A New Method of Channel Feedback Quantization
`for High Data Rate MIMO Systems
`
`Mehdi Ansari Sadrabadi, Amir K. Khandani and Farshad Lahouti
`Coding & Signal Transmission Laboratory (www.cst.uwaterloo.ca)
`Dept. of Elec. and Comp. Eng., University of Waterloo
`Waterloo, ON, Canada, N2L 3G1
`Email: {mehdi, khandani, farshad}@cst.uwaterloo.ca
`
`Abstract— In this work, we study a Multiple-Input Multiple-
`Output wireless system, where the channel state information is
`partially available at the transmitter through a feedback link.
`Based on Singular Value Decomposition, the MIMO channel
`is split into independent subchannels, which allows separate,
`and therefore, efficient decoding of the transmitted data signal.
`Effective feedback of the required spatial channel information
`entails efficient quantization/encoding of a Haar unitary matrix.
`The parameter reduction of an n× n unitary matrix to its n2− n
`basic parameters is performed through Givens decomposition.
`We prove that Givens matrices of a Haar unitary matrix are
`statistically independent. Subsequently, we derive the probability
`distribution function (PDF) of the corresponding matrix elements.
`Based on these analyses, an efficient quantization scheme is
`proposed. The performance evaluation is provided for a sce-
`nario where the rates allocated to each independent channel
`are selected according to its corresponding gain. The results
`indicate a significant performance improvement compared to the
`performance of MIMO systems without feedback at the cost of
`a very low-rate feedback link.
`
`I. INTRODUCTION
`Multiple-Input Multiple-Output
`(MIMO) communication
`systems have received considerable attention in response to the
`increasing requirements of high spectral efficiency in wireless
`communications. In fact,
`the capacity of MIMO systems
`equipped with Mt transmit and Mr receive antennas scales
`up almost linearly with the minimum of Mt and Mr in flat
`Rayleigh fading environments [1] [2].
`In recent years, researchers have examined the transmission
`strategies for MIMO systems, in which the transmitter and/or
`the receiver have full or partial knowledge of the Channel State
`Information (CSI). In [3], it has been shown that the achievable
`bit rate when perfect CSI is available both at the transmitter
`and the receiver is significantly higher than that when CSI is
`only available at the receiver. Due to practical restrictions such
`as an imperfect channel estimation and a limited feedback data
`rate, CSI is not perfect at the transmitter. However, unlike the
`single antenna systems, where exploiting CSI at the transmitter
`does not significantly enhance the capacity, in multiple antenna
`systems, the capacity is substantially improved through even
`partial CSI [4].
`When CSI is available at the transmitter of a Multiple-Input
`Single-Output (MISO) system, beamforming can be used to
`exploit transmit diversity through spatial match filtering. In
`the context of MISO systems, several quantization schemes
`
`have been suggested to feed back instantaneous CSI to the
`transmitter. A simple and effective scheme has been suggested
`for a 3G wireless standard [5]. In [6] the authors have designed
`a codebook of beamformer vectors with the objective of
`minimizing the outage probability. Similar works, titled in
`single-substream precoding, have been reported in [7] [8],
`where the codebook design criterion is derived to maximize
`the received Signal to Noise Ratio (SNR).
`For MIMO systems, the problem of quantizing CSI is more
`involved than for MISO systems. In [9] a precoder is combined
`with space-time encoder. The precoder is designed so as to
`reduce an upper bound on the worst pairwise codeword error
`probability conditioned on imperfect CSI at the transmitter.
`In [10], by assuming the availability of partial CSI at the
`transmitter of a MIMO system, a criterion has been presented
`to design a precoder based on the capacity maximization.
`However, [10] has not provided a practical approach to design
`such a precoder when the number of receive antennas is more
`than one. In this paper, we present a technique to address the
`need for a practical feedback scheme for a MIMO system (as
`opposed to a MISO). After we accomplished this work [11],
`we became aware of a similar work in quantizing the spatial
`information of the channel [12]. Specifically, the authors in
`[12] have come up with the same idea of using Givens
`rotations to reduce the number of parameters which need to be
`quantized. They use the time-dependency of the corresponding
`parameters of adjacent frames in slowly time-varying channels
`and employ a differential quantization for each parameter.
