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Prony, Padé, and Linear Prediction for the Time and Frequency Domain Design of IIR Digital Filters and Parameter Identification

Module by: C. Sidney Burrus. E-mail the author

Summary: Model based signal processing or signal analysis or signal representation has a rather different point of view from the more traditional filtering and algorithm based approaches. However, in all of these, the names of Prony, Padé, and linear prediction come up. This note examines these ideas with the goal of showing they are all based on the same principles and all can be extended and generalized.

Introduction

Model based signal processing or signal analysis or signal representation has a rather different point of view from the more traditional filtering and algorithm based approaches. However, in all of these, the names of Prony, Padé, and linear prediction come up. This note examines these ideas with the goal of showing they are all based on the same principles and all can be extended and generalized.

Prony originally posed the problem of separating the individual characteristics of gases from measurements made on a combination of them in 1795 [58]. We pose this problem using modern terminology and notation as follows. A signal is modeled as y(n)y(n) which consists of equally spaced samples of x(t)x(t) which is a finite sum of NN exponentials,

x ( t ) = k = 1 N K k e α k t . x ( t ) = k = 1 N K k e α k t .
(1)

The sampling is described by

y ( n ) = x ( T n ) = k = 1 N K k e α k T n = k = 1 N K k λ k n y ( n ) = x ( T n ) = k = 1 N K k e α k T n = k = 1 N K k λ k n
(2)

where λk=eαkTλk=eαkT. Prony's problem is to calculate the 2N2N parameters, KkKk and αkαk (or λkλk), from the samples, y(n)y(n).

Prony's original approach [58] to solving this problem is straight-forward but awkward, requiring two different procedures for even order and odd order systems. A more flexible and general method can be derived using the z-transform of Prony's formulation.

Padé's method [55] is a rational approximation of a polynomial. It has been shown [80] to be the z-transformed version of Prony's method. The z-transform of Equation 2 can be written as an NthNth order rational function of zz

Y ( z ) = b 0 + b 1 z - 1 + + b M - 1 z - M + 1 1 + a 1 z - 1 + + a N - 1 z - N + 1 Y ( z ) = b 0 + b 1 z - 1 + + b M - 1 z - M + 1 1 + a 1 z - 1 + + a N - 1 z - N + 1
(3)
with M=N-1M=N-1. The problem is to find the 2N+12N+1 parameters, bkbk and akak, from the appropriate samples y(n)y(n). This can be expressed as a matrix convolution and solved [5], [57]. The equivalence of Prony's method and Padé's method is discussed in [80]. Soewito made use of this equivalence [71].

Linear prediction uses a model of the signal which assumes that a particular value of the signal can be expressed as a weighted linear combination of the NN previous values of the signal [48], [15]. Finding those weights turns out to be exactly the same problem as finding the denominator coefficients in Equation 3 and equivalent to finding the αkαk or λkλk in Equation 2.

These methods can all be extended from an interpolation problem to a least equation error method by using more samples y(n)y(n) than the number of unknown coefficients, anan and bnbn or KkKk and αkαk. This is discussed in [5], [57]. Iterative methods have been developed which solve an iterative reweighted least equation error to achieve a least solution error result [69], [68], [67]. McClellan has shown this to be equivalent to a well known adaptive filtering method [51].

The basic idea behind these three methods is based on the elimination or minimization of the equation error, which is a linear problem, rather than the usual minimization of the solution error, which is a nonlinear problem. Most of these formulations also uncouple the calculation of the numerator and the denominator coefficients [67]

It is possible to pose the problem in the frequency domain and also minimize the equation error [41], [57]. One advantage to doing that is better conditioning from the uniform sampling of the frequency response between zero and ππ rather than using the KK samples at the beginning of an infinite time interval [57], [34], [70], [13] and to deal with the case where the design criteria are given in the frequency domain.

Prony's method (or Padé's method or linear prediction) in both the time and frequency domain is not only important for its own sake, but also as a method to start other iterative methods or being the base of other iterative methods that minimize other error criteria. There is much still to be discovered about this important, interesting, and powerful set of methods.

