to the traditional projections = and = =2 are presented along side the lines corresponding to the calculated projections along and, corresponding to = and?, respectively. In the final column the and projections are shown. These projection clearly demonstrate the variation in spread between the three ambiguity functions which was not detected using existing methods.

6 References [1] D.L. Donoho, S. Mallat, and R. von Sachs, Estimating covariances of locally stationary processes: Rates of convergences of best basis methods, Tech. Rep., Stanford University, February [2] W. Kozek, On the underspread/overspread classification of random processes, eitschrift Für angewandte Mathematik und Mechanik, vol. 76, no. 3, [3] W. Kozek, F. Hlawatsch, H. Kirchauer, and U. Trautwein, Correlative time-frequency analysis and classification of nonstationary random processes, in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, 1994, pp [4] R. A. Hedges and B. W. Suter, The introduction of numerical spread as a measure of nonstationarity, in 32nd Symposium on the Interface of Computing Science and Statistics, 2, vol. 33. [5] G. Matz, F. Hlawatsch, and W. Kozek, Generalized evolutionary spectral analysis and the weyl spectrum of nonstationary random processes, IEEE Transactions on Signal Processing, vol. 45, no. 6, pp , June [6] L. Cohen, Time-Frequency Analysis, Prentice Hall Signal Processing Series. Prentice Hall PTR, [7] S.R.Deans, The Radon Transform and Some of Its Applications, John Wiley & Sons, [8] E. J. Candès and D. L. Donoho, Ridgelets: A key to higher-dimensional intermittency?, in Wavelets: the Key to Intermittent Information, London, February 1999, The Royal Society of London. [9] B. W. Suter, Multirate and Wavelet Signal Processing, vol. 8 of Wavelets and its Applications, Academic Press, [1] R. A. Hedges, Hybrid Wavelet Packet Analysis: Theory and Implementations, Ph.D. thesis, Arizona State University, 1999.