IEEE WIRELESS COMMUNICATIONS LETTERS, ACCEPTED FOR PUBLICATION
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A Differential Codebook with Adaptive Scaling for Limited Feedback MU MISO Systems Jawad Mirza, Student Member, IEEE, Pawel A. Dmochowski, Senior Member, IEEE, Peter J. Smith, Senior Member, IEEE, and Mansoor Shafi, Fellow, IEEE
Abstract—In this letter, we propose an adaptive scaling technique for a differential codebook (DCB) in multiuser (MU) multiple-input single-output (MISO) systems operating under spatially and temporally correlated channels. The proposed adaptive scaling technique depends on the mean channel variation due to spatial/temporal correlation in the channel and the mean quantization error associated with the previous feedback, hence avoiding the need to periodically reset the codebook. The performance of the proposed DCB is evaluated for both zeroforcing beamforming (ZFBF) and signal-to-leakage-and-noise ratio beamforming (SLNR BF) schemes. Simulation results show the effectiveness of the proposed adaptive scaling and also the superiority of the SLNR BF scheme over the ZFBF scheme. Index Terms—Zero-forcing, signal-to-leakage-and-noise ratio, differential codebook, multiple-input single-output (MISO).
I. I NTRODUCTION
T
HE feedback of channel state information (CSI) to the transmitter is an important component of multiple-input multiple-output (MIMO) systems as it is critical for transmit beamforming and interference mitigation [1]. However, it is difficult to provide perfect CSI to the transmitter and a low-rate feedback link is commonly used by the receiver to send the quantized CSI to the transmitter. For this purpose, both transmitter and receiver maintain a codebook containing quantized entries representing the channel information. Such systems are known as limited feedback MIMO systems. The performance of limited feedback MIMO systems depends heavily upon the codebook design. If the codebook is well designed and matches the propagation environment then quantization errors are reduced, resulting in a smaller capacity loss. In temporally correlated channels, the use of a differential codebook (DCB) that exploits the temporal correlation has been shown to improve the capacity of a MIMO system [2]. A polar-cap DCB design for a temporally and spatially correlated MU MISO channel is discussed in [3]. This codebook is rotated to be centered on the previously selected codeword after each feedback and outperforms the rotation-based codebook [2]. The disadvantage of [3] is that it requires the process to be reset after a few feedback intervals, especially when using the adaptive scaling method. The adaptive scaling in [3], [4] continually shrinks the radius of the codebook as the channel evolves. After a while, the codebook radius becomes
Manuscript received August 5, 2013. The associate editor coordinating the review of this letter and approving it for publication was K. Huang. J. Mirza and P. A. Dmochowski are with the School of Engineering and Computer Science, Victoria University of Wellington, New Zealand (e-mail:
[email protected]). P. J. Smith is with the Department of Electrical and Computer Engineering, University of Canterbury, New Zealand. M. Shafi is with Telecom New Zealand. Digital Object Identifier 10.1109/WCL.2013.13.130565
too small to successfully track the channel and this issue is handled by resetting the codebook to the original version. This letter provides new results, as we propose a different and improved adaptive scaling technique. In this paper we make the following contributions: • Motivated by [3], we propose a new adaptive scaling method for the DCB operating under spatially and temporally correlated MISO channels. The adaptive scaling takes into consideration the effect of spatial/temporal correlation and quantization errors and thus does not shrink beyond a certain limit. Therefore, in our proposed method the codebook is not set to a base codebook after every few feedback intervals hence, the gains offered by the DCB are conserved. • We use SLNR BF for MU transmission. By comparing with the DCB using ZFBF, we show the superiority of , SLNR BF. To the best of the authors knowledge, there is no study that evaluates the performance of SLNR BF system with DCBs. Notation: We use AH , AT , A−1 , and A⊥ to denote the conjugate transpose, the transpose, the inverse and the null space of the matrix A, respectively. . and |.| stand for vector and scalar norms respectively. E[.] denotes expectation. II. S YSTEM M ODEL Consider a MU MISO system with a single base-station (BS) equipped with nT transmit antennas, serving K users simultaneously. Each user has only a single antenna (nR = 1). The received signal at the ith user is given by y i = hi x + n i ,
(1)
where hi is a channel of size 1 × nT between the BS and the ith user. x is the transmitted signal from the BS such that x = K i=1 wi si , where wi and si are the normalized beamforming vector and data symbol of the ith user, respectively. The noise at user i is denoted by ni and is assumed to be complex Gaussian with variance, σi2 . The data symbols are assumed to be normalized such that E[|si |2 ] = 1. The received signal-tonoise ratio (SNR) of the user i is defined by SNRi = 1/σi2 . The signal-to-interference-plus-noise ratio (SINR) for the ith user is given by SINRi =
σi2 +
|hi wi |2 K k=1,k=i
|hi wk |2
,
(2)
and treating interference as Gaussian, the sum-capacity of the system can be written as
c 2013 IEEE 2162-2337/13$31.00
Csum =
K i=1
log2 (1 + SINRi ).