Literature on Prony's Method and Extensions

Original [58], [55]

Basic [28], [75], [80], [5], [27], [19], [12], [2], [16]

Related [46], [49], [37], [10], [31][65], [50], [47], [4], [76], [73]

Applications[44], [66], [6], [7], [45], [33], [79], [56], [6]

Iteration [81], [69], [54], [23], [17], [51], [25], [62], [71], [29], [77]

The Exact Solution or the Interpolation Problem in the Time Domain

The formulation of the time-domain IIR filter design problem and the methods for its solution are a version of Prony's (actually Padé's) method.

The z-transform transfer function for an IIR filter [57] is given by

H ( z ) = B ( z ) A ( z ) = b 0 + b 1 z - 1 + + b M z - M 1 + a 1 z - 1 + + a N z - N = h 0 + h 1 z - 1 + h 2 z - 2 + H ( z ) = B ( z ) A ( z ) = b 0 + b 1 z - 1 + + b M z - M 1 + a 1 z - 1 + + a N z - N = h 0 + h 1 z - 1 + h 2 z - 2 +
(4)

In the time domain, this becomes a convolution

n = 0 N a n h i - n = b i n = 0 N a n h i - n = b i
(5)

where a0=1a0=1 and i=0,1,,Mi=0,1,,M which can be expressed in matrix form by

b 0 b 1 b M 0 0 = h 0 0 0 0 h 1 h 0 0 h 2 h 1 h 0 h L h 0 1 a 1 a N 0 0 b 0 b 1 b M 0 0 = h 0 0 0 0 h 1 h 0 0 h 2 h 1 h 0 h L h 0 1 a 1 a N 0 0
(6)

Note that the hnhn in Equation 6 are the infinitely enduring causal impulse response values of the filter, not the aliased version of it used in [13] and Equation 23. A more compact matrix notation is

b 0 = H a 0 b 0 = H a 0
(7)

where HH is (L+1)(L+1) by (L+1)(L+1), bb is length-(M+1)(M+1), and aa is length-(N+1)(N+1). Because the lower L-NL-N terms of the right-hand vector of Equation 6 are zero, the HH matrix can be reduced by deleting the right-most L-NL-N columns to give H0H0 which causes Equation 7 to become

b 0 = H 0 a b 0 = H 0 a
(8)

Because the first element of aa is unity, it is partitioned to remove the unity term and the remaining length-NN vector is denoted a*a*. The simultaneous equations represented by Equation 8 are uncoupled by further partitioning of the HH matrix as shown in

b 0 = H 1 h 1 H 2 1 a * b 0 = H 1 h 1 H 2 1 a *
(9)

where H1H1 is (M+1)(M+1) by (N+1)(N+1), h1h1 is length-(L-M)(L-M), and H2H2 is (L-M)(L-M) by NN. The lower (L-M)(L-M) equations are written

0 = h 1 + H 2 a * 0 = h 1 + H 2 a *
(10)

or

h 1 = - H 2 a * h 1 = - H 2 a *
(11)

which must be solved for a*a*. The upper M+1M+1 equations of Equation 9 are written

b = H 1 a b = H 1 a
(12)

which allows the calculation of bb.

If L=N+ML=N+M, H2H2 is square. If H2H2 is nonsingular, Equation 11 can be solved exactly for the denominator coefficients in a*a*, which are augmented by the unity term to give aa. From Equation 12, the numerator coefficients in bb are found. If H2H2 is singular [42] and there are multiple solutions, a lower order problem can be posed. If there are no solutions, the methods of the next section can be used and/or the assumed order increased.

Note that any trade-off in the order of numerator and denominator can be prescribed. If the filter is in fact an FIR filter, aa is unity and a*a* does not exist. Under these conditions, Equation 12 states that bn=hnbn=hn, which is one of the cases of FIR frequency sampling covered in Section 3.1 of [57]. Also note that there is no control over the stability of the filter designed by this method.

