As suggested in [1] and [16], massive MIMO operating in the FDD mode is unrealistic due to the fact that: i) a DL pilot of duration equals to the RRH antenna number is called for; ii) a vector CSI of the same length needs to be fed back from each UE. In this subsection, we develop an RR DL SSFC estimator and its corresponding pilot design that makes significant amount of feedback reduction possible.
While shadow fading is frequency-invariant [43]– [46], unlike in the TDD mode, there is no reciprocity between UL and DL pathloss in the FDD mode. Fortunately, a translation can be done to obtain DL pathloss with the estimated UL ones.
Figure 5.3: The flow chart of closed-loop transceiver design in FDD mode. Colored blocks are done by MSs while others done by BS.
Remark 25. Suppose the frequency bands used for UL and DL transmission are respec-tively ωULand ωDL. The frequency dependence of pathloss is approximately characterized by
Therefore, for a same link using different bands, βUL
βDL ≈ ωDL2
ωUL2 . (5.6)
Besides, the fact that UE k’s UL mean AoA φk and SSFC modeling order mk equal to the DL counterparts enables RR DL SSFC estimation and even feedback reduction.
Specifically, UE k receives the pilot Pk∈ CMn×T emitted by its serving RRH n:
xHk =
However, as βkDL and φk are not known to UE k, the following DL pilot structure is proposed with T = mk:
Pk= 1
pβkDLW(φk)Qmk (5.8)
which results in ˆbk = xk, simply the received signal that requires minimal extra com-putation. Note that in (5.8), the multiplication of W(φk) can be regarded as RRH beamforms DL pilot (or signal) to the φk direction for better signal power concentra-tion. This RR DL SSFC estimate potentially reduce the amount of CSI needed to be fed back since vector gk of dimension M is represented by length-m ˆbk. While for practical systems as LTE, vector quantization (or codeword selection) is done to the feedback
vector [28], we assume in the following that the vector is fed back without quantization and, furthermore, via a noise-free channel.
F1 All K UEs broadcast their mutually orthogonal pilot sequences to all N RRHs simultaneously.
F2 All N RRHs estimate K UL LFSCs with (3.7) and translate them into βˆDL= ˆβULωUL2
ω2DL, (5.9)
information intended for the CBS.
F3 The CBS performs cluster selection algorithm given in [4] with the gathered DL LSFC estimates.
F4 Each selected RRH chooses a basis (DCT or polynomial-based) for RR UL SSFC estimation.
F5 Each selected RRH determines the modeling order ˆm?k for each served UE k via the IMOD algorithm and obtains its RR UL SSFC and mean AoA estimates using (4.10)–(4.11) with this order.
F6 RRHs determine whether or not to perform antenna selection with (5.3).
F7 Each RRH beamforms the DL pilot intended for a served UE to AoD ˆφk obtained in F4 and compensate for the DL LSFC ˆβkDL in F2, i.e., (5.8).
F8 All served UEs feed their received signal xk back to serving RRHs.
F9 RRHs recover DL SSFC estimates via ˆgHk = xHk QHmˆ?
kWH( ˆφk). (5.10)
F10 Each RRH performs either MRT or ZF precoding with DL LSFC and SSFC esti-mates obtained in F2 and F9, respectively.
F11 Each RRH doing power allocation via compensating the LSFCs for all serving UEs.
5.5 Numerical Results and Discussion
In this section, we present the simulation results about sum rate performance of antenna selection algorithms (CS and GNS) and different precoding methods (ZF and MRT). Here we denote MT the total number of RRH antennas and MF the number of selected RRH antennas. From Fig. 5.4-5.5, we can see that when MF is about more than two times larger than K with K the number of MSs, GNS algorithm has almost the same sum rate performance as CS as we have discussed in Lemma 5.1.1. In other words, we only have to use low complexity GNS algorithm to do TAS in massive MIMO system. Moreover, both figures tell us the fact that when MF is finite, or MKF < ∞, the achievable sum rate of MRT is strictly inferior than ZF precoder, as we have shown in Lemma 5.1.2.
