On beamforming with finite rate feedback in multiple ... · beamforming.1 Further, the gain over no...

2562 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 10, OCTOBER 2003

On Beamforming With Finite Rate Feedback inMultiple-Antenna Systems

Krishna Kiran Mukkavilli, Member, IEEE, Ashutosh Sabharwal, Member, IEEE, Elza Erkip, Member, IEEE, andBehnaam Aazhang, Fellow, IEEE

Abstract—In this paper, we study a multiple-antenna systemwhere the transmitter is equipped with quantized informationabout instantaneous channel realizations. Assuming that thetransmitter uses the quantized information for beamforming, wederive a universal lower bound on the outage probability for anyfinite set of beamformers. The universal lower bound provides aconcise characterization of the gain with each additional bit offeedback information regarding the channel. Using the bound,it is shown that finite information systems approach the perfectinformation case as( 1)2 , where is the number offeedback bits and is the number of transmit antennas. Thegeometrical bounding technique, used in the proof of the lowerbound, also leads to a design criterion for good beamformers,whose outage performance approaches the lower bound. Thedesign criterion minimizes the maximum inner product betweenany two beamforming vectors in the beamformer codebook, and isequivalent to the problem of designing unitary space–time codesunder certain conditions. Finally, we show that good beamformersare good packings of two-dimensional subspaces in a2 -dimen-sional real Grassmannian manifold with chordal distance as themetric.

Index Terms—Beamforming, feedback, multiple antennas, out-age probability, transmit diversity, unitary codes.

I. INTRODUCTION

RECENTLY, multiple-antenna systems have received agreat deal of attention, mainly due to their potential to

meet the growing demand of data rates for current and futurewireless systems. Current cellular standards have already madeprovisions for two antennas on the handsets while allowingfor more at the base stations. The efficient deployment ofmultiple antenna elements in practical systems requires algo-rithms which achieve the desired spectral efficiency at a lowcomputational cost.

In this paper, we derive a universal bound on the performanceof a beamforming system, in which the transmitter is provided

Manuscript received October 30, 2002; revised April 28, 2003. This worksupported in part by Nokia, TATP under Grant 1999-003604-080, and the Na-tional Science Foundation under Grant CCR-0311398. The work of E. Erkip wassupported in part by the National Science Foundation under CAREER GrantCCR-0093163. The material in this paper was presented in part at the IEEEInternational Symposium on Information Theory, Yokohama, Japan, June/July2003.

K. K. Mukkavilli is with Qualcomm Inc., San Diego, CA 92121 USA (e-mail:[email protected]).

A. Sabharwal and B. Aazhang are with the Department of Electrical and Com-puter Engineering (MS-366), Rice University, Houston, TX 77005 USA (e-mail:[email protected]; [email protected]).

E. Erkip is with the Department of Electrical and Computer Engineering,Polytechnic University, Brooklyn, Brooklyn, NY 11201 USA.

Communicated by B. M. Hochwald, Guest Editor.Digital Object Identifier 10.1109/TIT.2003.817433

with bits of information regarding the instantaneous channelvector. By assuming perfect channel information at the receiverand noiseless, zero-delay feedback link, we clearly quantify theimpact of quantization on the outage probability of resultingbeamforming system. Interestingly, the bounding techniquealso results in a geometrically intuitive construction criterionto design finite-rate beamformer codebooks. We show severalconstructions whose performance follows closely the universallower bound, thereby showing tightness of the lower bound.

Our primary motivation for the proposed problem formula-tion with limited transmitter information stems from severalinterrelated reasons. First, the outage probability when bothtransmitter and receiver are equipped with perfect channelinformation is much lower than that with no information atthe transmitter. The gain due to transmitter information isavailable even when the transmitter does not perform channeladaptive rate control, and relies only on power control andbeamforming.1 Further, the gain over no transmitter informa-tion is significant ( decibels, where is the numberof transmit antennas) even when the transmitter does not usepower control over time.

Fig. 1 shows the large performance gains of the power con-trol and the beamforming methods, made possible by channelstate information at the transmitter. Note thatif neither trans-mitter nor receiver know the channel, then a noncoherent trans-mission is optimal. However, with receiver feedback regardingthe channel, estimating and feeding back a few bits can appre-ciably outperform a noncoherent system [2], especially if thecoherence time of the channel is sufficiently large.

Second, the proposed formulation directly captures the finiterate nature of the feedback link along with the available feed-back channel capacity. In the process, it also acts as a model forseveral practical communication systems, which do have feed-back mechanisms to assist the transmitter [3]. Third, for a largenumber of antennas, systems which use channel knowledge atthe transmitter are typically lower in decoder complexity thanthose which do not. For example, in our case, beamforming con-catenated with a scalar codebook (rank one signaling) requireslower decoding complexity than a typical space–time code (fullrank codebook).

A similar quantized feedback model was first analyzed in [4]to maximize average mutual information and average receivedsignal-to-noise ratio (SNR), for a multiple transmit and singlereceive antenna system. The use of Lloyd algorithm for quan-tization was suggested as a possible solution for the design of

1If the transmitter also performs rate control, then the outage can bemade arbitrarily small at the cost of increasing delay [1].

0018-9448/03$17.00 © 2003 IEEE

MUKKAVILLI et al.: ON BEAMFORMING WITH FINITE RATE FEEDBACK IN MULTIPLE-ANTENNA SYSTEMS 2563

Fig. 1. Performance gains of feedback-based schemes compared to space–time codes for four transmit antennas. The plot shows the predicted performance intheory when optimal codes are used with each of the transmission schemes. Space–time code refers to full rank transmission with independent and identicallydistributed (i.i.d.) Gaussian codes and beamforming refers to rank one transmission with Gaussian code. Power control refers to full rank space–time code withtemporal power control while the last case refers to unit rank transmission with temporal power control.

Fig. 2. A feedback-based wireless system witht transmit antennas and a single receive antenna. The feedback channel is error free and has capacityB bits perframe

beamformers with quantized channel information at the trans-mitter. While the approach of [4] gives useful results in theasymptotics of the number of transmit antennas, it does not pro-vide insight into the structure and performance of good beam-former codebooks with a finite number of antennas or feedbackbits. Design of quantized power control for multiple antennaswas considered in [2], with a full-rank code (like space–timecode). Analytical and numerical tools were used to design thequantizers to minimize outage and it was shown that substantialgains in outage can be obtained by employing a few bits of feed-back for power control. The problem of designing beamformersto minimize the pairwise error probability of codewords withcertain structure imposed on the feedback channel was also ad-dressed in [5], [6].