An Approximate Solution or the Least Equation Error Problem in the Time Domain

In order to obtain better practical filter designs, the interpolation scheme of the previous section is extended to give an approximation design method [57]. It should be noted at the outset that the method developed in this section minimizes an equation-error measure and not the usual solution or signal error measure.

The number of samples specified, L+1L+1, will be made larger than the number of filter coefficients, M+N+1M+N+1. This means that H2H2 is rectangular and, therefore, Equation 7 cannot in general be satisfied. To formulate an approximation problem, a length-(L+1)(L+1) error vector εε is introduced in Equation 7 and Equation 8 to give

b 0 = H 0 a + ε b 0 = H 0 a + ε
(13)

Equation Equation 11 becomes

h 1 - ε = - H 2 a * h 1 - ε = - H 2 a *
(14)

where now H2H2 is rectangular with (L-M)>N(L-M)>N. Using the same methods as used to derive Equation 11, the error εε is minimized in a least-squared error sense by the solution of the normal equations [42] which occurs when the error is orthogonal to the impulse response.

H 2 T h 1 = - H 2 T H 2 a * H 2 T h 1 = - H 2 T H 2 a *
(15)

If the equations are not singular, the solution is

a * = - [ H 2 T H 2 ] - 1 H 2 T h 1 . a * = - [ H 2 T H 2 ] - 1 H 2 T h 1 .
(16)

which uses the so-called pseudo-inverse [42], [11]. If the normal equations are singular, the pseudo-inverse can be used to obtain a minimum norm or reduced order solution.

The numerator coefficients are found by the same techniques as before in Equation 12

b = H 1 a b = H 1 a
(17)

which results in the upper M+1M+1 terms in εε being zero and the total squared equation error being minimum.

As is true for LS-error design of FIR filters, Equation 15 is often numerically ill-conditioned and Equation 16 should not be used to solve for a*a*. Special algorithms such as those used by Matlab and LINPACK [20] should be employed.

Prony's Method in the Frequency Domain gives the Frequency-Sampling Design of IIR Filters

In this section a frequency-sampling design method is developed such that the frequency response of the IIR filter will interpolate or pass through the given samples of a desired response. This development is parallel to that used for Prony's method in the time domain. Since an IIR filter cannot have linear phase, the sampled response must contain both magnitude and phase. The extension of the frequency- sampling method to a LS-error approximation can be done as for the FIR filter [57]. The method presented in this section uses a criterion based on the equation error rather than the more common error between the actual and desired frequency response. Nevertheless, it is a useful noniterative design method. Finally, a general discussion of iterative design methods for LS-frequency response error is given.

The method for calculating samples of the frequency response of an IIR filter can be reversed to design a filter much the same way it was for the FIR filter using frequency sampling [57]. The z-transform transfer function for an IIR filter is given by

H ( z ) = B ( z ) A ( z ) = b 0 + b 1 z - 1 + + b M z - M 1 + a 1 z - 1 + + a N z - N . H ( z ) = B ( z ) A ( z ) = b 0 + b 1 z - 1 + + b M z - M 1 + a 1 z - 1 + + a N z - N .
(18)

The frequency response of the filter is given by setting z=e-jωz=e-jω. Using the notation

H ( ω ) = H ( z ) | z = e - j ω . H ( ω ) = H ( z ) | z = e - j ω .
(19)

Equally-spaced samples of the frequency response are chosen so that the number of samples is equal to the number of unknown coefficients in Equation 18. These L+1L+1 = M+N+1M+N+1 samples of this frequency response are given by

H k = H ( ω k ) = H ( 2 π k L + 1 ) H k = H ( ω k ) = H ( 2 π k L + 1 )
(20)

and can be calculated from the length-(L+1)(L+1)) DFTs of the numerator and denominator (padded with zeros to the proper length).

