1108 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL....

1108 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 3, MARCH 2008

Quantization Bounds on Grassmann Manifolds andApplications to MIMO Communications

Wei Dai, Member, IEEE, Youjian (Eugene) Liu, Member, IEEE, and Brian Rider

Abstract—The Grassmann manifold Gn;p ( ) is the set of allp-dimensional planes (through the origin) in the n-dimensionalEuclidean space n, where is either or . This paper con-siders the quantization problem in which a source in Gn;p ( ) isquantized through a code in Gn;q ( ), with p and q not necessarilythe same. The analysis is based on the volume of a metric ballin Gn;p ( ) with center in Gn;q ( ), and our chief result is aclosed-form expression for the volume of a metric ball of radiusat most one. This volume formula holds for arbitrary n, p, q, and

, while previous results pertained only to some special cases.Based on this volume formula, several bounds are derived for therate–distortion tradeoff assuming that the quantization rate issufficiently high. The lower and upper bounds on the distortionrate function are asymptotically identical, and therefore preciselyquantify the asymptotic rate–distortion tradeoff. We also showthat random codes are asymptotically optimal in the sense thatthey achieve the minimum possible distortion in probability as nand the code rate approach infinity linearly. Finally, as an appli-cation of the derived results to communication theory, we quantifythe effect of beamforming matrix selection in multiple-antennacommunication systems with finite rate channel state feedback.

Index Terms—Finite-rate feedback, Grassmann manifolds, lim-ited feedback, multiple-input multiple-output (MIMO) communi-cations, quantization, rate distortion function.

I. INTRODUCTION

THE Grassmann manifold is the set of all -dimen-sional planes (through the origin) in the -dimensional Eu-

clidean space , where is either or . It forms a compactRiemann manifold of real dimension , wherewhen and when . The Grassmann manifoldprovides a useful analysis tool for multiple-antenna communi-cations (also known as multiple-input multiple-output (MIMO)

Manuscript received August 18, 2005; revised October 11, 2007. This workwas supported by the National Science Foundation under Grants DMS-0505680,CCF-0728955, and ECCS-0725915 and the Junior Faculty Development Award,University of Colorado at Boulder. The material in this work was presentedin part at the IEEE Globe Telecommunications Conference, St. Louis, MO,November/December 2005, and at IEEE International Symposium on Informa-tion Theory, Nice, France, June 2007.

W. Dai was with the Department of Electrical and Computer Engineering,University of Colorado at Boulder, Boulder, CO 80309 USA. He is now withthe Department of Electrical and Computer Engineering, University of Illinoisat Urbana-Champaign, Urbana, IL 61801 USA (e-mail: [email protected]).

Y. (E.) Liu is with the Department of Electrical and Computer Engineering,University of Colorado at Boulder, CO 80309 USA (e-mail: [email protected]).

B. Rider is with the Department of Mathematics, University of Colorado atBoulder, Boulder, CO 80309 USA (e-mail: [email protected]).

Communicated by B. Hassibi, Associate Editor for Communications.Color versions of Figure 1 in this paper is available online at http://ieeexplore.

ieee.org.Digital Object Identifier 10.1109/TIT.2007.915691

communication systems). The capacity of noncoherent MIMOsystems at high signal-to-noise ratio (SNR) region was derivedby analysis in the Grassmann manifold [1]. The well-knownspherical codes for MIMO systems can be viewed as codes inthe Grassmann manifold [2]. Further, for coherent MIMO sys-tems with finite-rate feedback, the quantization of eigenchannelvectors is related to the quantization on the Grassmann mani-fold [3]–[7].

This paper studies quantization on the Grassmann manifold.Roughly speaking, a quantization is a representation of a source:it maps an element in (the source) into a subset

, which is often discrete and referred to as a code. Whileit is traditionally assumed that [8]–[11], we are interestedin a more general case where may not necessarily equal .The performance limit of quantization is given by the so-calledrate–distortion tradeoff. Let the source be randomly distributedand define a distortion metric between elements in and

. The rate–distortion tradeoff is described by the min-imum average distortion achievable for a given code size, orequivalently, the minimum code size required to achieve a par-ticular average distortion. The major focus of this paper is tounderstand this tradeoff.