0 50 100 150 200 250 300 350 400 450
5 10 15 20 25 30 35 40 45
Total number of RRH antennas M T
Sum capacity(bps/Hz)
ZF(GNS) ZF(CS) MRT(GNS) MRT(CS)
Figure 5.4: Sum rate performance of antenna selection algorithms (CS and GNS) versus MT provided that ZF and MRT precoding are used, where MT = 3MF represents the number of RRH antennas, MF is the number of selected antennas, number of users is 4, and SNR= 11.76 dB.
0 20 40 60 80 100 120 140 160 180 200 5
10 15 20 25 30 35 40 45
Number of selected antennas M F
Sum capacity (bps/Hz)
ZF(GNS) ZF(CS) MRT(GNS) MRT(CS)
Figure 5.5: Sum rate performance of antenna selection algorithms (CS and GNS) versus MF provided that ZF and MRT precoding are used, where MT = 200 represents the number of RRH antennas, MF is the number of selected antennas, number of users is 4, and SNR= 11.76 dB.
Chapter 6 Conclusion
Due to the effect of noise reduction in massive MIMO systems, a novel LSFC es-timator for both i.i.d. and spatial correlated channel is proposed. This eses-timator can extended to the one considering multiple pilot blocks to improve performance. Advan-tages of the proposed LSFC estimator includes: low complexity and no prior knowledge of SSFCs and spatial correlation is needed. Furthermore, it can be shown that the proposed LSFC estimator has asymptotically zero-MSE and hence approaches to the MMSE estimator when the number of BS antennas is large. The fact that it even sig-nificantly outperforms the conventional LSFC estimator with known SSFCs is revealed by the simulation results.
By using the estimated LSFCs, a estimator incorporating individual estimation of mean AoA and SSFC combined with rank-reduced channel model is also presented. The estimated mean AoA is helpful for downlink beamforming, while the RR characteristics enables us to reduce the amount of feedback overhead, pilot dimension and hence pilot transmission time in FDD mode. We analysis the effect of modeling order on MSE performance of the proposed estimator by mathematical approaches and find out that the
“bias matrix” is an important metric in modeling order selection. Moreover, we present some candidate for basis selection as KLT, DCT, and polynomial basis, after connecting the RR model to image signal processing, we show that the DCT-II basis is a practical, low computational complexity and near-optimal choice owing to the so-called energy
compaction property. Eventually, IMOD algorithm is devised to determine the optimal modeling order used in the SSFC estimator, it offers a low computational-complexity approach to quickly determine the optimal modeling order (or, rank indicator) for each user as shown in our simulation results.
Numerical results show the effectiveness of the proposed estimator to enhance accu-racy by reducing the number of parameters needed to be estimated and that an optimal modeling order is related to the value of AS and SNR. In addition, massive MIMO system is shown to be helpful in terms of reducing the NMSE of the proposed SSFC estimator. After that, with the aid of the large aperture offered by a massive MIMO BS, a precise estimation of mean AoA is seen, even when the AS is not small.
Finally, we present a closed-loop transceiver design of distributed massive MIMO sys-tem for both TDD and FDD mode. The transceiver combines the estimators proposed in the previous sections with a practical MIMO system including antenna selection, cluster selection and transmit beamforming. With the aid of massive number of antennas, the antenna selection metric has been simplified a lot even if we consider the LSFC and spatial correlation into this metric. Finally, we show that in FDD mode, all MSs need to do is to feedback their received signal instead of doing channel estimation and a limited feedback effect can help us to construct a smaller codebook.
Appendix A
Proof of Theorem 3.2.1
Lemma 3.3.1 implies that if
lim sup
Note that Assumption 1 is equivalent to condition (A.3) and can imply condition (A.1) because if ∀i, kΦ
Appendix B
Proof of Theorem 4.4.1
We first derive the variance term in MSE. Substitute (4.32) into (4.25) yields h = Eˆ
where step (a) follows from the fact that property that the matrices in a trace of a product can be switched, step (c) follows from the equality W(φ)WH(φ) = IM, and step (d) utilizes the orthogonality of a basis matrix.
The bias term, (4.26), renders
b(ˆh) = E
In the above derivation, step (a) is obtained by substituting (4.32) into (4.26), step (b) follows from the fact that trace of a scalar is equal to the scalar itself and step (c) invokes the property that the vectors (matrices) in a trace of a product are commutative.
