On the Achievable Rates of Multiple Antenna Broadcast Channels with Feedback-Link Capacity Constraint

In this paper, we study a MIMO fading broadcast channel where each receiver has perfect channel state information (CSI) while the channel state information at the transmitter (CSIT) is acquired by explicit channel feedback from each receiver through capacity-constrained feedback links. Two feedback schemes are considered, i.e., the analog and digital feedback. We analyze the achievable ergodic rates of zero-forcing dirty-paper coding (ZF-DPC), which is a nonlinear precoding scheme inherently superior to linear ZF beamforming. Closed-form lower and upper bounds on the achievable ergodic rates of ZF-DPC with Gaussian inputs and uniform power allocation are derived. Based on the closed-form rate bounds, sufﬁcient and necessary conditions on the feedback channels to ensure non-zero and full downlink multiplexing gain are obtained. Speciﬁcally, for analog feedback in AWGN feedback channels, it is sufﬁcient and necessary to scale the average feedback link SNR linearly with the downlink SNR in order to achieve the full multiplexing gain. While for digital feedback, in order to achieve the downlink multiplexing gain of α M (0 < α ≤ 1) , it is sufﬁcient and necessary to scale the number of feedback bits B per user as B = α ( M − 1) log 2 PN 0 where M is the number of transmit antennas and PN 0 is the average downlink SNR.


I. INTRODUCTION
The multiple antenna broadcast channels, also called multiple-input multiple-output (MIMO) downlink channels, have attracted great research interest for a number of years because of their spectral efficiency improvement and potential for commercial application in wireless systems.Initial research in this field has mainly focused on the information-theoretic aspect including capacity and downlink-uplink duality [1]- [4] and transmit precoding schemes [5]- [9].These results are based on a common assumption that the transmitter in the downlink has access to perfect channel state information (CSI).It is well known that the multiplexing gain of a point-to-point MIMO channel is the minimum of the number of transmit and receive antennas even without CSIT [10].On the other hand, in a MIMO downlink with single-antenna receivers and i.i.d.channel fading statistics, in the case of no CSIT, user multiplexing is generally not possible and the multiplexing gain is reduced to unity [11].As a result, the role of the CSI at the transmitter (CSIT) is much more critical in MIMO downlink channels than that in point-to-point MIMO channels.
The acquisition of the CSI at the transmitter is an interesting and important issue.For time-division duplex (TDD) systems, we usually assume that the channel reciprocity between the downlink and uplink can be exploited and the transmitter in the downlink utilizes the pilot symbols transmitted in the uplink to estimate the downlink channel [12].The impact of the channel estimation error and pilot design on the performance of the MIMO downlink in TDD systems has been studied in [13]- [18].For frequency-division duplex (FDD) systems, no channel reciprocity can be exploited and thus it is necessary to introduce feedback links to convey the CSI acquired at the receivers in the downlink back to the transmitter.
There are generally two kinds of CSI feedback schemes applied for MIMO downlink channels in the literature.The fist scheme is called the unquantized and uncoded CSI feedback or analog feedback (AF) in short, where each user estimates its downlink channel coefficients and transmits them explicitly on the feedback link using unquantized quadrature-amplitude modulation [12][19] [24] [25].The performance of the downlink linear zero-forcing beamforming (ZF-BF) scheme with AF was evaluated through simulations in [19] and analytical results were given later in [24] and [25].The second feedback scheme is called the vector quantized CSI feedback or digital feedback (DF) in short, where each user quantizes its downlink channel coefficients using some predetermined quantization codebooks and feeds back the bits representing the quantization index [20]- [27], [37].The MIMO broadcast channel with DF has been considered in [23]- [27].The vector quantization scheme based on the distortion measure of the angle between the codevector and the downlink channel vector was adopted in [23]- [25] and a closed form expression of the lower bound on the achievable rate of ZF-BF was derived.The results there also showed that the number of feedback bits per user must increase linearly with the logarithm of the downlink SNR to maintain the full multiplexing gain.Further, the authors in [26] pointed out that in the scenario where the number of users is larger than that of the transmit antennas, with simple user selection, having more users reduces feedback load per user for a target performance.A different vector quantization approach was adopted in [27] where the quantization is based on the distortion measure of mean-square error (MSE) between the codevector and the downlink channel vector.Under such quantization codebook, the authors analyzed both the linear ZF-BF and nonlinear zero-forcing dirty-paper coding (ZF-DPC) [1] and derived loose upper bounds of the achievable rates with limited feedback.In [37], the authors considered the regularized block diagonalization (RBD) with a random vector quantization (RVQ) for MIMO downlink channels and derived the bound for the throughput.
In this paper, we consider both analog and digital feedback schemes and study the achievable rates of a MIMO broadcast channel with these two feedback schemes, respectively.Different from most of previous work where the analysis was focused on ZF-BF or RBD as introduced above, ZF-DPC is analyzed in our work which is inherently superior to ZF-BF and RBD due to its nonlinear interference precancelation characteristic and is optimal in terms of the sum-capacity as [1] proved.For DF, we adopt the vector quantization distortion measure of the angle between the codevector and the downlink channel vector, and perform RVQ [23]- [25] for analytical convenience.Our main contributions and key findings in this paper are as follows: • A comprehensive analysis of the achievable rates of ZF-DPC with either analog or digital feedback is presented, and closed-form lower and upper bounds on the achievable rates are derived.For fixed feedback-link capacity constraint, the downlink achievable rates of ZF-DPC are bounded as the downlink SNR tends to infinity, which indicates that the downlink multiplexing gain with fixed feedback-link capacity constraint is zero.
• In order to achieve full downlink multiplexing gain, it is sufficient and necessary to scale the average feedback link SNR linearly with the downlink SNR for AF in AWGN feedback channels.While for DF in an error-free feedback link, in order to achieve the downlink multiplexing gain of α M (0 < α ≤ 1), it is sufficient and necessary to scale the feedback bits per user as B = α(M − 1) log 2

