Analysis of vector quantizers using transformed codebook with application to feedback-based multiple antenna systems

Transformed codebooks are often obtained by a transformation of a codebook, potentially optimum for a particular set of statistical conditions, to best match the statistical environment at hand. The procedure, though suboptimal, has recently been suggested for feedback MISO systems because of their simplicity and effectiveness. We first consider in this paper the analysis of a general vector quantizer with transformed codebook. Bounds on the average distortion of this class of quantizers are provided to characterize the effects of sub-optimality introduced by the transformed codebook on system performance. We then focus our attention on the application of the proposed general framework to providing capacity analysis of a feedback-based MISO system over correlated fading channels using channel quantizers with both optimal and transformed codebooks. In particular, upper and lower bounds on the channel capacity loss of MISO systems with transformed codebooks are provided and compared to that of the optimal quantizers. Numerical and simulation results are presented which confirm the tightness of the theoretical distortion bounds.


INTRODUCTION
Communication systems using multiple antennas have recently re ceived much attention due to their promise of providing significant capacity increases. The performance of the multiple antenna sys tems depends heavily on the availability of the channel state infor mation (CSI) at the transmitter (CSlT) and at the receiver (CSIR). Most of the MTMO system design and analysis adopt one of two extreme CSTT assumptions, complete CSJT and no CSJT. Tn this pa per, we consider systems with CSI assumptions in between these extremes. We assume perfect CSIR is available at the receiver, and focus our attention on MTMO systems where CST is conveyed from the receiver to the transmitter through a finite-rate feedback link. Recently, several interesting papers have appeared, proposing de sign algorithms as well as analytically quantifying the performance of the finite-rate feedback multiple antenna systems.
Most past works on the analysis of finite-rate feedback MIMO systems have adopted one of three approaches. The first is to ap proximate the channel quantization region corresponding to each code point based on the channel geometric property. Mukkavilli et. al. [1] derived a universal lower bound on the outage probability of quantized MTSO beamforming systems with arbitrary number of transmit antennas t over i.i.d. Rayleigh fading channels. Love and et. al. [2] related the problem to that of Grassmannian line pack ing and provided corresponding performance bounds of multiple antenna systems with finite-rate feedback. The second approach is based on approximating the statistical distribution of the key ran dom variable that characterizes the system performance. This ap proach was used by Xia et. al. in [3] and Roh et. al. in [4], where the authors analyzed the performance of MTSO systems over i.i.d. Rayleigh fading channels, and obtained closed form expressions of the capacity loss (or SNR loss) in terms of feedback rate B and an tenna size t. The third approach adopted by Narula et. al. in [5] is based on relating the quantization problem to the rate distortion the ory, where the authors obtained an approximation of the expected loss of the received SNR due to finite rate quantization of the beam forming vectors in an MISO system. Moreover, Love and Heath in [6] and Xia et. al. in [3] extended the beamforming codebook design algorithms to correlated MTMO fading channels by using transformed codebooks obtained by a rotation-based transformation on an optimum codebook designed assuming i.i.d Rayleigh fading channels.
Most of the analytical results available to date are case specific and limited to i.i.d. MISO channels, and the approaches are hard to extend to more complicated schemes. Tn this paper, we con sider the analysis of CST-feedback-based multiple antenna system from a source coding perspective. We do this by using the general framework developed in [7] wherein channel quantization is formu lated as a general vector quantization problem with encoder side information, constrained quantization space and non-mean-squared distortion function. The analysis was developed for optimal quan tizers with perfect statistical knowledge. This paper first extends the general distortion analysis to sub-optimal quantizers with trans formed codebooks. Distortion bounds of this class of quantizers are provided to characterize the effects of sub-optimality introduced by the transformed codebook on system performance. As an utiliza tion of the general framework, this paper further investigates the effects of finite-rate CSI quantization on MISO systems over cor related fading channels. Tn particular, upper and lower bounds on the system capacity loss due to the finite-rate channel quantization are provided for MISO systems with transformed codebooks. Per formance comparisons between MISO CSI quantizers with optimal and transformed codebooks are also provided under different chan nel correlations. Numerical and simulation results are presented which confirm the tightness of the theoretical distortion bounds.

