Power-Controlled Feedback and Training for Two-way MIMO Channels

Transcription

1 1 Power-Controlled Feedback and Training for Two-way MIMO Channels Vaneet Aggarwal, and Ashutosh Sabharwal, Senior Meber, IEEE arxiv: v3 [csit] 9 Jan 010 Abstract Most counication systes use soe for of feedback, often related to channel state inforation The coon odels used in analyses either assue perfect channel state inforation at the receiver and/or noiseless state feedback links However, in practical systes, neither is the channel estiate known perfectly at the receiver and nor is the feedback link perfect In this paper, we study the achievable diversity ultiplexing tradeoff using iid Gaussian codebooks, considering the errors in training the receiver and the errors in the feedback link for FDD systes, where the forward and the feedback are independent MIMO channels Our key result is that the axiu diversity order with onebit of feedback inforation is identical to systes with ore feedback bits Thus, asyptotically in SNR, ore than one bit of feedback does not iprove the syste perforance at constant rates Furtherore, the one-bit diversity-ultiplexing perforance is identical to the syste which has perfect channel state inforation at the receiver along with noiseless feedback link This achievability uses novel concepts of power controlled feedback and training, which naturally surface when we consider iperfect channel estiation and noisy feedback links In the process of evaluating the proposed training and feedback protocols, we find an asyptotic expression for the joint probability of the SNR exponents of eigenvalues of the actual channel and the estiated channel which ay be of independent interest Index Ters Channel state inforation, diversity ultiplexing tradeoff, feedback, ultiple access channel, outage probability, power-controlled, training I INTRODUCTION Channel state inforation at the transitter has been well established to iprove counication perforance, easured either as increased capacity see eg [1 6], iproved diversity-ultiplexing perforance [7 18] or higher signalto-noise ratio at the receiver [, 19, 0] aong any possible etrics Several of the aforeentioned works have considered the ipact of incoplete channel knowledge at the transitter, by considering quantized channel inforation at the transitter which can be visualized to be ade available by the receiver through a noiseless finite-capacity feedback link While the odel and subsequent analysis clearly shows that reduced channel inforation at the transitter can lead to significant perforance gains due to channel knowledge, a key requireent is that the receiver knows what the transitter knows even if there is an error in feedback link In practice, the counicating nodes are distributed and have no way of aligning their channel knowledge perfectly V Aggarwal is with the Departent of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA eail: vaggarwa@princetonedu A Sabharwal is with the Departent of Electrical and Coputer Engineering, Rice University, Houston, TX, 77005, USA eail: ashu@riceedu In this paper, we systeatically analyze the ipact of isatch in channel knowledge at the transitter and receiver For clarity of presentation, we will largely focus on a singleuser point-to-point link one transitter and one receiver and only at the end of the paper, extend the results to the case of ultiple-access channels any transitters and one receiver The key departure fro prior work is that we explicitly odel both the forward and feedback links as fading wireless channels A little thought iediately shows that all practical wireless networks have two-way counication links, that is, the nodes are transceivers such that all the received and transitted packets control, data, feedback travel over noisy fading channels The two-way odel to analyze feedback was first proposed in [10] for TDD systes, where it was shown to enable accurate resource accounting of the feedback link resources power and spectru and analyze the iportant case of transitter and receiver channel knowledge isatch A key essage fro [10] was that in the general case when the transitter and receiver knowledge is isatched, both feedback and forward counication has to be jointly designed The siple two-way training protocol proposed in [10] led to the concept of power-controlled training which enabled joint estiation of transitter s intended power control and actual channel realization We will continue the line of thought initiated in [10] for the case of FDD Frequency Division Duplex systes and ore iportantly, systeatically show how isatch between inforation at the transitter and receiver ipacts the overall syste perforance For the case of FDD systes, we will continue to odel the forward and feedback links as fading channels However, unlike [10], the forward and feedback channel will be assued to be copletely uncorrelated since they use different frequency bands In this case, the two-way training protocol proposed in [10] will not be applicable and we will use the quantized feedback odel, in which the receiver sends a quantized version of its own channel inforation over the feedback link Our ain result is an achievable diversity-ultiplexing tradeoff for an n MIMO Multiple Input Multiple Output syste, with easured channel inforation at the receiver which is used to send quantized channel inforation via a noisy link using iid Gaussian codebooks We show that, in the scope of odeling assuptions, diversity-ultiplexing tradeoff with one-bit of noisy feedback with estiated CSIR channel state inforation at receiver is identical to diversityultiplexing tradeoff with perfect CSIR and noiseless one-bit feedback Even ore iportantly, ore bits of feedback do not iprove the axiu diversity order in the iperfect