P N0
where M is the number of transmit antennas and P N0 is the average downlink SNR.
We note that although ZF-DPC with DF has been considered in [27], our work also differs from it in several aspects.First, a different distortion measure for channel vector quantization is applied in our work compared to that in [27] as stated earlier.Second, a more thorough analysis about the downlink achievable rates and multiplexing gain is presented in this paper than that in [27], covering both AF and DF.We also find out that the feedback scheme with the codebook adopted in [27] is actually equivalent to the AF scheme analyzed in our work, therefore can be easily incorporated into our analysis framework.
The remainder of this paper is organized as follows.We give a brief introduction to ZF-DPC with perfect CSIT in Section II.Comprehensive analysis of achievable rates and multiplexing gain for both AF and DF are presented in Section III and IV, respectively.A rough comparison of AF and DF is also given in Section IV.Finally, conclusions and discussions for future work are given in Section V.
Throughout the paper, the symbols (•) where h i ∈ C 1×M is the complex channel gain vector between the BS and user i, x ∈ C M ×1 is the transmitted signal with a total transmit power constraint P , i.e., E x H x = P , and v i is the complex white Gaussian noise with variance N 0 .For analytical convenience we assume spatially independent Rayleigh fading channels between the BS and the users, i.e., the entries of h i are i.i.d.CN (0, 1), and where In this paper we focus on the case K = M .If K < M , there will be a loss of multiplexing gain.The case K > M will introduce multi-user diversity gain and we will leave it for future work.
We first give a brief introduction of ZF-DPC under perfect CSIT in this section.
In the ZF-DPC scheme, the BS performs a QR-type decomposition to the overall channel matrix H denoted as H = GQ, where G is an M × M lower triangular matrix and Q is an M × M unitary matrix.We let x = Q H d and the components of d are generated by successive dirty-paper encoding with Gaussian codebooks [1], then the resulting signal model with the precoded transmit signal can be written as: From (3) the received signal at user i is given by where g ij = [G] i,j and d i , the i-th entry of d, is the output of dirty-paper coding for user i treating the term j<i g ij d j as the non-causally known interference signal.
From the total transmit power constraint E x H x = P we have E d H d = P .If the transmit power is uniformly allocated to each user, i.e., d i ∼ CN (0, P/M ), then for i.i.d.Rayleigh flat fading channel, the closed form expression of the achievable ergodic sum rate using ZF-DPC is given by [1], [27]: and where dt is the exponential integral function of order n [28]1 .The multiplexing gain [10] of ZF-DPC under perfect CSIT is M , i.e., lim which is the full multiplexing gain of the downlink [1] [23].