The quantization problem for Grassmann manifolds has beensolved previously for some special cases, and, to the authors’knowledge, always under the assumption that . In [8],an isometric embedding of into a sphere in Euclideanspace is given. Then, using the Rankin bound inEuclidean space, the Rankin bound in is obtained. Un-fortunately, this bound is not tight when the code size is large.Instead of resorting to a specific isometric embedding, the quan-tization problem can also be attacked by volume calculation inthe Grassmann manifold directly. Let denote a metric ballof radius in . The exact volume formula for ain with and is provided in [3]. When isfixed and , an asymptotic volume formula is derived byLaplace method in [9]. For general and , when the radius ofthe ball is small, a tool using Jacobi vector fields is employedto estimate the volume [10] but no closed-form formula is pro-vided. These volume evaluations provide different versions ofGilbert–Varshamov and Hamming bounds [9], [10]. While the

case has been intensively studied, unequal dimensionalquantization does arise in some multiple-antenna communica-tion systems, see [7] for an example. It is thus worthwhile to gobeyond .

The main contribution of this paper is to accurately quan-tify the rate–distortion tradeoff for arbitrary

and when quantization rate is sufficiently high. The keyis a closed-form formula for the volume of a metric ball in

0018-9448/$25.00 © 2008 IEEE

DAI et al.: QUANTIZATION BOUNDS ON GRASSMANN MANIFOLDS AND APPLICATIONS TO MIMO COMMUNICATIONS 1109

Grassmann manifolds when the radius is small. Specifically, wepresent the following.

1) An explicit volume formula for a metric ball is derived forarbitrary , , , and when the radius .1 For the case

, a metric ball in whose center is inis taken into consideration. The result takes a simple formin which the main order term is for a con-stant depending only on , , , and . Usefullower and upper bounds on the volume are also presented.

2) The improved version of Gilbert–Varshamov and Ham-ming bounds are obtained by a direct application of theabove.

3) Tight lower and upper bounds are derived for the rate–dis-tortion tradeoff. In particular, for fixed and with andthe code rate (logarithm of the code size) approaching in-finity linearly, the bounds become identical, and so pre-cisely quantify the asymptotic rate–distortion tradeoff. Wealso show that random codes are asymptotically optimal inthe sense that they achieve the minimum achievable distor-tion in probability in our asymptotic region.

Finally, as an application of the derived quantization bound toMIMO communications, we detail the intimate relationshipbetween beamforming matrix selection and quantization onGrassmann manifolds. The derived distortion–rate tradeoff isessential for characterizing the effect of beamforming matrixselection.

The paper is organized as follows. Section II provides somepreliminaries on the Grassmann manifold. Section III derivesthe explicit volume formula for a metric ball in the Grassmannmanifold. The corresponding sphere-packing bounds are ob-tained and the rate–distortion tradeoff is accurately quantified inSection IV. An application of the quantization bounds to MIMOsystems with finite-rate channel state feedback is detailed inSection V. Section VI contains the conclusions.

II. PRELIMINARIES

This section presents a brief introduction to the Grassmannmanifold. A metric and a measure on the Grassmann manifoldare defined, and the problems relevant to quantization on theGrassmann manifold are formulated.

A. Metric and Measure on

For the sake of applications [3]–[5], the projection Frobeniusmetric (chordal distance) is employed throughout the paper. Forany two planes and (without loss ofgenerality, we assume that ), we define the principalangles and the chordal distance between and as follows.Let and be the unit vectors such that ismaximal. Inductively, let and be the unit vectorssuch that and for all and is

1The general case for an arbitrary radius was solved by the authors [12]. Themechanism is fundamentally different and the results hold in an asymptotic re-gion quite different from here.

maximal. The principal angles are defined asfor [8], [13]. The chordal distance between and

is given by

(1)

The invariant measure on is defined as follows. Letand be the groups of orthogonal and unitary

matrices, respectively. Let and when, or and when . For any

measurable set and arbitrary and

The invariant measure defines the isotropic distribution on[13].