2 syste estiated CSIR with noisy feedback in contrast of the idealized syste where the axiu diversity order increases exponentially [8, 13, 17] The conclusion holds for ultiuser systes in general The encouraging news fro our analysis is two-fold First, even noisy and isatched inforation about the channel at both transitter and receiver is sufficient to iprove the whole diversity-ultiplexing tradeoff when copared to the case with no feedback Second, very few bits of feedback, in fact one-bit, is all one needs to build in a practical syste However, to achieve any diversity order gain over the nofeedback syste, a straightforward application of the Genieaided feedback analysis [8, 13, 17] fails That is, conventional training followed by conventionally coded quantized feedback signal leads to no iproveent in diversity order A cobination of power-controlled training uch like in [10] and the new concept of power-controlled feedback proposed in this paper appear essential for significant iproveent in diversity-ultiplexing gain copared to the no-feedback syste We will build our ain result in three ajor steps In the first step, we assue that the receiver knowledge of the channel is perfect and the feedback channel is errorprone Thus, the transitter inforation about the channel is potentially different fro that sent by the receiver In this case, the coding of the feedback signal becoes iportant If the feedback inforation about the channel is sent using a codebook where each codeword has equal power, then the axiu diversity order is n irrespective of the aount of feedback inforation In contrast, b-bit noiseless feedback leads to a axiu diversity order grows exponentially in nuber of bits [8, 13] The reason for such draatic decrease fro unbounded growth with b in the noiseless case to axiu n in the noisy case is that the transitter cannot distinguish between rare channel states, especially those where it is supposed to send large power to overcoe poor channel conditions As a result, it becoes conservative in its power allocation and the gain fro power control does not increase unboundedly as in the case of noiseless feedback A careful exaination of the outage events on the feedback path naturally points to unequal error protection in the for of power controlled feedback coding schee, where the rare states are encoded with higher power codewords With power controlled feedback, the diversity order can be iproved to n + n, a substantial increase copared to constant power feedback As our second step, we assue that the channel knowledge at the receiver is iperfect but the feedback path is error-free We first analyze the coonly used protocol setup, where the receiver is trained at the average available power SNR assuing noise variance is one to obtain a channel estiate and then a quantized version of the estiated channel is fed back to the transitter While both the transitter and receiver have identical inforation, the error in receiver inforation leads to no gain in the diversity order copared to a no-feedback syste We take a cue fro an earlier work in [10] and the power-controlled feedback echanis, and propose a tworound training protocol, where the receiver is trained twice First round training uses an average power SNR and then after obtaining the channel feedback, the receiver is trained again with a training power dependent on the channel estiate This iplies that in poor estiated channel conditions, second round training power is higher than the first and in good conditions, it is lower The adaptation of training power is labeled power-controlled training and allows a substantial increase in axiu diversity order n +n copared to no-feedback case n For general ultiplexing gains, the achievable diversity order is identical to that obtained when the receiver knows perfect channel state inforation with 1 bit of perfect feedback in [13] Further, additional bits of feedback do not help in iproving the axiu diversity order at constant rates when the receiver is trained to obtain a channel estiate Finally, we put the first two steps together to analyze the general case of noisy channel estiates with noisy feedback The key conclusion is that the receiver channel estiate errors are the bottleneck in the axiu achievable diversity order Thus, the axiu achievable diversity order at ultiplexing gain r 0 is n + n, which is less than n + n obtained in the case of perfect receiver inforation For general ultiplexing, we get the sae tradeoff as with 1 bit of perfect feedback and the receiver having perfect channel state inforation as in [13] This diversity ultiplexing tradeoff can be achieved with a single bit of noisy feedback fro the receiver The results on MIMO point-to-point channels are extended to the ultiple access channel in which all the transitters have antennas each and the receiver has n antennas, and all the conclusions drawn earlier for the single user also holds in a ultiple access syste The channel inforation for each transitter is easured at the receiver and a global quantized feedback is sent fro the receiver to all the transitters We use the siilar cobination of power control training and feedback as in single-user systes to achieve a diversity order of n +n with a single bit of power-controlled feedback in the ain case in which the errors in the channel estiate and the feedback link are accounted We note that the iproveent in diversity order is copletely dependent on our use of power control as adaptation echanis For exaple, in MIMO systes, feedback can be used for beaforing eg [19, 0] However, beaforing does not lead to any change the diversity order and only increases the receiver SNR by a constant aount Since we have focused on the asyptotic regie, we do not consider schees like beaforing which have identical diversityultiplexing perforance as a non-feedback based syste In this paper, we find new achievable schees to iprove the diversity ultiplexing tradeoff which are better than the traditional approaches but do not clai globally optiality of these schees The rest of the paper is organized as follows Section II describes preliinaries on channel odel and diversity ultiplexing tradeoff Section III suarizes the known results for the case when the receiver knows perfect channel state inforation and the feedback is sent over a noiseless channel [13] Section IV describes the diversity order when the receiver knows the channel perfectly while the transitter

3 3 receives feedback on a noisy channel Section V describes the diversity ultiplexing tradeoff when the receiver is trained to get channel estiate while the feedback link to the transitter is noiseless Section VI consider both the above errors, ie, it considers receiver estiating the channel and the feedback link is also noisy Section VII presents soe nuerical results We consider the extension to ultiple access channels in Section VIII Section IX concludes the paper A Two-way Channel Model II PRELIMINARIES We will priarily focus on single-user ultiple input-output channel with the transitting node denoted by T and receiving node denoted by R Later, we will extend the ain result to the ultiple access channel in Section VIII For the single-user channel, we will assue that there are transit antennas at the source node and n receive antennas at the destination node, such that the input-output relation is given by T R : Y HX + W, 1 where the eleents of H and W are assued to be iid with coplex Noral distribution of zero