III. ACHIEVABLE RATES OF ZF-DPC UNDER ANALOG FEEDBACK
In this section we consider the analog feedback (AF) scheme, where each user estimates its downlink channel coefficients and transmits them explicitly on the feedback link without any quantization or coding.In order to focus on the impact of feedback link capacity constraint, we assume perfect CSI at each user's receiver (CSIR), and no feedback delay, i.e., the downlink CSI is fed back instantaneously in the same block as the subsequent downlink data transmission.For ease of analysis, we also impose two restrictions on the transmission strategy: 1) the total transmit power is equally allocated to the users, and 2) independent Gaussian encoding is applied for each user at the transmitter side.

A. Analog Feedback
The M users estimate and feed back their complex channel coefficients using orthogonal feedback channels.A simplifying assumption of our work is that we consider the AWGN feedback channels, i.e., no fading in the feedback links.Each user takes β f b M (β f b ≥ 1 and β f b M is an integer) channel uses to feed back its M complex channel coefficients by modulating them with a group of orthonormal spreading . Then the received signals of the feedback channel from user i over β f b M channel uses can be written in a compact form: where h i,m ∼ CN (0, 1) denotes the downlink channel gain from the m-th transmit antenna of the BS to user i, the 1 × β f b M vector w f b i with i.i.d.entries each distributed as CN (0, 1) denotes the additive white Gaussian noise on the feedback channel and SNR f b represents the average transmit power (and also the average SNR in the feedback channel).
After despreading, the sufficient statistic for estimating h i,m is obtained as written below: where n i,m is the equivalent noise distributed as CN (0, 1).MMSE estimation is performed to estimate h i,m .We denote the MMSE estimate of h i,m as ĥi,m and the corresponding estimation error h i,m − ĥi,m as δ i,m .Since h i,m ∼ CN (0, 1), ĥi,m and δ i,m are also circularly symmetric complex Gaussian random variables with zero mean, and their variances are: Moreover, ĥi,m and δ i,m are independent from each other.
The vector quantization scheme using the distortion measure of MSE in [27] leads to the same statistics of the channel error as the AF scheme introduced above, so it is equivalent to the AF scheme.Therefore the following analysis framework developed for AF can be readily applied to the case studied in [27].

B. Lower Bound on the Achievable Rate of ZF-DPC With AF
The BS collects the channel estimates ĥi,m (i, m = 1, • • • , M ) to form the estimated channel matrix H, then simply we have the following relationship between H and H [36]: where Here, H and ∆ are mutually independent [36].
The BS performs ZF-DPC treating the estimated channel matrix H as the true one.The QR decomposition of H can be written as H = G Q, where G is a lower triangular matrix and Q is a unitary matrix.The received signal is modeled as: From the above equation we can extract the received signal at user i as listed below: where ĝij = [ G] i,j and ∆ i is the ith row of ∆.
We have the following theorem that gives a lower bound on the achievable ergodic rate of ZF-DPC under AF.
Theorem 1: If the downlink channel is i.i.d.Rayleigh fading and the feedback channels are AWGN channels, then the achievable ergodic rate of ZF-DPC with AF is lower bounded as: where and Proof: We first consider the lower bound on the achievable rate under given H. Recall ( 14) and introduce three notations: Then we have the following signal model: With uniform power allocation among the M users and independent Gaussian encoding, d i ∼ CN (0, P M ), d i and d j (i = j) are independent of each other.So x i and s i are mutually independent, but n i is no longer Gaussian and is not independent of x i , so we cannot directly apply the result of dirty-paper coding in [30] to derive the capacity of this channel.
As s i is still known at the transmitter, from [31] we know that the achievable rate of this kind of channel can be formulated in the form of mutual information as shown below: where u i is an auxiliary random variable.Let u i = x i + αs i where α is called the inflation factor, then where the first "≥" follows from the fact that the entropy is larger than the conditional entropy, and the second "≥" follows from the fact that a Gaussian random variable has the largest differential entropy when the mean and variance of a random variable are given. Since Var Substituting ( 19) into (18) we have Var(xi)+Var(ni) maximizes the right hand side (RHS) of the inequality in ( 21) and thus we get The above inequality shows the lower bound on the achievable rate of user i under given H.In the following we derive closed form expression for the lower bound on the achievable ergodic rate under fading downlink channel.
Since ĥi,m ∼ CN (0, 1 − D i ), H can be decomposed as H = Υ H where the entries of H are i.i.d.
Then by taking the means of both sides of the inequality in (22) the achievable ergodic rate of user i is lower bounded as follows: where and E j (x) is the exponential integral function of order j.The closed form expression of the expectation in (23) follows from the results in [32].
Thus we have completed the proof.