In this paper, we consider a multiple transmit antenna systemalong with a single receive antenna. The channel is assumed tobe quasi-static in time and, hence, outage probability can beused as the relevant performance metric [7]. We assume thatthe channel realization is known perfectly at the receiver. Even

though this is not possible in practice, such an assumption al-lows us to focus on the effects of finite rate feedback on thesystem performance. We assume the existence of a feedbackchannel from the receiver to the transmitter as shown in Fig. 2.The capacity of the feedback channel is finite and limited to

bits per frame. Further, the feedback channel is assumed tobe free from errors. This feedback model is especially appro-priate for a frequency-division duplex (FDD) system where thechannels for the uplink and the downlink are different, so thattraining in the reverse direction is not possible. With the feed-back channel model under consideration, the transmitter knowl-edge comprises of the channel statistics along with limited in-formation ( bits) about the channel realization made availableby the feedback channel.

The design of the feedback channel is tightly coupledwith the transmission strategy. In this paper, the transmissionstrategy is fixed to be beamforming. Note that beamformingis optimal for a multiple-antenna system with a single receiveantenna when the channel is known perfectly at the transmitter.2


Further, beamforming with quantized feedback is a practicaltransmission scheme and is supported in the third-generationcellular standards employing two transmit antennas alongwith two and four bits of feedback [3]. We do not incorporatetemporal power and rate control in our analysis. The currentanalysis would be applicable for a system with short-termpower constraints where the power saved in one frame cannotbe used in another. It should be noted that additional gainsin outage can be obtained by channel adaptive rate controlmechanisms [1], [10].

The contributions of this paper are as follows.

1) Performance bounds on beamforming: We derive auniversal lower bound on the outage probability of anarbitrary beamformer with finite feedback bits, hencefinite number of beamforming vectors. The derived lowerbound is valid for all SNR and any number of transmitantennas. The bound is found to be fairly tight in allregimes of SNR, transmit antennas, and feedback bits,and is shown to be asymptotically tight in the number ofbeamforming vectors (or equivalently feedback bits). Weshow that the outage probability of beamforming withquantized feedback approaches the outage probability ofbeamforming with perfect channel feedback exponen-tially as , where is the number of transmitantennas and is the number of feedback bits per frame.Our analysis also allows us to define a distortion measurefor finite-rate feedback, based on the resulting outageprobability, which behaves similarly to the rate distortionmeasure in the theory of source coding.

2) Construction of beamformers: We derive the structureof the quantizer which minimizes the outage probabilitywith finite-rate feedback. We show that near-optimalbeamformer codebooks can be constructed by mini-mizing the maximum inner product between any twobeamforming vectors in the codebook. This designcriterion is similar to the design criterion of signalconstellations for Gaussian channels, which maximizesthe minimum Euclidean distance between any two signalpoints in the constellation.

The proposed beamformer design criterion turnsout to be equivalent to the design criterion for unitaryspace–time constellation. The equivalence between thetwo criteria holds with the following substitutions: a)coherence time for unitary space–time constellation is setequal to the number of transmit antennas for the beam-former design problem and b) the number of transmitantennas for unitary space–time constellation is set equalto one. Alternately, the beamformer design problem canbe interpreted as the problem of packing two-dimen-sional subspaces in -dimensional real Grassmannianmanifold with chordal distance as the metric, whereisthe number of transmit antennas.

It is interesting to note that a different approach, viz., SNRmaximization, also leads to the same design criterion for beam-

2The conditions for the optimality of beamforming with partial channel infor-mation at the transmitter is investigated in [4], [8], [9] with mutual informationas the performance metric.

former codebooks with single receive antenna [11]–[13]. How-ever, the two approaches lead to a different design criterion withmultiple receive antennas; while SNR maximization always re-sults in unit rank beamforming [5], [11], outage minimizationresults in generalized beamforming with spatial water-filling[14].

The rest of the paper is organized as follows. The systemmodel is described in Section II. In Section III, the beam-forming methodology is described and a universal lower boundon outage probability of beamforming with finite feedback isderived. The construction of good beamformer codebooks isdiscussed in Section IV while Section V concludes the paper.

II. SYSTEM MODEL

Consider a wireless communication system withtransmit antennas and a single receive antenna. We as-sume a narrow-band channel so that the channel is frequencynonselective. Let be the channelvector from the transmitter to the receiver. We assume thatthe entries in are i.i.d. and for each, is a circularlysymmetric complex Gaussian random variable with zero meanand unit variance per complex dimension. Letbe thevector transmitted along the transmit antennas at a giveninstant, where and represents thetransmission from the th antenna. The average power oftransmission is limited to so that , whererepresents the expectation operation while the superscriptrepresents conjugate transposition of a matrix. Ifdenotesthe corresponding signal received at the receiver whenistransmitted, then we have

(1)

where is the additive noise which is assumed to be circularlysymmetric complex Gaussian random variable with zero meanand unit variance.

With the above model for the channel statistics, the distribu-tion of conditioned on norm is given by

(2)

where is the matrix determinant and is the identitymatrix. Hence, conditioned on, is uniformly distributed onthe surface of a-dimensional complex hypersphere of radius

centered at origin.The channel statistics are assumed to be quasi-static in time

i.e., the channel realization is independent across differentframes of transmission while the channel does not vary withina given frame. The channel is assumed to be known perfectlyat the receiver while the transmitter hasbits of informationabout the channel provided by the feedback channel (see Fig.2). The transmission strategy is beamforming without powercontrol. The transmitted vector can be written as ,where is the beamforming vector while is the informa-tion-bearing scalar. The transmitter chooses the appropriatebeamforming vector from the set of beamforming vectors

based on the feedback bits.


III. PERFORMANCELIMITS ON BEAMFORMING WITH

QUANTIZED FEEDBACK

In this section, we introduce outage probability as our per-formance metric and explain the methodology of beamformingwith quantized channel information at the transmitter. We thenderive a lower bound on outage probability of beamformingwith finite rate feedback and demonstrate the tightness of thebound in several scenarios. The bound is also useful in char-acterizing the improvement of outage with increasing feedbackbits.

