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Quevedo, Daniel E., Boelcskei, Helmut, & Goodwin, Graham C.(2009)Quantization of filter bank frame expansions through moving horizon opti-mization.IEEE Transactions on Signal Processing, 57 (2), pp. 503-515.

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https://doi.org/10.1109/TSP.2008.2008259

1

Quantization of Filter Bank Frame Expansionsthrough Moving Horizon Optimization?

Daniel E. Quevedo*, Member, IEEE,Helmut Bolcskei†, Senior Member, IEEE,

Graham C. Goodwin*, Fellow, IEEE

Abstract— This paper describes a novel approach to quantiza-tion in oversampled filter banks. The new technique is based onmoving horizon optimization, does not rely on an additive whitenoise quantization model and allows stability to be explicitlyenforced in the associated nonlinear feedback loop. Moreover,the quantization structure proposed here includes Σ∆ and linearpredictive subband quantizers as a special case and, in general,outperforms them.

I. INTRODUCTION

Filterbanks (FBs) are widespread in many practical appli-cations, mainly in relation to subband signal processing andcoding of audio and image signals; see, e.g., [1]–[7]. Fig. 1depicts a FB with K-channels and decimation factor L in eachchannel, where K ≥ L. The FB is said to be critically sampledif K = L and oversampled if K > L. The main purpose ofa critically sampled FB is to compress a discrete-time inputsignal y into sequences ui, such that the number of bits usedto quantize the subband signals vi is small, whilst providing“good” reconstruction quality, i.e., whilst achieving y ≈ y. Forthat purpose, a series connection of an analysis FB, an abstractquantizer (or coder) Q and a synthesis FB is employed.

As can be seen in Fig. 1, a FB first decomposes the discrete-time input signal y into subband sequences vi through analysisfiltering and down-sampling. The abstract quantizer Q thenprocesses the resulting K subband sequences (in a linear ornonlinear fashion) and quantizes them, giving rise to the outputsequences ui. Finally, the synthesis FB up-samples and filtersthe sequences ui to provide an estimate of y.

Reconstruction errors in a FB basically arise from threesources [3]: aliasing and imaging (as a consequence of sam-pling rate conversions), phase and amplitude distortion (dueto the filters employed), and quantization effects. The effectsof aliasing, imaging, phase and amplitude distortion can bedealt with by proper design of the analysis and the synthesisfilters; see, e.g., [3]–[5]. On the other hand, quantization, ingeneral, incurs a loss of information, and therefore precludes yfrom being equal to y. Quantization effects are nonlinear (andnon-smooth) in nature. Nonetheless, often, linear analysis and

?This paper was presented in part at the 43rd IEEE Conference on Decisionand Control, Paradise Island, Bahamas, December 2004.

*School of Electrical Engineering & Computer Science, The Universityof Newcastle, Callaghan, NSW 2308, Australia; Emails: dquevedo@ieee.org,graham.goodwin@newcastle.edu.au†Communication Technology Laboratory, ETH Zurich, CH-8092, Zurich,

Switzerland; Email: boelcskei@nari.ee.ethz.ch

design methodologies can usefully be employed; see, e.g., [8]–[11]. Of course, these techniques are only approximate andone can expect linear design approaches to be, in general,outperformed by nonlinear ones.

While source coding applications typically employ criticallysampled FBs, oversampled FBs implement redundant signalexpansions in `2(Z) and have inherent noise reducing proper-ties rendering them attractive for applications requiring low-resolution quantizers. In the context of analog-to-digital (A/D)conversion, the practical advantages of using low-resolutionquantizers at the cost of increased (sampling) rate are indi-cated by the popular Σ∆ techniques [12], [13]. The use oflow-resolution quantizers increases circuit speed and reducescircuit complexity; 1-bit codewords, for example, eliminate theneed for word framing.

In [8], oversampling noise shaping (Σ∆) subband quan-tizers have been introduced as an extension of oversamplednoise shaping A/D converters. The approach in [8] has twoshortcomings:

1) a linear model for quantization is used, i.e., quantizationis modeled as introducing additive white noise indepen-dent of the signal, and

2) stability of the feedback loop cannot be guaranteed.In the present paper, we address these two issues by intro-

ducing a novel quantization method for oversampled FBs. Theapproach uses an exact nonlinear model of quantization andallows one to enforce stability. The new scheme is based onideas from finite-set constrained predictive control; see, e.g.,[14]. On a conceptual level, it extends previous work on scalarA/D conversion [15], [16]. (See also related work on discretecoefficient filter design [17] and on channel equalization [18])to the vector case.

In our approach (which we term a Moving Horizon SubbandQuantizer (MHSBQ)), at each time instant, the quantizedvalues are chosen to minimize the energy of a filtered ver-sion of the reconstruction error over a (corresponding) finitehorizon, together with a “final state weighting” term. TheMHSBQ looks ahead at a finite number of “future” samplesand effectively performs vector quantization (VQ) over themultiple time instants contained in the horizon. Thus, theMHSBQ amounts to (a special type of) space-time VQ ofthe K subband signals over multiple time instants.

The main design parameters in the MHSBQ are the horizonlength and the final state weighting term. We show that inthe special (and simplest) case of the horizon being equalto one and no final state weighting, the MHSBQ reduces

2

Synthesis filter bank

↓L

↓L

↓L RK

w1

w2

y

++

+

Q

↑L

↑L

↑L

Analysis filter bank

+y

E1

E2

EK

y1

y2

yK

v1

v2

vK

u1

u2

uK wK

R1

R2

Fig. 1. K-channel filter bank with decimation factor L.

to the linear feedback quantizers of [8]. Larger horizonslead to performance improvements. Another advantage of theapproach proposed in this paper, when compared to [8], isthat stability issues are addressed explicitly. More precisely, weshow that, in many situations, the final state weighting term ofthe MHSBQ can be chosen such that idle tones are suppressed,i.e., we can guarantee that, with a zero input signal, theoutput signal will decay to zero. Interestingly, these casesare associated with situations where the reconstruction errorover a finite horizon together with the final state weightingterm essentially amounts to the infinite horizon reconstructionmean-square-error (MSE).

Before proceeding, we note that the subject of scalar Σ∆quantization of redundant (i.e., oversampled) representationsusing exact quantization models and explicitly addressingstability issues has recently become a very active area ofresearch in the applied mathematics community; see, e.g. [19].For representative references the interested reader is referredto [20]–[24]. The approach proposed in the present paperencompasses scalar and subband Σ∆ quantization as a specialcase and appears to be an entirely new concept in the contextof quantization of overcomplete representations. Our stabilitynotion complements those used in [20]–[23]. Whilst in [20]–[23] stability refers to boundedness of signals, our results areconcerned with the stronger notion of signals, not only beingbounded, but also decaying to zero. In particular, the MHSBQguarantees absence of idle tones. This is not implied by theresults in [20] for scalar Σ∆ quantization. Nevertheless, ourresults do not resolve the stability problem completely, sincethey apply only to a specific class of input signals y.