C. Upper Bound on the Achievable Rate of ZF-DPC With AF
An upper bound of the achievable rate is derived by assuming a genie who can provide the encoders at the BS and the decoders at the users with some extra information.This upper bound is referred to as the genie-aided upper-bound.
October 26, 2010 DRAFT Recall equation ( 14) and rewrite it as follows: where qi is the ith column of Assume there is a genie who knows the values of ∆ i qi and |∆ i qm | (∀m = i) and tells these values to the encoder and decoder for user i, then with i.i.d.channel inputs ), n i is Gaussian distributed with zero mean and variance Var(n i ) = m =i |∆ i qm | 2 P/M + N 0 and is independent of x i .Hence the channel for user i in (25) will be recognized as a standard dirty-paper channel and its capacity is log 2 (1 + Var(x i )/Var(n i )) [30].Finally the downlink achievable ergodic rate can be upper bounded by the genie-aided upper bound as given in the following theorem: The achievable ergodic rate of ZF-DPC with AF is bounded by a genie-aided upper-bound as follows: It is difficult to derive a closed form expression for the right-hand side (RHS) in ( 26) so we use Monte Carlo simulations to obtain this upper bound.
We plot the lower and upper bounds on the achievable ergodic sum rate obtained in Theorem 1 and 2 with fixed feedback-link capacity constraint in Fig. 1.We set M = 4, β f b = 1 and SNR f b = 10, 15, 20 dB.Achievable rate of ZF-DPC with perfect CSIT is also plotted.An important observation from Fig. 1 is that there is a ceiling effect on the achievable rate of ZF-DPC under AF if the feedback-link capacity constraint is fixed, i.e., the achievable rate is bounded as the downlink SNR tends to infinity.
This can be explained intuitively that the power of the interference caused by imperfect CSIT always scales linearly with the signal power.A more rigid explanation is given in the following corollary: The achievable ergodic rate of ZF-DPC with AF and fixed feedback-link capacity is upper bounded for arbitrary downlink SNR: where γ is the Euler-Mascheroni constant [34] and . The proof of the corollary is in Appendix A. Although this upper bound is quite loose, it does predict the ceiling effect on the achievable rate with fixed feedback-link capacity.

D. Achievable Downlink Multiplexing Gain With AF
From Corollary 1 it is obvious that the downlink multiplexing gain with fixed feedback-link capacity is zero.In order to maintain a non-zero multiplexing gain, the feedback channel quality should improve at some rate as the downlink SNR increases, which is given in detail in the following theorem:  The curves coincide with the analytical results in Theorem 3. Note that increasing the value of a can further reduce the rate gap between the perfect CSIT case and the AF case.

IV. ACHIEVABLE RATES OF ZF-DPC UNDER DIGITAL FEEDBACK
We now consider digital feedback (DF), where the downlink CSI are estimated and quantized into several bits using a vector quantization codebook at each user and the quantization bits are fed back to the BS.The feedback channel is assumed to be capacity-constrained and error-free, i.e., as long as the number of feedback bits does not exceed the feedback-link capacity in terms of the maximum feedback bits per fading block, the feedback transmission will be error-free [21].We also assume perfect CSIR and no feedback delay as in Section III.Moreover, the same restrictions are imposed on the transmission strategy as in Section III.

