An Efficient Feedback Method for MIMO Systems
`with Slowly Time-Varying Channels
`
`June Chul Roh and Bhaskar D. Rao
`Department of Electrical and Computer Engineering
`University of California, San Diego
`La Jolla, CA 92093–0407, USA
`Email: jroh@ece.ucsd.edu, brao@ece.ucsd.edu
`
`Abstract— The capacity of a multiple-input multiple-output
`(MIMO) channel can be improved if the transmitter has knowl-
`edge of channel. In this paper, we propose an efficient and practi-
`cal feedback method based on parameterization and quantization
`of channel parameters. The spatial information of channel at
`transmitter, which is represented as a matrix with orthonormal
`columns, has a geometrical structure. In parameterization, the
`geometrical structure is exploited to extract a set of parameters
`that has a one-to-one mapping to the original matrix. In slowly
`time-varying channels, the parameters are also found to be
`smoothly changing in time. We employ adaptive Delta modulation
`to quantize and feed back each parameter. The results show that
`the proposed feedback scheme has a channel tracking feature
`and achieves a capacity very close to the perfect feedback case
`with a reasonable feedback rate.
`
`I. INTRODUCTION
`
`Multiple transmit and receive antenna system is considered
`as a strong candidate for future wireless systems because of
`potential improvement in channel capacity and link perfor-
`mance. A multiple antenna channel provides different capac-
`ities under different channel state information (CSI) assump-
`tions. The two common CSI assumptions are i) complete CSIT
`(channel state information at the transmitter) where perfect
`channel information is known to both the transmitter and the
`receiver, e.g., [1], [2]; and ii) no CSIT where perfect channel
`information is available only at the receiver, e.g., [1]. The
`former case, of course, provides a higher channel capacity than
`the latter, but the gain comes at an expense of the transmitter’s
`perfect knowledge of MIMO channel. However, since in
`many applications the channel information is provided to the
`transmitter through a dedicated feedback channel, it is almost
`impossible for the transmitter to have perfect information in
`time-varying channels. Many previous studies considered the
`above two extreme CSIT assumptions, and there are only a
`few studies dealing with how to feed back the MIMO channel
`information. Some researchers have worked on feedback of
`channel information in vector form, for example, for multiple-
`input single-output (MISO) channels [3] and for the principal
`eigen-mode of MIMO channels [4]. Onggosanusi and Dabak
`[5] studied feedback of matrix channel information for MIMO
`channels. They introduced a feedback scheme where among a
`set of unitary matrices for the channel spatial information, an
`index of the matrix minimizing error probability is fed back
`to the transmitter.
`
`The purpose of this paper is to provide a general framework
`for quantization of MIMO channel information and to develop
`a practical feedback method for slowly time-varying channels.
`The CSIT consists of the spatial information of channel and
`the power allocation over spatial channels. We first focus
`on quantization of the spatial
`information which can be
`represented as a matrix with orthonormal columns (a unitary
`matrix is an example). We notice a geometrical structure in
`the matrix. For example, the columns of a t× t unitary matrix
`V = [v1, . . . , vt] are all on the unit-norm sphere St ⊂ Ct
`and mutually orthogonal, i.e., v1 ∈ St, v2 ∈ (St ∩ v⊥
`1 ),
`
`
`v3 ∈ (St ∩ v⊥1 ∩ v⊥2 ), and so on, where v⊥
`is the orthogonal
`i
`complement of the space spanned by vi. In this paper, the
`geometrical structure is exploited in quantizing the spatial
`information. In particular, from the matrix with orthonormal
`columns, we extract a set of essential parameters that has a
`one-to-one mapping to the original matrix. The number of
`parameters equals the degree of freedom in the matrix. Then,
`instead of quantizing the original matrix, the parameters are
`quantized and fed back to the transmitter, and an approximate
`(quantized) version of spatial information is reconstructed at
`the transmitter. Although jointly quantizing the parameters
`(vector quantization) could be better choice, this paper consid-
`ers quantizing each parameter independently (scalar coding)
`because of its low complexity. More specifically, adaptive
`Delta modulation (ADM) [6] is employed from an observation
`that, in slowly time-varying channels, the extracted parameters
`are also smoothly changing. ADM is a practical low-rate scalar
`coding scheme that can track time-varying channels efficiently.
