Gradient-Adaptive Algorithms for Minimum Phase - All Pass Decomposition of an FIR System

Transcription

1 1 Gradient-Adaptive Algorithms for Minimum Phase - All Pass Decomposition of an FIR System Mar F. Flanagan, Member, IEEE, Michael McLaughlin, and Anthony D. Fagan, Member, IEEE Abstract Adaptive algorithms which perform minimum phase - all pass MP-AP) decomposition of a finite impulse response FIR) system are proposed. The first algorithm models the MP component of the system as a lattice filter cascaded with a gain stage. The algorithm has a low misconvergence probability, and is capable of detecting misconvergence during or after adaptation. Two further algorithms are proposed based on the theory of the bicepstrum. One adaptively solves a finite linear system of equations, and the other an augmented nonlinear system. The first has an error sensitive to the proximity of the system zeros to the unit circle, while the second, although more computationally intensive, may approach the exact MP-AP decomposition given a sufficient number of iterations. These real-time MP-AP decomposition algorithms have applications in the stabilization of compound precoding, which is a pre-equalization technique included as an option in the V.92 high-speed modem standard. Index Terms Minimum phase systems, all pass systems, adaptive filtering, lattice filters, bicepstrum. This paper is a preprint of a paper accepted by IET Signal Processing and is subject to Institution of Engineering and Technology Copyright When the final version is published, the copy of record will be available at the IET Digital Library M. F. Flanagan and A. D. Fagan are with the Department of Electronic and Electrical Engineering, University College Dublin, Belfield, Dublin 4, Ireland mar.flanagan@ieee.org, tony.fagan@ucd.ie). M. McLaughlin is with DecaWave Ltd., Digital Depot, Thomas St., Dublin 8, Ireland michael@decawave.com). March 13, 2009 DRAFT

2 2 I. INTRODUCTION It is well nown see e.g. [1]) that any finite impulse response FIR) system G z) of order p may be uniquely decomposed into the cascade of a minimum phase MP) component W z) FIR, of order p) and an all pass AP) component. This decomposition may of course be effected by invoing standard rootfinding algorithms; in this paper however, adaptive algorithms are proposed which may effect this decomposition in a real-time digital signal processing DSP) application. To the authors nowledge such algorithms have not been reported previously in the literature. Our motivation is the fact that in compound precoding, a power efficient method for combining decision-feedbac equalization DFE) with trellis coding in a modem transmitter, a real-time MP- AP decomposition of the precoder feedforward filter is necessary for implementation stability [2], [3]. Compound precoding is an option in the V.92 modem standard [4]. The lattice filter is a linear filter structure commonly used in linear prediction of signals see e.g. [5]), and the least-squares lattice predictor has been successfully applied to problems in speech coding [6] and blind equalization [7]. It may be shown [5] that every transversal implementation of a minimum phase FIR filter has an equivalent implementation in the form of a lattice filter followed by a gain stage. The bijective mapping from one filter implementation to the other may be achieved via the Levinson-Durbin and inverse Levinson-Durbin recursions. We derive and evaluate a gradient adaptive algorithm to achieve this decomposition where the MP component is implemented as a lattice filter followed by a gain stage, and updated accordingly. All stochastic updates are derived by substituting sample correlation for true correlation in the steepest descent updates, as in the ordinary least-mean-square LMS) algorithm. The lattice structure has the advantage that upon convergence we may easily test whether the filter W z) given by the algorithm is actually minimum phase. A necessary and sufficient condition for the filter to be minimum phase is that the reflection coefficients of the lattice filter all have magnitude less than unity; therefore failure of this condition, and hence misconvergence of the algorithm, may be detected at any stage during adaptation. In [8] a gradient-adaptive lattice filter was proposed for linear prediction; here the cost function was taen to be the sum of the forward and bacward prediction energies, which is specific to the linear prediction problem. The matching in the MMSE sense) of the lattice filter output to a desired output signal, which is proposed in this paper, is of more general application. DRAFT March 13, 2009