A. Digital Feedback
The downlink channel vector h i of user i can be expressed as h i = λ i hi , where λ i h i is the amplitude of h i and hi h i / h i is the direction of h i .Under the assumption that the entries of h i are i.i.d.CN (0, 1), we have λ 2 i ∼ χ 2 2M and hi is uniformly distributed on the M dimensional complex unit sphere [22].Moreover, λ i and hi are independent of each other [22].
The Random Vector Quantization (RVQ) [22][23] is adopted in our analysis due to its analytical tractability and close performance to the optimal quantization.The quantization codebook is randomly generated for each quantization process, and we analyze performance averaged over all such choices of random codebooks, in addition to averaging over the fading distribution.At the receiver end of user i, hi is quantized using RVQ.First, a random vector codebook for user i by selecting each of the N vectors independently from the uniform distribution on the M dimensional complex unit sphere, i.e., the same distribution as hi .The codebooks for different users are also independently generated to avoid the case that multiple users quantize their channel directions to the same quantization vector.The BS is assumed to know the codebooks generated each time by the users.
Then the code vector which has the largest absolute square inner product with hi is picked up as the quantization result, mathematically formulated as follows: The B = log 2 N quantization bits are fed back to the BS.Define ν i | ĥi hH i | 2 and θ i ∠ ĥi hH i , then we introduce two lemmas that are useful for further discussion: Lemma 2 ([33]): θ i is uniformly distributed in the interval (−π, π] and θ i is independent from ν i .
In the next subsection we will find that the information of θ i is necessary for phase compensation at user i's receiver.Therefore, we need to store the value of θ i at user i's receiver.Notice that the norm information of the channel vectors are not conveyed to the BS.

B. Lower Bound on the Achievable Rate of ZF-DPC With DF
Under the assumption that the feedback channel is error-free, the B bits conveyed by each user can be received by the BS correctly.The BS reconstructs the quantized channel vector ĥi using the B bits fed back from user i and treats ĥi as the true channel vector.Then the BS performs ZF-DPC using as did in Section III.B.The QR decomposition of H can be written as H = G Q, where G is a lower triangular matrix and Q is a unitary matrix.The received signal is modeled as: where October 26, 2010 DRAFT At each user's receiver, a phase compensation operation is carried out by multiplying e jθi to the received signal of user i, written in a compact form as follows: where Θ diag{e jθ1 , . . ., e jθM } is a diagonal matrix, w Θv has the same statistics as v.
Denote ∆ i e jθi hi − ĥi , then we can rewrite it in a compact form, i.e., Θ H = H + ∆, where . Equation ( 31) can be rewritten as: From the above equation we can extract the received signal at user i as listed below: We first give three lemmas useful for deriving the lower bound of the achievable rate of ZF-DPC under digital feedback.
Lemma 3: where The proofs of these three lemmas are in Appendix C, D and E, respectively.Then we have the following theorem on the lower bound of the achievable ergodic rate of ZF-DPC under DF.
Theorem 4: If the downlink channel is i.i.d.Rayleigh fading and the feedback channels are error-free, then the achievable ergodic rate of ZF-DPC with DF is lower bounded as: where ψ(x) is the Euler psi function [28] and E{∆ i ∆ H i } is given in Lemma 4.
Proof: Since λ i is known by the receiver of user i, the signal model in (33) can be transformed into: where Using the same methodology as in Section III.B, we arrive at the following inequality for the downlink achievable rate of user i under fixed H and Λ: With Gaussian inputs and uniform power allocation, ).In the digital feedback scheme, the channel norm information is not conveyed back to the BS, i.e., λ i is not known at the BS, so we are not able to adjust α according to Var(x i ) and Var(n i ).We just simply choose α = 1, then R DF i ( H, Λ) is lower bounded by: Substituting (39) into (38) we finally get the following lower bound under fixed H and Λ: Based on the above results, we can derive the lower bound for the achievable ergodic rate in the Rayleigh fading downlink channel.Taking the mean of both sides of the inequality in (40) we have where the second "≥" follows from the Jensen inequality of the concave function.
From Lemma 3, we can calculate the closed form expression for E log 2 |λ i ĝii | 2 : where ψ(x) is the Euler psi function [28].
Since λ 2 i ∼ χ 2 2M , the closed form expression for the third term in (41) can be calculated as shown below: where the closed form expression of E{∆ i ∆ H i } has been obtained in Lemma 4. Substituting (42) and ( 43) into (41) we finally get the conclusion.
Remark: From the above theorem and the monotony of e x E n (x) shown in Lemma 5, we can see that decreasing E{∆ i ∆ H i } will raise the lower bound on the achievable rate.Now we give an explanation on the necessity of the phase compensation operation at each receiver.In the absence of the phase compensation, the channel error vector would be ∆ i hi − ĥi .Then From Lemma 2, θ i is independent from ν i and E {cos θ i } = 0, then and thus the same lower bound remains no matter how many bits are used to quantize hi .Therefore the information of θ i and phase compensation play an important role in the DF scheme, which is different from the case in [23] where no phase compensation is needed.