`We use the following notations. A†
`and AT indicate the con-
`jugate transpose and the transpose of matrix A, respectively.
`In is the n × n identity matrix and 0m,n means the m × n
`zero matrix. diag(a1, . . . , an) is a square diagonal matrix with
`a1, . . . , an along the diagonal. The 2-norm of vector v is
`denoted by (cid:4)v(cid:4). E[· ] represents the expectation operator, and
`CN (µ, Σ) is circularly symmetric complex Gaussian random
`vector with mean µ and covariance Σ.
`
`II. SYSTEM MODEL AND MUTUAL INFORMATION
`A. Channel Model
`We consider a multiple antenna system with t antennas at
`the transmitter and r at the receiver. Assuming slow flat-fading,
`the MIMO channel is modeled by the channel matrix H ∈
`
`WCNC 2004 / IEEE Communications Society
`
`760
`
`0-7803-8344-3/04/$20.00 © 2004 IEEE
`
`Page 1 of 5
`
`SAMSUNG EXHIBIT 1008
`
`

`

`Cr×t. That is, the channel input x ∈ Ct and the channel output
`y ∈ Cr have the following relationship:
`
`y = Hx + η
`(1)
`where η ∈ Cr is the additive white Gaussian noise vector
`distributed by CN (0r,1, Ir). We denote the rank of H by m.
`And, the singular value decomposition (SVD) of H is given
`H, where UH ∈ Cr×r and VH ∈ Ct×t are
`by H = UHΣH V †
`unitary matrices and ΣH ∈ Rr×t contains the singular values
`σ1 ≥ . . . ≥ σm > 0 of H. We impose a constraint on the
`transmit power by E[x†x] ≤ PT .
`We assume that in all cases perfect CSI is known to the
`receiver. And, the first n (0 ≤ n ≤ m) columns of VH are
`to be quantized and fed back to the transmitter as channel
`spatial information. When we consider perfect feedback, i.e.,
`no quantization error, this setting includes the two extreme
`cases: i) n = m is the case that the transmitter has same
`spatial information as in the complete CSIT case; and ii) n = 0
`accounts for no spatial information at the transmitter as in the
`no CSIT case. And, when 0 < n < m, it corresponds to
`partial CSIT of [7], [8]. For notational convenience, let us
`define V = [v1, . . . , vn] where vi is the i-th column vector of
`VH.
`The CSIT consists of the spatial information of channel
`and the power allocation information. The matrix V conveys
`the spatial information that is needed at the transmitter. In
`[7], we discussed a multiple-antenna system concept in which
`the optimal power allocation is calculated at
`the receiver
`(cid:1)
`and provided to the transmitter as additional CSI. The power
`allocation information is represented by a real vector γ = [γi]
`i γi = 1 and 0 ≤ γi ≤ 1.
`where
`
`B. Feedback System Model
`
`This subsection describes a feedback system model for time-
`varying MIMO channels that accounts for the discrepancy
`between the real channel and the CSI at the transmitter. It
`will be used in performance evaluation in Section V. Figure 1
`depicts the block-fading model and the frame structure of the
`feedback system model. We assume that the channel matrix
`is not changing during a time block, which will be called
`channel block (with length TC). The channel matrix at k-
`th channel block is denoted by H[k], and V [k] and γ[k]
`are the corresponding CSI. The quantized version of the CSI
`( ˆV [k] and ˆγ[k] in the figure) is provided to the transmitter
`at a feedback rate of RF times per second via an error-free
`feedback channel. The time frame between two consequent
`channel updates is called feedback frame (with length TF =
`1/RF ). For simplicity, we assume there are M (an integer)
`channel blocks in a feedback frame, i.e., TF = M TC. In
`addition,
`in order to model composite delay, e.g., due to
`processing and propagation, we introduce an integer parameter
`D: at the starting point of each frame, the CSI corresponding
`to the D previous channel block is available at the transmitter.