`However, in this work, we quantize the parameters of the
`channel’s spatial information frame by frame. The quantization
`design and optimum bit allocation among the quantizers are
`accomplished based on the interference measure we define in
`Section III.
`Consider the situation in which a MIMO channel is split into
`several independent subchannels by means of Singular Value
`Decomposition (SVD) based on the CSI at the transmitter
`and the receiver. This allows independent decoding of the
`subchannels and results in a low decoding complexity. In
`general, the optimum Maximum Likelihood (ML) decoding
`in a MIMO system without feedback is equivalent to a lattice
`decoding problem, which incurs significant complexity. Lower
`complexity decoding algorithms can be devised by the proper
`design of a transmit strategy, e.g., the Bell Labs Layered
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`Space Time system [1]. However, this is achieved at the cost
`of degraded performance
`[13] [14]. This indicates that, in
`addition to the gain in the SNR performance, a reduction in
`the decoding complexity is another important advantage of a
`closed loop MIMO system based on the SVD.
`In this work, the modulation format is selected to match
`the subchannel SNR on each subchannel. In this scheme, the
`spatial information of the channel and the constellation index
`of each subchannel is needed at the transmitter. We develop an
`algorithm to quantize the spatial information of the channel,
`based on minimizing the interference between the subchannels.
`The rate allocation strategy is determined at the receiver and
`fed back to the transmitter by using an efficient low rate
`approach.
`The system model is described in Section II. In Section III,
`the parametrization and statistics of the right singular matrix
`of a Gaussian matrix is discussed. The feedback design is
`developed according to these properties, and the decoding
`strategy at
`the receiver. In Section IV, feedback scheme
`for transferring the rate information of each subchannel is
`discussed. In Section V, the simulation results are presented.
`Section VI concludes the paper.
`
`II. SYSTEM MODEL
`We consider an independent and identically distributed
`block fading channel model. For a multiple transmit antenna
`system with Mt transmit and Mr receive antennas, the model
`leads to the following complex baseband representation of the
`received signal:
`
`y = Hx + n,
`(1)
`where x is the Mt × 1 vector of the transmitted symbols, H
`is the Mr × Mt channel matrix, n is the Mr × 1 zero mean
`Gaussian noise vector with the autocorrelation σ2I where I is
`the identity matrix, and y is the received signal. The power
`constraint of the transmitted signal is defined as E(xx∗) =E I,
`where E represents the expectation and (.)∗
`is the hermitian
`of (.). The elements of the channel matrix H are circularly
`symmetric complex Gaussian distributed with zero mean and
`unit variance.
`The SVD of matrix H is defined as [15]
`∗,
`H = VΛU
`(2)
`where V and U are the unitary matrices, and Λ is a diagonal
`matrix. If U is available at the transmitter and the transmitted
`signal is prefiltered by U, then the received signal is given by
`
`y = HUx + n
`= VΛx + n.
`
`(3)
`
`,
`
`∗
`The receiver filters the received vector y by V
`∗
`y = Λx + n.
`r = V
`Therefore, a MIMO channel with Mt transmit antennas and
`Mr receive antennas is transformed to ‘rank H’ parallel
`subchannels. This transformation substantially reduces the
`decoding complexity. In the transition from (3) to (4), we take
`
`(4)
`
`advantage of the fact that the elements of n are statistically
`∗
`independent, and rotating n by the unitary matrix V
`does not
`change the distribution of the noise.
`As it can be seen in (4), the subchannels provide different
`gains corresponding to Λ. We consider a case in which data
`is transmitted and received separately in each subchannel with
`different rates and with equal energy. It can be shown that the
`use of equal energy maximizes the rate under the assumption
`of continuous approximation for a cubical shaping region
`(subject to a constraint on total energy). This method involves
`the allocation of an appropriate data rate to each subchannel,
`while a certain target error rate on each subchannel is met. By
`this assumption, the transmitter requires the rate information
`of each subchannel, in addition to the right singular matrix of
`the channel.
`
`(5)
`
`III. FEEDBACK DESIGN:CHANNEL SINGULAR MATRIX
`QUANTIZATION
`In the scenario described above, the transmitter needs to
`know the right singular matrix of the channel. We assume that
`a noiseless feedback link from the receiver to the transmitter is
`available. By the SVD of H at the receiver, the unitary matrix
`U is computed, quantized and sent to the transmitter.