Since
(B.3) can be simplified as
b(ˆh) = tr((W(φ)QmQHmWH(φ) − IM)2Φ)
def= tr(AΦ) (B.5)
Note that when full modeling order is used, m = M , and A = OM and the estimator becomes an unbiased estimator, which is consistent with Lemma 4.4.3. Nevertheless, for the general case that m ≤ M , we use the decomposition
QmQHm = Q
where Q ∈ CM ×M the “complete” basis matrix (i.e. no rank reduction), to express A as A = (W(φ)QmQHmWH(φ) − IM)2
The matrix Dm is an orthogonal projection and idempotent matrix since Dm =
Suppose ei is an all-zero vector except for the ith entry being 1, and W is a subspace of RM that is spanned by the orthonormal set of vectors em+1, · · · , eM, that is, W = span
Om×(M −m) IM −m
, then Dm is clearly the orthogonal projection matrix on W.
Hence, (B.7) and (B.5) can be further simplified as A = W(φ)QDmQHWH(φ) and
b(ˆh) = tr(AΦ)
= tr(W(φ)QDmQHWH(φ)Φ)
= tr(DmQHWH(φ)ΦW(φ)Q)
def= tr(DmB) (B.9)
where the bias matrix B is a positive semi-definite matrix because of the facts that BH = QHWH(φ)ΦW(φ)Q = B
and
xHBx = xHQHWH(φ)ΦW(φ)Qx
= xHQHWH(φ)Φ12Φ12W(φ)Qx
= kΦ12W(φ)Qxk22 ≥ 0, ∀ x (B.10) When substituting ei, ∀ 1 ≤ i ≤ M into the above x, it is obvious that all diagonal entries in B are non-negative. Furthermore, the definition of Frobenius inner product [33, Ch. 10.4] implies that
b(ˆh) = tr(DmB) = hDm, BiF (B.11) where h·iF denotes Frobenius inner product. The two equivalent expressions conclude that the bias term is the sum of the last (M − m) diagonal terms (which are all non-negative) of B.
Appendix C On Remark 14
If the mean AoA is not 0◦, the spatial correlation matrix is a complex matrix, that is, Φ ∈ CM ×M. However, [12] has shown that when the AoA spread is small (say, less than 15◦),
Φ ≈ W(φ) ¯ΦWH(φ) (C.1)
where ¯Φ ∈ RM ×M is a real matrix with Φ¯ij
= J0(|i − j|2πdλ ∆ cos φ), ∆ being the AS, λ being the wavelength and d being the antenna spacing. Hence, we can treat WH(φ)ΦW(φ) as an operation that rotate the phase of Φ to make it a real matrix.
From the “energy compaction property” [36] [41] of DCT-2, if we transform a length-M real sequence (or, real vector), x, by DCT-2, then in frequency domain, the larger coefficients (or strictly speaking, coefficients larger than a benchmark, η = 1) must be more highly concentrated at low indices much smaller than M . More specifically,
1 − Pmˆ?
`=1xl PM
`=1xl 1 (C.2)
with ˆm? M being defined as the index of first entry smaller than η on x, and all entries whose index larger than ˆm? are also smaller than η. Note that DCT-2 is a separable transform [34, Ch. 4], or, to put it in another way, multidimensional DCT-2 can be decomposed into successive application of one-dimensional DCT-2 in the appropriate directions.
Since B = QHWH(φ)ΦW(φk)Q, we can use a two-step interpretation to describe B:
Step 1 Phase-rotating Φ to be a real matrix, WH(φ)ΦW(φ).
Step 2 Using two-dimensional DCT-2, applied subsequently to rows and columns of the real matrix WH(φ)ΦW(φ), to transform it into frequency domain, B, where
1 −
where ˆm? << M , owing to the energy compaction property [36, Ch. 8] [41] of DCT-2.
