2. Downlink Data Transmission 15
2.4. Limited Channel State Information in Frequency Division Duplex (FDD)
2.4.2. Sub-Codebook Precoding - Single User
The basic principle of the idea on codebook splitting is published by the author of this thesis et. al. in [WKT+15] as a patent and in [KTH+16a] as a conference paper. In this section, further novel aspects and more details are provided.
As before, a codebook Ω of size N ×N(Ω) is considered and divided along the second dimension, the number of codewords N(Ω), intoN(SCB) sub-codebooks of sizeN ×N(Ω˜). Thus, N(Ω˜) ∈N+ is considered as a system design parameter that controls the number of precoded pilots per RB, assuming thatN(Ω˜) < N and N(Ω˜) < N(Ω). It follows that the number of sub-codebooks N(SCB) is obtained as
N(SCB) =
&
N(Ω) N(Ω˜)
'
. (2.73)
The N(SCB) sub-codebooks are assigned to disjunctive RBs. Possible mappings between sub-codebooks and RBs are discussed later. The n(SCB)-th sub-codebook, where n(SCB) ∈ n1, . . . , N(SCB)o, is denoted as Ω˜n(SCB). Using these sub-codebooks for precoded pilots such that B= ˜Ωn(SCB) yields the following system relevant aspects:
1. The effective channel ˆHk,l =Hk,lΩ˜n(SCB) in Eq. (2.13) is of dimensionM×N(Ω˜) and can be directly estimated by the user k which means that the sub-codebook is transparent to the users and can be changed by the network or BS without knowledge at user k. Consequently, no additional control information has to be signaled from the BS to the user. The complexity of the effective channel estimation is reduced by min(N,N(Ω))
N(Ω˜) .
2. With un-precoded pilots and shared codebooks, e.g. as in LTE, userkestimates the channel Hk,l and requires N(Ω) matrix multiplications of the channel and the codewords to find the best one. With the
8The data channel bandwidth is assumed to be 9 MHz that corresponds to 50 RBs. Note that the feedback rate is independent of the coherence timeT(C) and if the feedback interval is larger thanT(C) this results in loss in spectral efficiency.
9The feedback rate can be further decreased by mechanisms such as differential feedback, where a codeword index is selected over the entire bandwidth and per subband a difference value with a smaller value-range is reported.
precoded pilots the best codeword can be directly obtained from estimated effective channel ˆHk,l, e.g.
by selecting the column with the largest norm.
3. The precoded pilots can also be used for demodulation and no additional demodulation reference signals, as in current LTE, are required when transmitting data10.
4. The feedback bits to indicate the best codeword of user kreduce from log2
N(Ω)to log2
N(Ω˜). The mapping from the large codebookΩto sub-codebook ˜Ωn(SCB) can be described by an vector of indices acording to
Ω˜n(SCB) = [Ω]:,n(SCB)
n(SCB)
, (2.74)
where n(SCB)n(SCB) is of size N(Ω˜) ×1 and the elements are from the set of natural numbers in the interval [1, ..., N(Ω)]. In other words, the index vector n(SCB) selects N(Ω˜) codewords from the full codebook Ω for sub-codebook ˜Ωn(SCB). The sub-codebook splitting exploits the spatial beam structure discussed in the previous paragraph.
The optimum design of n(SCB)n(SCB) with respect to the spectral efficiency is a combinatorial problem with complexity NN((Ω)Ω˜) for a single sub-codebook designed independently from the others. However, theN(SCB) codebooks have to be designed jointly such that n(SCB)1 , . . .n(SCB)
N(SCB) are disjunctive sets. The disjunctive property ensures that all codewords and therefore implicitly beam directions have the same probability for the user to select from. Without prior knowledge about the user position distribution preferring certain directions is not necessarily beneficial in terms of spectral efficiency. Furthermore, a uniform and equally spaced codeword selection from the full codebook is assumed, such that sub-codebook ˜ΩN(SCB) equals the
“small” full codebookΩN(α)(Ω,SCB), N(β)(Ω,SCB)according to Eq. (2.56), whereN(α)(Ω,SCB) andN(β)(Ω,SCB) denote the number of horizontal and vertical directions such that N(α)(Ω,SCB)N(β)(Ω,SCB) = N(Ω˜). The other sub-codebooks ˜Ω1, . . . ,Ω˜N(SCB)−1 can be interpreted as angular shifted versions of ˜ΩN(SCB), with the same angular distance between the codewords.
The algorithm of the sub-codebook construction is straightforward and given as Matlab code in the appendix in Algorithm 4. The example in Fig. 2.24 withN(α)(Ω)=N(β)(Ω)= 16 and N(α)(Ω,SCB)=N(β)(Ω,SCB) = 4 illustrates how the sub-codebooks are constructed. The selection ofN(α)(Ω,SCB)andN(β)(Ω,SCB) in the example in Fig. 2.24 is based on the numerical results in Fig. 2.23 where maximum effective spectral efficiency is achieved with N(α)(Ω) =N(β)(Ω) = 4. Note that without limitation to generality codeword indices according to Fig. 2.25 are assumed for the sub-codebook construction in Algorithm 4.