`If we assume the quantization error ∆U for U, the received
`signal is
`∗∆Ux + n.
`r = Λx + ΛU
`The quantization scheme is based on minimizing the inter-
`ference between the parallel subchannels, since the receiver
`strategy is to detect the data in each subchannel independently.
`The variance of the interference signal is expressed as follows:
`∗∆Ux(cid:1)2)
`E((cid:1)ΛU
`∗
`
`∗∆Uxx∗∆U
`UΛ)
`= ETr(ΛU
`∗
`∗
`= λETr(∆U∆U
`)
`xx
`= λEETr((cid:1)∆U(cid:1)2),
`(6)
`where E(Λ2) =λ I, E(xx∗) =E I and Tr denotes the trace
`function. In (6), we use the property that the singular values
`of a Gaussian matrix are independent from the corresponding
`singular vectors [16], and also the equality Tr(AB) =Tr (BA).
`As a result, minimizing the mean of the interference power
`leads to the minimization of the Frobenius norm of ∆U. In
`order to minimize the interference, the unitary matrix U should
`be quantized, based on minimizing the expression in (6). In
`the following, we examine the statistical properties of the
`underlying unitary matrices.
`
`A. Statistics of Singular Matrices of a Random Gaussian
`Matrix
`In most analytic studies of MIMO systems, the channel
`between the transmitter and the receiver is assumed to be
`Rayleigh fading. This indicates that the entries of the channel
`matrix are statistically independent and identically distributed,
`and have a complex Gaussian distribution with a zero mean.
`We are interested in the probability distribution of the singular
`matrices1 of the mentioned channel matrix in the space of
`
`1The probability distribution of a matrix is the joint PDF of its elements.
`
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`

`M(n), namely the group of n×n unitary matrices. It is known
`that such a random unitary matrix takes its values uniformly
`from M(n) in the sense of the following property [17].
`
`Theorem 1 Let us assume that U is a singular matrix of a
`random Gaussian matrix. For all V ∈ M(n), the distribution
`of U and VU are the same.
`
`1 ≤ k < i ≤ n are statistically independent of each other.
`Moreover, the PDF of the elements of G(k, i) is
`i − k
`c2(i−k)−1,
`pk,i(c, ∠s) = pk,i(c)p(∠s) =
`π
`0 ≤ c ≤ 1, ∠s ∈ [−π, π].
`The proof is omitted because of the limited space. See [11]
`for the details.
`
`(8)
`
`B. Quantization of Unitary Matrices
`Based on the criterion presented for the quantizer design in
`(6), the distortion measure of the quantizer for matrix U is
`defined as follows:
`
`(9)
`
`1 2
`
`D(U) =
`
`Substituting (7) in (9), we derive the first order approximation
`of D(U) as follows:
`
`D(G(k, i)),
`
`(10)
`
`(11)
`
`k=1
`i=k+1
`where D(G) is defined as follows:
`
`1 2
`
`D(G) =
`
`G =
`
`s
`c
`
`c
`−s∗
`to refer to the non-trivial part of a Givens matrix.
`1) Method A: The basic parameters of the Givens matrix,
`
`(12)
`
`√
`
`,
`
`(13)
`
`ETr((cid:1)U −(cid:4)U(cid:1)2).
`n(cid:5)
`D(U) (cid:4) n−1(cid:5)
`ETr((cid:1)G −(cid:4)G(cid:1)2),
`and (cid:4)G is the quantized version of G. In the following, we use
`(cid:6)
`(cid:7)
`named c and θ = ∠s, are quantized as(cid:4)c and (cid:4)θ, independently.
`The transmitter uses(cid:4)c and (cid:4)θ to construct (cid:4)G as follows:
`(cid:8)
`(cid:9)
`(cid:4)c
`|(cid:4)s|ej(cid:4)θ
`(cid:4)G =
`−|(cid:4)s|e−j(cid:4)θ
`(cid:4)c
`where |(cid:4)s| =
`1 −(cid:4)c2. According to the construction scheme
`in (13) , (cid:4)G is also unitary. It can be easily demonstrated that
`(cid:6)
`(cid:7)
`(c −(cid:4)c)2
`+ E(1 − c2)E(θ −(cid:4)θ)2.