H k = DFT { b n } DFT { a n } = B k A k H k = DFT { b n } DFT { a n } = B k A k
(21)

where the indicated division is term-by-term division for each value of kk. Multiplication of both sides of Equation 21 by AkAk gives

B k = H k A k B k = H k A k
(22)

If the length-(L+1)(L+1) inverse DFT of HkHk is denoted by the length- (L+1)(L+1) sequence hnhn, equation Equation 22 becomes cyclic convolution which can be expressed in matrix form by

b 0 b 1 b M 0 0 = h 0 h L h L - 1 h 1 h 1 h 0 h L h 2 h 1 h 0 h L h 0 1 a 1 a N 0 0 b 0 b 1 b M 0 0 = h 0 h L h L - 1 h 1 h 1 h 0 h L h 2 h 1 h 0 h L h 0 1 a 1 a N 0 0
(23)

Note that the hnhn in Equation 23 are not the impulse response values of the filter (they are an aliased version of it) as used in the FIR case or in Equation 6. Using the same approach as used for Prony's method in the time domain, a more compact matrix notation is

b 0 = H a 0 b 0 = H a 0
(24)

where HH is (L+1)(L+1) by (L+1)(L+1), bb is length-(M+1)(M+1), and aa is length-(N+1)(N+1). Because the lower L-NL-N terms of the right-hand vector of Equation 23 are zero, the HH matrix can be reduced by deleting the right-most L-NL-N columns to give H0H0 which causes Equation 24 to become

b 0 = H 0 a b 0 = H 0 a
(25)

Because the first element of aa is unity, it is partitioned to remove the unity term and the remaining length-NN vector is denoted a*a*. The simultaneous equations represented by Equation 25 are uncoupled by further partitioning of the HH matrix as shown in

b 0 = H 1 h 1 H 2 1 a * b 0 = H 1 h 1 H 2 1 a *
(26)

where H1H1 is (M+1)(M+1) by (N+1)(N+1), h1h1 is length-(L-M)(L-M), and H2H2 is (L-M)(L-M) by NN. The lower (L-M)(L-M) equations are written

0 = h 1 + H 2 a * 0 = h 1 + H 2 a *
(27)

or

h 1 = - H 2 a * h 1 = - H 2 a *
(28)

which must be solved for a*a*. The upper M+1M+1 equations of (10) are written

b = H 1 a b = H 1 a
(29)

which allows the calculation of bb.

If L=N+ML=N+M, H2H2 is square. If H2H2 is nonsingular, Equation 28 can be solved exactly for the denominator coefficients in a*a*, which are augmented by the unity term to give aa. From Equation 29, the numerator coefficients in bb are found. If H2H2 is singular [42], [11] and there are multiple solutions, a lower order problem can be posed. If there are no solutions, the approximation methods must be used and/or the assumed order increased.

Note that any order numerator and denominator can be prescribed. If the filter is in fact an FIR filter, aa is unity and a*a* does not exist. Under these conditions, Equation 29 states that bn=hnbn=hn, which is one of the cases of FIR frequency sampling covered [57]. Also note that there is no control over the stability of the filter designed by this method.

Summary

This approach uses of the DFT therefore does not allow the possibility of unequally spaced frequency samples as was possible for FIR filter design.

The frequency-sampling design of IIR filters is somewhat more complicated than for FIR filters because of the requirement that H2H2 be nonsingular. As for the FIR filter, the samples of the desired frequency response must satisfy the conditions to insure that hnhn are real. The power of this method is its ability to interpolate arbitrary magnitude and phase specification. In contrast to most direct IIR design methods, this method does not require any iterative optimization with the accompanying convergence problems.

As with the FIR version, because this design approach is an interpolation method rather than an approximation method, the results may be poor between the interpolation points. This usually happens when the desired frequency-response samples are not consistent with what an IIR filter can achieve. One solution to this problem is the same as for the FIR case [57], the use of more frequency samples than the number of filter coefficients and the definition of an approximation error function that can be minimized. Another solution is choose another desired frequency that is closer to what can be achieved. There is also no simple restriction that will guarantee stable filters. If the frequency-response samples are consistent with an unstable filter, that is what will be designed.

Discrete Least-Squared Equation-Error IIR Filter Design in the Frequency Domain

In order to obtain better practical filter designs, the interpolation scheme is extended to give an approximation design method [57]. It should be again noted at the outset that the method developed in this section minimizes an equation-error measure and not the usual frequency-response error measure.