We finally note, that for the scalar Σ∆-based approaches in[20]–[24], it is often possible to compute the reconstructionMSE (or bounds thereon) explicitly as a function of the over-sampling factor, see also [25]. Obtaining such explicit resultsseems to be difficult for the MHSBQ approach presented inthis paper and is an important open problem.

The remainder of this paper is organized as follows: In Sec-tion II, we review basic concepts related to (oversampled) FBs.Section III presents the proposed MHSBQ. Corresponding de-

sign parameters and properties are investigated in Sections IVand V. Section VI documents a simulation study. We concludein Section VII.

Notation: We adopt vector-space notation to denotesignals. For example, in Fig. 1, y denotes the signal y =. . . , y(−1), y(0), y(1), . . . . We will use upper case bold-face letters for matrices and lower case bold-face letters forvectors and vector-signals; 0 denotes the all zeros vector.Unless explicitly specified, the dimension of vectors followsfrom the context. We use IL to denote the L×L identity matrixand define 0K to be the all zeros K×K matrix. Transpositionof a matrix is denoted via the superscript T ; the superscript∗ stands for complex conjugation; the superscript H denotesconjugate transposition; ‖x‖2P stands for the quadratic formxT Px, where x is any vector and P is a symmetric positivesemidefinite matrix; ‖e‖2 , eT e. Finally, we will use thefollowing definition of a VQ:

Definition 1 (Euclidean VQ): Consider a finite set of col-umn vectors B = bi, where bi ∈ RnB×1, ∀bi ∈ B, andwhere bi 6= bj , ∀i 6= j. A nearest neighbor VQ is definedas a mapping qB : RnB×1 → B which assigns to each vectorc ∈ RnB×1 the closest element of B (as measured by theEuclidean distance), i.e., qB(c) = bi ∈ B if and only if cbelongs to the region:

c ∈ RnB×1 : ‖c− bi‖2 ≤ ‖c− bj‖2, ∀ bj 6= bi, bj ∈ B

.

4Note that, in the special case when nB = 1, the VQ defined

above reduces to a scalar quantizer with a scalar output set. Inthe present work we shall, however, be interested in the moregeneral case where nB ≥ K.

II. BACKGROUND ON (OVERSAMPLED) FILTERBANKS

In this section, we give a brief overview of several funda-mental aspects of (oversampled) FBs. More thorough treat-ments can be found, e.g., in [2]–[5], [26], [27].

3

A. Basic Aspects

Referring to Fig. 1, the discrete-time input signal y issimultaneously filtered by the analysis filters

Ei(z) =∞∑

`=−∞

ei(`)z−`, i ∈ 1, 2, . . . ,K. (1)

The K signals yi are down-sampled [1] by a factor of L,i.e., only every L-th sample is retained. The resulting subbandsequences vi are then quantized leading to the K sequencesui. At each time instant k ∈ Z, we have

ui(k) ∈ Ui, ∀i ∈ 1, 2, . . . ,K, (2)

where each of the sets Ui, i ∈ 1, 2, . . . ,K, contains a finite(not necessarily the same) number of elements. The recon-struction error, y − y, clearly depends on how the sequencesui are chosen inside the abstract quantizer Q. The main focusof this paper is the design of Q for given sets Ui.1 A keyfeature of our approach (which will be presented in Section III)is that we are effectively performing VQ across subchannelsand across time. At each time instant k ∈ Z, the K quantizedvalues ui(k) in (2) are jointly designed such as to optimize thequantizer performance. Thus, our strategy is more general thanthose which simply choose each value ui(k) separately, i.e.,on a subchannel by subchannel basis, via scalar quantizers.

In the synthesis FB, the quantized sequences ui are up-sampled [1] by a factor of L resulting in the sequences wi.The K filters

Ri(z) =∞∑

`=−∞

ri(`)z−`, i ∈ 1, 2, . . . ,K (3)

then perform an interpolation/smoothing function, which fi-nally results in the estimate y.

We define the oversampling factor of the FB as:

r = K/L. (4)

B. Polyphase Representation

The oversampled FB of Fig. 1 is a multi-rate system, whichcan be represented conveniently in terms of the polyphasedescription as outlined below; see, e.g., [2], [3], [27].

We begin by defining the vector sequences

v(k) ,

v1(k)v2(k)

...

...vK(k)

, u(k) ,

u1(k)u2(k)

...

...uK(k)

,

d(k) ,

d1(k)d2(k)

...dL(k)

, d(k) ,

d1(k)d2(k)

...dL(k)

, (5)

1Of course, joint design of Q and the Ui would be interesting, but is beyondthe scope of the present paper.

where

di(k) , y(kL + i− 1), i ∈ 1, 2, . . . , L,di(k) , y(kL + i− 1), i ∈ 1, 2, . . . , L

(6)

are the polyphase components of y and y, respectively. Notethat the vectors defined in (5) are updated only every Lsampling instants of y.

By using the so-called noble identities, see, e.g., [3], directalgebraic manipulation yields that the FB of Fig. 1 can becharacterized via:2

v = E(z)d

d = R(z)u.(7)

Here, E(z) is the K × L analysis polyphase matrix, whileR(z) is the L×K synthesis polyphase matrix, defined as

[E(z)]i,n = Ei,n(z), i ∈ 1, 2, . . . ,K, n ∈ 1, 2, . . . , L,

and

[R(z)]n,i = Ri,n(z), i ∈ 1, 2, . . . ,K, n ∈ 1, 2, . . . , L,

respectively, with the polyphase filters

Ei,n(z) ,∞∑

`=−∞

ei(`L− n + 1)z−`,

Ri,n(z) ,∞∑

`=−∞

ri(`L + n− 1)z−`.

The resulting equivalent scheme is depicted in Fig. 2 show-ing the polyphase description of a multi-input multi-output(MIMO) single-rate system, which operates on the down-sampled signals.

Quantizer

R(z)

Synthesis FB

d v u dE(z)

Analysis FB

Q

Fig. 2. Oversampled FB in polyphase form.

C. Perfect Reconstruction

In the absence of quantization, perfect reconstruction, i.e.,y = y, is obtained by ensuring

R(z)E(z) = IL. (8)

FBs which satisfy (8) are termed perfect reconstructionFBs [3]. Note that choosing R(z) as any left-inverse of E(z)will result in a perfect reconstruction FB [27]. (Note that theleft-inverse of E(z) is non-unique only in the oversampledcase, where r > 1. When r = 1, (8) requires R(z) = E(z)−1;in which case E(z) needs to be invertible.) In particular, ifquantization effects are modeled as additive white Gaussiannoise, then choosing R(z) as the para-pseudo-inverse ofE(z), i.e., R(z) = (E(z)E(z))−1E(z),3 will minimize the

2Here, and in some expressions to follow, z should be interpreted as theforward shift operator. For example, v in (7) is the output sequence of theLTI system E(z) with input sequence d.

3Here, E(z) = EH(1/z∗) is the para-conjugate of E(z).