3 3 The bicepstrum is a signal representation with applications in blind deconvolution [9], [10] as well as signal time delay estimation and system identification [11]. In these applications, the algorithms usually involve estimation of the higher-order statistics of a possibly noisy) data sequence. We propose two gradient-adaptive algorithms for performing MP-AP decomposition of an FIR system based on the theory of the bicepstrum. Our present wor differs fundamentally from the previous wors described, in that we target decomposition of a nown system in the absence of noise. Thus, our problem lies in how to handle an infinitely large but exact system of linear equations, as opposed to problems of moment or cumulant estimation and noise suppression. The theory of the bicepstrum yields a relationship between the infinite set of differential cepstrum parameters and a certain finite set of third-order cumulants, in the form of an infinite system of linear equations, these cumulants being generated in a straightforward manner from the coefficients of the filter G z). If this system could be solved exactly, the exact MP-AP decomposition of G z) could be computed using simple recursion formulas involving only a small subset of the differential cepstrum parameters. The first algorithm solves the truncated linear system via steepest descent, and involves an approximation error dependent on the proximity of the filter zeros to the unit circle. The second algorithm, based on steepest descent solution of an nonlinearly augmented system, is more computationally intensive, but simulations show that it may converge to the exact MP-AP decomposition. The paper is organized as follows. Section II describes the proposed gradient adaptive lattice filtering algorithm for MP-AP decomposition, and Section III describes the bicepstrum-based algorithms. Section IV presents simulation results for all algorithms and Section V concludes this wor. II. MP-AP DECOMPOSITION BASED ON ADAPTIVE LATTICE FILTERING A. Adaptive Lattice Filtering for Desired Response First we address the general problem of updating the reflection coefficents of a lattice filter in order to obtain minimum mean-squared error MMSE) between the upper output signal and a desired response. Consider the circuit shown in figure 1. The equations governing the operation of the lattice filter are f m) b m) = f m) = γ m f m) + γ m b m) + b m) 1) March 13, 2009 DRAFT

4 4 for m = 1, 2,...p. Given the input signal {x } and the desired response {d }, the problem is to adapt the reflection coefficients γ j,j = 1, 2...p in order to minimize the mean-squared error MSE) where e = f p) J {γ j }) = E { } e 2 d. In order to derive a stochastic gradient algorithm, we need to estimate J for each γ j. It is easy to see that for any j {1, 2,...p}, e can be written as A + Bγ j, where A,B are independent of γ j. Therefore where we define s j) = fp) J = E { A + Bγ j ) 2} = 2E {AB} + 2γ j E { B 2} = 2E {A + Bγ j ) B} { } e = 2E e { } f p) = 2E e { } = 2E e s j) for each j {1, 2,...p}. Next, choose any j {1, 2,...p}. Observe that the sequences f j) and b j) of γ j. Therefore, putting m = j in 1) and differentiating with respect to γ j, we obtain are independent f j) = b j) and b j) = f j) Also, differentiating 1) with respect to γ j for any m {j + 1,j + 2,...p} yields f m) b m) = fm) f m) = γ m b m) + γ m + bm) DRAFT March 13, 2009

5 5 This means that each s j) can be computed using a sublattice, as shown in figure 2. Each of these sublattices is attached to the main lattice of figure 1 at the appropriate point. The steepest descent update for the {γ j } is then simply γ +1) j { = γ ) j µe e s j) } Replacing true correlation with sample correlation yields the adaptive algorithm γ +1) j = γ ) j µe s j) Note that the sublattice of figure 2 which computes s j) requires p j lattice stages. Therefore the entire lattice structure, including sublattices, requires p = pp+1) 2 lattice stages instead of the usual p stages. B. Gradient-Adaptive Lattice Based MP-AP Decomposition Algorithm The adaptive circuit for the MP-AP decomposition algorithm based on the gradient-adaptive lattice filter is shown in figure 3. The FIR filter Gz) has order p and models the system to be decomposed. The MP component W z) is implemented as a lattice filter with coefficients {γ j } followed by a gain stage β. The IIR section composed of the filters C z) and z p C z ) 1 represents the all pass component. All filters have order p in order to cater for the possibility of a maximum-phase G z). The coefficient c p is set to unity and is not updated by the adaptive algorithm. Note that the section comprised of the feedforward and feedbac filters with coefficients {c j } is guaranteed to be all pass for any {c j } and hence at any stage during adaptation) because of the reversal of the coefficients in the feedbac section. The MSE is ) defined as J {c j }, {γ j },β) = E {e 2 }. If the convolution of w = w 0 w 1... w p and ) c = c 0 c 1... c p is denoted h = h 0 h 1... h 2p ), then the MSE may be written J = h c T Q h c where Q is a matrix having as entries elements of the autocorrelation sequence R u i) = E {u u i } and the cross-correlation sequence R ud i) = E {u d i }. We assume that these March 13, 2009 DRAFT