The number of frequency samples specified, L+1L+1, is made larger than the number of filter coefficients, M+N+1M+N+1. This means that H2H2 is rectangular and, therefore, Equation 28 cannot in general be satisfied. To formulate an approximation problem, a length-(L+1)(L+1) error vector εε is introduced in Equation 25 and Equation 26 to give

b 0 = H 0 a + ε b 0 = H 0 a + ε
(30)

Equation Equation 28 becomes

h 1 - ε = - H 2 a * h 1 - ε = - H 2 a *
(31)

where now H2H2 is rectangular with (L-M)>N(L-M)>N. Using the same methods as used to derive Equation 28, the error εε is minimized in a least-squared error sense by the solution of the normal equations [42]

H 2 T h 1 = - H 2 T H 2 a * H 2 T h 1 = - H 2 T H 2 a *
(32)

If the equations are not singular, the solution is

a * = - [ H 2 T H 2 ] - 1 H 2 T h 1 . a * = - [ H 2 T H 2 ] - 1 H 2 T h 1 .
(33)

If the normal equations are singular, the pseudo-inverse [42], [11] can be used to obtain a minimum norm or reduced order solution.

The numerator coefficients are found by the same techniques as before in Equation 29

b = H 1 a b = H 1 a
(34)

which results in the upper M+1M+1 terms in εε being zero and the total squared equation error being minimum.

As is true for LS-error design of FIR filters, Equation 32 is often numerically ill-conditioned and Equation 33 should not be used to solve for a*a*. Special algorithms such as those used by Matlab and LINPACK [52], [20] should be employed.

The error εε defined in Equation 30 can better be understood by considering the frequency-domain formulation. Taking the DFT of Equation 30 gives

B k = H k A k + ε B k = H k A k + ε
(35)

where εε is the error in trying to satisfy Equation 25 when the equations are over-specified. This can be reformulated in terms of EE, the difference between the frequency response samples of the designed filter and the desired response samples, by dividing Equation 25 by AkAk to give

E k = B k A k - H k = ε k A k E k = B k A k - H k = ε k A k
(36)

where EE is the error in the solution of the approximation problem, and εε is the error in the equations defining the problem. The usual statement of a frequency-domain approximation problem is in terms of minimizing some measure of EE, but that results in solving nonlinear equations. The design procedure developed in this section minimizes the squared error εε, thus only requiring the solution of linear equations. There is an important relation between these problems. Equation Equation 36 shows that minimizing εε is the same as minimizing EE weighted by AA. However, AA is unknown until after the problem is solved.

Although this is posed as a frequency-domain design method, the method of solution for both the interpolation problem and the LS equation-error problem is the same as the time-domain Prony's method, discussed in Section 7.5 of reference [57].

Numerous modifications and extensions can be made to this method. If the desired frequency response is close to what can be achieved by an IIR filter, this method will give a design approximately the same as that of a true least-squared solution-error method. It can be shown that ε=0E=0ε=0E=0. In some cases, improved results can be obtained by estimating AkAk and using that as a weight on εε to approximate minimizing EE. There are iterative methods based on solving Equation 33 and Equation 34 to obtain values for AkAk. These values are used as weights on εε to solve for a new set of AkAk used as a new set of weights to solve again for AkAk[57]. The solution of Equation 33 and Equation 34 is sometimes used to obtain starting values for other iterative optimization algorithms that need good starting values for convergence.

To illustrate this design method a sixth-order lowpass filter was designed with 41 frequency samples to approximate. The magnitude of those less than 0.2 Hz is one and of those greater than 0.2 is zero. The phase was experimentally adjusted to result in a good magnitude response. The design was performed with Program 9 in the appendix of [57] and the frequency response is shown in Figure 7-33 of [57].

Summary

In this section an LS-error approximation method was posed to design IIR filters. By using an equation-error rather than a solution-error criterion, a problem resulted that required only the solution of simultaneous linear equations.