B. Quantization on

Given both a metric and a measure on , a quantiza-tion on the Grassmann manifold is defined as follows. Let be afinite-size discrete subset of . A quantization is a map-ping from the to the set (also known as a code)

An element in the code is called a codeword. Thus, roughlyspeaking, a quantization is to use a subset of to rep-resent the space . Here, different from traditional ap-proaches, and may not be the same.

Sphere packing/covering bounds relate the size of the codeto the minimum distance among the codewords. Let be theminimum distance between any two codewords of a code

and be a metric ball of radius in the . Ifis any positive integer such that , then there

exists a code of size with minimum distance . Thisprinciple is called as the Gilbert–Varshamov lower bound

(2)

On the other hand, for any code . TheHamming upper bound captures this fact as

(3)

For more information about the sphere packing/coveringbounds, see [9].

Rate–distortion tradeoff is another important aspect of thequantization problem. A distortion metric is a mapping

from the set of the element pairs in and into theset of nonnegative real numbers. Throughout this paper, we de-fine the distortion metric as the square of the chordal distance

. Assume that a source is randomly dis-tributed in . The distortion associated with a quantiza-tion is defined as


The rate–distortion tradeoff can be described by the infimumachievable distortion given a code size, which is called the dis-tortion–rate function, or equivalently, the infimum code sizerequired to achieve a particular distortion, which is called therate–distortion function. In this paper, the source is assumedto be uniformly distributed in . For a given code

, the optimal quantization to minimize the distortion isgiven by2

The distortion associated with this quantization is

For a given code size where is a positive integer, the dis-tortion rate function is then given by3

(4)

The rate–distortion function is given by

(5)

III. METRIC BALLS IN THE GRASSMANN MANIFOLD

This section derives an explicit volume formula for a metricball in the Grassmann manifold. It is the essential tool to quan-tify the rate–distortion tradeoff in Section IV.

The volume of a ball can be expressed as a multivariate inte-gral. Assume the invariant measure and the chordal distance

. For any given and , define

and

2The ties, i.e., the case that 9Q ;Q 2 C such that d (P;Q ) =mind (P;Q) = d (P;Q ), are broken arbitrarily as they occur with

probability zero.3The standard definition of the distortion–rate function involves the code rate,

which is log K . The definition in this paper is equivalent to the standard one.

It has been shown that and the valueis independent of the choice of the center [13]. For convenience,we denote and by without distinguishingthem. Then, the volume of a metric ball is given by

(6)

where are the principle anglesand the differential form is the joint density of the ’s [13],[14], explicitly given in (20) in Appendix A. Our basic result isa sharp estimate of this integral in a specified regime.

Theorem 1: When , the volume of a metric ballis given by

(7)

where we get (8) and (9) at the bottom of the page.Proof: See Appendix A.

The following corollary gives the two cases where the volumeformula becomes exact.

Corollary 1: When , in either of the following twocases:

1) and ,2) and ,

the volume of a metric ball is exactly

where is defined in (8) and (9).

We also have the general bounds.

Corollary 2: Assume . If and , the volumeof is bounded by

For all other cases

ifif

if

if ,(8)

and

(9)


Fig. 1. Volume of small balls in the Grassmann manifold. The integers besides curves are, from left to right, n, p, q, and �, respectively.

Proof: Corollary 1 and 2 follow the proof of Theorem 1 bytracking the higher order terms.

Theorem 1 is, of course, consistent with the previous resultsin [3] and [9], which pertain to special choices of , , , andand are stated below as examples.