6 6 two correlation sequences are stationary; hence so is the MSE. From figure 3, the estimation error at time step may be written as e = z d p p = c j q j c j d p+j d j=0 j=0 p = c j q j d p+j ) d q p ) j=0 The sequences {q } and {d } are independent of {c j }. Therefore the gradient of J with respect to {c j } is given by ) ) J = 2E {e q j d p+j)} c j Replacing true correlation by sample correlation, we obtain the stochastic gradient update for {c j } as c +1) j = c ) j µe q j d p+j ) An equivalent circuit to that of figure 3 is shown in figure 4. Since e = f p) a + d ), the update for the reflection coefficients {γ j } is as presented in the discussion of section II-A on adaptive lattice filtering for desired response, i.e. γ +1) j = γ ) j µe s j) j = 1, 2,...p where s j) is computed by the sublattices. To derive the update for the gain parameter β, note that if the order of the lattice filter and gain stage were switched in figure 4, the input to the gain stage would be f p) /β. Therefore the update for the gain parameter β may be obtained as simply the stochastic gradient update for a single-tap FIR filter, i.e. β +1) = β ) µe Finally note that if the circuit of figure 4 is implemented, the entire equivalent circuit of figure 3 need not be. The lattice filter without sublattices) and gain stage, followed by a p-element delay line is sufficient. Note that the MSE as a function of the coefficients to be updated) forms a multimodal surface. There are in general 2 p valid MMSE solutions, for each of which the lattice filter has f p) β ) ) DRAFT March 13, 2009

7 7 the same magnitude response. This is in contrast to the usual application of lattice filtering in linear prediction, where the lattice filter must be minimum phase. Only one of the 2 p MMSE solutions yields a minimum phase W z); therefore convergence of the adaptive algorithm to the MP-AP decomposition is not guaranteed, but depends strongly on the initialization of the algorithm. We advocate initialization of the reflection coefficients to γ 0) j = 0 for j = 1, 2,...p; this filter is minimum phase optimum in the sense that its zeros are at maximum distance from the unit circle while maintaining minimum phase. III. MP-AP DECOMPOSITION ALGORITHMS BASED ON THE BICEPSTRUM In this section we describe two algorithms to perform MP-AP decomposition of the FIR system G z), based on the theory of the bicepstrum. The pertinent details of this theory are presented in section III-A. This theory yields a relationship between the infinite set of differential cepstrum parameters and a certain finite set of third-order cumulants, in the form of an infinite system of linear equations, these cumulants being generated in a straightforward manner from the coefficients of the filter G z). If this system could be solved exactly, the exact MP-AP decomposition of G z) could be computed using simple recursion formulas involving only a small subset of the differential cepstrum parameters. The first presented algorithm solves the truncated linear system via steepest descent, and the second performs steepest descent solution of this system augmented with nonlinear equations. The proposed bicepstrum-based algorithms are generally more computationally intensive than the algorithm of Section II-B but do not suffer from the problem of misconvergence. A. Theory of the Bicepstrum In this section we develop the relevant relationships between the minimum and maximum phase components of G z), the set of third-order cumulants associated with G z), and the differential cepstrum parameters. This development borrows from [10] and [12]. For the sequence {g n)} with Z-transform G z), the complex cepstrum is defined as [1] g c n) = Z {log G z)} and the differential cepstrum as g d n) = Z { z log G z) } March 13, 2009 DRAFT