`Figure 1 is an example when D = 1. The CSIT is used in
`transmission during the frame before the next update arrives.
`
`TC
`
`Channel
`
`H[1]
`
`. . .
`
`H[M]
`
`H[M+1]
`
`. . .
`
`H[2M] H[2M+1]
`
`. . .
`
`Time
`
`Feedback
`
`CSI at Tx
`
`V^ [M], γ^[M]
`
`TF
`
`V^ [2M], γ [2M]
`^
`
`Fig. 1. Feedback system model (when D = 1).
`
`C. Mutual Information and Capacity
`Among many possible measures for evaluating the perfor-
`mance of the feedback method, we consider mutual informa-
`tion as performance measure in this paper. When the transmit
`signal x is distributed by CN (0t,1, Φx), the mutual information
`for a given channel realization H is given by I(x; y) =
`log det(Ir +HΦxH†) [1]. Since the covariance matrix Φx is a
`Hermitian positive semidefinite matrix, it can be decomposed
`with a unitary matrix W ∈ Ct×t and a
`as Φx = W ΦW †
`diagonal matrix Φ = diag(P1, . . . , Pt), Pi ≥ 0. From this, we
`notice that it is equivalent to transmitting x = W s, s ∈ Ct
`with E[ss†] = Φ over channel H, i.e., an equivalent channel
`is
`
`y = HW s + η.
`
`(2)
`
`(cid:1)
`
`This point of view is useful because each column of W can be
`interpreted as the beamforming vector for the corresponding
`symbol in s. And, in some cases, W and Φ can be adjusted
`by using the spatial and the power allocation information
`available at the transmitter. Let us define γ = [γ1, . . . , γt],
`where γi ∈ [0, 1] and
`i γi = 1, which is referred as
`power allocation information by setting γi = Pi/PT , i.e.,
`Φ(γ) = PT diag(γ1, . . . , γt). And, we denote the mutual
`information I(s; y) of the channel (2) when the transmitter
`uses beamforming matrix W and power allocation γ by
`Ψ(W, γ) = log det(I + HW Φ(γ)W †H†
`)
`= log det(I + V †
`H W Φ(γ)W †VHΣ2).
`(cid:1)m
`When the transmitter has perfect knowledge of channel (as
`in complete CSIT case), W is set to VH. Then, the mutual
`information is written as I(s; y) = Ψ(VH , γ) =
`i=1 log(1+
`Piλi), where λi = σ2
`i . With water-filling to maximize I(s; y),
`we have the channel capacity
`
`m(cid:2)
`
`i=1
`
`[ log(νλi) ]+
`CFull =
`(3)
`(cid:3)
`(cid:1)m
`where [a]+ is defined as max{a, 0} and ν is the water-filling
`ν − λ−1
`level satisfying the power constraint
`i
`i=1
`When we denote the optimum power allocation information
`by γwf, we can write CFull = Ψ(VH , γwf). On the other
`hand, when no information about channel is available at the
`transmitter (as in no CSIT case), the capacity is given by
`
`(cid:4)+ = PT .
`
`(cid:6)
`
`PT
`t
`
`λi
`
`(4)
`
`CNone = Ψ(It, γunif) =
`
`log
`
`1 +
`
`(cid:5)
`
`m(cid:2)
`
`i=1
`
`WCNC 2004 / IEEE Communications Society
`
`761
`
`0-7803-8344-3/04/$20.00 © 2004 IEEE
`
`Page 2 of 5
`
`

`

`method using Givens rotations in which the number of pa-
`rameters is equal to (8), the degree of freedom in the matrix.