4

reconstruction MSE [8]. Other approaches to the design ofR(z) taking into account quantization effects can be found,for example, in [9], [10], [28].

D. Quantization Aspects

Quantization commonly induces a loss of information whentransforming the sequence v into the sequence u. As aconsequence, the reconstructed sequence d will differ fromd, in general, i.e., the reconstruction error, which we defineas

ed , d− d (9)

will be nonzero. Note that, given (5) and (6), the sequence ed

can alternatively be characterized via:

ed(k) =

y(kL)

y(kL + 1)...

y(kL + L− 1)

−

y(kL)y(kL + 1)

...y(kL + L− 1)

, k ∈ Z.

(10)As stated in [3], the effect of quantization of the subband

sequences is very difficult to analyze exactly. The usualparadigm adopted models the quantizer by a linear gain-plus-noise model, as in the case of standard A/D conversion; see,e.g., [29]. This model is simple and may, in some cases, yielduseful results, see also [30]. Nevertheless, it fails to predictnonlinear effects, such as the appearance of idle tones orchaotic output sequences; see, e.g., [31].

A simple subband quantization method is described in [3],[32]. It uses one scalar quantizer for each subband, i.e., it sets

ui(k) = qUi(vi(k)), ∀k ∈ Z, ∀i ∈ 1, 2, . . . ,K (11)

where we have used the notation introduced in Definition 1,with nB = 1 and B = Ui. Based upon the linear quantizationparadigm mentioned above, the number of bits assigned toeach quantizer can be determined according to the energycontent of each subband sequence in an optimal manner. Thistype of methodology is widespread, especially in audio codingapplications; see, e.g., [6], [33].

E. Linear Feedback Subband Quantization

Oversampled FBs provide redundant expansions of theinput signal y, see [26], [27]. In [8] it was shown how thisredundancy can be exploited to reduce quantization errors in ythrough the use of linear feedback quantization methods, i.e.,the noise-shaping quantizer depicted in Fig. 3 and the signalpredictive quantizer in Fig. 4. As can be seen in those figures,linear feedback quantizers incorporate feedback loops whichcontain a linear MIMO filter G(z) of dimension K×K. Thesequantizers essentially correspond to MIMO versions of Σ∆-Converters and noise-shaping quantizers; see, e.g., [12], [13].4

The architecture in [8] (see Figures 3 and 4) amounts tosubchannel by subchannel quantization, i.e.,

u(k) ∈ U, ∀k ∈ Z, (12)

4Compare also to parallel Σ∆-Converter structures as described, forexample, in [34].

+

QIK −G(z)

−qU(·)

+−

v u

Fig. 3. Noise-shaping quantizer.

+

+

+

v u

Q

−qU(·)

IK −G(z)

Fig. 4. Signal predictive quantizer.

where the set U is defined via:

U , U1 × U2 × · · · × UK ⊂ RK , (13)

see (2). Based on the linear quantization model one can thendesign an optimal, in the sense of minimizing the reconstruc-tion MSE, filter G(z), see [8], [11].

In the next section, we propose a novel quantization methodfor oversampled FBs. As will be apparent, the new schemeembeds linear feedback quantization in a broader frameworkand contains noise shaping and signal predictive subbandquantizers as special cases. The method does not rely on anadditive white quantization noise model and allows one toenforce stability of the feedback loop.

III. MOVING HORIZON SUBBAND QUANTIZATION

The main aim of the abstract quantizer Q in Fig. 1 isto make the reconstruction error ed (defined in (9)) smallby appropriate selection of u, for fixed and given U. Inwhat follows, we will use methods stemming from the ModelPredictive Control framework, see, e.g., [35], [36], to designa subband quantizer, which minimizes a weighted measureof the reconstruction error. On a conceptual level, the pro-posed method extends our previous work on (scalar) signalquantization documented in [15], [16] to oversampled FBs or,equivalently, to non-square MIMO systems. The quantizationscheme to be presented will be seen to also outperform thelinear feedback subband quantizers of [8].

5

A. Frequency-Selective Subband QuantizationThe subband quantization problem outlined above can be

embedded into the more general problem of minimizing thefiltered distortion sequence:

ef , W(z)(H(z)d− u), (14)

subject to the vector sequence u, see (5), taking values in theset U defined in (13). In (14), H(z) is of dimension K×L andW(z) is a causal filter of dimension ne ×K, where ne ∈ N.

As an example of this setting, consider a perfect reconstruc-tion FB, see (8), and choose H(z) = E(z) and W(z) = R(z).In this case, ef corresponds to the reconstruction error ed

defined in (9).More generally, the two filters H(z) and W(z) allow the

designer to consider a (frequency-) weighted reconstructionerror. Indeed, if ef is minimized (in some appropriate sense),then u will approximate the sequence

a , H(z)d (15)

and W(z) in (14) weights the signal a − u in a frequency-selective manner. More details on how to choose W(z) andH(z) are provided later in Section IV-B. For now, we focuson the problem of minimizing ef for fixed filters W(z) andH(z).

For the ensuing discussion, it is convenient to use a state-space description and to characterize W(z) via:

W(z) = D + C(zIn −A)−1B (16)

where n ∈ N is the order of the chosen filter W(z), A ∈Rn×n, B ∈ Rn×K , C ∈ Rne×n, and where D ∈ Rne×K isnonzero. Based on (16), we have:

ef (k) = Cx(k) + D(a(k)− u(k))x(k + 1) = Ax(k) + B(a(k)− u(k)),

(17)

where x is the filter state.5

B. Optimization CriterionGiven the setting described above, the subband quantizer

can now be designed with an optimality criterion in mind.More precisely, we will focus on the energy of the filtereddistortion sequence ef . For that purpose, at every time instantk, we propose to minimize the following measure of ef

defined over a finite and fixed horizon N ∈ N:

VN (~u(k);x(k)) , ‖x′(k + N)‖2P +k+N−1∑

`=k

‖e′f (`)‖2, (18)

where P ∈ Rn×n is a given positive semidefinite matrix and

~u(k) ,

u′(k)

u′(k + 1)...

u′(k + N − 1)

∈ UN ⊂ RNK×1 (19)

5The state-space description adopted encompasses all causal rational- andFIR-filters with a nonzero direct feedthrough term. If W(z) is chosen to be arational filter, then n corresponds to its number of poles, i.e., to the number ofroots of the denominator polynomials in the Smith-McMillan form of W(z).If W(z) is chosen to be an FIR-filter, then n + 1 is the impulse responselength; see, e.g., [37].