ean and unit variance, CN0, 1 The atrices Y, H, X and W are of diension n T coh, n, T coh and n T coh, respectively Here T coh is coherence interval such that the channel H is fixed during a fading block of T coh consecutive channel uses, and statistically independent fro one block to another The transitter is assued to be power-liited, such that the long-ter power 1 is upper bounded, ie, T coh tracee [ XX ] SNR Since our focus will be studying feedback over noisy channels, we assue that the sae ultiple antennas at the transitter and receiver are available to send feedback on an orthogonal frequency band For the feedback path, the receiver will act as a transitter and the transitter as a receiver As a result, the feedback source which is destination for data bits will have n transit antennas and feedback destination which is source of data bits will be assued to have receive antennas Furtherore, a block fading channel odel is assued R T : Y f H f X f + W f, where H f is the MIMO fading channel for the feedback link, and the W f is the additive noise at the receiver of the feedback; both are assued to have iid CN0, 1 eleents The feedback transissions are also assued to be power-liited[ with ] a long-ter power constraint given by 1 T coh tracee X f X f SNR f Without loss of generality, we will assue the case where the transitter and receiver have syetric resources, such that SNR SNR f We note that a phase-syetric two-way channel odel with H Hf T was studied earlier in [10, 11] The phasesyetric two-way channel is a good odel for slow-fading tie division duplex TDD systes On the other hand, the above SNR-syetric SNR SNR f odel is well-suited for syetrically resourced FDD systes Sender antennas Forward Channel Feedback Channel Destination antennas Fig 1 Two-way MIMO channel The forward and feedback channels use the sae antennas If the forward and feedback are frequency-division duplex FDD, then H and H f are not equal On the other hand, if the forward and feedback channels are tie-division duplex TDD, then H Hf T is often a reasonable assuption B Obtaining Channel State Inforation The two-way channel odel allows two-way protocols, as depicted in Figure, where the transitter and receiver can conduct ultiple back-and-forth transactions to coplete transission of one codeword The odel was used in [10, 11] to study the diversity-ultiplexing perforance of twoway training ethod In this paper, we will focus on another instance of the two-way protocols which will involve channel estiation and quantized feedback However, unlike [10, 11], we will not account for resources spent on feedback path since the ethods developed in [10] directly apply to the current case Instead, we will focus on the ore iportant issue of understanding how the isatch in the transitter and receiver channel inforation affects syste perforance To develop a systeatic understanding of the two-way fading channel described in Section II-A, we will consider two fors of receiver knowledge about the channel H The receiver will either be assued to know the channel H perfectly or have a noisy estiate Ĥ obtained using a iniu eansquared channel estiate MMSE via a training sequence described in detail in Section V For the transitter knowledge, the receiver will quantize its own knowledge H or Ĥ and ap it to an index J R, where J R {0, 1,, K 1} where K 1 is the nuber of feedback levels used by the receiver for feedback The receiver then transits the quantized channel inforation J R over the feedback channel and the transitter knowledge of the index is denoted by J T For the case when the feedback channel is assued to be noiseless, J T J R, else J T J R with finite probability Here the error probability will be depend on the channel signal-to-noise ratio SNR and the transission schee Based on the quantized inforation about the channel, in the for of J T, the transitter adapts its codeword to iniize the probability of outage defined in the next section Sender antennas Destination antennas Fig Two-way protocols: In a two-way channel, the protocols can involve any exchanges between the transitter and receiver The arrows indicate the direction of transission

4 4 C Diversity-Multiplexing Tradeoff We will consider the case when then the codeword X spans a single fading block Based on the transitter channel knowledge J T i, the transitted codeword is chosen fro the codebook C i {X i 1, X i,, X i RT coh }, where the codebook rate is R All X i k s for 1 k RT coh are atrices of size T coh We assue that T coh is finite and does not scale with SNR In this paper, we will only consider single rate transission where the rate of the codebooks does not depend on the feedback index Furtherore, we will assue codebooks C i are derived fro the sae base codebook C by power scaling of the codewords In other words C i P i C, where the product iplies that each eleent of every codeword is ultiplied by P i where each eleent of codeword Xk C has unit power Thus, P i is the power of the transitted codewords Recall that there is an average transit power constraint, such that EP i SNR We further assue that the base codebook C consists of Gaussian entries, in other words we only focus on Gaussian inputs in this paper In point-to-point channels, outage is defined as the event that the utual inforation of the channel for a channel knowledge H ξ with soe distribution P H ξ for ξ a subset of all n atrices at the receiver, I ξ X; Y is less than the desired rate R [1] If ξ is a singleton set containing H, the channel H is known at the receiver in which case I ξ X; Y log det I + P J T HQH is the utual inforation of a point-to-point link with transit and n receive antennas, transit signal to noise ratio P J T and input distribution Gaussian with covariance atrix Q [] The dependence of the index at the transitter is ade explicit by writing the transit SNR as a function of J T Note that P i and P i will ean the sae thing in this paper Let ΠO denote the probability of outage, where O is the set of all the channels where the axiu supportable rate I ξ X; Y is less than the transitted rate R [1] The syste is said to have diversity order of d if ΠO SNR d 1 Note that all the index appings, codebooks, rates, powers are dependent on the average signal to noise ratio, SNR Specifically, the dependence of rate R on SNR is explicitly given by R r log SNR, where r is labeled as the ultiplexing gain The diversity-ultiplexing tradeoff is then described as the axiu diversity order dr that can be achieved for a given ultiplexing gain r The following result captures the diversity-ultiplexing tradeoff for the case of perfect receiver