8 8 For the case at hand, the finite sequence of filter coefficients g 0),g 1),...g p) has Z-transform p G z) = g n) z n n=0 This filter of order p has M zeros {a i } inside the unit circle and N zeros {c i } outside the unit circle, where M + N = p. G z) can thus be factored into minimum phase and maximum phase components as 1 G z) = AI z) Q B z) 2) where I z) and Q z) are both causal, monic and minimum phase, and have orders M and N respectively. We use the notation and I z) = Q z) = M i n) z n n=0 N q n) z n n=0 I z) has zeros {a i } and Q z) has zeros {b i }, and we define c i = 1/b i for each i. We also define the anticausal filter O z) = Q B z) z N, so the factorization of G z) becomes The quantities and G z) = AI z) O z) z N 3) A = B = for = 1, 2,... are related to the differential cepstrum parameters as A n n 2 i d n) = 0 n 1 M a i 4) i=1 N b i 5) i=1 6) 1 Here, to simplify the presentation, we adopt the notation T B z) = z t T `z where T z) is a filter of order t. DRAFT March 13, 2009

9 9 and B 1 n n 0 o d n) = 0 n 1 7) Therefore in what follows we shall also refer to {A } and {B } as the differential cepstrum parameters. Note that since Iz) and Qz) are both minimum phase, we have a i < 1 and b i < 1 for each i, and therefore the sequences {A } and {B } are both monotonically decreasing sequences which converge to zero as. Next, if the system G z) is driven by a zero-mean input x n) satisfying 2 1 i = j = E {x i)xj)x)} = 0 otherwise then the third-order cumulants of the output sequence y n) are C m,n) = E {y i)y i + n) y i + m)} { = E g r) x i r) g l)xi + n l) } g s)xi + m s) r Z l Z s Z = g r) g l) g s)e {x i r)xi + n l) x i + m s)} r Z s Z l Z = g r) g r + n) g r + m) r Z Note that if G z) is of order p then C m,n) = 0 outside the hexagonal region { m,n) Z 2 : m p, n p, m n p } The bispectrum of y n) is defined as the 2-dimensional Z-transform of the third-order cumulant sequence [12], i.e. S z 1,z 2 ) = m Z n Z C m,n) z1 m z2 n 2 The two-level i.i.d. sequence condition. n o xn) P xn) = 2 2/3 = xn) 13, P = 2 /3 = 2, for example, satisfies this 3 March 13, 2009 DRAFT

10 10 So S z 1,z 2 ) = m Z = Z = Z [ ] g ) g + n) g + m) z1 m z2 n n Z g ) Z [ m Z g + m)z m 1 g ) [ G z 1 )z 1] [ G z2 )z 2 ][ ] n Z g + n) z n 1 ] Note that the expression G z 1 )Gz 2 ) G z1 z2 = G z 1 ) G z 2 ) g )z1z 2 Z = G z 1 ) G z 2 )G ) z1 z2 ) = [ AI z1 )O z 1 )z N [ AI z1 z2 1 ] [ ] AI z2 )O z 2 )z2 N ) ] N ) O z 1 z2 ) z 1 z2 = A 3 I z 1 ) O z 1 )I z 2 ) O z 2 )I z1 z2 ) ) O z 1 z2 contains no term in N, the number of zeros of G z) outside the unit circle, even though G z) does 3. The bicepstrum of y n) is defined as c m,n) = Z {log S z 1,z 2 )} and can be shown to equal [13] c m,n) = log A 3 m = n = 0 A n /n m = 0,n > 0 B n /n m = 0,n < 0 A m /m n = 0,m > 0 B m /m n = 0,m < 0 A m /m m = n < 0 B m /m m = n > 0 0 otherwise. 8) 3 This leads to the property that the algorithms of sections III-B and III-C do not require nowledge of the value of N; this property is the reason for use of the bicepstrum rather than the complex cepstrum for this development. DRAFT March 13, 2009