Like the FIR filter version, the IIR frequency sampling design method and the LS equation-error extension can be used for complex approximation and, therefore, can design with both magnitude and phase specifications.

If the desired frequency-response samples are close to what an IIR filter of the specified order can achieve, this method will produce a filter very close to what a true least-squared error method would. However, when the specifications are not consistent with what can be achieved and the approximating error is large, the results can be very poor and in some cases, unstable. It is often difficult to set realistic phase response specifications. With this method, it is even more important to have a design environment that will allow easy trial-and-error procedure.

Other works on this problem are [36], [35], [26], [51], [34]. Other references can be found in [57], [36]. The Matlab command invfreqz() which is an inverse to the freqz() command gives a similar or, perhaps, the same result as the method described in this note but uses a different formulation [41], [70], [8], [9], [24]

More

Practical problems occur in the design of a filter to separate signals according to their energy. Because the energy content of a signal is the integral or sum of the square of the signal, a mean-squared-error measure is natural. Unfortunately, for the IIR filter design problem, the optimization procedure is nonlinear. This was pointed out in the last section where the equation error was used in order to have a linear problem.

Because of the nonlinear nature of the least-squared-error minimization, the method of solution becomes dependent on the desired frequency response, and therefore, there is no single method for design. The mean-squared error for magnitude approximation is defined as

q ( x ) = i = 0 L | H ( ω i ) | | H d ( ω i ) | 2 q ( x ) = i = 0 L | H ( ω i ) | | H d ( ω i ) | 2
(37)

where x x is a vector of filter parameters chosen to minimize q, and the error is sampled at L + 1 L + 1 frequencies ω i ω i . Steiglitz [72] chose the parameter vector x to be the coefficients of a cascade structure in order to best fit an iterative optimization scheme. He applied a standard optimization algorithm, the Fletcher-Powell method, to the minimization of Equation 37. Other methods which are more directly related to a squared-error measure can also be used.

Practical difficulties exist in solving this approximation problem. In some cases, local minima are found rather than the global minimum. In other cases, convergence of the minimization algorithm is slow or does not occur at all. Numerical problems can result from ill-conditioned equations, and there is no guarantee that the designed filter will be stable.

An important factor is the choice of a desired frequency- response function H d ( ω ) H d ( ω ) that does not result in the optimum approximation having a large error. This often means not having an abrupt discontinuity between the passband and stopband.

Another factor is the starting of the iterative optimization algorithm with a set of coefficients in x x that is close to the optimum. This can be accomplished by using the frequency sampling method to give a design that can be used to start a least-squares algorithm. Because the error defined in Equation 37 is in terms of magnitudes, an unstable design can be converted to a stable one by moving the unstable pole at a radius of r r in the z z -plane to a radius of 1 / r 1 / r . This does not change the magnitude frequency response and does stabilize the effect of that pole.

A generalization of the idea of a squared-error measure is defined by raising the error to the p p power where p p is a positive integer. This error is defined by

q ( x ) = i = 0 L | H ( ω ) H d ( ω ) | p q ( x ) = i = 0 L | H ( ω ) H d ( ω ) | p
(38)

Deczky [21] developed this approach and used the Fletcher-Powell method to minimize Equation 38. He also applied this method to the approximation of a desired group-delay function. An important characteristic of this formulation is that the solution approaches the Chebyshev or mini-max solution as p becomes large.

Iterative Algorithms using Prony's Methods

Other Optimal IIR Filter Design Methods

Martinez/Parks design [53], Jackson's improvement [30], others [64], [1], [60], [43], [39], [40], [38], [36], [35], [32], [61], [74], [59], [18], [3], [72], [21], [14], [22], [71].

Summary

This note has developed a time-domain and a frequency-domain method to design an IIR digital filter that interpolates desired samples or that gives an optimal, least squared equation error approximation. These methods are directly related to Prony, Padé, and linear prediction. In addition, they can be used to obtain good starting values for other iterative algorithms or iterated themselves to obtain optimal approximations with other criteria [41], [68], [69], [57], [29].

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