Next, the sub codewords ˜Ωn(SCB) are frequency multiplexed as shown in Fig. 2.26. In principle the sub-codebooks can also be distributed into the time dimension within the coherence time of the channel, however here only frequency multiplex is considered due to simulation complexity11. Note that the multiplexing of sub-codebooks to orthogonal time-frequency resources can be subject to further optimization depending on the user density and traffic distribution, however, it is not in the focus of this thesis and therefore only the distributed mode as shown in Fig. 2.26 is used. The number of orthogonal frequency resource is denoted by N(RB), e.g. in LTE the smallest unit for user selection in the frequency dimension is a physical RB with 180 kHz bandwidth. The same resolution is assumed in this thesis for sub-codebook resolution in the frequency domain. If the number of sub-codebooksN(SCB)is smaller than the number of physical RBsN(RB) such that N(SCB) < N(RB), several or all sub-codebooks appear multiple times. For example in Fig. 2.25 sub-codebook ˜Ω1 is assigned to RB 1 and again on RB 17 assumingN(SCB)= 16 sub-codebooks. With this the term sub-codebook block is introduced. A sub-codebook block is a complete set of all sub-codebooks
10Demodulation reference signals are precoded reference signals with the precoder used for downlink data transmission that is determined by the BS after scheduling.
11Time evolving channels would linearly increase the simulation complexity and storage.
Figure 2.24.: Example of sub-codebook construction ˜Ωn(SCB) of sizeN(α)(Ω,SCB)=N(β)(Ω,SCB) = 4 from a “large”
codebook of size N(α)(Ω) =N(β)(Ω) = 16. The green filled squares are codewords of sub-codebook n(SCB), while gray filled squares represent codewords from sub-codebooks with a lower index n(SCB). Codewords have maximum average angular distance from each other to keep inter-stream interference low.
1
Figure 2.25.: Codeword indexing in the full codebook with respect to the number of horizontal and vertical codewordsN(α)(Ω) andN(β)(Ω), respectively.
Figure 2.26.: Sub-codebook multiplexing in frequency domain to provide user with selection diversity.
Table 2.6.: Codebook specific simulation assumptions for performance evaluation of sub-codebook splitting.
Parameter Value
SINR bounds, γ(min), γ(max) γ(min) =−5 dB ,γ(max)= 100 dB
Number of available users 1, or given
Number of selected users 1, or according to Algorithm 1
N(α), N(β), N 10,10,100
Full “large” codebook: N(α)(Ω), N(β)(Ω), N(Ω) 16,16,256 Full “small” codebook: N(α)(Ω), N(β)(Ω), N(Ω) 4,4,16
16,16,256
α(CW,min), α(CW,max) −50◦,50◦
β(CW,min), β(CW,max) −40◦,8◦
Sub-codebooks: N(α)(Ω,SCB), N(β)(Ω,SCB), N(Ω˜) 4,4,16
N(SCB) According to Algorithm 4
from 1, . . . , N(SCB), e.g. with the settings in Table 2.6 with N(SCB) = 16 sub-codebooks and N(RB) = 50 physical RBs, there are 3 sub-codebook blocks.
A summary of the codebook specific parameters and their setting for the following performance evaluation is given in Table 2.6. The parameter values are derived based on numerical simulations as explained above.
Fig. 2.27a compares the following four cases:
1. All sub-codebooks: This means that the 16 sub-codebooks are distributed in the frequency domain according to Fig. 2.26. Each user selects on each RB his “best” codeword of the sub-codebook according to Eq. (2.57). This BS than applies the selected codeword for transmission to the user.
2. Max. of sub-codebook block: Again the 16 sub-codebooks are distributed in the frequency domain according to Fig. 2.26. In contrast to case 1, each user selects his best codeword according to Eq. (2.57) per sub-codebook block and not per RB. With the parameters in Table 2.6 the user selects his first codeword from RBs 1-16, the second codeword from RBs 17-32, the third codeword from RB 33-48 and the forth from RBs 49-50. While the last two RBs are not a complete sub-codebook block the user can report a codeword in order not to waste these resources for downlink transmission.
3. Full codebook 16×16: This case is the reference and the spectral efficiency of the full codebook N(α)(Ω) =N(β)(Ω) = 16, N(Ω) = 256 that is used for the sub-codebook splitting in the previous two cases is given. This codebook is used on all RBs.
