1 Design of Discrete-time Matrix All-Pass Filters Using Subspace Nevanlinna Pick Interpolation

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Design of Discrete-time Matrix All-Pass Filters

Using Subspace Nevanlinna Pick Interpolation

Agulla Surya Bharath, Devanshu Singh Gaharwar, Kumar Appaiah and Debasattam Pal

Abstract—Unitary matrix-valued functions of frequency are ma-

trix all-pass systems, since they preserve the norm of the

input vector signals. Typically, such systems are represented

and analyzed using their unitary-matrix valued frequency do-

main characteristics, although obtaining rational realizations for

matrix all-pass systems enables compact representations and

efﬁcient implementations. However, an approach to obtain matrix

all-pass ﬁlters that satisfy phase constraints at certain frequencies

was hitherto unknown. In this paper, we present an interpolation

strategy to obtain a rational matrix-valued transfer function

from frequency domain constraints for discrete-time matrix all-

pass systems. Using an extension of the Subspace Nevanlinna

Pick Interpolation Problem (SNIP), we design a construction

for discrete-time matrix all-pass systems that satisfy the desired

phase characteristics. An innovation that enables this is the

extension of the SNIP to the boundary case to obtain efﬁcient

time-domain implementations of matrix all-pass ﬁlters as matrix

linear constant coefﬁcient difference equations, facilitated by a

rational (realizable) matrix transfer function. We also show that

the derivative of matrix phase constraints, related to the group

delay at the interpolating points, can be optimized to control the

all-pass transfer matrices at the unspeciﬁed frequencies. Simula-

tions show that the proposed technique for unitary matrix ﬁlter

design performs as well as traditional DFT based interpolation

approaches, including Geodesic interpolation and the popular

Givens rotation based matrix parameterization.

1 INTRODUCTION

Filtering signals is among the most fundamental operations in

signal processing. In general, ﬁltering scalar signals is well

understood, and there is mature theory that discusses ﬁlter

design and implementation for both analog and digital scalar

ﬁlters. However, with the increased interest in multiple-input

multiple-output systems in several allied areas, the concept of

ﬁltering vector signals has gained importance. Designing pre-

cise ﬁlters for vector signals (wherein the signal at each time

instant is a real or complex vector) under various constraints

is also interesting from the point of view of several practical

applications, although there has not been much work in the

past in this direction. In this paper, we focus on the design of

discrete-time “matrix” all-pass ﬁlters, that transform a vector

signal’s phase while ensuring that their norm is not altered

for all frequencies. In particular, unlike the standard practice

of using frequency domain transform techniques for ﬁltering,

we present an interpolation based ﬁlter design technique that

produces matrix all-pass ﬁlters for practical realizability. This

idea has several applications, such as combined left and right

audio signals in case of stereo audio as well as for feedback

in control and communication systems etc. As an example to

show the effectiveness of the proposed techniques, we consider

the MIMO precoders for wireless communication systems

which employ orthogonal frequency division multiplexing

(OFDM). These precoders can be accurately and efﬁciently

realized using time domain techniques, as opposed to the

traditionally used approaches [1, 2].

Matrix ﬁltering with a norm preservation constraint is typi-

cally accomplished using frequency domain techniques [3, 4].

Speciﬁcally, this involves computing the Fourier transform

of the signal, performing the all-pass ﬁltering on a per-

frequency basis, and using the inverse Fourier transform, as is

common in the case of vector communication systems [4, 5].

However, when the matrix all-pass ﬁlter lends itself to a time

domain realization, this method is not ideal. In particular,

when the matrix all-pass ﬁlter has an efﬁcient linear constant

coefﬁcient difference equation (LCCDE) realization, the ﬁlter

realized using frequency domain techniques will be inaccurate,

and will also result in less efﬁcient realizations. To address

this, we present an interpolation based matrix all-pass ﬁlter

design technique that results in a realizable ﬁlter (that can

be implemented in the time domain as an LCCDE) while

satisfying the frequency domain constraints. Our approach

extends the classical Subspace Nevanlinna Pick Interpolation

(SNIP) method [6] that is well-known in the context of control

systems to the “boundary” case to obtain matrix ﬁlters that

satisfy some prior constraints, while ensuring that the Fourier

transform of its system function is a unitary matrix at all

frequencies, thus obtaining norm-preserving (matrix all-pass)

ﬁlters.