inforation and no transitter inforation For a given rate R r log SNR and power P SNR p, define the outage set OR, P {H : I + P HQH < R} Denote the result diversity order as Gr, p, which is ΠOR, P SNR Gr,p The following result copletely characterizes Gr, p and is a straightforward extension of the result in [] Lea 1 [16] The diversity-ultiplexing tradeoff for P 1 We adopt the notation of [] to denote to represent exponential equality We siilarly use <, >,, to denote exponential inequalities SNR p and r p in, n is given by the r, Gr, p where in,n Gr, p inf i 1 + ax, n in, nα i where the inf is over all α 1,, α in,n satisfying in,n {α 1 α in,n 0, p α i + < r} i0 Reark 1 Gr, p is a piecewise linear curve connecting the points r, Gr, p kp, p kn k, k 0, 1,, in, n for fixed, n and p > 0 This follows directly fro Lea of [13] Gr, 1 is the diversity ultiplexing tradeoff with perfect CSIR and the transit signal to noise ratio SNR [] Further, Gr, p pg r p, 1 D Suary of Results We will study the following four systes with different accuracy of channel state inforation CSI at the transitter and receiver 1 CSIRT q : In this case, the receiver knowledge about H is assued to be perfect and the transitter is assued to receive a noiseless quantized feedback fro the receiver about channel H This case was first studied in [3] and the diversity order increase was first proved in [17] with later extensions in [13] The channel quantizer aps H to an index J R which is then counicated over the noiseless feedback channel Since the feedback is noiseless, J T J R Based on the index J T, the transitter adapts its transission power as described in Section III The diversity order for K levels of feedback is defined recursively as d RTq r, K Gr, 1 + d RTq r, K 1 where d RTq r, 0 0 and grows exponentially in nuber of feedback bits CSIR T q : Our first set of results analyze the case of iperfect channel knowledge at the transitter, where the errors are caused by errors in the received quantized feedback index In this case, J T J R with finite probability We show that if a MIMO schee optiized for equally likely input essages is used to send the feedback inforation, then the diversity order is liited to d R T q r, K given in Table I However, the feedback inforation is not equally-likely Hence, we propose a natural unequal error protection ethod labeled power controlled feedback, where the power control is perfored based on input probabilities The power-controlled feedback results in a diversity-order d R T q r, K, specified in Table I With power-controlled feedback, the axiu diversity order increases fro d R T q 0, K n to d R T q 0, K nn + for all K > For K, the axiu diversity order increases fro d R T q 0, K n to d R T q 0, K nn CSI RT q : Our next step will be to isolate the effect of errors in receiver knowledge of the channel Thus, the receiver will estiate Ĥ, which will be apped to J R Since the feedback is assued to be perfect,

5 5 the transitter knowledge is sae as that of receiver, J T J R We will show that for a three-phase powercontrolled training based protocol can achieve a diversity order of d RT q r, K Gr, 1 + Gr, 1 In fact, if the power-controlled training is not perfored then channel estiation errors copletely doinate and feedback is rendered useless; the resultant tradeoff collapses to d RT q r, K Gr, 1 for all K So analogous to powercontrolled feedback, power-controlled training appears essential to iprove the diversity order in this case 4 CSI R T q : Finally, we put the above two cases together and derive the axiu achievable diversity order as d R T q r, K Gr, 1 + Gr, 1, which is equivalent to the case with only receiver errors Thus, we conclude that the receiver errors doinate the achievable diversity order Reark An iportant word of caution for all the results new and known suarized in Table I Unlike the previous work in [10, 1], we do not account for the resources spent in channel training and feedback in this paper Resource accounting can be perfored using the procedure developed in [10], by scaling the ultiplexing r appropriately More precisely, the ultiplexing r should be replaced by rt coh /T coh 1 for CSIR T q, rt coh /T coh for the three-phase protocol in power-controlled training of CSI RT q, rt coh /T coh for two-phase protocol in constant power training of CSI RT q, and rt coh /T coh 1 for the three-phase protocol of CSI R T q The resource accounting ultipliers assues that the feedback requires one channel use and the training requires channel uses The details are further explained in Rearks 3 and 5 Further, the use of the nuber of antennas to use and the protocols would need to be optiized for each ultiplexing as in [10, 1] and is oitted in this paper for readability III CSIRT Q : PERFECT CSIR WITH NOISELESS QUANTIZED FEEDBACK The diversity-ultiplexing tradeoff for the case of receiver with perfect inforation and noiseless quantized inforation has been extensively studied in [8, 13, 14] In this section, we will discuss the ain ideas for the single user MIMO channel odel stated in Section II-A We start off with an exaple to illustrate the ain idea Exaple 1 SISO: Consider the case where n 1 Without any feedback, the axiu diversity at r 0 is 1 The space of channels {H} C can be divided into { two sets: the outage set O 0 H : H < R 1 SNR } and its copleent O 0 {H}\O 0 ; see Figure 3a The probability of the set ΠO 0 SNR 1 and ΠO 0 1 SNR 1 With one-bit of noiseless feedback, the receiver can convey whether the current channel H belongs to O 0 or O 0 Since the outage event is rare, the transitter can send uch larger power than usual to reduce the outage probability as follows When the feedback index is J R 1 representing event O 0, the transitter will use transit power 1 SNR/ΠO 0 1 SNR For the feedback index J R 0, representing O 0, power 1 SNR is used The average power used in the above twolevel power control is 1 SNR/ΠO 0 ΠO SNR 1 SNR 1 SNR With the above one-bit power control, the outage probability is SNR { since the set of channels } in outage is reduced to O 1 H : H < R 1 ΠO 0 SNR { H : H < R 1 }; see Figure 3b SNR If the feedback rate was log 3 bits/channel state, then the receiver could convey inforation about three events {O 1, O 0 \O 1, O 0 } In state O 1, the transitter can send power SNR 3 since ΠO 1 SNR and thus reduce the outage probability to SNR 3 If we had two bits of feedback or equivalently K 4 levels, then the diversity order will be 4 and the outage region will be given by O 3 which will be the set of all those channels which could not support rate R with power SNR 4 ; see Figure 3c Using the above recursive