contains the decision variables (quantized values). The con-straint set U underlying the cost function (18) is definedin (13); primed variables refer to “potential values” of thefiltered distortion ef and the final state x(k+N), which wouldresult if the quantizer outputs u(k),u(k+1), . . . ,u(k+N−1) were set equal to the corresponding values in ~u(k). Inaccordance with (17), these potential trajectories are formedas:

e′f (`) = Cx′(`) + D(a(`)− u′(`))x′(` + 1) = Ax′(`) + B(a(`)− u′(`)),

(20)

with ` ∈ k, k + 1, . . . , k + N − 1.The initial condition in (20) is:

x′(k) = x(k)

which is uniquely determined from past values of u andof d. To be more precise, due to (15), the sequenced(0),d(1), . . . ,d(k − 1) gives rise to the sequencea(0),a(1), . . . ,a(k − 1). The vector sequence x can thenbe calculated via the recursion, see (17),

x(k) = Ax(k − 1) + B(a(k − 1)− u(k − 1)) (21)

initialized with x(0) = 0.Minimization of VN in (18) at time instant k gives rise to

the optimizer

~u?(k) , arg min~u(k)∈UN

VN (~u(k);x(k)). (22)

Clearly, (22) amounts to (a special type of) space-time VQof the K subband signals over the multiple time instantscontained in the horizon window. Indeed, the solution ~u?(k) ofthe optimization problem contains values which, in principle,could be used in the sequence u (see Fig. 2) at instantsk, k+1, . . . , k+N−1, see (19). To obtain better performance,in Section III-C, we will present an alternative strategy, whereminimizations are carried out at every time instant, thus,providing a sequence of optimizers ~u?(0), ~u?(1), . . . . Thesequence u is then obtained by setting each u(k), k ∈ N equalto the first element of ~u?(k) (which is of dimension K×1 andcontains values for all K subchannels for instant k, see (19)).

Remark 1 (Decision Delay): An important feature of (22)is that ~u?(k) is obtained through “looking ahead” at the signald up until time instant k+N−1. Thus, the quantizer incurs adecision delay of N time steps (or, equivalently, NL samplesof y) plus the time required to compute the optimizer in (22).4

Before proceeding, we emphasize that a key aspect of ourapproach is the inclusion of the final state weighting term‖x′(k + N)‖2P in VN . As is known from the systems andcontrol literature, see, for example, [36], final state weightingcan, at times, be used as a mechanism to approximate infinitehorizon behavior via a finite-horizon cost function. In fact,the following result shows that (in the case of stable filtersW(z)), through proper choice of the matrix P, we can, indeed,approximate the minimization of the energy of ef over aninfinite horizon through the use of the finite-horizon costfunction (18):

6

Lemma 1 (Infinite Horizons): Suppose that W(z) is a sta-ble filter, i.e., all eigenvalues of A in (16) lie strictly inside theunit circle, and that u′(`) is equal to a(`) for every ` ≥ k+N .If the matrix P in (18) satisfies the Lyapunov Equation

AT PA + CT C = P (23)

we have

VN (~u(k);x(k)) =∞∑

`=k

‖e′f (`)‖2. (24)

Proof: The proof uses basic properties of linear systems,as described, for example, in [38]. We first note that, since (16)is a minimal realization of the stable system W(z), we havethat CT C is positive definite and the Lyapunov Equation (23)has a unique solution; see, e.g., [37].

On the other hand, it follows directly from (20) that, ifu′(`) = a(`), ∀` ≥ k + N , then:

∞∑`=k+N

‖e′f (`)‖2 =∞∑

`=k+N

‖Cx′(`)‖2

= (x′(k + N))T

( ∞∑i=0

(AT )iCT CAi

)x′(k + N).

Direct substitution confirms that P =∑∞

i=0(AT )iCT CAi,

indeed, satisfies (23). Thus,

‖x′(k + N)‖2P =∞∑

`=k+N

‖e′f (`)‖2.

The result (24) now follows from (18).We note that the quantized nature of u′ will generally

preclude u′(`) from being equal to a(`) for every ` ≥ k + Nand (24) is therefore only approximate. Nevertheless, as willbecome clear in Section IV-C, choosing P as the uniquesolution of the Lyapunov Equation (23) helps to ensure idletone suppression.

C. Moving Horizon Optimization

The basic idea of this paper is to use the moving horizonprinciple as described, for example, in [36]. In this paradigm,at every time instant k the finite horizon cost function VN

in (18) is minimized over a corresponding horizon. The outputof the abstract quantizer is set to:

u(k)←− u?(k) (25)

where the vector

u?(k) ,[IK 0K . . . 0K

]~u?(k), (26)

is obtained from the solution of (22).In our approach, u?(k) is then used in (17) to deliver the

successor state x(k + 1) according to

x(k + 1) = Ax(k) + B(a(k)− u?(k)). (27)

In the next time instant, x(k+1) is used to minimize the costfunction VN (~u(k+1);x(k+1)), yielding u(k+1) = u?(k+1),see (26). This procedure is repeated ad-infinitum. As illustratedin Fig. 5, for a horizon of N = 5, the time window used in the

minimization of VN slides forward as k increases. The past ispropagated forward in time via the state sequence x.6

u?(k)

k + 2

k + 1

u?(k + 1)

u?(k + 2)

k + 1k

k + 2 k + 3 k + 4

x(k + 1)

x(k + 2)

k + 2 k + 3 k + 4 k + 5

k + 3 k + 4 k + 6k + 5

Fig. 5. Moving horizon optimization, N = 5.

Fig. 6 schematizes the proposed subband quantizer. Wewill term it a Moving Horizon Subband Quantizer (MHSBQ).Clearly, the MHSBQ is a time-varying subband quantizationscheme.

Remark 2 (Relationship to Model Predictive Control):The MHSBQ can be regarded as a particular MIMO ModelPredictive Controller, see, e.g., [35], [36], with plant W(z),controlled input u and reference signal W(z)H(z)d. Aspecial feature of the particular situation at hand is that uis restricted to satisfy the quantization constraint (12). Thisputs the subband quantization problem into the finite setconstrained predictive control framework developed recentlyin [14]. 4

IV. DESIGN PARAMETERS

In this section, we will provide some details on the role ofthe MHSBQ design parameters N , H(z), W(z) and P.

A. Optimization Horizon N

Given (18), it is easy to see that, in general, larger valuesfor the optimization horizon N provide better performancesince more information about the associated filtered distortionis taken into account when choosing the quantized subbandsequences. In addition, one can expect that, for large enoughN , the effect of u(k) on ef (`) for ` ≥ k + N will be

6Note that due to the Markov-property (17), the effect of u(`),a(`)`<k

on ef (k + j)j≥0 is summarized by the state x(k).

7

Eqs. (25) – (27)(zIn −A)−1B

−

+

a

dH(z)

[IK zIK . . . zN−1IK

]u

QFig. 6. The Moving Horizon Subband Quantizer.

negligible and, consequently, the MHSBQ will give the sameperformance as if an infinite horizon were used. (Note that, asshown in Section III-B, the final state weighting term can beused to emulate infinite horizon behavior for finite N .)

On the other hand, as already mentioned in Remark 1,choosing large horizons incurs large decision delays. Solvingthe combinatorial program needed to obtain ~u?(k) in (22) forrecursive filters W(z), results in exponential complexity inN .7,8 If W(z) is FIR, then the optimization can be carriedout following a Viterbi Algorithm-like approach [41], and hascomputational complexity which is only linear in N , see also[18].