A. Performance Metric: Outage Probability

In this work, the channel is assumed to be quasi-static and theergodic capacity of such a channel is zero [14], since for anydesired transmission rate, there is a finite probability of ex-periencing a channel realization that cannot support the rate.The concept of outage probability was introduced for delay-lim-ited applications by Ozarowet al. [7] where ergodic capacity isnot applicable. Outage is defined as the event that the channelrealization cannot support the desired rate of transmission, theoutage probability measures the frequency of such an event. Fur-ther, outage probability gives a lower bound for frame error ratesin practical coded systems and hence yields useful bounds onsystem performance. We use outage probability as the perfor-mance metric for the design and analysis of the beamformingschemes in this paper.

B. Outage Probability with Beamforming

The channel realization is known perfectly at the receiver. Thereceiver quantizes the channel vector and transmits the informa-tion about the quantized channel vector to the transmitter overthe feedback channel. Since the rate of feedback channel is lim-ited to bits per frame, the set of quantized vectors can com-prise of at most vectors. Further, we can assume thatthe set of quantized vectors at the receiver is the same as the setof beamforming vectors at the transmitter.

Let be an arbitrary beamformercodebook, where along with for each

. Note that is also the set of quantizers used atthe receiver for quantizing the channel vector. The beamformingvector will be chosen for transmission when the feedback in-formation corresponds to. For each , let be theset of all channel realizations which will be quantized toand,hence, the beamforming vector will be chosen for transmis-sion whenever . We will address the problem of quan-tizing the channel vector to one of the vectors in the codebook

in the next section. The received signal when is givenby (using for beamforming in (1))

(3)

where : and is given by . Sincethis is a Gaussian channel for a given beamforming vectorand channel realization, the mutual information conditionedon and is maximized by choosing Gaussian distribution for

. Hence,

(4)

Also, note that the event of is the same as the event ofchoosing the beamforming vector for transmission. Hence,the outage probability conditioned on choosingfor transmis-sion is given by

(5)

Thus, the outage probability while using the beamformer code-book is given by the total probability rule

(6)

where is the probability of choosing for transmissionand .

C. Quantization Metric

From the previous section, we observe that the outage proba-bility depends on the inner product of the channel vectorandthe beamforming vector . It turns out, as we show in the fol-lowing lemma, the quantizer which minimizes the outage proba-bility is related to the inner product and is the one which choosesthe quantization vector with the smallest inner product norm.It should also be pointed out that the beamformer codebookis fixed for the analysis in this section while we address theproblem of assigning the channel realizationto the vectorsin . We address the problem of designing good beamformercodebooks in Section IV.

Lemma 1 (Optimal quantizer):For any beamformer code-book , the outage probability is min-imized by choosing for each channel realization, the vector

which maximizes .Proof: Let be any quantizer such that: . Fur-

ther, let be a quantizer such that

By definition, we have

(7)

Let be the outage probability resulting from usingthe beamformer codebookalong with the quantizer , while

is the resulting outage probability of usingalongwith . It suffices to show that .Define the indicator function : such that

ifotherwise.

(8)

Equivalently, we have

ifotherwise.

(9)


Similarly, define using the quantizer as

ifotherwise.

(10)

Then, from (7), we have

(11)

Finally, the outage probability of the codebookwith the quan-tizer is given by

using (11)

(12)

where stands for expectation operation with respect to thedistribution of .

Hence, for each realization, choosing such thatis maximized, minimizes the outage probability of the

beamformer . In the sequel, the quantizer used will be un-less otherwise mentioned. With this quantizer, the Voronoi re-gion corresponding to a beamforming vector and channelnorm is given by

(13)

Since the beamforming vectors have unit norm, the quantizationmetric is independent of the norm ofand depends only on the“direction” of . Hence, if a particular belongs to , thenfor any , belongs to . Letdenote the surface area (sometimes referred to as content[15]) of .

We will now characterize the subsets of which can sup-port a given rate of transmission (no-outage regions) and thosethat cannot support the given rate (outage regions). We willshow that the no-outage regions in particular have a nice struc-ture which permits the computation of their measure, thus en-abling the evaluation of the complement of the required outageprobability.

D. No-Outage Regions

Consider the beamformer codebookcomprising of thebeamforming vectors . Let

be the corresponding Voronoi regions on the surface ofthe hypersphere , as described in (13). Our beam-forming methodology is such that is chosen for transmissionif and only if . By Lemma 1, this minimizes outageprobability for beamformer . We will first characterize theevent where the transmission along is not in error in orderto evaluate the outage probability of the codebook. Let

be the subset of such that for all , thedesired transmission ratecan be supported (without outage).We will refer to the regions for different and asno-outage regionsin the sequel. Recall from (5) that outagedoes not occur when is chosen for transmission if and onlyif , where . Hence, the no-outageregion corresponding to is given by

(14)

Finally, the probability of successful transmission, denoted by, when is chosen for transmission is given by

(15)

For small values of , i.e., , all the channel realizationswill be in outage for all . This is true irrespective of

the number of beamforming vectors, i.e., there is a sphere ofradius centered at origin, which is always in outage. How-ever, as we will show later, the volume of this sphere goes tozero as the number of dimensions (or equivalently the numberof transmit antennas) goes to infinity.3 . As a result of the pre-ceding observation, we have for . As in-creases beyond , nonempty no-outage regions start formingaround each of the beamforming vectors. We will useto denote the surface area of .

E. Outage Probability Conditioned on

The probability of successful transmission with the beam-former transmitting at rate , conditioned on , canbe calculated as

(16)

where is the probability of choosing thebeamforming vector for a channel realization with thesquared norm given by, while isthe probability that the resulting transmission along the vector

will not result in outage. Since is distributed uniformly fora given norm from (2), is given by the fractionof the total surface area of the-dimensional hypersphere

occupied by , i.e.,

(17)

where denotes the surface area of a-dimensional com-plex hypersphere of radius . The surface area of a complexhypersphere is given by the following lemma.

Lemma 2: The surface area of a-dimensional complex hy-persphere of radius is given by

(18)

Proof: See Appendix I.

The uniform distribution of for a given norm also simplifiesthe calculation of . Noting that a)conditioned on , the event of choosing for trans-

3One of the outcomes of this observation is that for any finite transmissionrateR, the probability of outage can be driven to zero by providing enoughtransmit antennas and feedback bits.


mission is the same as the event and b) when ischosen, outage does not occur for , we have

since

(19)

From (19), we see that the distribution ofremains uniformconditioned on .