C. Upper Bound on the Achievable Rate of ZF-DPC With DF
The upper bound on the achievable rate of ZF-DPC with DF can be obtained in a similar way as in Section III.C.
Recall equation (36) and rewrite it as follows: where qi is the ith column of We also assume there is a genie who knows the values of λ i , ∆ i qi and |∆ i qm | (∀m = i), then following the methodology in Section III.C, we can see that the channel for user i is also recognized as a standard dirty-paper channel.Therefore the downlink achievable ergodic rate can be upper bounded by the genie-aided upper bound as shown below: Theorem 5: The achievable ergodic rate of ZF-DPC with DF is bounded by a genie-aided upper-bound, i.e., As Theorem 2, it is difficult to derive a closed form expression for the right-hand side in 47, numerical simulations are also necessary to calculate the upper bound given in Theorem 5.
The lower and upper bounds on the achievable ergodic sum rate obtained in Theorem 4 and 5 with fixed feedback-link capacity constraint are plotted in Fig. 3.We set M = 4 and calculate three groups of curves where the number of feedback bits per user, i.e., B, is 12, 16 and 20 respectively.Achievable rate of ZF-DPC with perfect CSIT is also plotted.The curves in Fig. 3 reveal the ceiling effect on the achievable rate which is just the same as the AF case.
From Theorem 5 we also derive a closed-form upper bound for the achievable rate with DF as shown below.

Corollary 2:
The achievable ergodic rate of ZF-DPC with DF and a fixed number of feedback bits per user B is upper bounded for arbitrary downlink SNR: The proof is in Appendix F. Although this upper bound is quite loose, it does predict the ceiling effect on the achievable rate with fixed feedback bits per user.

D. Achievable Downlink Multiplexing Gain With DF
The multiplexing gain of the downlink with DF and fixed feedback bits per user is zero due to the ceiling effect.In order to maintain non-zero multiplexing gain, the feedback bits per user should scale with the downlink SNR.With Theorem 4 and Corollary 2 we can derive the following sufficient and necessary conditions on the scaling to ensure non-zero and full multiplexing gain: Theorem 6: For DF and error-free feedback channels, assume that the number of feedback bits per user scales according to: then we have the following conclusions: 1) A sufficient and necessary condition for achieving the downlink multiplexing gain of α 0 M (0 < α 0 < 1) is that α = α 0 .
2) A sufficient and necessary condition for achieving the full downlink multiplexing gain of M is that α ≥ 1.
3) If α > 1, then The proof of Theorem 6 is in Appendix G.Note that the same conclusion has been drawn for ZF-BF in [23].Fig. 4 illustrates the conclusions in Theorem 6.We set M = 4 and α = 0.5, 1, 1.5.The curves coincide with the analytical results in Theorem 6.

E. Comparison of AF and DF
In order to compare DF with AF, we need to relate the number of feedback bits B per user with SNR f b and the number of channel uses in the feedback link, or equivalently β f b .In this paper, we make an idealistic assumption that the AWGN feedback link can operate error-free at its capacity, i.e., it can reliably transmit log 2 (1 + SNR f b ) bits per channel use.This assumption describes the maximum possible number of bits that can ever be conveyed correctly through the AWGN feedback channel.In [25], the Then from Theorem 3 we can see that as long as β f b > 1, only a multiplexing gain of less than M can be achieved for AF.This means the DF scheme is asymptotically superior to the AF scheme when It should be noted that the feedback link is assumed error-free, therefore true performance of DF may degrade when some specific digital modulation and coding scheme is used.A thorough comparison when using practical modulation and coding schemes for DF, such as uncoded QAM modulation discussed in [25], is for future work.