(3)
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IEEE WIRELESS COMMUNICATIONS LETTERS, ACCEPTED FOR PUBLICATION
The receiver estimates the channel via reference signals transmitted from the BS. Each user has its own codebook denoted by Fi = [fi1 , fi2 , . . . , fiN ], where N is the number of codebook entries or codewords which are nT × 1 column vectors. Thus, for each feedback, a user sends B = log2 N bits to the transmitter via a feedback link, corresponding to the selected ¯ H fj |, where h ¯ i = hT /hi . codeword ¯fi = argmax1≤j≤2B |h i i At the BS, the selected codeword is multiplied by the channel quality indicator (CQI), CQI= hi . We assume that perfect CQI is available at the transmitter for all the users. Therefore, the reconstructed quantized version of the ith user channel, hi , is given by ˆfi = hi ¯fiT . The BS uses the quantized channel to find beamforming vectors for each user. In LTE Rel. 9, the demodulation reference signal (DM-RS) is introduced that is user specific and delivers the beamforming vector information to the user. III. M ULTIUSER T RANSMISSION S TRATEGIES In this section we briefly explain the two well-known beamforming techniques namely, ZFBF and SLNR BF [5]. A. ZFBF In ZFBF, the beamforming vectors are formed such that the product hi wk is zero, for all k = i. In the case of a limited feedback system, the ZFBF cannot cancel the multiuser interference completely but is still able to reduce it significantly [6]. At the BS, the quantized K × nT channel matrix, H, is T T ] . A beamforming weight constructed as H = [ˆf1T · · · ˆfK matrix is obtained by taking the pseudo-inverse of H given by W = HH (HHH )−1 . The ZFBF vector for the ith user, wi , is the normalized ith column of W. B. SLNR BF The SLNR BF solution is provided in [5] for the perfect CSI case. In limited feedback systems, the leakage from user K i is approximated by k=1,k=i |ˆfk wi |2 . The SLNR BF vector, wi , for the ith user is the normalized maximum eigenvector ˜ i )−1 ˆf H ˆfi , where H ˜ HH ˜ i is a quantized channel of (σi2 I + H i i matrix of size (K − 1) × nT for the ith user constructed from the codebook entries fed back by the other users such that, ˜ i = [ˆf T · · · ˆf T ˆf T · · · ˆf T ]T . H 1 i−1 i+1 K IV. D IFFERENTIAL C ODEBOOK (DCB) In this section, we explain the DCB design where each user generates its codebook by modifying the rank-1 Grassmannian codebook (GCB) denoted by C. In order to generate the DCB, we alter the GCB such that its center is at [1, 0, . . . , 0]T . This alteration involves selecting the Grassmannian codeword, cj , 1 ≤ j ≤ 2B , that is closest to the vector [1, 0, . . . 0]T in terms of chordal distance and then obtaining the rotation matrix, Θ, between them to rotate all the GCB entries. We denote this c2 , . . . , c˜2B ]. After this adjustment adjusted GCB by C˜ = [˜ c1 , ˜ we scale the rotated codewords by a factor α that defines the radius1 of the DCB in order to bring the codewords closer for tracking the spatially/temporally correlated channel. This 1 Note
that after scaling, all the codewords lie inside the radius, α.