4. Full codebook 4×4 - This case is for comparison with the sub-codebook splitting and the spectral efficiency of the “small” codebook N(α)(Ω) = N(β)(Ω) = 4, N(Ω) = 16 having the same size as the
sub-codebook is given. The the same “small” sub-codebook is applied on all RBs.
The first observation in Fig. 2.27a is that the spectral efficiency of the sub-codebooks (case 1) overlaps with case 4 the “small” full codebook of size 4×4, that is of the same size as the sub-codebooks. This verifies that the proposed sub-codebook splitting strategy achieves similar performance with each sub-codebook.
Second, the spectral efficiency of the “large” full codebook of size 16×16 achieves a higher spectral efficiency than case 1 and case 4, e.g. 1 bit/s/Hz more at the 50 %-ile. However, the main observation is that the sub-codebook splitting with codeword selection out of a sub-sub-codebook block achieves a higher spectral efficiency than the “large” full codebook over the complete range of the CDF, e.g. 0.2 bit/s/Hz at the 50 %-ile. Note, that the original motivation for sub-codebook splitting was the reduction of pilot overhead, that can be achieved without performance loss compared to the full codebook. However, some discussion is required to interpret this result correctly. In the full codebook of size 16×16 a single user is randomly assigned to each RB and receives data on his selected codeword. In contrast to this, in case 2 where each user selects the best codeword out of the sub-codebook block, there are now 16 users that can select their best codeword over 16 RBs. Thus, the gain for case two is actually a frequency diversity gain inherently utilized by the codeword selection over the sub-codebook block in case 2. Therefore, the frequency diversity of the simulated scenario is investigated next for clarification.
Fig. 2.27b shows the frequency selectivity of the effective channel for codeword 1 of the full 16×16 codebook in comparison to MRT precoding for the [10,10] UPA. This frequency selectivity of the codebook contradicts the findings of “channel hardening” in Section 2.3. Channel hardening with a practical number of antennas requires phase adapted precoding, however with DFT based codebooks only “directions” are approximated.
Thus a part of the frequency selectivity of the channel remains, e.g. the variance with respect to the median value over 50 subcarriers is 19 dB and 1 dB for codebook and MRT precoding, respectively. A ULA with 10 elements has a half-power beam-width in the main radiation direction of approximately 10°. With 80°
azimuth spread and 52° elevation spread at the BS in the simulated urban macro NLoS scenario [3GP17e]
a lot of the multiple path components remain with codebook precoding. Increasing the number of antennas the beam-width becomes smaller and the effective channel with codebook precoding becomes less frequency selective. In the extreme case the beam-width is so small that only a single path will remain visible above the noise threshold. One finding of this is, that frequency selective scheduling is still beneficial w.r.t. spectral efficiency in massive MIMO FDD systems using codebook based precoding.
Therefore, the sub-codebook precoding is next combined with frequency selective opportunistic scheduling to further increase spectral efficiency. In Fig. 2.28 the number of available users is increased and on each sub-codebook the user with largest effective channel power according to Eq. (2.57) over all codewords is selected. Note, that the assignment of sub-codebooks to RBs is still the same as in Fig. 2.26. Only the assignment of users to the sub-codebooks and codewords is done by the BS. The assumption here is that each users reports per RB for the respective sub-codebook the power values of the best codeword. That is similar to PMI and CQI reporting in LTE. This means, that per RB the BS searches the maximum value out the ˜K feedback values. It can be observed in Fig. 2.28 that the sub-codebook splitting utilizes the same user diversity gains as MRT with an approximately constant gap. Overall, the sub-codebook splitting can achieve the same beamforming gain as MRT due to the high channel quantization but with less than 10 % pilot overhead, assuming one pilot per codeword on a RB resolution in LTE.
Remark. It is shown that with the proposed sub-codebook splitting scheme beamforming gains similar to MRT can be achieved in a FDD system while keeping the pilot overhead low, e.g. in the shown example by less than 10 %.
Remark. With codebook based precoding, frequency aware user scheduling is still required in massive MIMO FDD systems, because the channel hardening effect does not kick in without channel aware phase adaptation in the precoder.
(a) Spectral efficiency performance of codebook splitting, where the reference cases 4×4 full codebook and 16×16 full codebook consider no sub-codebook split-ting. 4×4 corresponds to the same size as the sub-codebooks and 16×16 to the full codebook before splitting. In case of maximum of sub-codebook block, the best codebook out of 16 sub-codebooks in the block is selected.
0 10 20 30 40 50
Subcarrier index -110
-105 -100 -95 -90 -85 -80
10log10
1 ˜h˜hH
2
Codebook MRT
(b) Channel hardening comparing codebook and MRT precoder which correspond to channel phase adaptive and non-phase adaptive precoders, respectively.
Figure 2.27.: Sub-codebook splitting evaluation.
Figure 2.28.: Spectral efficiency over the number of available users in the cell.