As a consequence, the optimization horizon parameterN allows the designer to trade-off performance for on-linecomputational effort. Fortunately, as will be demonstrated insimulation studies (see Sections V-B and VI), excellent perfor-mance can often be achieved with relatively small horizons.

B. Weighting Filters W(z) and H(z)As already outlined in Section III-A, the frequency weight-

ing filter W(z) and the pre-filter H(z) in (14) can be utilizedto (implicitly) specify spectral characteristics of the subbandsequences ui and of the resultant reconstruction error ed.

One possible design choice motivated by the specific prob-lem considered in this paper is:

H(z) = E(z), W(z) = z−δR(z), (28)

where δ is the unique integer such that z−δR(z) is causalwith non-zero direct feedthrough. In this case, the input to thequantizer in Fig. 6 is given by a = v. Moreover, (7) and (14)yield

ef = z−δR(z)(E(z)d− u) = z−δ(R(z)E(z)d− d

), (29)

i.e., ef corresponds to the difference between (possibly de-layed versions of) the reconstructed sequence d and the re-constructed sequence which would be obtained in the absenceof quantization effects.

7Computation times to solve (22) are, in general, also exponential in thenumber of subband sequences K.

8Note that by adapting recent developments in the general VQ framework,see, e.g., [39], [40], optimization algorithms may be conceived, which, onaverage, require less computational complexity than explicit exhaustive search.

Furthermore, if E(z) and R(z) are chosen to correspond toa perfect reconstruction FB, see (8), then (29) yields

ef = z−δ(d− d) = z−δed. (30)

Consequently, with (28) and R(z)E(z) = IL, the MHSBQminimizes the energy of ed via a moving horizon approach.As already mentioned in Sections III-B and IV-A, the movinghorizon strategy serves as a practical approximation to infinite-horizon optimal designs. Lemma 1 gives an indication of thequality of the approximation.9

Certainly, other design choices for W(z) and H(z) arepossible. For example, one can incorporate psycho-acousticalaspects into the subband quantizer, by choosing W(z) accord-ingly. Also, in Section V, we will outline an alternative designprocedure for W(z) and H(z), which will relate the proposedapproach to linear feedback quantization principles and showthat Σ∆ and signal predictive subband quantizers are specialcases of the MHSBQ.

C. Final State Weighting Matrix P

A very significant, but largely unsolved, issue which ariseswhen including a quantizer in a feedback loop is that ofstability. Poor stability properties manifest themselves in theappearance of nonlinear phenomena such as limit cycles (inparticular, idle tones) and chaotic trajectories, see also [14],[31], [42], [43]. In particular, even with zero input, architec-tures such as (undithered) scalar and subband Σ∆-convertersand also the MHSBQ introduced in Section III, may producenon-zero outputs (idle tones) that do not decay over time.In general, it cannot be guaranteed that quantization-inducederrors will decay.

We will next show how the final state weighting term inthe cost function (18) can be used to suppress idle tonesin the MHSBQ for any horizon N , pre-filter H(z) andstable weighting filter W(z). The following result generalizesTheorem 2 of [16] to the MIMO case.

9It is worth mentioning here that in [8] a linear quantization model is usedto find feedback filters in linear feedback subband quantizers such that thevariance of ed is minimized. In contrast to [8], the design choice (28) isimmediate and does not rest upon a linear quantization model. Furthermore,in Section V we will show that the MHSBQ generalizes and, in general,outperforms the linear feedback quantizers in [8].

8

Theorem 1: Suppose that the sequence a is such that, for afinite value k ∈ N, we have

a(`) ∈ U, ∀` ≥ k. (31)

Then, if P is chosen to satisfy the Lyapunov Equation (23),we have

lim`→∞

ef (`) = 0.

Proof: The proof is included in the Appendix.While Theorem 1 is applicable to a very restricted class of

input signals, it does guarantee that idle tones will not occur,provided 0 ∈ Ui, ∀i ∈ 1, 2, . . . ,K. It should be emphasizedthat, case studies, such as those included in Section VI,indicate that setting P equal to the stabilizing choice givenin (23) tends to avoid limit cycles even for more general inputsignals.

Remark 3 (Relationship to other stability results): If K =L = 1, then the MHSBQ scheme is conceptually equivalent tothe multistep optimal A/D converter presented in [16]. If, inthis scalar case, the optimization horizon is chosen as N = 1,then it was shown in [16] that Σ∆-converter architecturesare obtained. Thus, Σ∆-converters are contained within ourframework as a very particular and simple special case. Con-sequently, it is worth putting our result into perspective withrespect to existing results on the stability of Σ∆-converters.10

Theorem 1 concerns all MHSBQs, such that W(z) is stable,P solves (23) and the input signal and H(z) are such that (31)is satisfied, in which case we have lim`→∞ x(`) = 0. (Notethat, in this case, it directly follows that the state sequence xis bounded, since the MHSBQ corresponds to a stable systemdriven by bounded input signals, see Fig. 6.) Thus, the resultdoes not apply to designs where W(z) has integrators in one(or more) of the scalar subchannels.11 It should also be notedthat the condition (31) is rather restrictive and seems to beuseful only for idle tone suppression

Most existing analytical stability results available in thescalar Σ∆-converter literature, see, e.g., [20]–[23], only con-cern specific designs of the feedback filter, typically thosewith integrators. The stability notion employed in [20]–[23]amounts to boundedness of state and converter output tra-jectories for bounded inputs. This is useful for establishingthe behavior of the reconstruction MSE as a function ofoversampling ratio, but does not address the issue of idle tones.In summary, Theorem 1, complements existing results. Theclass of systems considered is broader and the stability notionis different. 4

V. RELATIONSHIP TO LINEAR FEEDBACK SUBBANDQUANTIZERS

As stated in Section III-A, the frequency weighting filterW(z) is of dimension ne ×K. In Section IV-B we proposedto choose W(z) via (28), in which case ne = L. In the present

10In Section V-B, we show that for N = 1 the MHSBQ is equivalent tothe linear feedback subband quantization architectures presented in [8].

11Whilst stability cannot be ensured in these cases, simulation studies haveshown that (in the related context of scalar A/D conversion) the movinghorizon optimization approach gives significant performance gains whencompared to standard Σ∆-conversion approaches [16].

section, we will focus on designs where ne = K and show thatMHSBQs generalize the linear feedback subband quantizers in[8].

A. Closed Form Solution

The subband quantizer proposed in Section III requires thatone solve the non-convex finite-set-constrained optimizationproblem (22) at every time step (of d). In the particular caseof a square filter W(z) of dimension K×K, i.e., of choosingne = K, see Section III-A, we can extend the results in [16],to the vector-valued case, in order to interpret the MHSBQ interms of a VQ immersed in a feedback loop. For that purposeit is convenient to define the matrices:12

Γ ,

C

CA...

CAN−1

, Φ ,

W0 0 . . . 0

W1 W0. . .

......