Substituting (17) and (19) in (16), we have

(20)

F. Spherical Caps on Hyperspheres

As (20) suggests, we need to evaluate the area of the no-outage regions in order to compute the probability ofoutage for a particular beamformer. In general, it is not easy tocompute the area of the no-outage regions. For this reason, wedefinespherical capson the surface of the hypersphere, whichclosely approximate the no-outage regions. These spherical capspossess enough structure to allow area computation, which inturn yields a good approximation for the area of the no-outageregions.

For each and , consider the regionsgiven by

(21)

is the portion of the hypersphere cut off by thesubspace . For the sake of illustration, if we let

, then will be a spherical cap on the surface of thesphere as shown in Fig. 3. Following this intuition, we will referto as a spherical cap in the general-dimensional complexcase. Note that is congruent to for all and , i.e., if

is rotated to coincide with , then will coincide with.

From the definitions given in (14) and (21), it is clear that. On the other hand, it can be shown that the

union of the no-outage regions actually equals the unionof the spherical caps . This is an interesting property sincethe no-outage regions are always nonoverlapping while thespherical caps will overlap for large. Hence, the no-outageregions partition the region formed by the union of the spher-ical caps into nonoverlapping regions leading to the followinglemma.

Lemma 3: For all , .Proof: For , both and are empty sets

for all and hence holds trivially.

Fig. 3. The nature of the regionS ( ) for the illustrative case of . Note thatS ( ) is a spherical cap denoted by the shaded region.

Suppose . Then, from the definition ofand , we know that for each so that

. Now consider .Then, for some . From the definition of , it fol-lows that . Hence,so that for some .

Therefore, and .Hence, the result follows.

We can rewrite (20) in the light of the above lemma. Sinceare nonoverlapping for all, we have

using Lemma 3 (22)

Hence, (20), i.e., the probability of successful transmission, fora given channel norm, can also written as

(23)

We will now relate to the area of individualspherical caps for different values of . We noted ear-lier that are all congruent so that is the same forall and, hence, it suffices to compute the area of . To sim-plify the computation of , we can assume, without lossof generality, that coincides with one of the coordinate axes,i.e., . If is different from this, thenby multiplying the vectors in with a suitable unitary matrix,

can be forced to coincide with one of the coordinate axes.


The performance of the beamformer codebookis invariant ofsuch rotations since the distribution ofis symmetric about theorigin.

With our choice for , we have

and its area is given by the following lemma.

Lemma 4: The surface area of the spherical cap formed bythe intersection of the subspace and the complexhypersphere is given by

(24)

Proof: See Appendix II.

For all those for which no two overlap, Lemma 4characterizes the exact area of the region corresponding to suc-cessful transmission in (23). If overlaps for some

and , then we can arrive at an upper bound on the area ofthe region corresponding to successful transmission via Lemma4. We defineoverlap radiusof a beamformer as the channelnorm at which this transition takes place.

Definition 1: For any beamformer codebook, theoverlapradiusis defined as the largest channel norm squared for whichno two spherical caps overlap. It is denoted by .

If

then . Clearly, . In the sequel, wewill use the short notation to denote whenever there isno ambiguity about the beamformer codebook under discussion.The overlap radius is a key parameter which will be used inthe derivation of the lower bound on outage probability as wellas in the design of beamformer codebooks.

The preceding observations can be used to bound the numer-ator in (23). In particular, when , the spherical caps areall empty sets so that

(25)

When , the spherical caps form disjoint sets. Inthis case, we have

(26)

Finally, when , the spherical caps are no longer disjointand hence we have

(27)

Substituting (18) and the above set of relations in (23), we get

(28)

Let be such that , so that

(29)

Considering that we will eventually derive a lower bound onoutage probability of the beamformer, we can rewrite the setof inequalities in (28) as

(30)

Note that the set of relations given in (28) are still useful andwe will revisit them while discussing the design criterion forbeamformers.

G. Lower Bound on Outage Probability With Feedback

In this subsection, we will derive a lower bound on the outageprobability of beamformer codebooks with finite number ofbeamforming vectors. It should be noted that the lower boundderived holds for an arbitrary beamformer codebook. Thus,the quality of any beamformer codebookcan be determinedby measuring its performance against the lower bound.

We will make use of the distribution of along with condi-tional distribution of successful transmission given by (30) toarrive at the lower bound on the probability of outage for thebeamformer codebook. Note that , where each

is distributed as complex proper Gaussian with zero mean andunit variance. Therefore, is given by the sum of the squaresof Gaussian random variables with zero mean and varianceequal to . Hence, has a central chi-square distribution with

degrees of freedom. The probability density function (pdf) ofis given by [16]

(31)

The cumulative distribution function (cdf) of is given by [16]

(32)

We can now integrate the conditional distribution of suc-cessful transmission given in (30) using the distribution ofgiven in (31) to arrive at the complement of the probability ofoutage of the beamformer. The first main result of this paper isgiven in the following theorem.

Theorem 1 (Outage Lower Bound):Consider a wirelesssystem with transmit antennas along with a single receiveantenna transmitting at rate bits/s/Hz and SNR over aquasi-static Rayleigh-fading channel. The outage probability


of such a system employing a beamformer codebook withbeamforming vectors (corresponding to bits of

feedback per frame) is bounded below as

(33)

where

and (34)

Proof: Let be an arbitrary beamformer codebook withbeamforming vectors. Let be the probability of

no-outage when is used for beamforming. We have

Using the upper bound on from (30)and the distribution of in (31), we get

(35)

The second term in (35) can be simplified as

(36)

where is given by (32). The first term in (35) can be sim-plified as follows. We have

(37)

(38)

where we have used the substitution in (37). Using(36) and (38) in (35) and using ,we get

The lower bound given above is valid at all SNR and for an ar-bitrary beamformer comprising of beamforming vectors. Wehave observed that the bound is fairly tight for various cases of

and which were investigated (see Section IV-B). Notethat even though the lower bound is valid when (i.e., fewerbeamforming vectors than the transmit antennas), it is very loosein that case. In particular, with the beamforming methodologyusedinthispaper, therateofdecayofoutageprobabilitywithSNRforbeamformingwillbe while thebounddecaysat therate of as we show in the next section. In the next section, wealso show that the bound is asymptotically tight in the number ofbeamforming vectors. The bound will be used to explicitly char-acterize the behavior of outage probability of good beamformersas a function of the number of transmit antennas and the numberof beamforming vectors. Finally, the procedure used in arrivingat the bound will also prove useful in characterizing near-optimalbeamformer codebooks and will be used to derive a design crite-rion for good beamformer codebooks.