rotation and scaling is similar to the rules described in [7]. This step completes the generation of the DCB at both BS and user i. The feedback steps are as follows: • For the first transmission, user i selects an appropriate codeword ¯fi from a base codebook (e.g. RVQ) and feeds back the index of this selected codeword to the BS. • For the second transmission, both BS and user i use DCB where the index of the selected codeword for the first feedback (from the base codebook) becomes a new center of the DCB. • The process continues with the DCB with adaptive scaling, discussed later, where the main idea is to vary the value of the scaling parameter, α, depending on mean channel variation and quantization error. A. Codebook Rotation and Scaling The codebook rotation and scaling follow the principles explained in [7]. We drop the user index i from the expressions and introduce the time index t from this point onwards for clarity. In order to rotate the previously selected codeword ¯ft−1 ¯ to the current codeword Hft , the rotation matrix is given selected ⊥ ⊥ ¯ ¯ ¯ ¯ by Θt = ft ft ft−1 ft−1 . Once the rotation matrix Θt is calculated, all the codewords in the codebook are rotated using this rotation matrix. Each codeword of the adjusted ˜ is scaled individually, so that the j th codeword, c˜j is GCB, C, scaled to T jθ1 jθ2 jθnT 2 2 s(˜cj ) = 1 − α (1 − r1 )e , αr2 e , . . . , αrnT e , (4) where rm ejθm is the polar form of the mth entry of the original codeword and α is a scaling parameter, 0 < α < 1. We divide the scaled codeword by its norm in order to maintain a unit norm codeword. B. Adaptive Scaling In this paper, we assume the channel is spatially and temporally correlated with a spatial correlation coefficient ρ and temporal correlation coefficient following the Jakes’ model = J0 (2πfD τ ), where fD is the Doppler frequency. The spatial correlation is modeled by a Kronecker model. If long term channel statistics and ρ are shared with the BS [2], then adaptive scaling can be used. The base station does not need to know the exact value of ρ. It can be assumed with the correct antenna placement, ρ is small. We later show that the ρ value has a minimal effect on the adaptation process. The proposed adaptive scaling method depends on mean quantization error and mean channel variation (i.e the amount of spatial/temporal correlation in the channel). The mean channel variation is defined as the average chordal distance between the previous and current channel directions, given by 2 H ¯ ¯ 1 − ht−1 ht . (5) dmean = E Using Jensens’ inequality in (5) we have
¯ 2 . dmean ≤ 1 − E ¯ hH t−1 ht
(6)
If we assume that channel is modeled as a first-order GaussMarkov (FOGM) process with both spatial and temporal
MIRZA et al.: A DIFFERENTIAL CODEBOOK WITH ADAPTIVE SCALING FOR LIMITED FEEDBACK MU MISO SYSTEMS
correlation [3], then 2
2 H 1/2 2 = E h (h + 1 − R g ) , E hH ht−1 t−1 t t−1 t (7) where gt is a CnT ×1 vector having i.i.d. CN (0, 1) entries and R is a spatial correlation matrix (assumed to follow an exponential model [3]). Simplifying (7) and setting the expectation of cross terms to zero, we get 2 2 H 2 4 2 1/2 = E hH +(1− h E h )E R g . h t−1 t t−1 t t−1 (8) Using the independence between amplitude and direction of ht 2 H for spatially and ht−1 [8], we can approximate E ht−1 ht and temporally correlated channels with close to 1 by 2 2 H 4 ¯ ¯ . E hH ≈ E h (9) E h h h t−1 t−1 t t−1 t Substituting (9) in (8) and simplifying gives ¯ t 2 ≈ 2 + (1 − 2 )Ψ E ¯ hH h t−1 where Ψ=
1/2 2 E hH R g t t−1 E [ht−1 4 ]
.
(10)
(11)
1/2 Noting the fact that hH u)H , where u is distributed t−1 = (R according to CN (0, 1) and substituting in (11), we get 2 E uH Rgt . Ψ= (12) E (R1/2 u2 )2
Using the eigenvalue decomposition R = ΦDΦH , where Φ is an nT ×nT matrix of eigenvectors of R and D is a diagonal matrix containing the eigenvalues (d1 , d2 , . . . , dnT ) of R, (12) can be written as 2 ˜ H D˜ gt E u , (13) Ψ= E [(˜ uH D˜ u)2 ] ˜ H = uH Φ and g ˜ t = ΦH gt are distributed according where u to CN (0, 1). The numerator of (13) can be simplified to H 2 nT 2 ˜ D˜ ˜tH E D˜ ˜H gt g E u uu = gt = Tr E D˜ z=1 dz H ˜t ) and by using the fact that off diagonal entries of (D˜ gt g 2 2 ˜ H ) are zero mean and |˜ u | and |˜ g | are exponentially (D˜ uu z t,z gt,z |2 = 1. Similarly, the dedistributed with E |˜ uz |2 = E |˜ H nT 2 2 2 nominator E (˜ u D˜ u)2 = E d |u | . Expanding z=1 z z nT 2 2 2 and using E |uz |4 = 2, we can write (13) z=1 dz |uz | as n T d2z z=1 Ψ= , q = 1, . . . , nT . (14) nT nT 2 d2z + 2 dz dq z=1
z=1 z 1.