`
`the first order approximation of D(G) is
`D(G) (cid:4) E
`
`1 − c2
`We apply (8) to simplify the following expression,
`E(1 − c2
`1
`2(i − k) + 1
`k,i) =
`(cid:8)
`n(cid:5)
`By applying (10) and (14), and (15), we write,
`
`Such a distribution is called the Haar distribution and the cor-
`responding unitary matrices are called Haar unitary matrices
`[17]. We refer to this property as the right invariance property.
`A complex n × n matrix can be described by 2n2 real
`(cid:2)
`(cid:1)
`parameters. However, the definition of a unitary matrix implies
`there is a dependency between these parameters. The number
`of equations describing this dependency for an n × n unitary
`(cid:2)
`(cid:1)
`matrix is n + 2
`(as the norm of each column is unit and
`every two columns are orthogonal to each other). Therefore,
`a unitary matrix U has n2 = 2n2 − (n + 2
`) independent
`parameters. Here, for the purpose of matrix decomposition
`using SVD, n out of n2 parameters are also redundant, since
`SVD can be performed such that the diagonal elements of U
`in (2) are set to be real. Several different approaches such
`as the Cayley transform, Householder reflection, and Givens
`rotations can be used to parameterize a complex n× n unitary
`matrix U in its n2 − n real parameters [15].
`In this work, we consider the matrix decomposition using
`Givens matrices. Besides their ability to decompose the unitary
`matrix to the minimum number of parameters, the resulting
`parameters are statistically independent (Theorem 2). The
`independence property facilitates the quantization procedure.
`A complex unitary matrix U can be decomposed in terms
`of the products of Givens matrices [15], i.e.,
`
`n 2
`
`n 2
`
`n−1(cid:3)
`
`n(cid:3)
`
`U =
`
`G(k, i),
`
`(7)
`
`k=1
`
`i=k+1
`
`where each G(k, i) is an n × n unitary matrix with two
`parameters, c, and, s. Parameter c is in the position (k, k)
`and (i, i), s is in (k, i) and −s∗
`is in (i, k), k < i. The
`other diagonal elements of the matrix G(k, i) are 1 and the
`remaining elements are zero. Since G(k, i) is a unitary matrix,
`then |c|2 + |s|2 = 1. In this work, we can assume that c is
`real since the SVD operation allows U to be multiplied by an
`arbitrary diagonal unitary matrix
`In the following, the statistical properties of Givens matrices
`corresponding to a Haar unitary matrix U is derived. This will
`be later used to determine the quantization strategy. The key
`point of the codebook design for a Haar unitary matrix is the
`following result2.
`Theorem 2 Let us assume that U is an n × n unitary matrix
`with a Haar distribution which is decomposed into Givens
`matrices as in (7). The set of Givens matrices {G(k, i)} for
`
`(14)
`
`(15)
`
`(16)
`
`D(U) (cid:4) n−1(cid:5)
`
`k=1
`
`+
`
`.
`
`E
`
`1 − c2
`
`k,i
`
`(cid:9)
`(ck,i −(cid:4)ck,i)2
`E(θk,i −(cid:4)θk,i)2.
`(cid:9)
`(ck,i −(cid:4)ck,i)2
`
`i=k+1
`1
`2(i − k) + 1
`(cid:8)
`We design Linde-Buzo-Gray (LBG) quantizers for different
`ck,i and θk,i to minimize
`
`2As we mentioned earlier, after we accomplished this work, we became
`aware of [12] which independently proves a similar result.
`
`E
`
`1 − c2
`
`k,i
`
`,
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`

`

`and,
`
`E(θk,i −(cid:4)θk,i)2,
`
`respectively.
`We utilize dynamic programming to find the optimum
`allocation of bits among the quantizers. We use a trellis
`diagram with B +1 states and n2 − n stages to allocate B bits
`to the quantizers of the independent parameters ck,i and θk,i,
`1 ≤ i < k ≤ n of the n×n unitary matrix. The lth state in jth
`stage corresponds to the distortion caused by the jth parameter
`using l − 1 bits. In the trellis diagram, each branch represents
`the difference between the number of bits corresponding to the
`two ending states on the branch. The search through the trellis
`determines the path with minimum overall distortion and the
`corresponding number of bits for each parameter.