Besides, the concept of using polynomial basis is to approximate the smooth spatial waveform by a polynomial with lower order, m, than the full order, M . When the polynomial basis is used, we can replace Step 2 as
Step 2 Using two polynomial to describe respectively the rows and columns of the smooth real matrix WH(φ)ΦW(φ). Because the spatial waveform (either along rows or columns of WH(φ)ΦW(φ)) is smooth, the larger coefficients are also highly concentrated at low polynomial orders (i.e. low indices), while the coefficients corresponds to higher polynomial order are all small than η (since we do not need such high-order polynomial to describe the smooth spatial waveform). In other words,
Consequently, the diagonal entries in B must also be highly concentrated at low in-dices much smaller than M as in the DCT-2 basis case. We can say that, the polynomial basis we used, which is a discrete polynomial transform, also has the so-called “energy compaction property”. And the compaction performance of it is only slightly worse than that of DCT-2 owing to Remark 11.
Appendix D
Proof of Theorem 4.5.1
Above all, we rewrite ˆh(I) in (4.44) as hˆ(I) = 1
γQmQHmYp
= 1
γQmQHm(γh + Np)
= QmQHmh + 1
γQmQHmNp (D.1)
Take expectation to (4.44) yields
E
nhˆ(I)o
= QmQHmh + E
1
γQmQHmNp
= QmQHmh (D.2)
Substituting (D.2) into (D.1), we rewrite ˆh(I) as hˆ(I) = En
hˆ(I)o + 1
γQmQHmNp.
Thus,
where step (a) follows from the property (B.2).
After that, the bias term in the MSE can be derived:
b(ˆh(I)) = E
in step (a), we substitute (D.2) into (4.26), step (b) follows from the fact that trace of a scalar is equal to the scalar itself, step (c) utilizes the property that the vectors (matrices) in a trace of a product can be switched, and step (d) is according to (B.4).
Similar to (B.7), ˜A can be rewritten as
A˜ = (QmQHm− IM)2
= (IM − QmQHm)2
= (Q
IM −
Im Om×(M −m)
O(M −m)×m O(M −m)
| {z }
Dm
QH)2
= QDmQHQDmQH
= QD2mQH
(a)= QDmQH (D.5)
where step (a) uses the idempotent property of Dm in (B.8).
Accordingly, substitute (D.5) into (D.4) yield
b(ˆh) = tr(AΦ)
= tr(QDmQHΦ)
= tr(DmQHΦQ)
def= tr(DmB),˜ (D.6)
thus proves Theorem 4.5.1. It is worth mentioning that ˜B is the positive semi-definite bias matrix of the SSFC-2 estimator.
Appendix E On Remark 17
If the mean AoA is not 0◦, Φ ∈ CM ×M and, according to (C.1), can be regarded as a phase-rotated version of a real matrix, ¯Φ ∈ RM ×M.
As DFT{ejω0nx[n]} = X(ej(ω−ω0)), where X(ejω) =DFT{x[n]}, performing two-dimensional (2D) DCT-2 on the phase-rotated matrix Φ, the η−support also shifted to some medium frequency indices. In other words, suppose that x is a length-M com-plex vector, and after transformed by the DCT-2,
1 − Pmˆ?
`=mr−1xl
PM
`=1xl 1 (E.1)
with ˆm? being defined as the index of first entry smaller than η on x, and all entries whose index larger than ˆm? are also smaller than η, mr being defined as the index of first entry larger than η on x, and ˆm?− mr M .
Consequently, ˜B can be decomposed into the following two steps:
Step 1 Rotating the phase of a real matrix ¯Φ to obtain the phase-rotated complex matrix, Φ.
Step 2 Using the two-dimensional DCT-2 (or discrete polynomial transform, applied subsequently to rows and columns of the phase-rotated matrix W(φ) ¯ΦWH(φ).
Owning to the energy compaction property of DCT-2 (or discrete polynomial transform) [34, Ch. 4] [35, Ch. 2], the resulting frequency domain coefficients larger than a fixed value, η = 1, are highly concentrated; however, they are mostly
concentrated within some medium frequency indices due to the frequency transla-tion property. More precisely,
1 − Pmˆ?
`=mr−1[ ˜B]ll PM
`=1[ ˜B]ll = 1 − Pmˆ?
`=mr−1[ ˜B]ll
M 1 (E.2)
with ˆm? − mr M .
Note that when the polynomial basis is used, we can treat it as a kind of discrete polynomial transform which also has energy compaction property as mentioned in Ap-pendix C, and the above interpretation is still valid. Therefore, it would lead to the same result as the DCT-2 basis case.