11 11 A relationship may be derived between the third-order cumulants C m, n) and the differential cepstrum parameters as follows. Defining s z 1,z 2 ) = log S z 1,z 2 ) we have s z 1,z 2 ) z 1 = 1 S z 1,z 2 ) S z 1,z 2 ) z 1 [ ] [ ] s z 1,z 2 ) S z 1,z 2 ) z 1 S z 1,z 2 ) = z 1 z 1 z 1 Taing the 2-dimensional inverse Z-transform, [ mc m,n)] C m,n) = mc m,n) 9) Substituting 8) for the bicepstrum parameters c m, n) in the convolution sum yields =1 { A [C m,n) C m +,n + )] } + B [C m,n ) C m +,n)] = mc m,n) 10) This equation relates the third-order cumulants to the differential cepstrum parameters. The differential cepstrum parameters then relate to the minimum and maximum phase components of G z) via n A i n ) = ni n) 11) =1 and n B q n ) = nq n) 12) =1 for n 1. We conclude that the theory of the bicepstrum yields a relationship via 10)) between the set of third-order cumulants {C m,n) : m,n) Z 2 } and the set of differential cepstrum parameters {A : 1} {B : 1}, and also a relationship via 11) and 12)) between a subset of the differential cepstrum parameters {A : 1 n} {B : 1 n} and the maximum and minimum phase components of Gz), i.e. {in) : n 1} {qn) : n 1}. March 13, 2009 DRAFT

12 12 B. Bicepstrum-Based Gradient-Adaptive Algorithm 1: Linear System Solution First note that 10) holds for each m,n) Z 2. We consider only equations corresponding to m,n) pairs which lie in the union of three sets which are subsets of three lines in the plane 4 : S 1 = { m,n) Z 2 m = 1, 0 n p + 1 } S 2 = { m,n) Z 2 n = 0,m 2 } S 3 = { m,n) Z 2 m = n 2 } Note that when m > p, the set S 2 yields difference equations in the {A }: A C m, 0) = 0 m > p =1 and when m > p, the set S 3 yields difference equations in the {B }: B C m,m ) = 0 m > p =1 We have an infinite set of linear equations; therefore we consider only the equations for m E, for some positive integer E. These correspond to the set S = {m,n) S 1 S 2 S 3 m E}. This yields 2E + p equations in 2 E + p) unnowns {A 1,A 2,...A E+p,B 1,B 2,...B E+p }. This system is underdetermined by p equations; therefore we mae the approximation A i = 0 for i > R = E + p r and B i = 0 for i > S = E + p s, where r = [ p 2] and r + s = p. By 4) and 5), the accuracy of this approximation increases with increasing distance of the zeros of G z) from the unit circle; also, since {A } and {B } are monotonically decreasing sequences, the accuracy of the approximation increases with increasing E. The resulting reduced 2E + p) 2E + p) system where a = Ja + Qb = u ) T ) T A 1 A 2... A R, b = B 1 B 2... B S, J is an R + S) R matrix and Q is an R + S) S matrix, is in general ran-deficient; its solution via singular value 4 The choice of subsets S 1, S 2 and S 3 is motivated by the symmetry properties of the third-order cumulant sequence. The thirdorder cumulant sequence has a sixfold symmetry in the m, n) plane c.f. [14]), i.e. Cm, n) = Cn, m) = C n, m n) = C m, n m) = Cm n, n) = Cn m, m). This symmetry implies that many of the equations in the system of 10) are linearly dependent. The choice of this particular subset of equations avoids linearly dependence of equations due to these symmetries. DRAFT March 13, 2009