`In this method, we quantize each Givens
`2) Method B:
`matrix as a unit and define a new parameterization for this
`purpose. The non-trivial part of a Givens matrix can be shown
`as follows:
`
`(cid:7)
`
`(cid:6)
`
`ejθ sin(η)
`cos(η)
`−e−jθ sin(η)
`G =
`,
`(17)
`cos(η)
`where 0 ≤ θ ≤ 2π and 0 ≤ η ≤ π. The distortion measure
`for G, relative to a reference matrix with parameters η0 and
`θ0, is
`D0(G) = 1 − E(cos(η) cos(η0) + sin(η) sin(η0) cos(θ − θ0)).
`(18)
`We use the LBG algorithm to determine the regions and
`centroids of the two-dimensional quantizers corresponding to
`various (η, θ). The distortion function is
`
`(cid:10)
`
`M(cid:5)
`
`Rm
`
`D =
`
`Dm(G)p(η, θ)dηdθ,
`
`(19)
`
`V. PERFORMANCE EVALUATION
`
`(cid:8)
`
`(cid:12)
`
`−1
`ηm = tan
`
`(cid:10)
`(cid:10)
`
`Rm
`
`where
`
`and,
`
`γm =
`
`m + γ2m
`ς 2
`Rm cosl+1(η) sin(η)dηdθ
`
`
`
`,
`
`(21)
`
`cosl(η) sin2(η) cos(θ)dηdθ,
`
`(22)
`
`Rm
`
`cosl(η) sin2(η) sin(θ)dηdθ,
`ςm =
`(23)
`and l = 2(i − k) − 1, in the case of quantizing G(k, i) in (7).
`By applying the above algorithm, we design codebooks of the
`matrices for different rates. In this method, a trellis diagram
`with the same structure as method A trellis diagram is used
`for optimum bit allocation. The trellis diagram contains n2−n
`stages, each corresponds to a Givens component of an n × n
`unitary matrix, and B + 1 states (B is the number of bits).
`
`2
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`The lth state in jth stage corresponds to the distortion caused
`by the jth Givens matrix using l − 1 bits.
`
`IV. FEEDBACK DESIGN:ENCODING OF RATE ALLOCATION
`INFORMATION
`
`Besides the quantized right singular matrix of the channel
`that is fed back to the transmitter, information pertaining the
`rate that will be allocated to each subchannel is also fed
`back. This indicates a set of Mt indices from a set of NR
`predetermined rates, e.g., the different modulation schemes.
`Obviously, the total rate is bounded, and since we can perform
`the SVD of the channel matrix so that the singular values
`become ordered, the Mt indices correspond to an ordered
`set of increasing positive integers (rates). To encode this
`information, we can use a trellis diagram with NR states and
`Mt stages. The states correspond to the set of possible rates
`in an increasing fashion, and there is a branch from each state
`to another state in the next stage, only if the entering state is
`located at the same or at a lower level position. Each path in
`the trellis then corresponds to a set of subchannel rates, whose
`index is chosen by the receiver and fed back to the transmitter.
`The trellis structure exploits the ordering property of the rates,
`and therefore, allows their efficient coding at a rate of [18]
`Mt + NR − 1
`NR − 1
`in
`The complexity of this algorithm is very low and is,
`fact, proportional to the number of states. Similar structures
`have been used to address the points of a block-based trellis
`quantizer in [18], or a pyramid vector quantizer in [19].
`
`(cid:7)(cid:14)
`
`.