Example 1: The volume formula for a whereand is derived in [3] as

agreeing with Theorem 1, where and .

Example 2: For the case that are fixed and ,an asymptotic volume formula for a is derived by Bargand Nogin [9], which reads

(10)

On the other hand, an asymptotic analysis of the result of The-orem 1 gives

for fixed and asymptotically large .This follows from Stirling’s approximation applied to

. In this setting, thus, Theorem 1provides refinements to the previously known (10).

Again, Theorem 1 is distinct from the above results in that itholds for arbitrary , , , and . For a metric ball with param-eter not asymptotically large, i.e., and are comparable to ,it is not appropriate to use (10) to estimate the volume. A trivial

example arises in the case. If , the exactvolume of for is the constant . The formula inTheorem 1 gives and . How-

ever, the approximation (formula (10)) produces anestimate much smaller than one for small .

For engineering purposes, it is often satisfactory to approxi-mate the volume of a metric ball by whenthe radius is sufficiently small. Fig. 1 compares the actualvolume (6) to this simple approximation. As it is typically diffi-cult to evaluate the integral (6) directly, we use Monte Carlo tosimulate

by fixing a and generating isotropically dis-tributed . The simulation results show that theapproximation (solid lines) is close to the(simulated) volume (circles) for all . For comparison, wealso depict the Barg–Nogin approximation (10) for the case

and , seeing that for those parameters thatapproximation (dash–dot lines) is not even of the correct order.

IV. QUANTIZATION BOUNDS

This section derives the sphere-packing bounds and quanti-fies the rate–distortion tradeoff. The results hold for Grassmannmanifolds with arbitrary , , , and .

A. Sphere Packing/Covering Bounds

The Gilbert–Varshamov and Hamming bounds are given inthe following corollary.

Corollary 3: When is sufficiently small ( necessarily),there exists a code with size and the minimumdistance between codewords such that


For any code with the minimum distance betweencodewords

Proof: The corollary is proved by substituting the volumeformula (7) into (2) and (3) and setting .

Remark 1: Applying Corollary 2 would provide sharper in-formation on the higher order term. But we omit this.

B. Quantization: The Rate–Distortion Tradeoff

Recall the distortion rate function defined in (4).

Theorem 2: When is sufficiently large( necessarily), the distortionrate function is bounded as in

(11)

Remark: Once again, for engineering purposes, the mainorder terms in (11) are usually sufficiently accurate to charac-terize the distortion rate function. The details of thecorrection are spelled out in Theorem 4.

The lower and the upper bounds are established in Appen-dices B and C, respectively, but we provide here a sketch of theproofs.

The lower bound follows from a sphere packing/covering ar-gument. We construct an ideal quantizer, which may not exist,to minimize the distortion. Suppose that there exists metricballs (centered in ) of the same radius packing andcovering the entire at the same time. Then the quan-tizer which maps each of those balls into its centerwould give the minimum distortion among all quantizers. Whilethe corresponding distortion may not be achievable, it is cer-tainly a lower bound on the distortion rate function.

Next, the upper bound is obtained by calculating the averagedistortion of random codes. As the distortion of any particularcode is an upper bound of the distortion rate function, so is theaverage distortion of an ensemble of random codes. A randomcode is generated by drawing thecodewords ’s independently from the isotropic distributionon . Denoting the average distortion of random codesby , extreme order statistics, see for example[15], reduce the calculation of to that of thevolume (6). Thus, our explicit volume formula (7) leads to theasymptotic value of and so also the upperbound.

Being dual to the distortion rate function, lower and upperbounds are constructed for the rate distortion function.

Corollary 4: When the required distortion is sufficientlysmall ( necessarily), the rate distortion function satisfiesthe following bounds:

(12)

To investigate the difference between the lower and upperbounds in (11), proceed as follows. Since the exponential termsare the same in both bounds, focus on the coefficients. The dif-ference between the two bounds depends only on .There are two cases to consider.