The classical Nevanlinna interpolation problem has its roots

in the problem of synthesis of dynamical systems as passive

electrical networks (see [7]). This, as well as all the subsequent

extensions of it, however, considers only the situation wherein

the interpolating frequencies and the prescribed values of

the desired transfer function are strictly within the respective

critical regions. For example, in the scalar version of the SNIP

dealt with in [6], the polar plot of the transfer function must lie

strictly within the unit disk, and the frequencies that are given

lie on the open right-half of the complex plane. It is important

to note that the solution of the classical SNIP crucially depends

on these strictness assumptions. In this paper, we push the

SNIP to its boundary: we deal with the case wherein the

desired transfer function’s polar plot is on the unit disk (i.e.,

all-pass), and the frequencies, too, are given on the boundary

(the unit circle because we consider discrete time systems).

Our key contributions in this paper are as follows:

•We present an approach to realize a discrete-time matrix

arXiv:2210.14015v1 [eess.SP] 25 Oct 2022

all-pass ﬁlter, when given a feasible set of frequency

responses (unitary matrices) and group delay matrices for

a ﬁnite set of frequencies {ωi}. Speciﬁcally, our solution

yields a rational matrix z-transform for the required all-

pass ﬁlter that satisﬁes all the given frequency domain

conditions, and its transfer function matrix is unitary

valued for all ω∈(−π, π]. This can be viewed as a gen-

eralization of the Blaschke interpolation based approach

that is speciﬁc to scalar all-pass ﬁlter design [8, 9] to the

matrix case.

•We obtain this matrix all-pass ﬁlter by extending the

SNIP technique to the boundary case. Speciﬁcally, since

the Pick matrix in the case of standard SNIP [6] becomes

ill-deﬁned when we demand a unitary valued solution,

we provide a modiﬁed approach using the modiﬁed

Pick (Schwarz-Pick) matrix to generalize the SNIP ﬁlter

realization to the boundary case in discrete-time setting.

•Finally, we also present an optimization based approach

that tunes the slopes of the matrix phase response at spe-

ciﬁc frequencies to obtain realizable ﬁlters with desirable

characteristics.

The proposed approach for ﬁltering is both novel as well

as efﬁcient in terms of implementation. In particular, prior

approaches to perform all-pass matrix ﬁltering in the frequency

domain have used DFT based techniques that involve at least

NFFT multiplications [10, 11]. In addition, these techniques

have largely relied on frequency domain interpolation of

precoders interpolation on manifolds [12, 13] or interpolation

of parameterized unitary matrices [14, 15], a technique that

is employed in recent wireless OFDM based standards as

well [11]. However, as we show in this paper, in situations

where the all-pass ﬁlter has an impulse response that can be

characterized using fewer coefﬁcients, signiﬁcant savings in

terms of computations can be realized using the SNIP based

approach, while faithfully capturing the frequency domain

precoder characteristics.

The rest of the paper is organized as follows: Section 2

outlines the importance of matrix all-pass ﬁlter design problem

statement, Section 3 brieﬂy describes the classical SNIP,

Section 4 describes the discrete-time matrix all-pass ﬁlter

design problem and its solution, Section 5 contains the simu-

lations results and interpretations for some practical purposes,

Section 6 provides some concluding remarks and discusses

future directions.

Notation: Unless otherwise speciﬁed, bold capital symbols re-

fer to matrices, bold smallcase symbols correspond to vectors,

Imrefers to an m×midentity matrix and 0m×mrefers to an

m×mall-zero matrix. We also use the abbreviations CT for

continuous-time and DT for discrete-time. For any matrix A,

A∗is the conjugate transpose of A.