. . . . . . 0WN−1 . . . W1 W0

,M ,

[AN−1B AN−2B . . . AB B

]. (32)

In the above definition,

W0 , D, Wi , CAi−1B, i ∈ 1, 2, . . . , N − 1,

see (16). Thus, the columns of Φ are truncated versions of theimpulse response corresponding to W(z).

Furthermore, we introduce the NK ×K filters:

Υ(z) , Ψ−T(ΦT Γ + MT PAN

)(zIn −A)−1B,

Ω(z) , Ψ[IK zIK . . . zN−1IK

]T + Υ(z),(33)

where the matrix Ψ ∈ RNK×NK is obtained from thefactorization:

ΨT Ψ = ΦT Φ + MT PM. (34)

With this, we obtain the following result:Lemma 2: Suppose that UN = s1, s2, . . . , sr, where13

r = |U|N and that W(z) has realization (16), with D ∈RK×K , then the output of the MHSBQ at any time instantk is given by:

u(k) =[IK 0K . . . 0K

]Ψ−1qeUN (c(k)) (35)

where the signal c satisfies:

c , Ω(z)H(z)d−Υ(z)u (36)

and qeUN (·) is a Euclidean VQ (see Definition 1) with imagebeing given by the set:

UN , s1, s2, . . . , sr (37)

which is obtained, term-wise, from UN via:

sj = Ψsj , sj ∈ UN , j ∈ 1, 2, . . . , r.

Proof: The proof of Lemma 2 follows closely that ofTheorem 1 in [16] and is omitted here for the sake of brevity.

12For further details, the interested reader is referred to [16].13|U| denotes the cardinality of U.

9

u

Υ(z)

c+

−

a

Q

Ω(z)[IK 0K . . . 0K

]Ψ−1qeUN (·)

Fig. 7. Implementation of the MHSBQ as a feedback loop, ne = K.

The above result leads directly to the closed loop imple-mentation of the MHSBQ depicted in Fig. 7. It should beemphasized that, if W(z) is not square, then Fig. 7 does notapply.

B. Relationship to Linear Feedback Quantization

The developments in the previous section set the stage forshowing that subband linear feedback quantization methods,as introduced in [8], are contained in the class of MHSBQsstudied in this paper.

To establish this relationship more formally, consider thespecial (and simple) case of horizon N = 1, and a squarefrequency weighting filter W(z) with ne = K and whereD = IK , see (16). In this case, the matrices defined in (32)reduce to Γ = C, Φ = IK , and M = B. Thus, the filters Υ(z)and Ω(z) in (33) are of dimension K ×K and are given by:

Υ(z) = Ψ−T(C + BT PA

)(zIn −A)−1B,

Ω(z) = Ψ + Υ(z)(38)

where Ψ ∈ RK×K is obtained from the factorization ΨT Ψ =IK +BT PB. As a consequence, the dynamics of the MHSBQin Fig. 7 are governed by:

u(k) = Ψ−1qeU(c(k))

where:

c , Ψ−T(W(z)H(z)d− (W(z)−ΨT Ψ)u

),

W(z) , ΨT Ψ +(C + BT PA

)(zIn −A)−1B.

(39)

If we furthermore use no terminal state weighting, i.e., P =0n, then Ψ = IK , U = U, and W(z) = W(z) so that thedynamics of the MHSBQ are described by:

u(k) = qU(c(k))

withc = W(z)H(z)d− (W(z)− IK)u.

On the other hand, the noise-shaping quantizer of Fig. 3 ischaracterized via:

u(k) = qU(b(k))

where:b , G−1(z)v − (G−1(z)− IK)u.

Thus, the two schemes are algebraically equivalent, if we set:

W(z) = G−1(z) and H(z) = E(z). (40)

With respect to the signal predictive quantizer of Fig. 4, di-rect algebraic manipulation yields that equivalence is obtainedif we set:

W(z) = G−1(z) and H(z) = G(z)E(z).

We can therefore conclude that the subband MHSBQ is ageneralization of both the noise-shaping quantizer, and thesignal predictive quantizer proposed in [8]. In particular, theMHSBQ extends the linear feedback subband quantizers in [8]in two ways:

1) It allows for multi-step optimality by choosing N > 1.2) Stability concepts can be directly included into the

design via the final state weighting matrix P.As documented in the sequel, these two aspects can be used

to give enhanced performance of the MHSBQ, when comparedto [8].

Remark 4 (Resultant Design Procedure): It is interestingto note that the relationships established above suggest adesign procedure for the filters W(z) and H(z) which can beused in MHSBQs with any horizon N ≥ 1 and any final stateweighting matrix P. Indeed, given a design for G(z) obtained,for example, via the technique in [8], the filters W(z) andH(z) can be synthesized according to (40). We note, however,that, whilst this is certainly possible, the MHSBQs so obtainedwill, in general, give worse performance than those which arebased on filters W(z) and H(z) designed via (28). This willbe illustrated in a simple example in Section V-C. 4

Remark 5 (Stable Linear Feedback Subband Quantization):If 0 ∈ Ui, ∀i ∈ 1, 2, . . . ,K, and if P in (39) is chosen tosatisfy (23), then Theorem 1 states that the resultant MHSBQwith horizon N = 1 will have no idle tones. This subbandquantization architecture can regarded as a stabilized linearfeedback quantizer. Note that for N = 1 the cardinality ofthe image set of the VQ underlying the MHSBQ, namelyqeU(·), is equal to the sum of the cardinalities of the imagesets of the K scalar quantizers used in the linear feedbackquantizers in [8], i.e., of qUi

(·), i ∈ 1, 2, . . . ,K. Thus,idle tone suppression can be guaranteed at the expense ofonly a small increase in on-line computational effort/circuitcomplexity. 4

C. Example: Two-channel Haar FB

As a simple example, adopted from [8], consider a two-channel oversampled FB with L = 1, corresponding to an

10

oversampling factor of r = 2, see (4). The analysis FB isformed by the Haar filters:

E1(z) =1√2(1 + z−1), E2(z) =

1√2(1− z−1),

and the synthesis FB (corresponding to the para-pseudo-inverse of the analysis FB) is characterized by:

R1(z) =1

2√

2(1 + z), R2(z) =

12√

2(1− z).

This choice corresponds to the polyphase description

E(z) =1√2

[1 + z−1

1− z−1

], R(z) =

12√

2

[1 + z 1− z

]and, thus, satisfies the perfect reconstruction condition (8).

We consider four situations: First, we design the MHSBQfollowing (28), with δ = 1:

H(z) =1√2

[1 + z−1

1− z−1

],W(z) =

12√

2

[1 + z−1 −1 + z−1

].

(41)A state-space description of the filter W(z) in (41) is givenby (16) where n = 1 and

A = 01, B =1

2√

2

[1 1

], C = I1, D =

12√

2

[1 −1

].

We set P as in Theorem 1, i.e., as the solution to (23), whichhere simply gives P = I1. We also analyze the case of nofinal state weighting, i.e., P = 01.