arguent, for K levels of feedback, a diversity order of K b can be achieved, where b is the nuber of feedback bits per channel realization The above exaple captures the essence of the general result for the MIMO channels, given by the following theore Theore 1 [13] Suppose that K 1 and r < in, n Then, the diversity-ultiplexing tradeoff for the case of perfect CSIR with noiseless quantized feedback is d RTq r, K defined recursively as where d RTq r, 0 0 d RTq r, K Gr, 1 + d RTq r, K 1, The ain idea of the proof is along the lines of Exaple 1 and is suarized as follows Based on its knowledge of H, the receiver decides the feedback index J R J T {0, 1,, K 1}, where K b is the nuber of quantization levels and b is the nuber of feedback bits If the transitter receives feedback index J T, it sends data at power level P JT Without loss of generality, P 0 P 1 P The optial deterinistic index apping has the following for [13], arg in i I i, I { k : log deti + HH P k R, J R k {0,, K 1}} 0, if the set I is epty Based on the above assignent of feedback indices, the optial power levels can be found out as P i SNR 1+p i where p i are recursively defined as: p 0 0, p j Gr, 1+p j 1 j 1 Using the above recursion, the optial diversity is given by Gr, 1 + p which reduces to d RTq r, K, thus proving Theore 1 We note that special cases of Theore 1 were also proved in [8] We note that event that {H : log deti +HH P < R} corresponds to the outage event, because none of the power levels can be used to reach a utual inforation of R Thus any state can be assigned to the outage event and in fact, the diversity-ultiplexing tradeoff is unaffected by which index is used to represent this state Since the receiver knowledge is perfect, using the index J R 0 ensures that the overall power consuption is iniized However, when the receiver

6 6 TABLE I SUMMARY OF DIVERSITY-MULTIPLEXING TRADEOFFS SEE REMARK FOR A CAUTION IN USING THIS TABLE Case Main Characteristic D-M Tradeoff CSIRT Perfect Inforation at T and R d RT r r < in, n CSIRT q Quantized Inforation at T d RTq r, K Gr, 1 + d RTq r, K 1, d RTq r, 0 0 CSIR T q Noisy inforation at T d R T q r, K inb K r, n + Gr, 1 CSIR T q Constant Power Feedback Noisy inforation at T d R T q r, K in d RTq r, K, ax qj 1+d RTq r,j Power-controlled Feedback in nq i + q i d RTq r, i CSI RT q Noisy inforation at R d RT q r, K Gr, 1 Constant Power Training CSI RT q Noisy inforation at R d RT q r, K Gr, 1 + Gr, 1 Power-controlled training CSI R T q No Genie-aided inforation d R T q r, K Gr, 1 + Gr, 1 Power-controlled training & feedback a b c Fig 3 Exaple 1: Channel events for the case of a no feedback O 0 is the set of channels in outage and b one-bit feedback O 1 is the outage set and c two-bit feedback O 3 is the outage set or transitter knowledge is not perfect which will be the case in the rest of the paper, we will assign the perceived outage state the highest power level P, which helps reduce the outage probability due to isestiation of channel The axiu diversity order in Theore 1 increases very rapidly with the nuber of feedback levels K The axiu diversity order [13] d RTq 0, K K g1 ng, which grows exponentially fast in the nuber of levels K In fact, as K, the diversity order also increases unboundedly Since as K, the feedback approaches the perfect feedback and as a result, perfect channel inversion becoes possible In [3], it was shown that for perfect channel state inforation at transitter and receiver, zero outage can be obtained at finite SNR if ax, n > 1 Perfect channel inversion essentially converts the fading channel into a vector Gaussian channel, whose error probability goes to zero for all rates less than its capacity The unbounded growth of diversity order is rather unsettling and is in fact, a fragile result as shown by our results in the following sections IV CSIR T Q : PERFECT CSIR WITH NOISY QUANTIZED FEEDBACK In this section, we will analyze the case when the receiver knows the channel H perfectly but the quantization index J R is conveyed to the transitter over a noisy feedback link, resulting in the event J T J R with non-zero probability We will consider two feedback designs In the first design, the feedback channel will use a constant power transission schee designed for equally-likely sybols which is the coonly studied case for iid data Learning fro the liitations of constant power feedback transission, we will then construct a new power-controlled feedback strategy which will exploit the unequal probability of different channel events quantized at the receiver A Constant Power Feedback Transission First, we observe the ipact of using the power control described in Section III by considering Exaple 1 Exaple Ipact of Feedback Errors, SISO: Consider the case of b feedback bits which allows K 4 feedback indices In this case, the receiver can convey four events, which

7 7 we choose to be O {O 0, O 0 \ O 1, O 1 \ O, O } shown in Figure 3c, where the events are described as O i { H : log 1 + H SNR 1+i < R }, i 0, 1, 3 As discussed in the previous section, the power control in the forward channel uses four different power levels, P JT SNR 1+J T for J T 0, 1,, 3 The probability of each of the above events and the associated power control is defined in Table II The last colun in Table II shows the ipact of probability of events as seen by the transitter when there are errors in the feedback link such that ΠJ T j J R i SNR 1 for j i as shown in Figure 4b For exaple, with noisy feedback, ΠJ T 3 3 j0 ΠJ T 3 J R jπj R j SNR 1 + SNR 1 SNR 1 + SNR 1 SNR + SNR 3 SNR 1 In fact, there are two doinant error events, first being O 0 J R 0 being confused as either O 0 \ O 1 or O 1 \ O or O J T 1, or 3 which aounts to liiting the axiu power that can be used and the second being the probability that O 1 J R 1 being confused as O 0 J T 0 The second error event has approxiate probability of SNR which liits the axiu diversity to Thus, the event probabilities at transitter are no longer the sae in the presence of the feedback errors As a result, the transitter cannot use the power control P JT SNR 1+J T without exceeding the average power constraint In fact, the highest power the transitter can send is of the order SNR, and the axiu diversity order of the constant power feedback echanis is liited to irrespective of the nuber of feedback bits We now generalize the above exaple to MIMO for arbitrary ultiplexing gain Define B j r be defined by the recursive equation { 0, j 0 B j r Gr, 1 + inn, B j 1 r, j 1 Theore [16] Suppose that K > 1 and r < in, n Then, the diversity-ultiplexing tradeoff for the constant power feedback