`Theorem 1 (Parameterization): A matrixV ∈ Ct×n(t ≥
`(cid:9)
`(cid:7)
`n(cid:8)
`t−k(cid:8)
`n) with orthonormal columns can be decomposed as
`
`V =
`
`Dk(φk,k, . . . , φk,t)
`
`Gt−l,t−l+1(θk,l)
`
`˜I
`
`(9)
`
`k=1
`l=1
`where t dimensional diagonal matrix
`Dk(φk,k, . . . , φk,t) = diag(1k−1, ejφk,k , . . . , ejφk,t)
`1k−1 is (k−1) 1’s; and Gp−1,p(θ) is the Givens matrix which
`operates in the (p − 1, p) coordinate plane of the form
`Ip−2
`c −s
`s
`c
`
`
`
`Gp−1,p(θ) =
`
`(10)
`
`
`
`,
`
`It−p
`c = cos θ and s = sin θ; and t × n matrix ˜I = [In, 0n,t−n]T .
`Let us explain the above parameterization procedure with an
`example. Consider 4× 3 matrix V with orthonormal columns.
`× × ×
`| × | × ×
`1
`0
`0
`× × ×
`| × | × ×
`0 × ×
`× × ×
`| × | × ×
`0 × ×
`× × ×
`| × | × ×
`0 × ×
`1
`0
`0
`1
`0
`0
`| × | ×
`0
`1
`0
`0
`0×
`| × |
`| × | ×
`0
`0
`0×
`| × |
`| × | ×
`0
`0
`where | × | represents the magnitude of a particular element.
`The procedure is similar to the QR decomposition using
`Givens matrices. In the first step, we want to make all the
`entries in the first column under the first component all zeros.
`To do that, we first extract the phases from the first column
`by pre-multiplying V by D†
`1 to have a real-valued column,
`and then apply a series of Givens matrices with appropriate
`parameters to make all entries under the (1, 1) element zeros.
`Since the Givens rotation preserves the length of vector, the
`(1, 1) element becomes 1. At the same time, all the entries
`in the first row except the (1, 1) element also become zeros
`because of the orthogonality between columns. We carry out
`similar procedures on the remaining columns sequentially, and
`then finally we have a diagonal matrix ˜I. Since a Givens matrix
`is an orthogonal matrix, the matrix V can be factored as
`V = D1(φ1,1, . . . , φ1,4) G3,4(θ1,1) G2,3(θ1,2) G1,2(θ1,3)
`· D2(φ2,2, φ2,3, φ2,4) G3,4(θ2,1) G2,3(θ2,2)
`· D3(φ3,3, φ3,4) G3,4(θ3,1) ˜I.
`Therefore, once we have a set of parameters, the phases {φk,l}
`and the rotation angles {θk,l}, the original matrix V can be
`exactly reconstructed.
`Now, we will show that the number of parameters obtained
`by the proposed parameterization is equal to the degree of
`freedom in V with the following with the following Lemma
`and Theorem.
`Lemma 1: Define as ˜V the resulting matrix after applying
`D1 and the first l Givens rotations G†
`t−q,t−q+1(θ1,q), q =
`
`†2
`
`D
`
`−−→
`
`
`
`†3
`
`G
`
`,4−−−→ ˜I
`
`
`
`
`
`†3
`
`G
`
`
`
`0
`0
`
`
`
`
`
`†1
`
`†2
`
`1
`0
`0
`0
`
`0
`1
`0
`0
`
`
`
`†3
`
`D
`
`−−→
`
`−−−−−−−−−−−→
`,4,G
`,3,G
`,2
`
`
`
`†1
`
`D
`
`−−→
`
`
`
`†3
`
`G
`
`−−−−−−−→
`,4,G
`,3
`
`†2
`
`
`
`
`
`
`
`where γunif = [1/t, . . . , 1/t]. These two capacities for the
`extreme cases will be used as references in comparing perfor-
`mances.