In a second setup, we utilize the noise-shaping filter of [8],namely:

G(z) = I2 − z−1

[0.5 0.5−0.5 −0.5

],

and compute W(z) and H(z) according to (40):

H(z) =1√2

[1 + z−1

1− z−1

],

W(z) = G−1(z) =[1 + 0.5z−1 0.5z−1

−0.5z−1 1− 0.5z−1

].

(42)

A state-space realization of W(z) in (42) is (16) with n = 1and

A = 01, B =1√2

[1 1

], C =

1√2

[1−1

], D = I2.

Again, we analyze two cases, namely no final state weighting,i.e., P = 01, and P chosen according to (23), which, in thissituation, gives P = I1. It is worth noting that the noise-shaping quantizer of [8] is equivalent to the MHSBQ withdesign parameters N = 1, P = 01, and W(z) and H(z) asin (42).

To compare the relative merits of these design choices, wefix the constraint sets according to U1 = U2 = −1, 0, 1, anduse, as input signal y, Tf = 1000 samples of an i.i.d. real-valued Gaussian process having zero-mean and unit variance.Fig. 8 shows the values of the reconstruction error variance,defined as:14

S2 ,1Tf

Tf−1∑`=0

(y(`)− y(`))2. (43)

0 1 2 3 40.1

0.15

0.2

0.25

0.3

0.35

N

S2

filters as in (41); P solves (23)filters as in (41); P = 0filters as in (42); P solves (23)filters as in (42); P = 0

noise!shapingquantizer of [8]

Fig. 8. Reconstruction error variance provided by the Haar FB with MHSBQusing the filters given in (41) (solid and dotted lines) and in (42) (dashedand dash-dotted lines). The noise-shaping quantizer of [8] corresponds to theMHSBQ designed according to (42) with N = 1 and P = 01.

As can be seen from Fig. 8, performance improves in allfour cases as the horizon N is increased beyond N = 1.15 Inparticular, all MHSBQs with P chosen as in Theorem 1 andfor horizon N ≥ 2, outperform the noise-shaping subbandquantizer proposed in [8]. Given that by setting the weightingfilter W(z) as in (41) the MHSBQ aims at minimizing thereconstruction error ed, see (30), we observe in Fig. 8 thatchoosing W(z) as in (41) and P to solve (23) gives the bestresults.

The benefits of choosing P as in Theorem 1 are apparentwhen W(z) is chosen as in (41) and for horizon N = 1.Interestingly, in this specific example, with W(z) as in (42)and N ∈ 1, 2, choosing P = 01 gives better performancethan using the value of P which solves (23).

VI. DESIGN STUDY: COSINE-MODULATED FBS

Cosine-modulated FBs (CMFBs) [44]–[47] have receivedsignificant attention as they can be designed with relativeease (all subband filters correspond to modulated versionsof a single prototype filter) and allow for a computationallyefficient implementation (based on the DCT). In the following,we investigate a six-channel (K = 6) even-stacked CMFB [47]with oversampling factor 3 (i.e., L = 2).16 We utilize an FIRprototype filter of the form

Hp(z) =Lh−1∑m=0

hp(m)z−m (44)

14We average over 100 realizations of the input sequence.15Here and in the sequel, N = 0 denotes direct quantization as in (11).16Even-stacked CMFBs possess the additional property of all subchannel

filters being linear-phase, a property useful, for example, in image codingapplications [48]. Further details on (oversampled) CMFBs can be found in[47].

11

0 50 100!0.2

00.20.40.60.8

1

Initial filter: Impulse response

m

Initial filter

10!1 100!100

!80

!60

!40

!20

0

Frequency (rad/sec)

0 50 100!0.2

00.20.40.6

0.81

Orthogonal filter: Impulse response

m

Orthogonal filter

Frequency (rad/sec)10!1 100

!100

!80

!60

!40

!20

0

Fig. 9. Result of the orthogonalization procedure in [49]: Upper plots show impulse response (LHS) and magnitude of transfer function (RHS) of initialfilter; lower plots show corresponding orthogonal filter leading to a paraunitary even-stacked CMFB.

where Lh is even (Lh = 102), and which satisfies the linearphase property, i.e.,

hp(m) = hp(Lh− 1−m), ∀m ∈ 0, 1, . . . , Lh− 1. (45)

We design the prototype filter Hp(z) by applying the or-thogonalization method proposed in [49] to an initial lowpassfilter (designed with bandwidth 1/K). Fig. 9 shows the initialfilter and the resulting orthogonal filter. As can be seen,the orthogonalized prototype filter Hp(z) exhibits time- andfrequency- characteristics that are similar to those of theoriginal (non-orthogonal) filter. It is worth noting that theimpulse response of Hp(z) is negligible for m ≤ 0 and form ≥ 100. The resulting analysis FB is paraunitary, see [47],[49]. We can therefore synthesize a perfect reconstruction FB,see Section II-C, by choosing the synthesis FB to be an even-stacked CMFB with prototype filter

F p(z) =0∑

m=−Lh+1

fp(m)z−m

wherefp(m) = hp(−m). (46)

Simulations were carried out with binary sets Ui = −1, 1,∀i ∈ 1, 2, . . . , 6, and the input y was chosen as Tf = 1000samples of an i.i.d. real valued Gaussian random processhaving zero-mean and variance 4.

The following design choices were considered: H(z) =E(z) and W(z) = z−51R(z), see (28); and H(z) = E(z)and W(z) = G−1(z), see (40).17 Moreover, we investigatedhorizons N ∈ 0, 1, 2, 3, final state weighting as in (23) aswell as no final state weighting, i.e., P = 0n, where n is theorder of the filters W(z) used in each case. Again, we pointout that, as shown in Section V-B, the noise shaping structureinvestigated in [8] corresponds to the MHSBQ with designparameters N = 1, P = 0n and W(z) = G−1(z).

Figures 10 and 11 contain the sample reconstruction errorvariances obtained, see (43). It can be seen that the benefitof choosing W(z) = z−51R(z) and employing final stateweighting according to (23) is remarkable. With these designparameters, larger horizons give better performance and also

17Here, G(z) was chosen as an FIR filter of order 2, designed accordingto the procedure proposed in [8].

12

with a modest value of N , say N = 2, we obtain excellentresults when compared to the designs where P = 0n. Theseobservations are representative of many experiments that wehave performed.

0 1 2 30.5

0.55

0.6

0.65

0.7

0.75

0.8

N

S2

W(z) = z!51R(z)W(z) = G!1(z)

Fig. 10. Reconstruction error variance as a function of N , P solves (23).

0 1 2 30.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

N

S2

W(z) = z!51R(z)W(z) = G!1(z)

noise!shapingquantizer of [8]

Fig. 11. Reconstruction error variance as a function of N , P = 0n.

It is interesting to note that, as can be seen in Fig. 11, inthe example at hand, all designs where P = 0n give worseperformance than direct quantization following (11) (whichis here denoted as N = 0). We can therefore conclude thatfinal state weighting is crucial to achieve good performance.18

Finally, comparing the case N = 2 and P solving (23) tothe noise-shaping quantizer in [8] (i.e., N = 1 and P = 0n),we can see that MHSBQ with final state weighting leads to areduction in reconstruction MSE of more than 80%.