Case 1: . This only occurs when . Thenthe whole contains only one element and all planes in

are mapped to this element. This results in a trivialquantization.

Case 2: . An elementary calculation showsthat

with the difference between these two coefficients vanishing as.

As a result, the asymptotic rate–distortion tradeoff is exactlyquantified.

Corollary 5: Suppose that and are fixed, and the coderate approach to infinity simultaneously with

If the normalized code rate is sufficiently large (necessarily), then

On the other hand, if the required distortion is sufficientlysmall ( necessarily), then the minimum code size requiredto achieve that distortion satisfies

Proof: The leading order is read off from


and

That the multiplicative errors fall into place followsfrom Theorem 4.

While the asymptotic rate–distortion tradeoff is preciselyquantified, the next question could be how to achieve it. As onemight anticipate, random codes are asymptotically optimal inprobability.

Theorem 3: Let be a code randomly gener-ated from the isotropic distribution and with size . Fix and. Let with . If the normal-

ized code rate is sufficiently large ( necessarily),then for

Proof: See Appendix D.

We finally detail the errors reported in (11) and(12).

Theorem 4: Let be an arbitrary real number such thatand be sufficiently large (

necessarily). If and , then

If and , or and , then

If and , or and , then we get theexpression at the bottom of the page.

Proof: The proof is given in Appendices B and C.

C. Discussion and Empirical Comparisons

As a comparison, consider the distortion–rate function ap-proximation derived in [16]. For the case whereand , [16] offers the approximation

(13)

by asymptotic arguments ( is large here). According to ourresults in Theorem 2, the approximation (13) is indeed a lowerbound for the distortion–rate function and valid for all possible

’s. For the and case, a lower bound of an upperbound on the distortion–rate function is given in [16] based onan estimation of the minimum distance of a code. Being neithera lower bound nor an upper bound and holding only for

, it is less robust than our Theorem 2 bounds—see alsoFig. 2 for an empirical comparison.

Besides characterizing the rate–distortion tradeoff, we arealso interested in designing a code to minimizedistortion for a given finite code size . Generally, it is com-putational intensive to design a code to minimize distortiondirectly. In [4] and [17], a suboptimal design criterion—maxi-mization of the minimum distance between codeword pairs—isproposed. We refer to this suboptimal criterion as the max-mincriterion. The following can be verified for the case basedon the analysis of this paper. Let the minimum distance of acode be . Note that the metric balls of radius and centeredat are disjoint. Then the corresponding distortion isupper-bounded by

(14)

An elementary calculation along with (7) shows that the firstderivative of the upper bound is negative whenever

So, the upper bound (14) is a decreasing function of forsmall enough, and the max-min criterion is an appropriate de-sign criterion to obtain codes with small distortion. Since thiscriterion only requires the calculation of the distance betweencodeword pairs, the computational complexity is less than thatof designing a code to minimize the distortion directly.

Fig. 2 compares the simulated distortion rate function (theplus markers) with its lower bound (the dashed lines) and upperbound (the solid lines) in (11). Here, is assumed. To sim-ulate the distortion rate function, we use the max-min criterion


Fig. 2. Bounds on the distortion–rate function.

Fig. 3. System model for a MIMO system with finite rate feedback.

to design codes and use the distortion of the best designed codeas the simulated distortion rate function. Simulation resultsshow that the bounds in (11) hold for large . When isrelatively small, the formula (11) can serve as good approxi-mation to the distortion–rate function as well. Simulations alsoverify the previous discussion on the difference between thetwo bounds. The difference between the bounds is small and itbecomes smaller as increases. In addition, we compare ourbounds with the approximation (the “ ” markers) derived in[16]. Simulations show that approximation is neither an uppernor a lower bound. It appears to work well in the caseand , but fails for and . Once again, thebounds (11) derived in this paper hold for arbitrary and .