13 13 decomposition taes significant computational effort, especially for large values of E. Therefore a steepest descent algorithm is used in order to avoid the computational complexity associated with computation of the pseudoinverse. The error vector is defined as e = Ja + Qb u 13) The sum of error squares is then I = e T e = I A 1 m,n) S e m,n) 2 and the gradient of I with respect to the differential cepstrum parameters is given by ) T I I A 2... A = 2J T e 14) R and I B 1 I B 2... I B S ) T = 2Q T e 15) yielding a steepest descent update for the differential cepstrum parameters as a +1) = a ) µj T e b +1) = b ) µq T e The minimum and maximum phase components of G z) are then solved for using the recursions 11) and 12). Finally the gain parameter A is solved for as follows. First, the estimates of i n) and q n) yielded by 11) and 12) are combined to form x n) = i n) q B n), where q B n) is the inverse Z-transform of Q B z), and is equal to q N n). Then, in light of 2), A is estimated as the mean value of g n) /x n) over n {0, 1,...p}, i.e. Â = 1 p + 1 p n=0 g n) x n) C. Bicepstrum-Based Gradient-Adaptive Algorithm 2: Nonlinear System Solution A problem with the previous algorithm is that it can at best yield only an approximate solution for the differential cepstrum parameters. This is because a finite subset of the equations given by 10) does not contain enough information to solve for the {A } and {B } exactly, and is also because certain of the {A } and {B } are approximated to zero. March 13, 2009 DRAFT

14 14 ) T The algorithm of this section solves iteratively for the vectors a = A 1 A 2... A E+p ) T and b = B 1 B 2... B E+p. The linear system 10) with m,n) S yields 2E + p equations which must be satisfied; as in the algorithm of Section III-B this yields a cost function I 1 = e T e where the error vector e is given by 13). This time however none of the variables {A } or {B } are approximated to zero, i.e. r = s = 0. Also, in light of 11), defining n τ n = A i n ) + ni n) 16) =1 we must have τ n = 0 for n 1. We may obtain τ n = 0 for n = 1, 2,...p by regarding the {i n) n = 1, 2,...p} as functions of the {A = 1, 2,...p} via 16). We can then form another cost function: I 2 = τ T τ ) T where τ = τ p+1 τ p+2... τ E+p. To obtain an estimate of the gradient of I2 with respect to the A i s, we use the fact that from 16), so we have τ n = A j i n j) n j 0 otherwise I 2 A 1 I 2 A 2... ) T I 2 A = 2M T E+p τ τ 17) where M τ is the E E + p) Toeplitz matrix M τ = τ p+1 A 1 τ p+1 A 2... τ p+2 A 1 τ p+2. A τ p+e A τ p+1 A p+e.. τ p+e A p+e The gradient of I 2 with respect to the {B i } is zero. A similar argument applies with σ n = i p) i p 1) i p).... = i 0) n B q n ) + nq n) =1 DRAFT March 13, 2009

15 15 The equations σ n = 0 for n = 1, 2,...p allow the recursive solution for the q n), n = 1, 2,...p. We form another cost function where σ = I 3 = σ T σ σ p+1 σ p+2... σ E+p ) T, and the gradient of I3 with respect to the {B i } is I 3 B 1 I 3 B 2... ) T I 3 B = 2M T E+p σ σ 18) where M σ is the E E + p) Toeplitz matrix q p) q p 1) q p).... M σ = q 0) The gradient of I 3 with respect to the {A i } is zero. We see the solution a, b) which minimizes the overall cost function I = 3 =1 It may be deduced from 14), 15), 17) and 18) that a gradient-based solution to this problem is achieved via the steepest descent updates a +1) = a ) µj T e µm T τ τ I and b +1) = b ) µq T e µm T σσ where at every iteration the final terms in the two equations are computed by using the a ) and b ) vectors to solve for {i n) n = 1, 2,...p} and {q n) n = 1, 2,...p} via 11) and 12). IV. SIMULATION RESULTS The lattice based adaptive algorithm of Section II-B was tested on a set of 5000 filters G z) of order p = 9, the coefficients of G z) in each case being taen from a uniform probability distribution on the interval [, 1]. Since the MSE function forms a multimodal surface, the March 13, 2009 DRAFT