Appendix F
Proof of Lemma 4.5.3
Since
tr(DmB) − tr(D˜ mB)
= tr(DmQHΦQ) − tr(DmQHW(φ)HΦW(φ)Q)
= tr(DmQH(Φ − W(φ)HΦW(φ))Q)
(a)= tr(D2mQH(Φ − W(φ)HΦW(φ))Q)
(b)= tr(DmQH(Φ − W(φ)HΦW(φ))QDm) (F.1)
where step (a) utilizes the idempotent property of Dm in (B.8), and step (b) follows from the fact that trace of product of matrices is commutative.
We need the following two lemmas from [32, Appendix 1].
Lemma F.0.1. A square matrix A ∈ CM ×M is positive semi-definite if and only if the principal minors are all nonnegative.
Lemma F.0.2. A square matrix A ∈ CM ×M is positive semi-definite if and only if it can be written as
A = ZZH (F.2)
where Z ∈ CM ×M may not be full rank.
It is clear that the principal minors are all zero, hence nonnegative, due to the all-zero diagonal terms of (Φ − W(φ)HΦW(φ)). (Φ − W(φ)HΦW(φ)) is thus positive
semi-definite by Lemma F.0.1. Applying Lemma F.0.2 to (Φ − W(φ)HΦW(φ)), we obtain
Φ − W(φ)HΦW(φ) = ZZH (F.3)
which, when substituting into (F.1), gives
tr(DmB) − tr(D˜ mB) = tr(DmQHZZHQDm) = kZHQDmk2F ≥ 0 (F.4)
and completes the proof.
Appendix G
Proof of Lemma 5.1.2
We focus on RRH-` and omit the superscript ` of M` and K` henceforth. Let S be the current antenna set, and MF = |S|.
(Capacity of MF precoder)
Without loss of generality, we assume equal power allocation and denote by P , hH` , WS = GHS, and x, the total power, `th row of HS, precoding matrix and transmitted signal, respectively. We then have
√pk =
and the (composite) received signal from all K serving users is given by
y = GSWSPSx + n −→a.s.
The signal received by the ith user is yi with the corresponding SINR given on the top of next page.
SINRi,M F −→a.s.
Thus the capacity is given by
C = X
i
log(1 + SINRi,M F) (G.2)
Theoretically, the hHi hj, ∀j 6= i terms in (G.1), which are the non-diagonal terms of HHH, approach to zero as MK → ∞ in according with Lemma 3.3.1. However, our numerical experiment indicates that even if MK = 1000000, and the there is no spatial correlation, the non-diagonal terms of HHH are still large (about several hundreds to several thousand). These large non-diagonal terms result in strong interference in (G.1), thus, SINRi,M F becomes much smaller than expected.
SINRi,M F P M βi2
tr(Dβ) = P M
tr(Dβ)/βi2 (G.3)
From the above derivation and simulation results, we conclude that in practice, MF precoder is not a good precoder in massive MIMO system.
(Capacity of ZF precoder)
Similar to the MF case, we assume the system employs equal power allocation with total power P . With the precoding matrix ˜WS = G†S = GHS(GSGHS)−1, we have so that the (composite) received vector become
y = GSW˜SPSx + n
−→a.s.
s P M
tr(D−1β )x + n (G.4)
whose ith component represents the received signal of the ith user yi −→a.s.
s P M
tr(D−1β )xi+ ni (G.5)
The corresponding SINR renders the asymptote SINRi,ZF −→a.s. P M
tr(D−1β ) (G.6)
The resulting capacity has a similar expression as (G.2).
As the LSFC for all K serving users are approximately equal due to the fact that an RRH only serves the nearby MSs, we obtain from (G.3) and (G.6)
SINRi,M F a.s. P M
tr(Dβ)/βi2 ≈ P M
K/βi (G.7)
SINRi,ZF −→a.s. P M
tr(D−1β ) ≈ P M
K/βi, (G.8)
which causes
SINRi,M F a.s. SINRi,ZF (G.9)
This means ZF precoding can remove all interference and offer an SINR much higher than that achievable by using an MF precoder even in a practical massive MIMO system where MK < ∞.
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