We will now evaluate the outage probability of beamformingwhen perfect channel information is available at the transmitter.Note that thiscanbetreatedasthecasewhenismadevery large.The case of perfect channel information at the transmitter servesasausefulreferenceinevaluatingtheperformanceofbeamformercodebooks with finite number of beamforming vectors.

1) Perfect Feedback Bound:Suppose that the channel isknown perfectly at the transmitter and the receiver. It wasshown in [17] that the optimal scheme for minimizing outageconsists of a Gaussian code followed by a beamformer. Theoptimal beamformer is given by a unit vector in the directionof , where is the channel realization. The received signal inthis case is given by

(39)

The mutual information with Gaussian codes is given by

(40)

and the outage probability for a rateis given by

(41)

where we have used the central chi-squared distribution ofgiven in (31) and also substituted for .

The bound given in (33) can be used to understand the effectof increasing the capacity of feedback channel. Fig. 4 comparesthe performance predicted by the lower bound in (33) with theoutage probability of beamforming with perfect feedback givenin (41), for the case of four transmit antennas and a single re-ceive antenna at a fixed SNR of 10 dB. It can be seen that theperformance gap between 2-bit quantized feedback and perfectfeedback is significant. But the gap is considerably reduced byabout 8 bits of feedback. The bound suggests that about 8 bits offeedback is sufficient to perform very close to the case of per-fect channel information with four transmit antennas and furtherincrease in the number of feedback bits will result in very littleimprovement in the outage performance.

We will now illustrate the behavior of the bound as a functionof the number of beamforming vectors and SNR.

2) Asymptotic Tightness in : In this subsection, we willinvestigate the effect of increasing on the lower bound on


Fig. 4. Improvement in outage probability with increasing feedback bits for four transmit antennas with a single receive antenna at SNR= 10 dB, as predictedby Theorem 1.

outage probability given in (33). Note that perfect channel in-formation is the limiting case of quantized channel informationas grows without bound. In particular, we will show that thelower bound in (33) converges to (41) for large.

Proposition 1 (Asymptotic Tightness in): The lower boundon the outage probability of beamforming with finite rate feed-back given by Theorem 1 is asymptotically tight in the numberof beamforming vectors .

Proof: Let Then,, so that the lower bound on outage in (33) can be

written as

(42)

Using the series expansion of, the right-hand side of (42) canbe written as

(43)

The second and third terms in (43) can be simplified as

(44)

Now consider the term in (44). Substituting for, we have

for large

for (45)

Hence, the expression in (44) goes tofor large . Finally,noting that , for large , we see that (43) can be sim-plified to give the lower bound asymptotically as

(46)

which is the same as the outage probability of beamforming withperfect feedback given in (41).

Therefore, the lower bound given by Theorem 1 is asymptot-ically tight as the number of feedback bits gets large. This alsosuggests that, for sufficient number of feedback bits, the lowerbound can be used as a good approximation to the outage proba-bility of “good” beamformers. We will discuss the constructionof good beamformers whose outage performance will be closeto that predicted by Theorem 1 in Section IV.

3) High-SNR Approximation:In this subsection, we will in-vestigate the behavior of the outage probability of good beam-formers with increasing SNR. From the preceding discussionand Theorem 1, the outage probability of good beamformers,for sufficiently large , can be approximated by

(47)

(48)

where we have used the series expansion oftwice in (47).Note that so that as increases, goes to zero.


Substituting for in terms of in (48) and ignoring the higherpowers of , we get

(49)

We can do a similar high-SNR analysis of outage probabilityof beamforming with perfect feedback. In particular, from (41),we get

for large (50)

where we have neglected the higher order terms in (50).The high-SNR approximation of outage probability of beam-

forming with finite feedback given by (49) can be reinterpretedin the light of (50). In particular, (49) consists of two terms,where the first term is the same as the outage probability ofbeamforming with perfect channel information as given by (50).The second term in (49) accounts for the finite nature of the feed-back and captures the loss incurred by finite rate feedback. Theloss in outage performance with finite rate feedback comparedto perfect feedback is characterized by

(51)

Note that and decays to with increasing for afixed . In the next section, we define a distortion measure basedon the above high-SNR analysis of outage probability which hasa close resemblance to the rate distortion function in the theoryof source coding.

4) Distortion Measure for Quantization:We have seen thatthe problem of feedback design for beamforming is nothing butdesigning a quantizer for the channel with finite number of bits.In the traditional source coding problem, the objective of a quan-tizer is to describe the source as well as possible (i.e., with least“distortion”) with the available bits. On the other hand, the goalof channel feedback design is to minimize the outage proba-bility, which does not necessarily arise from the best descriptionof the channel in the mean squared sense. For the description ofa Gaussian source (such as the channel vectorin our case), therate distortion function [18] characterizes the difference in per-formance of a finite bit quantizer and an infinite bit quantizer;where infinite bits lead to zero mean squared distortion. Moti-vated by the rate distortion function, we define a distortion mea-sure for the feedback design problem based on the outage prob-ability. In particular, we measure the fractional loss in outageperformance of finite bit channel quantizer compared to perfectchannel information at the transmitter. Note that, in the absence

of channel adaptive rate control, perfect channel information atthe transmitter does not necessarily result in zero outage.

Let denote the outage probability of beam-forming with perfect channel information at the transmitter, forlarge . Also, let denote the outage probability ofa good beamformer with bits of feedback, for large . Wedefine a measure of distortion, , as follows.

Definition 2: The distortion measure is defined as therelative loss in outage performance of the best beamformer with

bits of feedback when compared to a beamformer designedwith perfect channel information at the transmitter, for largeSNR.

Using the notation developed earlier, we have

(52)

Substituting (49) and (50) in (52), we get

for sufficiently large

using (53)

Hence, the performance loss of quantized beamforming com-pared to perfect channel feedback decays exponentially with thenumber of feedback bits for sufficiently large and SNR.Fig. 5 plots as a function of the number of feed-back bits for different number of transmit antennas. The slopeof the curve is inversely proportional to the number oftransmit antennas. This means that the marginal performancegain obtained by the addition of a feedback bit reduces as weincrease the number of transmit antennas. This is not surprisingsince the number of channel coefficients to be quantized in-creases with the transmit antennas. It is interesting to note thatwith the mean squared error as distortion metric, betweenand , the classical rate distortion theory predicts that

. Hence, our distortion measure based on the outage proba-bility behaves similar to the distortion metric based on the meansquared error in , though not in .