16 16 choice of initial filter coefficients is important. The reflection coefficients were initialized to γ 0) j = 0 for j = 1, 2,...p as described in Section II-B. The gain parameter was initialized to β 0) = 10. Small values of β 0) 1) were found to result in more probable convergence to a nonminimum phase W z), while large values of β 0) 100) were found to result in numerical instability. C z) was initialized as a comb filter, i.e. c 0) 0 = c 0) p = 1 and c 0) j = 0 for j / {0,p}, so that at initially the all pass section has cancelling pole-zero pairs evenly spaced on the unit circle. The excitation {u } was chosen as a white sequence with elements equiprobable in {, +1}) in order to identify the system G z) at all frequencies. The step size was chosen as µ = 10 3, small enough so that of the 5000 simulation runs, each of 20,000 iterations, none diverged. As the filter W z) converges to the MP component of G z), the zeros of the all pass filter move out from the unit circle and the poles of the all pass filter which are the reciprocals of its zeros) move inside the unit circle to provide the correct compensation. It was found that in 4628 of the 5000 cases, the filter given by the algorithm was minimum phase 92.6% success rate), and the average final MSE was Figure 5 shows the evolution of the ensemble averaged MSE with the number of iterations. Also, by way of illustration, z-plane results are shown in figures 6 and 7 for one of these filters G z). Figure 6 shows the pole-zero diagram of the original filter to be decomposed. Figure 7 shows the pole-zero diagrams of the MP and AP components given by the adaptive algorithm. The adaptive algorithm of Section III-B was tested on a variety of systems G z) of different orders. The value of E required for adequate performance was found to increase with the filter order p, in general. Note however that for any p, the possibility of the zeros of G z) being very close to the unit circle means that good performance cannot be guaranteed no matter how large a value of E is chosen. As a general illustration of performance, we demonstrate the results where the algorithm is applied to the FIR filter of figure 8 with p = 18 and M = N = 9, which is particularly challenging in that it exhibits both a large value of p and a number of zeros close to the unit circle. The step size µ = was found to give good convergence, and the truncation depth was chosen as E = 60 lower values of E caused noticeable performance degradation). Because of the rapid initial convergence of the algorithm, only 1000 iterations were necessary to achieve a cost function value below The algorithm was terminated at this point, although as in any other steepest descent algorithm) the cost function becomes arbitrarily small given DRAFT March 13, 2009

17 17 a sufficient number of iterations. Figure 9 compares the z-plane diagram of the estimated MP component with the actual z-plane diagram. It may be seen that the positions of the estimated zeros are indistinguishable from their actual positions in the complex plane, even though Gz) has zeros close to the unit circle. The adaptive algorithm of Section III-C was found to exhibit much slower initial convergence than the algorithm of Section III-B, in general. As an illustration of performance, the algorithm was applied to the FIR system shown in figure 8. The simulation parameters were E = 60 and µ = After iterations all estimated coefficients of the polynomials I z) and Q z) were found to agree with their actual values to 3 decimal places the corresponding z- plane diagrams are indistinguishable and are therefore omitted). In all cases tested, no flooring effect of the cost function was exhibited. It is left as an interesting open problem to prove that the proposed algorithm has the property of guaranteed converence to the exact MP-AP system decomposition. V. CONCLUSION Three new adaptive algorithms which perform real-time MP-AP decomposition of an FIR system have been proposed and evaluated. The first algorithm, based on adaptive lattice filtering, models the minimum phase component of the system as a lattice filter cascaded with a gain stage. With this adaptive filter structure, algorithm misconvergence is of low probability and is immediately detectable. The other two algorithms are based on the theory of the bicepstrum. The second algorithm solves a system of linear equations via steepest descent, and incurs an approximation error dependent on the proximity of the system zeros to the unit circle. The third algorithm solves a companion system of nonlinear equations via recursion and has the distinct advantage of being capable of converging to the exact MP-AP decomposition. As well as having general application, these algorithms may be used to perform the MP-AP decomposition required for stability of compound precoding in the V.92 modem. VI. ACKNOWLEDGMENT The authors would lie to acnowledge the support of Enterprise Ireland and Lae Communications Ltd. March 13, 2009 DRAFT

