Signal Approximation Using the Bilinear Transform

SIGNAL APPROXIMATION USING THE BILINEAR TRANSFORM Archana Venkataraman, Alan V. Oppenheim MIT Digital Signal Processing Group 77 Massachusetts Avenue, Cambridge, MA 02139 [email protected], [email protected] ABSTRACT The analysis presented in this paper has application in contexts This paper explores the approximation properties of a unique basis where only a fixed number of DT values can be used to represent a expansion, which realizes a bilinear frequency warping between a CT signal. For example, in a binary detection problem, one might continuous-time signal and its discrete-time representation. We in- want to limit the number of digital multiplies used to compute the vestigate the role that certain parameters and signal characteristics inner product of two CT signals. Numerical simulations of this sce- have on these approximations, and we extend the analysis to a win- nario suggest that the bilinear expansion achieves a better detection dowed representation, which increases the overall time resolution. performance than Nyquist sampling for certain signal classes. Approximations derived from the bilinear representation and from Nyquist sampling are compared in the context of a binary detection 2. THE BILINEAR REPRESENTATION problem. Simulation results indicate that, for many types of signals, the bilinear approximations achieve a better detection performance. As derived in [1], the network shown in Fig. 1 realizes a one-to-one Index Terms— Signal Representations, Approximation Meth- frequency warping between the Laplace and Z-transform domains ods, Bilinear Transformations, Signal Detection according to the bilinear transform. Specifically, the Laplace transform F (s) of the signal f(t) and the Z-transform F (z) of the se- f[n] 1. INTRODUCTION quence are related through the change of variables a + s Although Nyquist sampling is commonly used in practice, there re- z = (1) main several drawbacks to this representation. For example, the a − s continuous-time (CT) signal must be appropriately bandlimited in where a is the real valued parameter indicated in the cascades of order to avoid frequency aliasing distortions. Additionally, if the Fig. 1. The relationship in (1) maps the entire range of CT frequen- number of time samples used in a particular computation is con- jω strained, the Nyquist approximation may do a poor job of represent- cies ( -axis) onto the range of unique DT frequencies (unit circle). ing the original signal. For these reasons, it is useful to consider Fig. 1 can also be viewed as a basis expansion of the CT signal f(t) alternative signal representations. in which the Laplace transforms of the basis functions are √ „ « In the 1960’s a basis expansion was proposed [1, 2], imple- n−1 menting a nonlinear frequency warping between a CT signal and 2a a − s Λn(s)= ,n≥ 1 (2) its discrete-time (DT) representation according to the bilinear trans- a + s a + s form. Since there is a one-to-one relationship between the two frequency domains, this bilinear expansion theoretically avoids both As given in [1], the corresponding time-domain expressions are the bandlimited requirement and the frequency aliasing distortions √ n−1 −at associated with Nyquist sampling. Furthermore, the DT expansion λn(t)= 2a(−1) e Ln−1(2at)u(t),n≥ 1 (3) coefficients can be obtained using a cascade of first-order analog systems. Modern-day integrated circuit technology has made it practi- where Ln(·) represents a zero-order Laguerre polynomial. It is shown {λ (t)}∞ cal to compute these coefficients through conventional circuit design in [1] that n n=1 is an orthonormal set of functions which span techniques. Consequently, the bilinear expansion can be considered the space of causal, finite-energy signals. Fig. 2 depicts the bilinear as an alternative to Nyquist representations in various applications. basis functions as the index n and the parameter a vary. In this paper, we explore the approximation performance of the By applying properties of Laguerre polynomials [3, 4] and by bilinear expansion presented in [1] by drawing from properties of defining ξ =2n − 3, the bilinear basis functions can be bounded the corresponding basis functions. We consider how various signal according to the expression characteristics, such as a rational Laplace transform and the energy 8 distribution over time, affect the approximations. Additionally, we > 1, 0 ≤ t ≤ 1 > 4aξ examineamodified version of the bilinear representation, in which <> (4aξt)−1/4, 1 ≤ t ≤ ξ 4aξ 2a “ h i” 1 the CT signal is segmented using a short-duration window. This |λn(t)|≤C − > 2ξ (2ξ)1/3 + |2ξ − 2at| 4 , ξ ≤ t ≤ 3ξ segmentation procedure helps to improve the overall approximation > 2 2 :> a a quality by exploiting properties of the representation. e−βt,t≥ 3ξ 2a This work was supported in part by the MIT Provost Presidential Fellow- (4) ship, the NDSEG Fellowship, The Texas Instruments Leadership University where β is a positive constant and a>0. Eq. (4) will be useful Program, BAE Systems PO 112991, and Lincoln Laboratory PO 3077828. when analyzing the approximation properties of this representation. 