Narulaet al. [4] have arrived at a similar result with receivedSNR as the performance metric. We wil recast their result in aform suitable for easy comparison with (53). If we defineas the fraction loss in SNR with quantized beamforming com-pared to a beamformer designed with perfect channel informa-tion, we have from [4], after suitable manipulations

(54)

Hence, the SNR metric also displays an exponential decay withrespect to the number of feedback bits similar to the outagemetric given by (53), though the behavior is different in.

We should also point out the different conditions under which(53) and (54) are attained. The distortion rate function


Fig. 5. Improvement in outage with feedback bits for two, four, six, and eight transmit antennas given by (53). The slope of the fraction loss plot is inverselyproportional to the number of transmit antennas.

defined in this paper is valid for high SNR and large number ofbeamforming vectors. We do not need the number of transmitantennas to be large. On the other hand, the distortion ratefunction in (54) is derived using the mean squared distortionrate function of the classical rate distortion theory. Thebounds of the rate distortion theory can be approached bycoding over large blocks of source outputs (channel is thesource to be encoded in this problem) with independent realiza-tions. Since, quasi-static fading assumptions for the channel donot allow for independent realizations over time, Narulaet al.[4] resort to independent realizations over space (obtained vialarge number of transmit antennas). Hence, (53) requires highSNR while (54) requires large number of transmit antennas.

IV. CONSTRUCTION OFBEAMFORMERS

In the previous section, we presented a lower bound onthe outage probability of beamformer codebooks with a finitenumber of vectors. In this section, we will discuss the con-struction of beamformer codebooks approaching the bound intheir outage performance. We will see that the performanceof the beamformer codebooks thus constructed comes veryclose to the lower bound presented in the last section. Thedesign criterion for beamformer codebooks is also related tothe problem of designing unitary space–time codes for multipleantennas as well as the problem of subspace packing in realGrassmannian spaces.

A. Design Criterion

We can draw inferences about the structure of good beam-former codebooks from the proof methodology of the lowerbound discussed in the last section. Recall from Section III-Fthe probability of successful transmission with a beamformer

codebook , conditioned on the channel norm is given as

(55)

where was defined as the overlap radius of the beamformercodebook . For , the spherical caps do not overlap andhence the total area corresponding to successful transmissionsis given by the sum of the areas of individual spherical caps. Onthe other hand, for , some of the spherical caps overlapand, as a result, the area corresponding to the successful trans-missions is less than that of the sum of the individual sphericalcaps. This leads to the inequality for in (55). It is clearthat the performance of a beamformer can be improved by in-creasing its overlap radius so that the spherical caps do not inter-sect up to a larger radius. The lower bound in Theorem 1 can beachieved with equality if the beamformer is such that the spher-ical caps do not overlap till the entire surface area is covered fora certain . For such a beamformer, , where was de-fined such that

(56)

Hence, the design criterion for a good beamformer will be tomaximize the overlap radius . The design of good beam-formers is reduced to the following optimization problem:

(57)

We will now use the definitions of the spherical caps andin particular to calculate the overlap radius for a given beam-


former codebook. We will show that the overlap radius is in-versely proportional to the angle between the two closest vec-tors in the codebook, where the angles are defined via the innerproduct norm. Hence, the design criterion for good beamformercodebook seeks to maximize the angular distance between anytwo vectors in the codebook. In this respect, we will see that(57) is similar to the design criterion for signal constellationsfor Gaussian channels. In both cases, the goal is to maximize theminimum distance between any two members in the set. WhileEuclidean distance is the relevant metric for signal constellationdesign, the inner product norm serves the purpose of metric inthe case of beamformer codebook design. We show the calcula-tion of for a beamformer codebook in the following lemma.

Lemma 5 (Overlap Radius):For any beamformer codebook, we have

(58)

Proof: Let be any beamformercomprising of beamforming vectors, with foreach . Following the notation developed earlier, for a givenchannel norm , let and be the Voronoi regionand no-outage region corresponding tofor each . Further,let be the spherical cap around as described in SectionIII-E. Let be the smallest channel norm for whichand intersect. Then, it is clear that can be calculatedas . Hence, we will first evaluate fora given and and then calculate .

Consider the smallest channel norm at which in-tersects . By the definition of , if is a vector whichlies on the boundary of , then satisfies .Let be a vector which lies on the boundary of and .Also, let . Then, is the smallest value of whichsatisfies the following conditions:

(59)

(60)

(61)

(62)

where is a unit vector orthogonal to both and .Also, and are complex scalars.

On equating the left-hand side of (61) and (62), we get

(63)

(where we have ignored the less interesting case ofwhich happens when is a scalar multiple of . In this case,

we can drop one of the two vectors fromwithout affecting theoutage performance of).

Without loss of generality, let be real and . Sub-stituting this in (61), we get

(64)

Similarly, substituting for and in (59), we get

(65)

Now, substituting the value of from (65) in (64) and simpli-fying, we get

(66)

Finally, note that is the minimum value of which satis-fies this equation. To evaluate , we minimize with respectto and . Clearly, is minimized when (i.e., the spher-ical caps and first intersect in the subspace formedby and ). Let assume its minimum value at . Dif-ferentiating with respect to and setting it to zero at ,we get

(67)

where and is any integer.Evaluating the second derivative and setting it greater than

zero at (for a minima at ), we get

(68)

From (67) and (68), we see thatis minimized at ,for any integer . Set to give . Substituting thisvalue of in (66), we get

(69)

Finally, is given by

(70)

Hence, the overlap radius is a function of the two closest vec-tors in the beamformer. The design criterion for beamformers

, given in (57), is now equivalently given by thefollowing lemma.

Lemma 6 (Design Criterion):The design criterion for a goodbeamformer codebookcomprising of beamforming vectorsfor transmit antennas is given by

(71)

The norm of the inner product is inversely propor-tional to the angular “distance” betweenand . In particular,

attains a maximum value ofwhen is aligned with(corresponding to minimum angular distance), while the min-

imum value of is attained when is orthogonal to (the cor-responding maximum angular distance). Hence, the design crite-riongiven in (71)seeks tomaximize theangulardistancebetweenthe two closest vectors in the codebook. In this respect, the designcriterionissimilartothecriterionofmaximizingtheminimumEu-clideandistanceused for signal constellationdesign for Gaussianchannels.However,theEuclideandistancemetricismarkedlydif-ferent from the norm of the inner product distance.