`Now, we consider the following two scenarios where some
`non-perfect channel information is available at the transmitter.
`The first case is when the transmitter uses the quantized
`and/or delayed version of spatial information ˆVH and power
`allocation information ˆγ. Then, the mutual information can be
`written as
`
`(5)
`
`I ˆVH ,ˆγ = Ψ( ˆVH , ˆγ).
`Note that the subscripts in mutual information and capacity
`notations indicate the CSIT. The second case is when the
`transmitter has only spatial information ˆVH and no power
`allocation information. In this case, one easy choice of power
`allocation is uniform allocation. Then, the mutual information
`is given by
`
`I ˆVH = Ψ( ˆVH , γunif) =
`
`log
`
`1 +
`
`m(cid:2)
`
`i=1
`
`(cid:5)
`
`(cid:6)
`
`PT
`t
`
`λi(HeqH†
`eq)
`
`(6)
`
`where Heq = H ˆVH and λi(HeqH†
`eq) is the i-th largest
`eigenvalue of HeqH†
`eq.
`We expect that channel feedback has more gain when t > r
`[7]. In this case, since the rank of channel m < t, we need to
`feedback only first m columns of VH, i.e., V = [v1, . . . , vm]
`(n = m). And, if ˆV is reasonably close to V , the optimum
`power allocation will has nearly zeros in the last t− m entries
`in γ. Therefore, when only spatial information ˆV is available
`at the transmitter, a reasonable uniform power allocation is
`γ = [1/m, . . . , 1/m, 0, . . . , 0].
`
`III. PARAMETERIZATION OF CHANNEL INFORMATION
`In this section, we focus on how to extract essential param-
`eters from the spatial information denoted by V . Since the
`columns in spatial information V are geometrically structured,
`the degree of freedom in the matrix is much smaller than the
`number of real-number entries in the matrix. The degree of
`(cid:6)
`(cid:5)
`freedom in V ∈ Ct×n can be expressed as
`= 2tn − n2 (real numbers)
`N = 2t · n − n − 2
`
`(7)
`
`n 2
`
`where the first term is the number of real-number entries in
`the matrix, and second term accounts for reductions from
`unit-norm property of each column, and third term from
`orthogonality in each pair of columns. For example, a t × t
`unitary matrix has 2t2 real-number entries, but its degree of
`freedom is only t2. Furthermore, one phase in each column
`can be made fixed (e.g., the first row has all nonnegative real
`numbers), which gives n additional reductions. Then,
`N = (2t − 1)n − n2 (real numbers)
`Now, we want to extract a set of essential parameters that
`has a one-to-one mapping to the matrix V . There are several
`possible ways such as using Givens rotations or Householder
`reflections. In this paper, we propose a parameterization
`
`(8)
`
`WCNC 2004 / IEEE Communications Society
`
`762
`
`0-7803-8344-3/04/$20.00 © 2004 IEEE
`
`Page 3 of 5
`
`

`

`3) Reconstruct the spatial information ˆV from ˆΘ (Recon-
`struction): ˆV = T −1( ˆΘ).
`The proposed methodology for quantization has many advan-
`tages. Some of them are as follows. The number of parameters
`to quantize is minimal since it equals the degree of freedom in
`the spatial information. The parameters, which are phases and
`angles, are all bounded quantities. The reconstructed matrix
`ˆV has the same geometrical structure as V , i.e., ˆV † ˆV = In.
`In addition, the methodology is general and can be applied to
`any multiple antenna scenario: MISO systems (when n = 1)
`and MIMO systems with partial feedback (when 1 < n < m)
`as well as MIMO systems with full feedback (n = m).
`
`V
`
`T−−−−−→ Θ(cid:22)Q
`
`T −1←−−−−−−− ˆΘ
`
`ˆV
`
`Fig. 2. Quantization in the parameter domain.