BACKGROUND INFORMATION
The finite-rate feedback-based multiple antenna system can be for mulated as a generalized fixed-rate vector quantization problem [7] and analyzed by adapting tools from high resolution quantization theory. In this section, we briefly describe the generalized high rate quantization theory provided in [7]. Extension and application of the distortion analysis to CST-feedback-based MTSO systems in the context of correlated channels are provided in later sections.

General Vector Quantization Framework
The multiple antenna systems with finite-rate feedback can be mod eled as a generalized vector quantization problem with additional attributes such as encoder side information, constrained quantiza tion space and non-mean-squared distortion measures. To be spe cific, the source variable x = (y, z ) is a two-vector tuple with vec tor y E Q representing the actual quantization variable of dimension kq and z E ;Z being the additional side information of dimension kz.
The encoder side information z is available at the encoder but not at the decoder. Based on a particular source realization x, the en coder (or the quantizer) represents vector y by one of the N vectors Yl, Y2, ... ,Y N, which form the codebook. The encoding or the quantization process is denoted as Y = Q(y, z ) . The distortion of a where DO (y, y; z) is a general non mean-squared distortion func tion between y and y that is parameterized by z. It is further as sumed that function DO has a continuous second order derivative (or Hessian matrix w.r.t. to y) Wz(y) with the (i,j/ h element given by (2) y, YJ y=y

Distortion Analysis of the General Vector Quantizer
Under high resolution assumptions, large N, the average asymptotic distortion can be represented by the following form, which is similar to the Bennett's integral provided in [8] D=2 -�� r r I (y; z;lEz(Y))p(y, Z) A(y ) --lq d y d z, (3) .Jz.J Q where IEz(y) denotes the asymptotic projected Voronoi cell that contains y with side information z. In equation (3), A(y) is a func tion representing the relative density of the codepoints, which is called point density, such that A(Y) d y is approximately the frac tion of quantization points in a small neighborhood of y. Function I ( y; z; IE ) is the normalized inertia profile that represents the rela tive distortion of the quantizer Q at position y conditioned on side information z with Voronoi shape IE. Both A(Y) and I(y; z; IE ) are the key performance determining characteristics that can be used to analyze the effects of different system parameters, such as source distribution, distortion function, quantization rate etc., on the finite rate q uantizer.
Note that if the source variable (vector) y is further subject to kc constraints given by the vector equation g(y) = 0, the asymp totic distortion integral given by (3) is still valid under some minor modifications. In these cases, the actual degrees of freedom of the quantization variable reduce from kq to k� = kq -kc, and the average asymptotic distortion decays exponentially with rate 2 -2B/k� .

QUANTIZERS WITH TRANSFORMED CODEBOOK
In certain situations, the underlying source distribution p(y, z) or the distortion function DO may vary continuously during the quan tization process. However, it is practically infeasible to design sep arate codebooks optimized for every different source distribution and distortion function. In these cases, quantizers constructed by transforming another code book based on the current statistical dis tribution of the source variable is a promising alternative.

Problem Setup
It is first assumed that all the code books are generated from one fixed codebook Co which is designed to match source distribu tion Po (y, z), and distortion function Do, O with sensitivity matrix Wo,z(Y). Codebook Co has a point density given by AO(y), and a normalized inertial profile Io(y;z;IEO, z(Y)) that is optimized to matches the distortion function Do, O' with lEo, z (y) representing the asymptotic Voronoi cell that contains y with side information z. If the source distribution changes from PO (y, z) to p(y, z) and the distortion function becomes DO instead of Do, 0 with sensitivity matrix W z (y) instead of W 0 , z(y), the encoder and decoder will correspondingly adopt a transformed codebook C from Co by a gen eral one-to-one mapping F ( . ) with both of its domain and codomain in space 1Ql, i. e.
Two types of sub-optimality arise when the transformed codebook is used instead of the optimal one. One comes from the sub-optimal point density Atr(Y) of the transformed codebook, which can be derived from AO(y) by the following transformation AO (F-l(y)) Atr(Y) = IFd (F-l(y)) I ' F ( ) = 8F(y) d y 8y ' (5) If the source variable is subject to kc constraints given by the vector equation g(y) = 0, the transformed point density is given by where V 2 (y) is an orthonormal matrix with its columns constitut ing an orthonormal basis for the orthogonal compliment of the range space R ( � g(y) ) . Compared to the optimal point density A * (y) that matches to p(y, z) and DO, Atr (y) given by equation (5) is sub optimal and will lead to performance degradation. However, given no restrictions on the transformation, there always exists an F that makes Atr(Y) exactly equal to A*(y).
The other sub-optimality arises from the fixed location of the code points in the transformed codebook C, in the sense that the Voronoi shape of the transformed codebook does not match the dis tortion function DO and hence is not optimized to minimize the inertial profile. However, the Voronoi region IEtr, z(Y) of the trans formed code book is hard to characterize and depends both on the transformation F as well as the distortion function DO ' Fortunately, the approximated inertial profile Yr. r (F(y) ; z ) of the transformed code book can be upper and lower bounded by equation (7). Fur thermore, if the source variable is subject to kc constraints given by the vector equation g(y) = 0, the constrained inertial profile Ietr (F(y); z) can be similarly bounded by equation (8).