V. AN APPLICATION TO MIMO SYSTEMS WITH FINITE-RATE

CHANNEL STATE FEEDBACK

As an application of the derived quantization bounds on theGrassmann manifold, this section discusses the information-the-oretical benefit of finite-rate channel-state feedback for MIMOsystems using power on/off strategy. In particular, we show thatthe benefit of the channel state feedback can be accurately char-acterized by the distortion–rate tradeoff on the Grassmann man-ifold. The effect of finite-rate feedback on MIMO systems usingpower on/off strategy has been widely studied. MIMO systemswith only one on-beam are discussed in [3] and [4], where the

beamforming codebook design criterion and performance anal-ysis are derived by geometric arguments in the Grassmann man-ifold . MIMO systems with multiple on-beams are con-sidered in [18], [19] [16], [20], [21]. Criteria to select the beam-forming matrix are developed in [18] and [19]. The SNR lossdue to quantized beamforming is discussed in [16]. The cor-responding analysis is based on Barg’s formula (10) and onlyvalid for MIMO systems with asymptotically large number oftransmit antennas. The effect of beamforming quantization oninformation rate is investigated in [20] and [21]. The loss in in-formation rate is quantified for high-SNR region in [20]. Thatanalysis is based on an approximation of the logdet function inthe high-SNR region and a metric on the Grassmann manifoldother than the chordal distance. In [21], a formula to calculatethe information rate for all SNR regimes is proposed by lettingthe numbers of transmit antennas, receive antennas, and feed-back rate approach infinity simultaneously. However, this for-mula generally overestimates the performance.

The basic model of a wireless communication system withtransmit antennas, receive antennas, and finite-rate

channel state feedback is depicted in Fig. 3. The information bitstream is encoded into the Gaussian signal vectorand then multiplied by the beamforming matrixto generate the transmitted signal , where is thedimension of the signal satisfying and thebeamforming matrix satisfies . In power on/offstrategy, , where the constant denotes


Fig. 4. Performance of a constant number of on-beams versus feedback rate.

the on-power. Assume that the channel is Rayleigh flatfading, i.e., the entries of are independent and identicallydistributed (i.i.d.) circularly symmetric complex Gaussianvariables with zero mean and unit variance andis i.i.d. for each channel use. Let be the receivedsignal and be the Gaussian noise, then

where . We further assume that there is a

beamforming codebook de-clared to both the transmitter and the receiver before the trans-mission. At the beginning of each channel use, the channel state

is perfectly estimated at the receiver. A message, which isa function of the channel state, is sent back to the transmitterthrough a feedback channel. The feedback is error-free and rate-limited. According to the channel state feedback, the transmitterchooses an appropriate beamforming matrix . Let thefeedback rate be bits/channel use. Then the size of the beam-forming codebook . The feedback function is a map-ping from the set of channel state into the beamforming matrixindex set . This section will quan-tify the corresponding information rate

where and is the average received SNR.Before discussing the finite-rate feedback case, we consider

the case that the transmitter has full knowledge of the channelstate . In this setting, the optimal beamforming matrix is givenby where is the matrix composed bythe right singular vectors of corresponding to the largestsingular values [5]. The corresponding information rate is

(15)

where is the th largest eigenvalue of . In [5, Sec. III],we derived an asymptotic formula to approximate the quantity

for constant , and thus have asharp approximation of .

The effect of finite-rate feedback can be characterized by thequantization bounds in the Grassmann manifold. For finite-ratefeedback, we define a suboptimal feedback function

(16)

where and are the planes in the gen-erated by and , respectively. In [5], we showed that thisfeedback function is asymptotically optimal as andnear optimal when . With this feedback function andassuming that the feedback rate is large, it has also beenshown in [5] that

(17)

where

(18)

Thus, the difference between perfect beamforming case (15) andfinite-rate feedback case (17) is quantified by , which de-pends on the distortion rate function on the . Substi-tuting quantization bounds (11) into (18) yields an approxima-tion to the information rate as a function of the feedback rate