`
`One can apply some vector quantization (VQ) method to
`quantize the parameters. But, complexity issues and track-
`ing requirement motivate us to consider employing scalar
`quantization. Moreover, the independence of the parameters,
`Theorem 3, indicates that the overall loss is minimal. More
`specifically, adaptive Delta modulation (ADM) [6, ch. 8]
`is used to quantize each parameter. In slowly time-varying
`channels, the parameters are also slowly and continuously
`changing most of the time. The encoder of ADM consists
`of a simple accumulator and a one-bit quantizer. Basically, it
`quantizes the difference between the newly incoming sample
`and the previous quantized sample. For a parameter θ,
`ˆθ[k] = ˆθ[k − 1] ± ∆[k].
`And the step-size ∆[k] of the one-bit quantizer is adaptively
`changing to better track the dynamics of the signal. The ADM
`with one-bit memory [6] is an example. The step-size is
`increased if the consequent two encoded bits are same, and
`(cid:23)
`decreased otherwise, that is,
`if c[k] = c[k − 1]
`M1 ∆[k − 1],
`if c[k] (cid:7)= c[k − 1]
`M2 ∆[k − 1],
`where ∆[k] and c[k] is the step-size and the encoded bit for
`the k-th sample; and M1 > 1 and 0 < M2 < 1, usually
`M2 = 1/M1. Compared to VQ, ADM has considerably lower
`complexity. And it is a low-rate scalar quantization scheme (as
`low as one bit per parameter). Another important advantage is
`that ADM has inherently a channel tracking feature for slowly
`time-varying channels.
`
`(14)
`
`(15)
`
`∆[k] =
`
`V. NUMERICAL RESULTS
`We have performed simulations to investigate the perfor-
`mance of the proposed feedback method, especially in slowly
`time-varying MIMO channels. The components of the channel
`matrix are i.i.d. discrete-time random processes and each
`process models Rayleigh fading channel gain. The simulated
`
`1, . . . , l, in the procedure for the first column. Then, the (t −
`
`l, p) element of ˜V is given by
`(cid:4)
`(cid:4)v(l+1)
`1
`)†v(l+1)
`(v(l+1)
`1
`(cid:3)v(l+1)
`(cid:3)
`1
`
`if p = 1,
`if p = 2, . . . , n
`
`p
`
`(11)
`
`˜V [t − l, p] =
`
`i
`
`where v(l+1)
`is a vector defined as the last l + 1 elements in
`the i-th column vi, i.e., v(l+1)
`= [vt−l,i, vt−l+1,i, . . . , vt,i]T .
`i
`Proof: This can be proved by induction.
`Theorem 2: In the matrix factorization of Theorem 1, if
`orthonormal column matrix V has real-valued elements in the
`first row with alternating signs as + − + − . . ., then the first
`parameter of Dk is zero, i.e., φk,k = 0 for all k. Therefore, the
`number of parameters is (2t − 1)n − n2, which is the degree
`of freedom in V .
`Proof: This can be proved by using Lemma 1 and
`orthogonality between two columns of V . After applying D†
`
`and t − 1 Givens matrices G†t−l,t−l+1(θ1,l), l = 1, . . . , t − 1,
`1
`(cid:20)
`(cid:19)
`it can be shown that the resulting matrix is given by
`
`
`G†1,2 . . . G†t−1,t D†1 V =
`01,n−1
`1
`
`,
`V (cid:4)
`(12)
`0t−1,1
`is a (t−1)×(n−1) matrix with orthonormal columns
`and V (cid:4)
`that has same structure as V , i.e., alternating signs in the first
`row. Therefore, in a sequential way we can prove the first
`phase parameter of Dk is zero for all k. From this, we have n
`less parameters than (7), which results in the final conclusion.
`
`Now we find the distribution of the parameters and es-
`tablish their independence, a property useful for quantization
`purposes.