It is worthwhile to note that for the special case, where thenumber of beamforming vectors equals the number of transmitantennas, the design criterion results in selection diversity.


Fig. 6. Simulated outage performance of the beamformers constructed for four transmit antennas for rateR = 2 bits/s/Hz. For the sake of comparison, the lowerbound on outage probability for each number of feedback bits given in Theorem 1 is also plotted.

Corollary 1: When , the design criterion in (71) leadsto selection diversity with .

Proof: For any beamformer codebook

where the equality occurs if and only if is orthogonal tofor all . When , we can achieve the equality bychoosing as an orthonormal basis of . For example, couldbe standard basis of given by ,where the occurs for in the th position. This is nothing butchoosing the best antenna for transmission, which is commonlyreferred to asselection diversity.

Hence, the design rule of (71) is consistent with our knowl-edge for bits of feedback information. This design rulecan also be used to verify the asymptotic tightness of the lowerbound, given in Theorem 1, in the number of beamforming vec-tors. We have already mentioned that the performance given bythe lower bound can be achieved if . Using (56) and(58), the lower bound will be achieved if

(72)

As grows, the right-hand side in the above equation goesto . Hence, the equality holds if

which will be true for any beamformer codebook. Hence, thelower bound is achievable by any nondegenerate beamformercodebook (i.e., , ) as the number of beamformingvectors grows large.

B. Simulation Results

We present the performance results of beamformers con-structed using the design criterion given in (71). We use acomputer-aided search process to design the beamformers. Inparticular, we use the function in the optimizationtoolbox of MATLAB, to arrive at the beamforming vectors.We provide simulation results for the outage probabilityperformance of beamforming vectors in a quasi-static channel.For the sake of comparison, we also give the lower bound onoutage probability from (33). The performance of constructedbeamformers for four transmit antennas with 2, 3, and 4bits of feedback is presented in Fig. 6 for the case of2 bits/s/Hz. Note that 2 bits of feedback with four transmitantennas results in the selection diversity scheme as discussedabove. From Fig. 6, we see that the outage probability is veryclose to the corresponding lower bound predicted by (33),thus reflecting the tightness of the bound. The tightness of thebound improves as the number of feedback bitsis increased.Further, the outage probability gets closer to the case of perfectchannel information as increases. We also observe that a fewbits of feedback can achieve most of the gains promised bybeamforming with perfect channel information.


C. Similarity With Unitary Space–Time Constellations

Unitary space–time constellations for multiple-antennasystems were introduced by Marzetta and Hochwald in [19]. Inparticular, it was shown that unitary space–time constellationsachieve the capacity of multiple antenna systems when thechannel information is not available at either the transmitteror the receiver. The design criterion for unitary space–timeconstellations to minimize the pairwise error probability wasgiven in [20]. A unitary space–time constellation consists ofsignals , where , where is theblock length of the code (less than or equal to the coherencetime of the channel), is the number of transmit antennas,and is the number of codewords in the constellation. Also,the unitarity condition implies that for each . Itwas shown that the design criterion for unitary space–timeconstellations, which minimize pairwise error probability, is tominimize , where is defined as

(73)

where the norm used above is a scaled Frobenius norm of amatrix, the scaling factor being given by in this case.

Recall that, for the finite feedback problem consideredin this paper, if the beamformer codebookcomprises of

, then the design criterion for good beam-formers is

(74)

where . The criterion (73) is equivalent to thecriterion for beamformers given in (74) if we set and

in (73). We see an equivalence between coherence time,in the unitary constellation design problem, and the number oftransmit antennas, in the beamforming design problem. Hence,the design of beamformers is equivalent to the problem of thedesign of unitary space–time constellations under these condi-tions.

Hochwald et al. [21] addressed the problem of imposingstructure on the unitary space–time constellations for easierencoding/decoding as well as to reduce the dimensionalityof the search space during the design process. The equiva-lence of good beamformers with good unitary space–timeconstellations established in this paper allows the use of thesystematic unitary space–time constellations designed in [21]as beamformer codebooks. We reproduce the construction ofsystematic unitary space–time codes from [21] to explain theequivalence with an example.

Consider the set of linear block codes defined by thegenerator matrix , whose elements are in , ring of integersmodulo- . The code represented by comprises of code-words given by , where is a vector withelements taken from . Thus, the size of the codebook is givenby . The codewords are mapped into signals by map-ping the integers in the codeword into components of a complexsignal via the transformation

(75)

Finally, the unitary space–time constellation is given by

(76)

Suppose we wish to design a beamformer comprising ofvectors for transmit antennas and a single

receive antenna. Then the equivalent problem of designingunitary space–time constellation reduces to designing a code-book of size with a single transmit antenna (i.e.,

) and coherence time given by . One of thecodes designed for this problem in [21] is characterized by

, , and .The performance of this beamformer is given in Fig. 7. The

rate of transmission is 2 bits/s/Hz. The corresponding per-formance predicted by the lower bound on outage is also givenin the figure. It can be seen that the performance of the con-structed beamformer comes very close (less than 0.05 dB away)to the corresponding lower bound on outage probability. Theperformance of (full rank) space–times codes, which do not re-quire any feedback information, is also given in the plots forreference. The performance of the space–time code was ob-tained analytically assuming an optimal code for the case ofeight transmit antennas [14]. Note that there is a gain of about 4dB with 4 bits of feedback compared to the space–time codes.This example shows that the systematic unitary space–time con-stellations serve as very good beamformers. Some more con-structions of the beamformers for transmit antennas canbe obtained from [21] for the cases of 64, 133, 256, 529,1296, 2209 vectors.

D. Packings in Grassmannian Spaces

We conclude the discussion on the construction of beam-former codebooks by showing similarities to the problem ofpackings in Grassmannian spaces, which was discussed in detailby Conwayet al.in [22]. The similarity could potentially lead tonew construction methods for beamformer codebooks. Considerthe vector . We can write , whereand are the real and the imaginary parts of, respectively.Now consider the two-dimensional subspaceof formedby the span of the columns of the matrix given by

(77)

A measure of distance, called thechordal distance, between thesubspaces and is given by

(78)

where and are the principal angles betweenand [23].It can be shown that

(see Appendix III), so that the design criterion in (71) can berestated as

(79)


Fig. 7. Performance of systematic unitary space–time constellation as a beamformer. The beamformer consists of 16 vectors (4 bits of feedback) for eight transmitantennas transmitting at 2 bits/s/Hz. The lower bound on outage probability as well as the performances of beamformer with perfect channel information and (fullrank) space–time codes are given for comparison.

where is a set of cardinality , consisting of two-dimensionalsubspaces of formed by matrices of the nature given by (77).This is the packing problem discussed in [22].