transission with K indices of feedback is given by d R T q r, K inb K r, n + Gr, 1 Reark 3 Accounting for the feedback resources can be done as follows In the liit of high SNR, the feedback will consue one channel use and hence to get the rate R r logsnr, r should be replaced by rt coh /T coh 1 in the above expression However, the reader ust note that there are iplicit tie factor ters which can be easily integrated and one ay carry out an optiization on the diversity obtained as a function of these tie loss ters for each ultiplexing For exaple, for r > in, nt coh 1/T coh, the feedback would not be useful due to the tie spent and it is better to use a non-feedback based strategy Much like in Exaple, the axiu diversity order for r 0 is n for all K That is one-bit of feedback is sufficient to achieve the axiu diversity order with constant-power feedback However, for r > 0, as the nuber of feedback levels K increases, the diversity order d R T q r, K also increases, such that d R T q r, K Gr, 1 + n for all r Note that Gr, 1 is the diversity order without any feedback Coincidentally, the diversity order of the feedback link is n feedback link is non-coherent where the error probability can decay no faster than SNR n, and ends up deterining the axiu gain possible beyond Gr, 1 The key bottleneck in the above result is that the feedback link is using a transission schee optiized for equallylikely signals, which is appropriate if the feedback link was being used to send equally-likely essages like data packets However, the inforation being conveyed in the feedback link is not equiprobable and hence the usual MIMO schees optiized for equally-likely essages are not well suited Again, in the context of Exaple, we will first show an alternate design of power-controlled feedback can iprove the diversity order in the next section Before we proceed, we note that our analysis in [10] for a TDD two-way channel yields the sae axiu diversity order of n It is satisfying to see how two different feedback ethods have identical behavior; their relationship is further explored in [4] B Power-controlled Feedback In this section, we exploit the unequal probabilities of the outage events at different power levels to develop a power-controlled feedback schee to reduce overall outage probabilities Our feedback transission will be designed for a non-coherent channel, since the feedback channel H f is not known at the transitter or the receiver Exaple 3 Power-controlled Feedback: For the constantpower feedback transission schee described in the Section IV-A, each input is apped to a codeword with the sae power and is pictorially depicted in Figure 5a However, since the events in set O are not equi-likely, we assign different power levels to each event Now consider the power assignents shown in Table III, also depicted in Figure 5b for the feedback channel The average power over the feedback channel is SNR 0 1 SNR 1 + SNR SNR 1 + SNR 3 SNR +SNR 3 SNR 4 SNR Here the rare events are conveyed with ore power, which allows a ore reliable delivery of the feedback inforation without violating feedback power constraint Thus, the powercontrolled schee resebles an aplitude shift keying The reason that the probabilities at the transitter and the receiver are sae for power-controlled is because the events J T > J R occur with exponentially sall probability The outage probability can be lower bounded by ΠJ T 0, J R 1 which is an event when the receiver requested a higher power level than what it received Note that ΠJ R 1 SNR 1 and ΠJ T 0 J R 1 SNR 0 Hence, ΠJ T 0, J R 1 SNR 1 SNR SNR 3 is the probability of this doinant error event Note that both the forward and feedback channel power constraints are satisfied Hence, we can obtain a diversity order of 3, which is higher than the diversity order

8 8 TABLE II EXAMPLE : PROBABILITY OF EVENTS AT THE TRANSMITTER AND RECEIVER WITH AND WITHOUT NOISE IN THE FEEDBACK CHANNEL RECEIVER KNOWLEDGE IS ASSUMED TO BE PERFECT CAUTION: PROBABILITIES ARE ONLY REPORTED UP TO THEIR ORDER, AND CONSTANTS SUCH THAT THEY SUM TO ONE ARE OMITTED Event Feedback Prob of Event Prob at Transitter Prob at Transitter Index J R at Receiver noiseless feedback noisy feedback O SNR 1 1 SNR 1 1 SNR 1 O 0 \ O 1 1 SNR 1 SNR 1 SNR 1 O 1 \ O SNR SNR SNR 1 O 3 SNR 3 SNR 3 SNR 1 Event Input Prob Output Prob Input Prob Output Prob a b Fig 4 Exaple : Input-output probabilities of the feedback channel, when a the channel is noiseless and b when the channel is noisy causing isatch between transitter and receiver knowledge, assuing that a code optiized for equally-likely source is used The noise in the feedback channel changes the output probabilities, as shown by the circled output probabilities Note that the su of probabilities is ore than one since we have oitted the constants for ease of understanding TABLE III EXAMPLE 3: POWER ASSIGNMENT FOR NOISY FEEDBACK CHANNEL CAUTION: PROBABILITIES ARE ONLY REPORTED UP TO THEIR ORDER, AND CONSTANTS SUCH THAT THEY SUM TO ONE ARE OMITTED Event Feedback Prob at Receiver Feedback Transit Prob at Transitter Index Power noisy feedback O SNR 1 SNR 0 1 SNR 1 O 0 \ O 1 1 SNR 1 SNR SNR 1 O 1 \ O SNR SNR 3 SNR O 3 SNR 3 SNR 4 SNR 3 of obtained via constant-power feedback design Exaple but lower than 4 that can be achieved with a noiseless feedback channel Exaple 1 a b Fig 5 Exaples and 3: Doinant error events when a the feedback channel uses a transission suited for equally-likely essages and b a power-controlled feedback design where codewords are power-controlled based on their probability of occurrence We will generalize the above exaple to the case of general MIMO channels by using the following power-control for the feedback channel Let the feedback power level for different feedback essages be Q 0 SNR q 0, Q 1 SNR q 1,, Q SNR q with q j > q j 1 Assue for now that the set of powers {Q i } satisfy the feedback power constraint of SNR We will use the axiu aposteriori probability MAP detection rule to detect the essage transitted by the receiver on the feedback channel If the transitted signal is at signal to noise ratio level of Q, the received power is S f tracehh Q + W W + QHW + W H Since we are encoding the feedback index inforation in signal power, we will copare the received power S f with different threshold power levels We first note that for all q i 0, the received power will be doinated by the noise ter tracew W, which is 1 with high probability Thus, we assue that q j > 0 for j > 0 Suppose the eigenvalues of HH are λ 1,, λ N and λ i SNR α i where α i are the negative SNR exponents of the λ i, N in, n and α α 1,, α N Then, the distribution of α i is given by the following result Lea [] Assue α 1 α α N are the power