2 MOTIVATION

To motivate this problem, we ﬁrst pose the scalar all-pass ﬁlter

problem: if the frequency response of a discrete-time all-pass

ﬁlter is given for certain (ﬁnitely many) frequencies, how can

we obtain an all-pass ﬁlter that satisﬁes these constraints? In

general, there exist an inﬁnite number of ﬁlters that satisfy

these constraints. Recent work has shown that, if the group

delays are also known at the given frequencies, then a realiz-

able all-pass ﬁlter can be obtained as a Blaschke product [8].

However, the Blaschke product based approach is only suited

to the solution of the discrete-time scalar all-pass ﬁlter design

problem, and a direct extension of the same approach to the

case of matrix all-pass ﬁlters is not known.

Discrete-time all-pass ﬁlters are used for phase compensation

in various applications. A scalar all-pass ﬁlter can be used

to correct phase distortions in scalar signals. To the best of

our knowledge, this concept is yet to be extended to vector

signals (MIMO systems), wherein the input and output are

complex vector signals, and the transformation ﬁlter is a

matrix valued all-pass ﬁlter (unitary). These matrix all-pass

ﬁlters are commonly encountered in several situations, such as

MIMO-OFDM systems and stereo audio systems. In MIMO-

OFDM communication systems, when symbols are precoded

with unitary matrices at the transmitter, if the transmitter

possesses some channel state information (CSI), unitary matrix

precoding is typically performed using the right singular

vectors (or related unitary matrices) that are obtained from

the singular value decomposition (SVD) of the channel matrix

for every subcarrier (frequency band) [5]. Multiplying with a

unitary matrix in the frequency domain is norm preserving,

and thus, it can be thought of as a matrix all-pass ﬁltering

operation. In practice, having norm preserving matrix ﬁlters

is important in order to satisfy various constraints, such as

power in communication systems, or volume in the case of

audio signals, while only altering the matrix-valued “phase”.

Performing such a phase transformation using the coefﬁcients

of an appropriate discrete-time matrix all-pass ﬁlter with a

standard LCCDE implementation would obviate the need for

frequency domain processing, and result in a more faithful

realization, and can signiﬁcantly reduce the precoding com-

plexity in modern systems, such as those that use precoding

for millimeter wave wireless systems [16, 17].

The block diagram shown in Fig.1 depicts an unknown matrix

all pass ﬁlter with efﬁcient LCCDE representation, with x[n]

as input and y[n]as output. Our goal is to construct a system

that mimics the unknown matrix all-pass ﬁlter. One approach

is to use DFT based techniques, wherein output y1[n]is

produced for the input x[n]. Another method is to follow a

matrix ﬁlter design technique and construct a rational matrix

all pass ﬁlter with LCCDE representation, and this yields

output y2[n]for the input x[n]. In this situation only the latter

method is accurate.

It is evident from the block diagram Fig.1 that when the

matrix all-pass ﬁlter has an efﬁcient LCCDE realization, the

ﬁlter realized using frequency domain techniques may be

inaccurate, and may also result in less efﬁcient realizations.

An advantage of using the SNIP based LCCDE ﬁlter design

is that the computational complexity can be reduced with a

rational (realizable) all-pass ﬁlter method that has a compact

representation, being more amenable to time-domain LCCDE

implementations (details of the same are discussed further in

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1DesignofDiscrete-timeMatrixAll-PassFiltersUsingSubspaceNevanlinnaPickInterpolationAgullaSuryaBharath,DevanshuSinghGaharwar,KumarAppaiahandDebasattamPalAbstractUnitarymatrix-valuedfunctionsoffrequencyarema-trixall-passsystems,sincetheypreservethenormoftheinputvectorsignals.Typically,suchsystemsarer...

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