Distortion Integral of Transformed Codebooks
By substituting the transformed point density (5) and the bounds of the transformed inertial profile given by (7) into the distortion inte gration (3), we can upper and lower bound the asymptotic system distortion of a transformed quantizer by the following form   (6) and (8) into (3), the asymptotic distor tion bounds of a constrained quantizer with transformed codebook can also be obtained.
Similar to the case of the conventional product transformed code [9], there exist trade-offs between the two sub-optimalities: point density loss and Voronoi shape loss. To be specific, it is always possible to find a transformation F(·) such that the trans formed point density Atr(Y) matches exactly to the optimal point density A * (y). However, by doing so, the transformation may cause severe "oblongitis" of the Voronoi shape in some cases, which will lead to significant increment of the normalized inertial profile. Therefore, a compromised transformation that optimally trades off the two losses should be employed. This tradeoff is directly reflected in the distortion bound Dtr-upp that both hrupp (y; z ) and Atr(Y) in integration (9) depend on the transformation F(·).

OPTIMAL MISO CSI QUANTIZER
By utilizing the high-rate distortion analysis provided in Sec tion 2.1, this section provides a detailed investigation of the capacity loss of a finite-rate quantized MISO beamforming system over cor related fading channels.

System Model
We consider an MTSO system with t transmit antennas, one sin gle receive antenna, signaling through a frequency flat block fad ing channel. The channel impulse response h is assumed to be perfectly known at the receiver but partially available at the trans mitter through CST feedback. It is assumed that there exists a fi h H E C 1 x t is the MTSO channel response with distribution given by h � Nc(O, �h), and vector v is the channel directional vector given by v = h/l l hl l . The transmitted signal s is normalized to have a power constraint given by E [s 2 ] = p, with p representing the average signal to noise ratio at each receive antenna.
The performance of a finite-rate feedback MISO beamforming system can be characterized by the capacity loss CLoss. which is the expectation of the instantaneous mutual information rate loss Cdh, v) due to the finite rate quantization of the transmit beam forming vector. This performance metric was also used in [4] and is defined as

MISO Systems with Optimal CSI Quantization
By employing the general framework described in Section 2.1, the finite-rate quantized MTSO beamforming system can be formulated as a general fixed rate vector quantization problem. Specifically, the source variable to be quantized is the channel directional vector v of kq = 2t real dimensions. and the encoder side information is the channel power a = I l h1 1 2 . Moreover. under the norm and phase constraints, i.e. v is a unit norm vector and is invariant to arbitrary phase rotation ejl}, the actual free dimensions of vector v is re duced from kq to k� = 2t -2. The instantaneous capacity loss due to effects of finite-rate CST quantization is taken to be the system distortion function DO(v, v; a), which is given by the following form according to the definition given by (11) By utilizing the distortion analysis provided in [7], the normal ized inertia profile of the MTSO system is tightly lower bounded by where I t is a constant coefficient equal to I t = 7rt -1 / (t -I)!. The minimal distortion of the MISO system is hence achieved by using a codebook with an optimal point density given by where 2FO is the generalized hypergeometric function, and fJ 1 is a normalization constant that only depends on the antenna size t, channel correlation matrix �h and system SNR p. The average system distortion (or capacity loss) of the quantized MISO system is then tightly lower bounded by

MISO CSI QUANTIZER WITH TRANSFORMED CODEBOOK
In practically situations, it is impossible to design different code books optimized for every instantiation of the channel covariance matrix and use them adaptively. Tn these situations, it is convenient to use a channel quantizer whose codebook is generated from a fixed pre-designed code book through a transformation parameterized by the channel covariance matrix.