V. DISCUSSION ANDCONCLUSION

In this paper, we have provided a geometrical framework forthe design and analysis of beamformers for multiple-antennasystems. The framework was used to derive fundamental boundson performance as well as a design criterion for good beam-former codebooks.

Our framework of using finite number of bits to quantizethe channel vector naturally lends itself to rate-distortion-likeanalysis. The key difference from conventional rate-distortiontheory is that we are not interested in describing the channelvector accurately, but only in a description which leads to im-proved transmission efficiency in the forward channel. Thus, aquantizer which minimizes an norm (like in most practicalquantizer designs) to describe the channel isnot well suited tominimize outage performance of a beamformer-based system(see Lemma 1). But, even though our metric of optimization isdifferent from widespread quantization metrics, some of our re-sults have a rate-distortion-like flavor. We highlight two of ourresults which show similarities to conventional source coding: a)the exponential decay of the new distortion function (see (53)),much like in the source coding of Gaussian sources [18] and b)the equivalence of our quantization problem (or the beamformerdesign problem) with a known channel coding problem.

The high-SNR approximation of the outage probability (usedto derive the fractional loss from perfect information system)has behavior similar to the rate-distortion function of theGaussian sources. To achieve the rate-distortion bound requiresvector quantization using increasingly large blocks of input.But the fractional loss in outage result is asymptotic in SNR andnot the length of observation. In fact, the length of observation

is always one for beamformer design, since the channel infor-mation regarding the current block is useful for transmissionin the current block only and has to be thus received causally.The fact that we cannot impose asymptotics in block length toquantize the channel realizations is the major reason for thedifficulty in obtaining exact performance characterizations. Inthe light of the above comments, it is satisfying to see that thecurrent problem lends a nice geometrical structure and admitsa relatively tight finite block analysis.

It is well known that the coding for Gaussian channelsis dual to the problem of quantizing a Gaussian source withmean-squared distortion metric. In the same spirit, we show thatthe problem of source quantization (beamformer design withloss in outage probability as distortion metric) is closely relatedto noncoherent code design problem [20] for the same vectorchannel. This strong equivalence, though intuitively appealingonce known, came as a surprise to us. In the quantizationproblem for beamforming, the transmitter is only interestedin the direction of the channel vector and not in its channelnorm. Thus, all the relevant information regarding the sourcelies in its directional information. As a dual, in noncoherentcommunications, unitary space–time codes transmit alongdifferent directions (along different subspaces) and place noinformation in the magnitude of the signal.

Many interesting problems in beamforming remain to be ex-plored. We have addressed the problem of beamforming with asingle receive antenna only. With multiple receive antennas, wewill need to consider the multiple directions for transmission(given by the eigenvectors) along with the spatial power alloca-tion among these directions. It can be shown that beamformingalong multiple directions will result in additional gains overspace–time codes only when it is coupled with spatial powerallocation. Hence, multiple receive antennas present a differentset of problems which do not arise with a single receive antenna.


In this paper, we have addressed the design of feedback forbeamforming only. Power control mechanism can also resultin huge improvements in outage performance [2]. Joint designof power control and beamforming is an interesting problemwhich needs further investigation. It would also be interestingto see the conditions under which temporal power control andbeamforming can be decoupled with quantized channel infor-mation at the transmitter. Note that the optimal scheme with per-fect channel information at the transmitter does in fact decouplebeamforming and power control at the transmitter [17]. Finally,we addressed the case of quantized feedback which is appli-cable to an FDD system. The framework presented in this paperis general enough to extend to the case of noisy channel esti-mates also which is applicable to a time-division duplex (TDD)system.

APPENDIX IPROOF OFLEMMA 2

We will use the methodology given in [15] for the calculationof the surface area of a-dimensional real hypersphere to calcu-late the surface area of-dimensional complex hypersphere. Wewill first calculate the volume of the complex sphere ,where . Let , where for

. Further, for each, let , i.e., thereal and the imaginary parts of are given by and , re-spectively. Now consider the transformation of variables givenby

(80)

so that for each .

Proposition 2: The Jacobian for the transformation in (80) is.

Proof: The Jacobian of the transformation given in (80) isgiven by where

(81)

The determinant of this block-diagonal matrix can be shownequal to . On evaluation, we find thatis given by and, hence, the proposition follows.

Hence, the volume elements are related by

(82)

The volume of the hypersphere is given by

(83)

where we have integrated over the-variables in(83). Now consider another transformation given by

(84)

where we have used the notation and ,for . Note that goes only from tosince for each . Further, goes from to . The Jacobianof this transformation is given by [15]

(85)

Substituting (84) and (85) in (83), we get after simplification

(86)

(87)

Finally, the surface area is obtained by differentiating thevolume of the sphere. Hence,

(88)

APPENDIX IIPROOF OFLEMMA 4

Let denote the area of the spherical cap formed bythe intersection of the subspace and the -dimensionalcomplex hypersphere given by . We will use the sametechnique as in the proof of Lemma 2, where we first calcu-late the volume of a solid object and treat the surface area as adifferential element of the volume at the required radius. Nowconsider the region formed by the intersection of the subspace

with the interior of the hypersphere given by .In terms of the transformation given in (80), this region is givenby

The volume of this region, denoted by , is given by theintegral in (83) where the limits of are now different for each


. In particular, we have

(89)

Note that the multiple integral in brackets in (89) is nothing butthe volume of -dimensional complex hypersphere of radius

with a scaling factor of , which is given by(87). On substitution, we get

where

(90)

Finally, the surface area of the spherical cap is obtained by dif-ferentiating given by

(91)

APPENDIX IIIRELATIONSHIP BETWEEN CHORDAL DISTANCE AND

NORM OF INNER PRODUCT

In this appendix, we will show that

(92)

Following the notation introduced in Section IV-D, we see that

and (93)

form an orthonormal basis of the subspace. Similarly, we candefine and corresponding to . Define the matrix as

(94)

Then, the principal angles and between and are suchthat and , where and are thesingular values of [23].

Noting that and , we have

(95)

where the last equation follows from

Finally, we have

(96)

ACKNOWLEDGMENT

The authors wish to thank Mohammad Jaber Borran forhelpful discussions on unitary space–time constellations.

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