18Note that in the present case 0 /∈ Ui. Thus, we cannot guarantee absenceof idle tones.

VII. CONCLUSIONS

We proposed a novel method for quantizing filter bankframe expansions. Our strategy minimizes a filtered distortionsequence in a moving horizon fashion. The structure describedin this paper includes the linear feedback subband quantizersproposed in [8] as special cases. More precisely, while linearfeedback quantizers are only one-step optimal, the quantizerproposed in this paper is multi-step optimal. The new approachleads to enhanced performance, compared to the methodsdescribed in [8], and offers the possibility of guaranteeing theabsence of idle-tones. Open problems include characterizingthe MSE of the MHSBQ as a function of the oversamplingfactor, further establishing the relationship to [20]–[24], andextending the MHSBQ to general frame expansions.

APPENDIX

PROOF OF THEOREM 1The proof uses the sequence of optimal costs defined as

V ?N (k) , VN (~u?(k);x(k)), k ∈ Z, with ~u?(k) and x(k)

defined in (17) and (22), respectively.Suppose that, at time k, an optimal sequence

~u?(k) =

u?(k)uk+1

...uk+N−1

(47)

has been found. Thus, at sample k + 1, it follows from (27)that x(k + 1) = Ax(k) + B(a(k)− u?(k)) with u?(k) givenby (26). Next, consider the related sequence

us =

uk+1

uk+2

...uk+N−1

a(k + N)

∈ UN . (48)

Direct calculation then yields:

VN (us;x(k + 1)) = V ?N (k) + ‖Ax(k + N)‖2P

− ‖x(k + N)‖2P + ‖ef (k + N)‖2 − ‖ef (k)‖2

= V ?N (k) + ‖Ax(k + N)‖2P − ‖x(k + N)‖2P

+ ‖Cx(k + N)‖2 − ‖ef (k)‖2

where we have used (17). Since V ?N (k + 1) corresponds to

the optimal choice ~u?(k + 1), we have that V ?N (k + 1) ≤

VN (us;x(k + 1)). Hence, using (23), we obtain

V ?N (k + 1) ≤ V ?

N (k) + ‖Ax(k + N)‖2P − ‖x(k + N)‖2P+ ‖Cx(k + N)‖2 − ‖ef (k)‖2

= V ?N (k) + ‖x(k + N)‖2AT PA−P+CT C − ‖ef (k)‖2

= V ?N (k)− ‖ef (k)‖2 ≤ V ?

N (k), (49)

i.e., the sequence V ?N (k) is non-increasing. Since V ?

N (k) ≥ 0,∀k ∈ Z, it follows that the sequence V ?

N (k) is convergent,thus, limk→∞ V ?

N (k) exists.As a consequence of (49), we also have

V ?N (k + 1)− V ?

N (k) ≤ −‖ef (k)‖2 ≤ 0. (50)

13

Since V ?N (k) is convergent, it holds that

limk→∞

(V ?

N (k + 1)− V ?N (k)

)= 0

so that, by using (50), we obtain limk→∞ ‖ef (k)‖2 = 0 and,thus,

limk→∞

ef (k) = 0.

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14

PLACEPHOTOHERE

Daniel E. Quevedo received Ingeniero CivilElectronico and Magister en Ingenierıa Electronicadegrees from the Universidad Tecnica FedericoSanta Marıa, Valparaıso, Chile in 2000. In 2005,he received the Ph.D. degree from The Universityof Newcastle, Australia, where he currently holds aresearch academic position.

He has lectured at the Universidad Tecnica Fed-erico Santa Marıa and The University of Newcastle.He also has working experience at the VEW EnergieAG, Dortmund, Germany and at the Cerro Tololo

Inter-American Observatory, La Serena, Chile. His research interests coverseveral areas of automatic control, signal processing, and communications.

Daniel was supported by a full scholarship from the alumni associationduring his time at the Universidad Tecnica Federico Santa Marıa and re-ceived several university-wide prizes upon graduating. He received the IEEEConference on Decision and Control Best Student Paper Award in 2003 andwas also a finalist in 2002.

PLACEPHOTOHERE

Helmut Bolcskei was born in Modling, Austria onMay 29, 1970, and received the Dipl.-Ing. and Dr.techn. degrees in electrical engineering from ViennaUniversity of Technology, Vienna, Austria, in 1994and 1997, respectively. In 1998 he was with ViennaUniversity of Technology. From 1999 to 2001 he wasa postdoctoral researcher in the Information SystemsLaboratory, Department of Electrical Engineering,Stanford University, Stanford, CA. He was in thefounding team of Iospan Wireless Inc., a SiliconValley-based startup company (acquired by Intel

Corporation in 2002) specialized in multiple-input multiple-output (MIMO)wireless systems for high-speed Internet access. From 2001 to 2002 he wasan Assistant Professor of Electrical Engineering at the University of Illinoisat Urbana-Champaign. He has been with ETH Zurich since 2002, where he isProfessor of Communication Theory. He was a visiting researcher at PhilipsResearch Laboratories Eindhoven, The Netherlands, ENST Paris, France, andthe Heinrich Hertz Institute Berlin, Germany. His research interests includecommunication and information theory with special emphasis on wirelesscommunications, signal processing and quantum information processing.

He received the 2001 IEEE Signal Processing Society Young Author BestPaper Award, the 2006 IEEE Communications Society Leonard G. AbrahamBest Paper Award, the ETH Golden Owl Teaching Award, and was an ErwinSchrodinger Fellow (1999-2001) of the Austrian National Science Foundation(FWF). He was a plenary speaker at several IEEE conferences and servedas an associate editor of the IEEE Transactions on Signal Processing, theIEEE Transactions on Wireless Communications and the EURASIP Journalon Applied Signal Processing. He is currently on the editorial board of”Foundations and Trends in Networking”, serves as an associate editor forthe IEEE Transactions on Information Theory and is TPC co-chair of the2008 IEEE International Symposium on Information Theory.

PLACEPHOTOHERE

Graham C. Goodwin obtained a B.Sc (Physics),B.E (Electrical Engineering), and Ph.D from theUniversity of New South Wales. He is currently Pro-fessor of Electrical Engineering at the University ofNewcastle, Australia and holds Honorary Doctoratesfrom Lund Institute of Technology, Sweden and theTechnion, Israel.

He is the co-author of eight books, four editedvolumes, and many technical papers. Graham is therecipient of Control Systems Society 1999 HendrikBode Lecture Prize, a Best Paper award by IEEE

Trans. Automatic Control, a Best Paper award by Asian Journal of Control,and 2 Best Engineering Text Book awards from the International Federationof Automatic Control. He is a Fellow of the IEEE; an Honorary Fellow ofInstitute of Engineers, Australia; a Fellow of the Australian Academy ofScience; a Fellow of the Australian Academy of Technology, Science andEngineering; a Member of the International Statistical Institute; a Fellow ofthe Royal Society, London and a Foreign Member of the Royal SwedishAcademy of Sciences.