9 9 exponents as described above In the liit of high SNR, the probability density function of the SNR exponents, α, of the eigenvalues of HH is given by pα Π N SNR i 1+ n αi 1 inα 0 4 We now show that the thresholds for MAP decoding are SNR axq0,0+ɛ, SNR q1+ɛ,,snr qk +ɛ for soe sall ɛ > 0 This can be derived by observing reason is that ΠJ T > i J R i decays faster than polynoially in SNR and is hence 0 as long as the threshold for detecting J T i is above SNR qi+ɛ for any ɛ > 0 To see this, for 0 j < K 1, ΠJ T > j J R j Π tracehh SNR axqj,0 + 1 SNR axqj,0+ɛ Π SNR q j α i + 1 SNR axqj,0+ɛ Π SNR axqj in αi,0 SNR axqj,0+ɛ Π axq j in α i, 0 axq j, 0 + ɛ 5 We see that for the last expression to occur with finite probability, in α i < 0 which cannot happen with polynoially decreasing probability by Lea and hence ΠJ T > j J R j 0 irrespective of ɛ > 0 Now, for MAP detection, we would find a threshold between q i and q i+1 so as to iniize ΠJ T i+1, J R i+πj T i, J R i+1 Since the first ter 0 and the second ter is iniized by choosing ɛ as sall as possible, choosing ɛ sall enough gives the desired threshold for MAP decoding Further, for all Q >SNR, tracehh Q + W W + QHW + W H tracehh Q + 1 Hence, we obtain ΠJ T 0 J R 1 ΠtraceHH SNR p1 + 1 <SNR axp0,0+ɛ Let λ i be the eigenvalues of HH and λ i SNR α i Then, ΠJ T 0 J R 1 Π tracehh SNR q1 + 1 <SNR axq0,0+ɛ in,n Π SNR q1 αi + 1 < SNR axq0,0+ɛ Π SNR axq1 in αi,0 <SNR axq0,0+ɛ Π axq 1 in α i, 0 < axq 0, 0 + ɛ Π in α i > q 1 axq 0, 0 ɛ SNR nq1 axq0,0 where the last step follows fro Lea Since q Gr, 1 as ΠJ R 1 SNR Gr,1 and there is power constraint for feedback link of SNR, the outage probability is lower bounded by ΠJ T 0, J R 1 SNR Gr,1 SNR n1+gr,1 0 SNR n1+gr,1 Gr,1 Further, we can use the sae technique to find that for any j < i, ΠJ T j J R i SNR nqi axqj,0 6 Any function fsnr which decays faster than any polynoial in SNR, like exponential or super-exponential functions of SNR, are 0 since logfx li SNR logsnr 0 Thus we obtain the following result Theore 3 Power-controlled Feedback Suppose that K > 1 and r < in, n Then, the following diversityultiplexing tradeoff can be achieved with power-controlled feedback d R T q r, K in d RTq r, K, ax q j 1+d RTq r,j in nq i + q i d RTq r, i,, Proof: The proof is provided in Appendix B Corollary 1 The diversity ultiplexing tradeoff with one bit of iperfect feedback is sae as the diversity ultiplexing tradeoff with one bit of perfect feedback In other words, for K, d R T q r, Gr, 1 + Gr, 1 d RTq r, Proof: For K, the optial choice of q is to axiize q 1 + q+ 0 which gives optial choice of q 0 0 and q Gr, 1 Using this for one bit of feedback, diversity of Gr, 1 + Gr, 1 can be achieved with power-controlled feedback which is the optial considering the upper bound of perfect feedback is d RTq r, Gr, 1 + Gr, 1 Thus, we find that there is no loss of diversity with one iperfect bit of feedback copared to the case of perfect feedback Hence, power controlled feedback schee is better than the constant power schee which was liited to a axiu of n diversity even as r 0 Corollary As K and r 0, the axiu diversity that can be obtained with power-controlled feedback is nn +, ie, li d K,r 0 R T q r, K nn + Proof: Note that in this case, choosing q 0 0 and q i 1 + d RTq r, i 1 + n ni 1 n 1 1 n>1 + i1 n1 gives the optial diversity ultiplexing point for r 0 and thus, the diversity is nn+ We further note that K 3 is enough to get this point Thus, even with power-controlled feedback, arbitrary nuber of feedback bits do not yield unbounded increase in diversity order as in the case of CSIRT q, where the diversity order increases unbounded with K Note that however, we restricted our attention to an ordering q j > q j 1 which can be relaxed giving better results as in the following Lea However, for our objective of the achievability for iperfect channel state at the receiver and iperfect channel state at the transitter, the achievability strategy in Theore 3 is enough Further note that the new achievability strategy which relaxes the assuption of q j > q j 1 does give iproved diversity, but still the diversity reain bounded with increase of the feedback levels Lea 3 Let q i 0 and q i q j for any i j, 0 i, j K 1 Then, the following diversity can be achieved by power-controlled feedback

10 10 in d RTq r, K, ax q 0,,q Q in nq i q j + d RTq r, i, i,j 0,,:j<i and q j<q i where Q {q 0,, q : q j 1 + ind RTq r, j, in i {0,,j 1}:qi>q j d RTq r, i + q i q j n 0 j K 1} Proof: The proof is a siple generalization of the proof of Theore 3 The constraint on Q keeps the ΠJ T isnr qi SNR for the power constraint and the diversity expression records the possible doinant outage events C A Source Coding Interpretation The idea of power-controlled feedback transission is akin to source-coding the feedback inforation In conventional lossless source coding eg Huffan coding, the currency of representation is bits There, to iniize average code-length, the codeword length is approxiately inversely proportional to the probability of an event Rare events are represented by longer codewords while ore frequent events are represented by shorter length codewords, thereby iniizing the average codelength Analogously, our currency is average transit power and objective is to iniize average error probability Thus rare events get higher power and frequent events lower transit power Note that there are any power allocations which will eet the power constraint but they will all result in different error probabilities Our proposed feedback transit power allocation iniizes the error probability in asyptotic sense V CSI RT Q : ESTIMATED CSIR WITH NOISELESS QUANTIZED FEEDBACK In this section, we will consider the case when the receiver obtains its channel inforation fro an MMSE estiate, Ĥ, using training As a result, the receiver index J R is not always equal to the optial index, J, based on actual channel H We will odel the relationship between J and J R via an effective channel described in Section V-C The feedback link, on the other hand, is assued to be noiseless As a result, the transitter index J T J R We will analyze two protocols The first protocol, labeled constant-power training, trains the receiver once at the beginning using a constant power level We show that feedback, even if noiseless, is copletely useless in providing any gains in diversity order copared to a no-feedback syste Inspired by our understanding of the noisy feedback channel in Section IV, we propose a second protocol, labeled powercontrolled