1-4244-1484-9/08/$25.00 ©2008 IEEE 3729 ICASSP 2008 Authorized licensed use limited to: MIT Libraries. Downloaded on May 17,2010 at 19:14:02 UTC from IEEE Xplore. Restrictions apply. √ f(−t) 2a a−s a−s a−s a−s a+s a+s a+s a+s a+s 1.5 n = 1 t =0 t =0 t =0 t =0 t =0 n = 3 1 n = 5 f[1] f[2] f[3] f[4] f[5] n = 7 (a) Analysis Network 0.5 √ δ(t) 2a a−s a−s a−s a+s a+s a+s a+s 0 f[1] f[2] f[3] f[4] −0.5 0 5 10 15 20 time, t Σ (a) λn(t) for different index values; a =1 f(t) 1.5 a = 1 a = 2 (b) Synthesis Network 1 a = 3 Fig. 1. First order cascade derived from [1] and [2]. The analysis 0.5 network converts the CT signal f(t) into its DT representation f[n]. 0 The synthesis network reconstructs a CT signal from its expansion. −0.5 3. BILINEAR APPROXIMATION −1 0 5 10 15 In general, representing a CT signal through a basis expansion re- time, t λ (t) a n =5 quires an infinite number of (non-zero) expansion coefficients. We (b) n for different values of ; address this issue by retaining an appropriate subset IM of bilinear expansion terms to form an M-term signal approximation. The ap- Fig. 2. Dependence of the bilinear basis functions on the index n proximation error [M] is given by and the parameter a. 0 1 Z 2 ∞ X finite duration, successive delays of τ (ω ) will eventually shift the @ A g o [M]= f(t) − f[n]λn(t) dt (5) bulk of its energy beyond the sampling time t =0in Fig. 1. Once 0 n∈IM this occurs, the remaining expansion coefficients become very small. Therefore, in order to minimize the number of significant DT coeffi- We focus on a nonlinear approximation technique, in which the cients for a narrow-band signal, we should maximize τg(ωo) in (6). set IM is chosen based on the characteristics of f(t). Specifically, This is accomplished by setting a = ωo. because the bilinear basis functions are orthonormal, we retain the [M] Although many CT signals of interest are not narrow-band, the largest-magnitude DT coefficients to minimize . We qualita- above analysis suggests the following ‘maximin’ strategy to initial- tively compare approximation performances through the decay of ize the value of a: For a signal with most of its information content expansion coefficients when sorted by absolute value. As proposed effectively band-limited to |ω|≤ω , choose a = ω . for wavelet approximations in [5, 6], a faster decay corresponds to a M M Because the group delay in Equation (6) is monotonically de- smaller M-term approximation error. creasing in ω, this strategy guarantees a group delay greater than or equal to τg(ωM ) for all frequencies in the band [−ωM ,ωM ].Un- 3.1. Effect of the Parameter a like a Nyquist anti-aliasing filter, this algorithm does not eliminate As seen in Fig. 2(b), the behavior of the basis functions is affected by high-frequency information. Rather, it tries to capture as much sig- the parameter a. Predictably, this has a direct impact on the approxi- nal energy as possible in the earlier DT coefficients mation performance. As an example, we analyze the bilinear approximations of the windowed sinusoid f(t) ∝ sin(10t) for 0 ≤ t<1, 3.2. Signal Characteristics which Impact Approximation and then generalize from this information. The signal f(t) has been normalized for unit energy. It is straightforward to verify that signals which have rational Laplace Fig. 3(a)-(d) show the bilinear coefficients f[n] for a =1, 10, 100 transforms with all poles located at s = −a can be represented ex- and 1000, respectively. As seen, the fastest coefficient decay occurs actly using a finite number of bilinear coefficients. In the time do- when a =10, which is the carrier frequency of f(t). An intuitive main, this corresponds to polynomials in t weighted by the expo- −at basis for this result follows by considering the group delay of the nential decay e . The approximation performance worsens as the all-pass filters in the bilinear first-order cascades. pole location(s) move farther away from s = −a. This is confirmed » „ «– in simulation by examining the sorted coefficient decay of signals d a − jω 2a f (t) ∝ t3e−kt and f (t) ∝ e−at sin(pt) for different values of k τ (ω)= ∠ = (6) k p g dω a + jω a2 + ω2 and p, respectively.

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