18 18 REFERENCES [1] A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing. Englewood Cliffs, NJ.: Prentice Hall, [2] M. McLaughlin, Compound Precoding: Pre-equalization for channels with combined feedforward and feedbac characteristics, Irish Signals and Systems Conference, pp , [3] M. F. Flanagan, M. McLaughlin and A. D. Fagan, Compound Precoding: A Pre-equalization Technique for the Bandlimited Gaussian Channel", IET Communications, to appear. [4] ITU-T Recommendation V.92, Enhancements to recommendation V.90, [5] S. Hayin, Adaptive Filter Theory. Information and System Science Series, Prentice Hall, [6] J. Mahoul, Linear prediction: A tutorial review, Proceedings of the IEEE, vol. 63, no. 4, pp , April [7] J. Mannerosi and D. P. Taylor, Blind equalization using least-squares lattice prediction, IEEE Transactions on Signal Processing, vol. 47, no. 3, pp , March [8] J. Mahoul and R. Viswanathan, Adaptive lattice methods for linear prediction, IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 1978, pp , April [9] A. P. Petropulu and C. L. Niias, Blind deconvolution based on signal reconstruction from partial information using higherorder spectra, IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 1991, pp , April [10] S. Hayin, Blind Deconvolution. Prentice Hall, [11] C. L. Niias and F. Liu, Bicepstrum computation based in second- and third-order statistics with applications, IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 1990, pp , April [12] J. G. Proais, C. Rader, F. Ling and C. L. Niias, Advanced Digital Signal Processing. MacMillan, New Yor, [13] R. Pan and C. L. Niias, The complex cepstrum of higher-order cumulants and nonminimum phase system identification, IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 36, no. 2, pp , February [14] A. I. Hashad and C. W. Therrien, Applying the symmetry properties of third order cumulants in the identification of non-gaussian ARMA models, IEEE Signal Processing Worshop on Higher-Order Statistics, pp , June DRAFT March 13, 2009

19 19 f 0) f 1) f p) e γ 1 γ 2 γ p x d γ 1 γ 2 γ p b 0) b 1) b p) Fig. 1. Adaptive lattice filtering for desired response. The reflection coefficients of the lattice filter are adapted to minimize the mean-squared error between the lattice filter output f p) and the desired response d. b j) = f j) f j+1) f p) = s j) γ j+1 γ j+2 γ p γ j+1 γ j+2 γ p f j) = b j) b j+1) b p) Fig. 2. Sublattice for computation of s j), j {1, 2,... p}. These sequences are used to update the reflection coefficients of the adaptive lattice filter of figure 1. March 13, 2009 DRAFT

20 20 u LATTICE FILTER {γ j } β q Cz) z e Wz) z p Cz ) 1 d Gz) Fig. 3. Lattice-based adaptive circuit for MP-AP decomposition of an FIR system Gz). {s j) } u Cz) β LATTICE FILTER {γ j } f p) z e a z p Cz ) 1 d Gz) Fig. 4. j = 1, 2,... p. n Equivalent circuit to that of figure 3. The lattice filter in this circuit is equipped with sublattices to compute s j) o, DRAFT March 13, 2009

21 MSE Number of Iterations x 10 4 Fig. 5. MSE convergence, ensemble averaged over 5000 simulations of the adaptive lattice filtering algorithm of Section II-B Imaginary Part Real Part Fig. 6. Pole-zero diagram of the FIR system Gz) used to test the adaptive algorithm of Section II-B. March 13, 2009 DRAFT

22 Imaginary Part Imaginary Part Real Part Real Part Fig. 7. of Section II-B. Pole-zero diagram for the MP component left) and the AP component right) of G z) given by the adaptive algorithm DRAFT March 13, 2009

23 Imaginary Part Real Part Fig. 8. Pole-zero diagram for the FIR filter Gz) used to test the performance of the bicepstrum-based algorithms. March 13, 2009 DRAFT

24 Imaginary Part Imaginary Part Real Part Real Part Fig. 9. system Gz). Comparison of I z) zeros given by the algorithm of Section III-B left) with actual I z) zeros right) for the FIR DRAFT March 13, 2009