training which utilizes the feedback and trains the receiver twice, where the second training is power-controlled based on feedback inforation The second protocol has a higher diversity order than any non-feedback syste A Training the receiver In this subsection, we will consider MMSE channel estiation for a single user MIMO channel The channel is estiated using a training signal that is known at the receiver Fro the received signal, MMSE estiation is done as in [5] to get an estiate Ĥ of the original channel H Let X T be the training signal of size N for soe N < T coh that is known at the receiver and transitter The transitter sends X T and the destination receives Y T HX T + W T where Y T is a n N received signal and W T is the additive Gaussian noise with each entry fro CN0, 1 Following [5], the optial training signal is X T [ nµ I 0 N ], where µ 0 is tuned to satisfy power constraint, I denote identity atrix and 0 N represents N atrix having all entries 0 Hence, nµ 0 1 NSNR Further, the channel estiate is given by [ NSNR ] Ĥ Y T X T X T + I N 1 X T Y T which can be rewritten as 1 Ĥ H W NSNR I 1+ NSNR 0 N NSNR 1 + NSNR where W is the left n subatrix of W T We now note soe properties of MMSE estiate First, it is easy to see that expected value of H ĤĤ is zero confiring the orthogonality of the error with the unbiased estiate Further, the variance of any entry in H Ĥ is 1 Thus, H 1+ NSNR and Ĥ are atrices in which each corresponding eleent is highly correlated with the correlation coefficient between corresponding eleent of H and Ĥ is ρ 1 Further, 7 1+ NSNR note that any N channel uses are equivalent for analyzing the asyptotic perforance In general, if G and H are correlated with correlation coefficient ρ, the joint probability distribution function of eigenvalues of HH and GG is given by the following result: Lea 4 [6] Consider two n rando atrices H h ij and G g ij, i [1, n], j [1, ], each with iid coplex zero-ean unit-variance Gaussian entries, ie, E[h ij ] E[g ij ] 0, i, j, E[h ij h pq] E[g ij g pq] δ ip δ jq, where the Kronecker sybol δ ij is 1 or 0 when i j or i j respectively Moreover, the correlation aong the two rando atrices is given by E[h ij g pq] ρδ ip δ jq, i, j, p, q, where ρ ρ e jθ is a coplex nuber with ρ < 1 Let n and ν n The joint probability distribution function of the unordered eigenvalues of HH and GG is n k1 pλ, λ exp λ k+ λ k 1 ρ λ λ n!n!π n 1 j0 j!j + ν! ρ n n 1 ρ n Π n k1 λ k λk ν det I ρ λ k λl ν 1 ρ,8 where represents n diensional Vanderonde deterinant, I k denotes the k th order odified Bessel function of the first kind, the eigenvalues of HH and GG are given by

11 11 λ λ 1,, λ n and λ λ 1,, λ n respectively Note that although Lea 4 assued n, it can be extended to the other case of > n since nonzero eigenvalues of HH and H H are the sae Hence for all n and, let N in, n and ν n Then, the joint probability density function of the unordered eigenvalues of HH and GG is N k1 pλ, λ exp λ k+ λ k 1 ρ λ λ N! N!Π n 1 j0 j!j + ν! ρ n N Π N k1 λ k λk ν det I ρ λ k λl ν 1 ρ 1 ρ 9 N Recall that the eigenvalues of HH be λ 1,, λ N, λ i SNR α i and α α 1,, α N Siilarly, let the eigenvalues of ĤĤ be λ 1,, λ N, λ i SNR α i and α α 1,, α N The distribution of α i s is given earlier in Lea We will now find the joint distribution of α i s and α i s Let α 1 α α N and α 1 α α N Further, we define E k {α, α : inα i, α i 1 i 1,, k, and for all 0 k N 0 α i α i < 1 i > k} 10 Theore 4 Let H be the channel and Ĥ be the estiated channel In the liit of high SNR, the probability density function of the SNR exponents of the eigenvalues of HH and ĤĤ is given by pα, α N e k 1 Ek 11 k0 where α 1 α α N, α 1 α α N, and e k SNR k n +k Π k SNR i 1+ n αi Π N SNR i 1+ n αi 1 Proof: Since all the results were syetric about interchanging and n, without loss of generality, we take n for the purpose of this proof and the rest of the paper The proof is provided in Appendix A Reark 4 Since the receiver is trained at SNR, we note that for all α i < 1, α i α i with probability 1, which eans that the channel estiate is a reliable proxy for the actual channel On the other hand, if α i 1, all we can state is that α i 1 with probability 1 For exaple, if α i 100, all we can reliably say about α i is that it is 1 In SISO case, the above property of α i iplies that the channel cannot be resolved below the noise floor since the noise doinates the training signal The interesting iplication in MIMO is that this result of noise doinance holds for all the eigen-values None of the eigen-value of the channel can be resolved beyond α i 1 if α i 1 Exaple 4 Asyptotic Distribution: For the case of n 1, which is our running exaple, pα, α SNR α 1 0 α α<1 + SNR 1 α α 1 inα, α 1 This density function will be used to analyze the diversity tradeoffs in this section B Constant-power Training We first note the decoding schee with trained channel estiate at the receiver If the receiver has estiate Ĥ trained with the power of SNR p for soe p 0, the estiation error variance is SNR p As Y HX +W ĤX +H ĤX + W, a lower bound on utual inforation can be considered assuing that H ĤX + W is Gaussian noise and hence the expression for the coherent channel with actual channel as Ĥ can be used as a lower bound [1] The protocol is divided into three phases as described below: Phase 1: Training fro Tx to Rx: The training is done using transit power of SNR to obtain the channel estiate Ĥ On the basis of this training, the receiver decides a feedback level J R in the following way Suppose that there are K power levels P i SNR 1+p i, i {0,, K 1} The feedback index chosen at the receiver is given by arg in i I i, I {k : log deti + ĤĤ P k R, J R k {0,, K 1}} K 1, if the set I is epty Phase : The feedback index J R {0,, K 1} is sent to the transitter via the noiseless feedback channel Thus, the feedback index received by the transitter is J T J R Phase 3: The transitter receives a feedback power level J T and sends data at power level P JT SNR 1+p JT Exaple 5 Constant-power training: The only error in channel knowledge appears in the first phase, where the receiver is trained Note that for MMSE estiation Ĥ and H H Ĥ are uncorrelated and since the receiver is trained with power of SNR, the variance of H is 1/SNR Let Ĥ SNR α and H SNR 1 SNR α where the probability distribution function of α and α is p α, α SNR α α 1 α 0 1 α 0 The probability of outage is ΠO j0 ΠO, J T j ΠO, J T 1 ΠO, J R 1 ΠO, log1 + Ĥ SNR r log SNR, log1 + Ĥ SNR < r log SNR Π log 1 + Ĥ SNR 1 + SNR H < r log SNR, log1 + Ĥ SNR r log SNR, log1 + Ĥ SNR < r log SNR Π α 1 α + + < r, α + r, 1 α + < r SNR in α, α A α+ α, 13