.Simulations verify the above approximation. Let

. Fig. 4 compares the simulated (circles) andapproximated information rates as functions of . Theinformation rate approximated by the lower bound (solid lines)and the upper bound (dotted lines) in (11) are presented. Thesimulation results show that the information rate approximated


Fig. 5. Performance of a constant number of on-beams versus SNR. (a) Average mutual information. (b) Relative performance.

by the bounds (11) match the actual rate almost perfectly. As acomparison, the approximation proposed in [21], [22], whichis based on asymptotic analysis and Gaussian approximation,overestimates the information rate. Furthermore, we comparethe simulated information rate and the approximations for alarge range of SNRs in Fig. 5. Without loss of generality, weonly present the lower bound in (11) because it corresponds tothe random codes and can be achieved by appropriate code de-sign. Fig. 5(a) shows that the difference between the simulatedand approximated information rate is almost unnoticeable. Tomake the performance difference clearer, Fig. 5(b) presents the

relative performance as the ratio of the considered performanceand the capacity of a MIMO achieved by water-fillingpower control. The difference in relative performance is alsosmall for all simulated SNRs.

VI. CONCLUSION

This paper considers the quantization problem on the Grass-mann manifold. Based on an explicit volume formula for ametric ball in the Grassmann manifold, sphere-packing boundsare obtained and the rate–distortion tradeoff is accuratelycharacterized by establishing bounds on the distortion function.


Simulations verify the developed results. As an applicationof the derived quantization bounds, the information rate ofa MIMO system with finite-rate channel-state feedback andpower on/off strategy is accurately quantified.

APPENDIX

A. Proof of Theorem 1

The proof is divided into three parts, in which we calculatethe volume formula for the , ,and cases, respectively.

1) Case: First we prove the basic form

for . Afterward, we calculate the constants and.

The volume of a metric ball is given by

(19)

where the differential form is the joint density of ’s. Forconvenience, we introduce the following notations. Define

and order ’s such that if . Defineand also

Recall that for and for . With thesenotations, the invariant measure can be written as follows[14]:

(20)

where the constant is given by (21) shown at the bottomof the page.

To get the form (7), we perform the variable changefor . Under this transformation, the integral

domain

for

is changed to

for

for

where the last equation holds since . Thus

Next note that

and so we are able to express the volume of in the desiredform with constants defined in the second equation at the bottomof the page.

In order to calculate the constants and , weneed the following lemma [23].

Lemma 1: It holds that

(21)

and


where , , ,

and the integral is taken over , .Proof: This is Selberg’s second generalization of the beta

integral. See [23, Sec. 17.10] for a detailed proof.

According to Lemma 1, we have that

Substituting the formula (21) for into the integral ex-pansion of yields

after some simplifications.The constant can be calculated in a similar way.

Lemma 1 implies the equation at the bottom of the page.Therefore

which completes the proof of the case.

2) Case: This computation is closelyrelated to that for the case.

To see the connection, we define the generator matrix andthe orthogonal complement plane. For any given plane

, the generator matrix is the matrix whosecolumns are orthonormal and expand the plane . The gen-

erator matrix is not uniquely defined. However, the chordal dis-tance between and can be uniquelydefined by their generator matrices. Indeed

where and are generator matrices for the plane and ,respectively. It can be shown that the chordal distance is inde-pendent of the choice of the generator matrices. The orthogonalcomplement plane is defined as follows. For any given plane

, its orthogonal complement plane is the planein such that the minimum principal angle between

and is . It is straightforward that whereand are the generator matrices for and , respectively(the matrix is the matrix with all elements ).

With the definition of the orthogonal complement plane, thechordal distance between and can be related to that between

and . The relationship is given in the following lemma.

Lemma 2: For any given planes and, let and be their

orthogonal complement planes respectively. Then

Proof: This lemma can be proved by the generator ma-trices. Let , , , and be the generator matrices for ,

, , and respectively. Without loss of generality, we alsoassume that . Then

where the matrix is the one composed of and .Similarly


Then

where and are from the definition of the chordal distanceand the facts that and

.