Problem Setup
To be specific. suppose Co is the optimal codebook designed for the i.i.d. MTSO fading channels. When the elements of the fading chan nel response h are correlated, i.e. h � NC(�h), it is evident that co de book Co is no longer optimal. In order to compensate the mis match between Co and the current channel statistics, a transformed co de book C generated by a one-to-one mapping from codebook Co given by equation (4) can be used. Optimization of the transforma tion F(·) turns out to be difficult. and hence a simple sub-optimal transformation, F(v) = Gv/I I Gvl l , was proposed in [3] [6] where G E c t x t is a fixed matrix depends on the channel covariance matrix �h '

Capacity Loss Analysis of Transformed MISO Quantizers
First of all, according to the codebook transformation given by (16), the transformed point density function Actr(V) can be obtained as the following form. from equation (6).
which is equivalent to the PDF of a unit-norm complex vec tor x/l l xl l with x having complex Gaussian distribution x � Nc(O, �). It is evident that the transformed point density given by (17) does not match to the optimal point density function A * (v) given by (14) in the general case. However. for MTSO systems with a large number of antennas and in high-SNR and low-SNR regimes.
it can be shown that the optimal point density A * (v) reduces to be the source distribution p(v) of the directional vector v, given by 1 In this case, by choosing matrix G as a product G = U A 2" with matrices U and A form the eigen-value decomposition of the channel covariance matrix � h = U A U H , one can generate a trans formed code book with optimal point density Ae- Hence, there is no distortion loss caused by the point density mismatch, although the system suffers from the oblongitis of the Voronoi shape (or the shape loss). By substituting the transformation given by (16) into equation (8), the inertial profile of the transformed codebook can be upper and lower bounded by the following form where fe-opt ( v; Do) is the optimal inertia profile given by equation (13). It is evident from (19) that except unitary rotations of the i.i.d. codebook, any non-trivial transformation of the codebook will lead to mismatched Voronoi shape and hence causes inertial profile loss. Therefore, a code book transformation that compromises both the point density loss and the inertial profile loss is favored. However, the optimization of the overall distortion w.r.t. matrix G is diffi cult and beyond the scope of this paper. As a special case, when the transformation is chosen to match the point density only, i.e., 1 G = U A 2" , the average distortion bounds of a MISO system with transformed code book can be obtained (21) where the coefficients /32 and /33 are given by

Comparison with Optimal CSI Quantizers
Interestingly, in high-SNR and low-SNR regimes with a large num ber transmit antennas t, the average system distortion of CSI quan tizers with transformed codebook can be upper and lower bounded by some multiplicative factors of the optimal quantization distor tion, which is represented in equations (24) and (25). Note that the constant coefficients q and C2 can be viewed as the upper bounds of the penalty paid for using a transformed code book instead of op timal design. As verified by the numerical example shown in Sec tion 6, q and C2 are slightly greater than 1 for most channels that are not "highly" correlated. This means that the intuitive choice of F given in [3] [6] is a fairly good solution especially for cases when the channel covariance matrix has a relative small condition number (for channels not so "correlated"). Detr-Upp given by (21) as well as the system distortion by using the optimal code books are also included in the pl � . It can be observed from Fig. 2 that the distortion lower bound Dctr-Low is tight and the performance of the CSI quantizer with transformed codebook is close to that of the optimal codebooks. In order to see the effects of channel correlation on the system performance, we also plot in Fig. 3    from the plot that the performance degradation caused by the trans formed codebook is less than 10% in low-SNR regimes and 22% in high-SNR regimes for MISO systems with more than 10 transmit antennas.

CONCLUSION
We first investigated in this paper a general vector quantizer with transformed codebook. Bounds on the average distortion of this class of quantizers were provided to characterize the effects of sub optimality introduced by the transformed codebook on system per- Numer of Transmit Antennas, t Figure 4: Demonstration of the tightness of the distortion bounds.
formance. As an application of the proposed general framework, we provided in this paper a capacity analysis of the feedback-based MISO systems over correlated fading channels using channel quan tizers with both optimal and transformed codebooks. To be specific, upper and lower bounds on the channel capacity loss of MISO sys tems with transformed code books were provided and compared to that of the optimal quantizers. Numerical and simulation results were presented and further confirmed the tightness of the theoreti cal distortion bounds.