`Theorem 3 (Statistics of Parameters): When the channel
`matrix H has i.i.d. CN (0, 1) entries, then all the parameters
`from Theorem 1 are statistically independent. Moreover, the
`phase φk,j is uniformly distributed over (−π, π] for all k and
`j, and the rotational angle θk,l has probability density
`p(θk,l) = 2l sin2l−1 θk,l cos θk,l, 0 ≤ θk,l <
`Proof: The theorem can be proved using techniques
`for calculating the distribution of transformed random vec-
`tor/matrix similar to [9, Ch. 1–3]. Details are omitted due to
`(cid:1)t
`space limitations.
`The parameterization for power allocation information γ =
`[γ1, . . . , γt],
`i=1 γi = 1 is rather simple. We can see that γ
`has t − 1 of degree of freedom. And, the parameters can be
`constraint, the last one is determined as γt = 1 −(cid:1)t−1
`simply the first t− 1 elements, [γ1, . . . , γt−1]. Then, from the
`i=1 γi.
`
`.
`
`π 2
`
`(13)
`
`IV. QUANTIZATION IN PARAMETER DOMAIN
`The overall strategy for quantization is depicted in Figure
`2 and summarized below.
`1) From the spatial information V , extract a set of param-
`eters Θ (Parameterization): Θ =T (V ).
`2) Quantize the parameters Θ and feed back the quantized
`parameters ˆΘ (Quantization): ˆΘ = Q(Θ).
`
`WCNC 2004 / IEEE Communications Society
`
`763
`
`0-7803-8344-3/04/$20.00 © 2004 IEEE
`
`Page 4 of 5
`
`

`

`REFERENCES
`[1] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” AT&T Bell
`Labs Tech. Memo., 1995.
`[2] E. Biglieri, G. Caire, and G. Taricco, “Limiting performance of block-
`fading channels with multiple antennas,” IEEE Trans. Inform. Theory,
`vol. 47, no. 4, pp. 1273–1289, May 2001.
`[3] K. K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, “On beam-
`forming with finite rate feedback in multiple antenna systems,” IEEE
`Trans. Inform. Theory, vol. 49, no. 10, pp. 2562 – 2579, Oct. 2003.
`[4] D. J. Love, R. Heath, Jr., and T. Strohmer, “Grassmannian beamforming
`for multiple-input multiple-output wireless systems,” IEEE Trans. Inform.
`Theory, vol. 49, no. 10, pp. 2735 – 2747, Oct. 2003.
`[5] E. N. Onggosanusi and A. G. Dabak, “A feedback-based adaptive multi-
`input multi-output signaling scheme,” in Proc. Asilomar Conf. 2002,
`Pacific Grove, CA, Nov. 2002.
`[6] N. S. Jayant and P. Noll, Digital Coding for Waveforms: Principles and
`Applications to Speech and Video. Prentice-Hall, 1984.
`[7] J. C. Roh and B. D. Rao, “Multiple antenna channels with partial channel
`state information at the transmitter,” IEEE Trans. Wireless Commun., Mar.
`2004.
`[8] C. Murthy, J. C. Roh, and B. D. Rao, “Optimality of extended maximum
`ratio transmission,” in 6th Baiona Workshop on Signal Processing in
`Communications, Baiona, Spain, Sept. 2003.
`[9] R. J. Muirhead, Aspects of Multivariate Statsitical Theory.
`& Sons, 1982.
`
`John Wiley
`
`RF = 500, M = 4, D = 1
`
`PT=20 dB
`
`CFull
`CQ(V), Q(γ)
`CQ(V)
`CNone
`
`PT=−10
`
`PT=0 dB
`
`PT=10 dB
`
`1
`
`0.9
`
`0.8
`
`0.7
`
`0.6
`
`0.5
`
`0.4
`
`0.3
`
`0.2
`
`0.1
`
`Cumulative Distribution
`
`0
`
`0
`
`2
`
`4
`
`6
`
`8
`10
`12
`Capacity (bits/channel use)
`
`14
`
`16
`
`18
`
`20
`
`Fig. 3. Cumulative distribution of capacities when RF = 500/sec (t = 4
`and r = 2). Q(V ) = ˆV and Q(γ) = ˆγ.