(18)
Therefore, starting the process with the base codebook, we initially have the scaling parameter α0 = 1. For the second feedback, the scaling parameter is measured using (17) such −B that α1 = 2 2(nT −1) . The scaling parameter for the succeessive feedback intervals is calculated using (18). This adaptive scaling parameter can be shown to reach an asymptotic value −1 −B . (19) α∞ = dmean 1 − 2 2(nT −1) The adaptive scaling technique for time t − 1 and t is depicted in Fig. 1. In the proposed adaptive scaling technique, the periodic reset of the DCB to the base codebook is not required. V. P ERFORMANCE E VALUATION We use the WINNER II channel and FOGM channel to evaluate the performance of the proposed adaptive scaling method. For the FOGM channel, the value of ρ is set to 0.5, which is a practical value for widely separated transmit antennas. For a fair comparison, we assume the same value of ρ as an equivalent for the WINNER II model, which does not explicitly define this parameter. The BS is equipped with 4 antennas in a uniform linear array (ULA) setting with 0.5λ spacing among neighboring antennas. The scenario considered is urban macro (Uma) with non-line-of-sight (NLOS) propagation and the carrier frequency is 2.5 GHz. We assume that the
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IEEE WIRELESS COMMUNICATIONS LETTERS, ACCEPTED FOR PUBLICATION
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0.8 0.7
CDF
RVQ (WINNER II channel) Proposed DCB (v = 1 km/h) WINNER II channel
18
Proposed DCB (v = 1 km/h) FOGM channel Polar−cap DCB (v = 1 km/h) WINNER II channel
0.6 0.5 0.4 RVQ
0.3
Perfect CSI (WINNER II channel)
20
Perfect CSI SLNR BF Perfect CSI ZFBF Proposed DCB SLNR BF Proposed DCB ZFBF BF RVQ SLNR BF RVQ ZFBF
Sum Capacity [bps/Hz]
0.9
16
Proposed DCB (v = 5 km/h) FOGM channel Proposed DCB (v = 5 km/h) WINNER II channel Polar−cap DCB (v = 5 km/h) WINNER II channel
14 12 10 8
0.2
6
0.1
4 0 −20
−15
−10
−5
0 SINR [dB]
5
10
15
10
20
Fig. 2. SINR CDF for the 1st user at SNR = 10 dB with v = 1 km/h ( = 0.9987) in the WINNER II channel.
20
30 40 50 60 70 80 Time [in multiples of feedback interval]
90
100
Fig. 3. Sum-capacity vs time for SLNR BF at SNR = 10 dB with v = 1 km/h and v = 5 km/h.
20
VI. C ONCLUSION In this letter, we propose an adaptive scaling method for the DCB in spatially and temporally correlated MISO channels. The proposed scheme does not require periodic resets of the DCB after several feedback intervals and performs well for long transmission periods. Through simulation results, we show the superiority of SLNR BF over ZFBF. For typical values of the parameters ρ and , one can show that initially the quantization error term in (18) is the dominant factor in defining the scaling parameter for low speeds.
18 16
Sum−capacity (bps/Hz)
feedback link is lossless with zero delay. The feedback interval is 5 ms. The base codebook is a 4 bit rank-1 RVQ codebook [9] and the DCB design is based on a 4 bit rank-1 GCB. The total number of users are 4. In Fig. 2, the cumulative distribution function (CDF) of the SINR is plotted for one of four users when the user speed is 1 km/h. We compare three cases: perfect CSI, the proposed DCB, and the RVQ codebook for both ZFBF and SLNR BF methods. The SLNR BF outperforms the ZFBF scheme and with the proposed DCB it is even better than the perfect CSI ZFBF case at v = 1 km/h. Figure 3 shows the sum-capacity vs. time with v = 1 km/h and v = 5 km/h at SNR=10 dB. The time axis is in multiples of 5 ms feedback intervals. The sum-capacity with the DCB with proposed adaptive scaling is higher than a 4 bit RVQ codebook and the polar-cap DCB. It is seen that the sum-rate performance does not degrade over time with the proposed adaptive scaling technique and remains stable over time. The polar-cap DCB with adaptive scaling as discussed in [3] requires a reset to a base codebook after every Tmax = 9 feedback intervals. When the speed increases, the capacity loss also increases, as in the case of v = 5 km/h, implying that quantization errors are large and the DCB uses a large scaling parameter in order to track the varying channel. The performance gain of the proposed scheme also holds for the channel generated by the FOGM process. In Fig. 4 sumcapacity results are shown against the range of SNR values. The performance of the proposed DCB is superior than the polar-cap DCB because it achieves temporal stability in the sum-capacity and has a better codeword arrangement.
14
Perfect CSI Proposed DCB (v = 1 km/h) Polar−cap DCB (v = 1 km/h) Proposed DCB (v = 5 km/h) Polar−cap DCB (v = 5 km/h) RVQ
12 10 8 6 4 2 −5
0
5 SNR (dB)
10
15
Fig. 4. Sum-capacity performance vs. SNR for different user speeds with SLNR BF in the WINNER II channel.
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