V. CONCLUSION AND FUTURE WORK
We have investigated the performance of ZF-DPC in the multiuser MIMO downlink of a FDD system where the CSIT is obtained through capacity-constrained feedback channels.Two CSI feedback schemes, i.e., the analog and digital feedback schemes are considered in our work.Closed-form expressions for lower and upper bounds on the achievable ergodic rates of ZF-DPC with Gaussian inputs and uniform power allocation are derived.Based on the closed-form rate bounds, sufficient and necessary conditions on the feedback channels to ensure non-zero and full downlink multiplexing gain are obtained.Our primary results show that for AF in AWGN feedback channels, it is both sufficient and necessary to scale the average feedback link SNR linearly with the downlink SNR in order to achieve full downlink multiplexing gain.While for DF in an error-free feedback link, in order to achieve the downlink multiplexing gain of α M (0 < α ≤ 1), it is both sufficient and necessary to scale the feedback bits per user as B = where M is the number of transmit antennas and P N0 is the average downlink SNR.We also mention that there are several issues not considered in our work.In this paper we have assumed perfect CSI at the users' receivers.In a practical system, however, there are always channel estimation errors due to finite number of training symbols, which will further degrade the performance of ZF-DPC.
The impact of the feedback delay of the downlink CSI on the achievable rates is also not considered, which could be significant when the downlink channel is fast fading.AWGN and error-free feedback channels are assumed for AF and DF, respectively.Generalized types of feedback channels, e.g., Rayleigh fading feedback channels, need to be considered.For the DF scheme, we apply the RVQ for quantization of the channel vector in order to make the analysis easier.Generalization to arbitrary vector quantization codebooks is an interesting issue and we expect the same conclusions could be drawn.In the analysis of the DF scheme, we made an optimistic assumption that the AWGN feedback link can operate error-free at its capacity.This assumption can be removed by considering practical feedback transmission schemes, such as the uncoded QAM modulation discussed in [25].Throughout the paper, we have assumed that the number of users is equal to the number of transmit antennas.We conjecture that when the number of users is larger than that of transmit antennas and we properly design the user selection scheme, the feedback link quality (average feedback SNR for AF and the number of feedback bits for DF) per user could be less stringent while keeping the same performance.Finally, the analysis of the achievable ergodic rates are carried out with the restrictions of Gaussian inputs and uniform power allocation.Determining whether Gaussian input is optimal and the optimal power allocation scheme under imperfect CSIT is a challenging problem.

APPENDIX A PROOF OF COROLLARY 1
From Theorem 2 we have: From the proof of Theorem 1 we know that: Since ∆ is independent of Ĥ, ∆ i is also independent of ĝii and qi , so As for the term E log 2 |∆ i qm | 2 , because the entries of ∆ i are i.i.d. as CN (0, D i ) and qm = 1, Then we have: where γ is the Euler-Mascheroni constant [34].Substituting (A.2), (A. , a, b 0 > 0. Then the expression of β i is expanded as: Then we can find out that We now introduce two results about the exponential integral functions in [29]: where γ is the Euler-Mascheroni constant.From Theorem 1 and equations (B.3) and (B.4), the lower bound of R AF i will have the following asymptotic behavior: Therefore, From Corollary 1 we have: For the case b 0 > 1, the following holds: Therefore the asymptotic rate gap, i.e., the limit of From the definition of ∆ i we have where (•) stands for the real part of a complex number.We also have Therefore the following holds: Using Lemma 1, we can derive the closed form expressions for E{ √ ν i } as follows: . Moreover, we have the following integral [28]: Substituting the above results into (D.4) we finally get Thus we have completed the proof.