`
`PT=20 dB
`
`CFull
`CQ(V), Q(γ)
`CQ(V)
`CNone
`
`RF = 1000, M = 4, D = 1
`
`PT=−10
`
`PT=0 dB
`
`PT=10 dB
`
`1
`
`0.9
`
`0.8
`
`0.7
`
`0.6
`
`0.5
`
`0.4
`
`0.3
`
`0.2
`
`0.1
`
`Cumulative Distribution
`
`0
`
`0
`
`2
`
`4
`
`6
`
`8
`10
`12
`Capacity (bits/channel use)
`
`14
`
`16
`
`18
`
`20
`
`Fig. 4. Cumulative distribution of capacities when RF = 1000/sec (t = 4
`and r = 2).
`
`channel has the Doppler frequency fD = 7.4 Hz, which
`corresponds to a mobility of 4 km/h at carrier frequency of 2
`GHz. As for the frame structure of Section II-B, we considered
`the case of M = 4 and D = 1.
`Figure 3 shows the cumulative distribution of mutual infor-
`mation with different CSIT assumptions and various transmit
`power, PT = −10, 0, 10, 20 dB, with the feedback rate RF =
`500 per second (t = 4 and r = 2). CFull and CNone are
`calculated from (3) and (4), respectively. Note that the CSIT
`for CFull is perfect, that is, it involves neither quantization
`error nor channel tracking error. The performances of the
`proposed feedback method are shown as C ˆV ,ˆγ and C ˆV , which
`are calculated according to (5) and (6), respectively. These
`include the effect of quantization error and delay; therefore,
`they reflect more practical situations of feedback systems.
`From the results, we can see that, in low transmit power range,
`the two have some gap; but, in high transmit power range, the
`two have little difference. This means that power allocation
`information is important in low transmit power range, which
`can be understood from the water-filling argument. That is,
`when transmit power is low, the optimum transmission scheme
`is using only a few spatial channels that have high channel
`gains. Note that the feedback rate is corresponding to 5.5 kbps
`(for C ˆV ,ˆγ) and 5 kbps (for C ˆV ) of feedback bit-rate since we
`have 10 parameters for V and one for γ, and ADM encodes
`each parameter into one bit at each feedback instant.
`Figure 4 shows the results when the feedback rate is
`increased to RF = 1000 per second, which corresponds to
`11/10 kbps. We can see that the performances become much
`closer to CFull. This can be explained as follows. By increasing
`the feedback rate, the quantization error is reduced, since in
`ADM encoding the variations between the adjacent samples
`are reduced. Also the channel tracking error due to delay is
`lessened with increasing the feedback rate.
`
`VI. CONCLUSION
`We proposed a general framework for quantization of
`MIMO channel information, which involves parameterization
`of orthonormal column matrix and quantization of param-
`eters. We introduced a new parameterization method that
`uses Givens rotations and that provides minimal number of
`parameters. The distributions of the parameters were found and
`the independence between them was shown. In slowly time-
`varying channels, the extracted parameters are also slowly
`and continuously changing in time. This motivated employ-
`ing adaptive Delta modulation in quantizing the parameters.
`The adaptive Delta modulation is a simple and practical
`quantization method that has a channel tracking feature for
`slowly time-varying channels. The proposed feedback scheme
`requires (2t − 1)n − n2 bits to feedback V ∈ Ct×n. With the
`proposed feedback method, a performance close to the perfect
`feedback case can be achieved with a reasonable feedback rate.
`
`ACKNOWLEDGMENT
`This research was supported by CoRe grant No. 02–10109
`sponsored by Ericsson.
`
`WCNC 2004 / IEEE Communications Society
`
`764
`
`0-7803-8344-3/04/$20.00 © 2004 IEEE
`
`Page 5 of 5
`
`