`
`(24)
`
`(cid:13)
`
`(cid:6)
`
`Rrate =
`
`log2
`
`In this section, we present the performance results of the
`system, described in Section II. We assume that the precoding
`is performed by the quantized version of the right singular
`matrix of the channel by applying the quantization methods
`presented in Section III-B. For the different subchannels, we
`use different modulation schemes. The process of selecting
`the appropriate modulation scheme for each subchannel is
`accomplished at the receiver. We restrict the system to transmit
`t on each transmit antenna. It means
`data with the power
`that the power is equally distributed among data symbols,
`since we use an orthonormal precoder. Therefore the rate is
`maximized based on continuous approximation concept. At the
`receiver, the channel state information and the instantaneous
`quantization noise power is assumed to be available. For
`each subchannel,
`the probability of error is computed for
`different modulation schemes. The receiver selects a modu-
`lation scheme for each subchannel that achieves the target Bit
`Error Rate (BER) of the system and sends the indices of the
`corresponding modulation schemes to the transmitter through
`the feedback channel that was described in Section IV. The
`received SNR at the kth subchannel is,
`Eλ2
`
`Mt(σ2 +(cid:4)σ2
`
`k
`
`k)
`
`,
`
`(25)
`
`EM
`
`SNRk =
`
`m=1
`where Rm is the mth quantization region and M is the number
`of quantization partitions. The centroid (ηm, θm) is determined
`iteratively by minimizing the distortion function in the region
`Rm,
`ςm
`−1(θm = tan
`γm
`
`
`
`(cid:11)
`
`),
`
`(cid:9)
`
`(20)
`
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`

`

`where (cid:4)σ2
`
`k is the corresponding quantization noise variance of
`the kth subchannel. We consider a set of QAM modulation
`formats. At the receiver, the rate rk of kth subchannel is
`computed as follows,
`
`(cid:9)
`
`,
`
`(26)
`
`(27)
`
`rk,
`
`(cid:8)(cid:15)
`
`Q
`
`(cid:12) ∞
`
`max
`P (SNRk)≤Pb
`where Pb is the target BER of the system and P (SNR), the
`BER function of the modulation scheme with rate r, is [20]
`P (SNR) ≈ 4
`3rSNR
`2r − 1
`r
`e− x2
`where Q(x) = 1√
`2 dx.
`x
`2π
`The SVD and Givens decomposition are performed at the
`receiver. The number of computations required by the SVD
`and Givens decomposition for an n × n matrix are 21n3 and
`3n2(n − 1) flops, respectively [15].
`
`the system, if perfect channel information is available at the
`transmitter, is also depicted.
`
`VI. CONCLUSION
`In this work, we have presented efficient methods for the
`channel information quantization in a high data rate MIMO
`system. We have developed efficient algorithms for the quan-
`tization of the underlying unitary matrices. Also, we have pre-
`sented a low rate indexing of rate allocation information. The
`simulation results show a significant improvement compared
`to MIMO systems without feedback at the cost of a very low-
`rate feedback link.
`
`REFERENCES
`[1] G. J. Foschini and M. J. Gans, “On the limits of of wireless commu-
`nications in a fading environment,” Wireless Pres. Commun., vol. 6,
`pp. 315–335, Nov. 1998.
`[2] E. Telatar, “Capacity of multi-antenna Gaussian channels,” Bell Labs
`Journal, vol. 10, Nov/Dec 1999.
`[3] E. Biglieri, G. Caire, and G. Taricco, “Limiting performance of block
`fading channels with multiple antennas,” IEEE Trans. on Information
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`
`Rate Alloc. perfect CSI at the transmitter
`VBLAST
`Open Loop (Maximum Likliehood)
`Rate Alloc. 27 bits (Method B)
`Rate Alloc. 27 bits (Method A)
`
`20
`
`18
`
`16
`
`14
`
`12
`
`10
`
`8
`
`6
`
`4
`
`2
`
`Bite Rate
`
`0
`10
`
`15
`
`20
`
`SNR
`
`25
`
`30
`
`The average bit rate for different schemes where Mt = 3 and
`Fig. 1.
`Mr = 3. The target BER= 5 × 10−3.
`
`Figure 1 shows the average bit rate versus SNR for different
`MIMO systems with Mt = 3 and Mr = 3 at the target BER=
`5 × 10−3. We use 8 bits for the feedback of the right singular
`matrix. The modulation schemes we use are QAMs with bit
`rates between 1 and 7, inclusively, and then NR = 8. We use
`6 bits for the feedback of the rate allocation vector in each
`transmission block (the number of bits is derived by applying
`(24)). The two quantization methods presented in Section III-B
`are compared. Method B outperforms method A at the cost of
`complexity. The average bit rate of a 3×3 MIMO system with
`ML decoding is depicted. It can be seen that the performance
`gain, compared to the gain of the ML decoding of the open
`loop system is noticeable. For example, at the bit rate= 10 the
`system has a 3 dB improvement in comparison to the optimum
`open loop system. We also compare the performance of this
`system with that of a V-BLAST system which is proposed
`as a solution to overcome the complexity problem. Figure 1
`displays a significant improvement in comparison to the V-
`BLAST at the price of the feedback. The performance of
`
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