APPENDIX E PROOF OF LEMMA 5
Using the fact that E n (x) = −E n−1 (x) (n ≥ 0) [28], we can calculate the first order derivative as shown below: Using the definition of E n (x) we can expand the term E n (x) − E n−1 (x) as: From Theorem 5 we have For 1 ≤ i ≤ M − 1, we choose m > i, then it holds that ĥi qm = 0.The random variable hi can be written as hi = e −jθi √ 1 − Z • ĥi + √ Zs i where s i and Z are independent, with s i isotropically distributed in the nullspace of ĥi and Z distributed according to the distribution of the minimum of 2 B beta(M − 1, 1) random variables [23].Then |∆ i qm | 2 is given by where |s i qm | 2 is beta(1, M − 2) distributed and independent of Z [23].Then the following inequality holds [23]: For i = M , we choose m < i, then qm is independent of ĥi .The vector ∆ i can be expressed as Since ĥi is an isotropically distributed M -dimensional vector, ∆ i can be seen as the product of ( √ 1 − Z − 1) 2 + Z and an isotropically distributed M -dimensional vector t i which is independent of Z. Then |∆ i qm | 2 is given by respectively, then we have: Since 0 ≤ ν i ≤ 1, using the results about the average quantization error of RVQ in [23], the following holds: We consider the term P For the case that α 0 < 1, ω i → ∞ as P/N 0 → ∞.Substituting (G.(G.5) If α 0 = 1, the following holds: Hence from the monotony of e x E j (x) we have e M N 0 From ( 35) and (G.7) we have: The Euler-Mascheroni constant γ can be given by series [34]: We also have the series representation of the Euler psi function ψ(x) [28]: where the second equality follows from that ln(M − i + 1) = M −i k=1 ln 1 + 1 k .Substituting (G.16) into (G.13)we have:

Theorem 3 :
For AF and AWGN feedback channel, and β f b SNR f b scales as a P N0 b (a, b > 0), then a sufficient and necessary condition for achieving the multiplexing gain of b 0 M (0 < b 0 < 1) is that b = b 0 ; and a sufficient and necessary condition for achieving the full multiplexing gain of M is that b ≥ 1.Moreover, for b > 1, the asymptotic rate gap between the achievable rate of ZF-DPC with perfect CSIT and that under AF is zero as the downlink SNR goes to infinity.The proof of the theorem is in Appendix B.
authors have pointed out that for fair comparison, β f b M feedback channel uses in AF should correspond to β f b (M − 1) feedback channel uses in DF, since no channel norm information is fed back in DF and a system using DF could use one feedback channel symbol to transmit the norm information.Thus, the number of feedback bits per user in the AWGN feedback channel is B = β f b (M − 1) log 2 (1 + SNR f b ).Now we can make a comparison about the performance of AF and DF.Assume for DF, the feedback bits per user scale as B = (M − 1) log 2 P N0 , then from Theorem 6 we know that the multiplexing gain of M can be achieved.Using the relationship between B and SNR f b obtained above, we can derive the connection of SNR f b to the downlink SNR:

Fig. 5
compares the achievable rates under AF and DF for β f b = 1 and 2. As analyzed above, the asymptotic performance of DF is superior to AF when β f b = 2.

log 2
3) and (A.4) into (A.1)we arrive at the conclusion.APPENDIX B PROOF OF THEOREM 3 Sufficient Condition: Denote the RHS of the inequalities in Theorem 1 and Corollary 1 as R low i (P/N 0 ) .(B.1) Let β f b SNR f b = a P N0 b0

Fig. 1 .
Fig. 1.Lower and upper bounds on the achievable ergodic sum rate of ZF-DPC with AF in AWGN feedback channels.
0 (dB) Achievable rates (bits/s/Hz) Achievable rate with perfect CSIT Lower bound on the achievable rate Upper bound on the achievable rate b=0

Fig. 2 .
Fig. 2. Illustration of the achievable downlink multiplexing gain of ZF-DPC with AF in AWGN feedback channels.

B=20Fig. 3 .
Fig. 3. Lower and upper bounds on the achievable ergodic sum rate of ZF-DPC with DF in error-free feedback channels.

Fig. 4 .
Fig. 4. Illustration of the achievable downlink multiplexing gain of ZF-DPC with DF in error-free feedback channels.
is a monotonically decreasing function.
5)Since t i and qm are i.i.d.isotropic vectors in the M -dimensional vector space, the quantity |t i qm | 2 is beta(1, M − 1) distributed and independent of Z. Then similar to (F.4) we have −E log 2 |∆ i qm | 2 ≤ B + log 2 e M − 1 + log 2 (M − 1) + log 2 e.