By this lemma, the connection between thecase and case is clear. The volume formula forthe case can be calculated as follows:

where and are the metric balls in and, respectively. Therefore, the results for the

case can be directly applied by letting and. Finally, after some simplification, we have

where

and

3) Case: This computation isagain related to that for the case.

Similar to the case, the connection betweenthe case and case can be re-vealed by the generator matrix and the orthogonal complementplane. Let and . Then

. For any given planes and, let be the orthogonal comple-

ment plane of . Let , , and be the generator matricesfor , , and . Then

Therefore

Now use the volume formula. Note that

Then

where is the invariant measure with parameter , ,and . Substitute the form for (20) into the above for-mula. Then

where the first equation comes from the variable changes, is defined in (21), and , ,

and are given in expressions at the top of the followingpage. Applying Lemma 1 and some simplifications, we find that

and

To summarize, if , see the second set ofequations at the top of the following page.

B. Proof of the Lower Bound on

Assume a source is uniformly distributed in . Forany codebook , define the empirical cumulativedistribution function as


and

where

if

if ,

and

Then the distortion associated with the codebook is given by

(22)

The following theorem gives the empirical distribution tominimize the distortion.

Lemma 3: The empirical distribution function minimizingthe distortion for a given is

ififif

where satisfies .Proof: For any empirical distribution

Thus(23)

Therefore

where follows from integration by parts, and followsfrom (23).

From Lemma 3, it is clear that

(24)

where .The difficulty is that we do not know exactly for some

cases. To overcome this, we will construct a further lower boundon (24).

For all cases except the and case, alower bound on (24) may be constructed as follows. Let

and satisfy . Since(Corollary 2), . But

. We have . Therefore


For the case and , the computation is morecomplicated. The following lemma is helpful.

Lemma 4: Let . Letand satisfy . Let

and satisfy .Let and satisfy

. Then

Proof: Similar to the arguments for all the cases except theand case, it can be proved that and

Then for . It impliesfor . Therefore,

and

We calculate , where

, as follows:

C. Proof of the Upper Bound on

An upper bound on is arrived at by computingthe average distortion of random codes. Let

be a random code whose codewords’s are independently drawn from the uniform distribution

on . For any given element , define, . Then ’s are i.i.d. random

variables with distribution function

Define . Then

To calculate , we need a bound on the distributionfunction of .

Lemma 5: Let ’s be i.i.d. randomvariables with distribution function . Let

. Then for all

Proof: See [15, p. 10].

With the above upper bound on the distribution function of, we derive an upper bound on . In the following,

we use instead of for simplicity. Let be anarbitrary distribution function such that . It isclear that is zero if . Then

where follows from Lemma 5. Here and throughout

is an arbitrary real number and is large enough to guarantee.

If 1) and , or 2) and , thenwe can take (Corollaries 1 and 2).We have


(25)

where is from the variable change .Thus, for any given , , and

and so, when is large enough

If 3) and , or 4) and ,then whereand (Corollaries 1 and 2). Notethat for all . We take

. Then

where follows from the variable change. Once more, for any given

, , and

and

for sufficiently large .

D. Proof of Theorem 3

To simplify the notations, we define

for some positive constant . The choice of will bedetailed later. From Corollary 5, it can be verified that

and

Similar to Appendix C, for any given , definethe random variable

For all , let be defined as shown in theexpression at the bottom of the page. Then by Lemma 5,

for all .Now we upper bound the probability

. For all

where follows from the Chebyshev’s inequality, fol-lows from the Jensen’s inequality, and holds because

if and or andif and or and .


is independent of the choice of . Using the

same trick in (25), for all

Here, since , we assume that and arelarge enough so that .

If 1) and , or 2) and , then

Now choose such that . For andlarge enough, . Let . It can be verified that

If 3) and , or 4) and , then.

Note that

Let be such that and . Again

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