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A Weighted Least Squares Approach to Precoding with Pilots for
MIMO-OFDM
Tarkesh Pande, David J. Love, and James V. Krogmeier
School of Electrical and Computer Engineering
Purdue University
West Lafayette, IN 47907
{pande, djlove, jvk}@ecn.purdue.edu
December 14, 2005
Next generation wireless systems are expected to leverage multiple antenna, commonly called multiple-
input multiple-output (MIMO), technology transmitting over broadband channels via orthogonal frequency
division multiplexing (OFDM). It is now well-known that MIMO systems can obtain error rate and capacity
improvements by adapting the transmitter to the current channel conditions. Linear spatial precoding is
a popular technique for channel adaptation where the transmitted space-time signal for each subcarrier is
multiplied by a precoding matrix before transmission. Spatial precoding is complicated, however, in OFDM
because channel information is often only known for a small number of pilot tones. In this paper, we present
a subspace interpolation method for designing spatial precoders for all subcarriers by using the precoders
known on pilot tones. The non-pilot precoders are solutions to a weighted least squares design on the
Grassmann manifold. Simulation results show performance benefits over existing MIMO-OFDM techniques.
Index Terms -Closed-loop, Interpolation, Limited feedback, MIMO systems, Orthogonal frequency division
multiplexing, Subspace coding.
Permission to publish abstract separately granted.
This work was supported in part by the SBC Foundation, NSF under Grant CCF0513916 and the Indiana
Twenty First Century Research and Technology Fund.
1 Introduction
Recent work has shown that multiple antenna systems (often called multiple-input multiple-output (MIMO)
systems) exploiting channel knowledge at the transmitter, commonly called closed-loop systems, can obtain
large benefits in error rate reduction and capacity improvements compared to non-adaptive systems. In
narrowband systems that experience flat-fading, methods have been devised allowing both time division
duplexing (TDD) and frequency division duplexing (FDD) to efficiently use channel knowledge. Because
of the substantial data rate requirements, next generation communication systems will transmit over large
bandwidths with signals that experience frequency-selective fading. One effective method for dealing with
frequency-selectivity is orthogonal frequency division multiplexing (OFDM). OFDM constructs a broadband
signal using an orthogonal transformation applied to a multitude of narrowband signals. The end result is
that instead of one high-rate multi-tap channel we are left with a number of lower-rate single-tap channels.
Applying closed-loop techniques is challenging because of the number of different channels corresponding
to different subcarriers. Channel estimation for OFDM is commonly done using pilots [1]. The general idea
is for the OFDM symbol to interlace Kpilots tones within the Nsubcarriers. This means that each OFDM
symbol still has N−Ksubcarriers transmitting user data. Because the data conveyed on the pilot tones is
predetermined, the transmitted symbol can be divided out of the received signal to yield a flat-fading channel
estimate at each pilot. Thus, the receiver will have knowledge of the Kchannels corresponding to the pilots.
Common OFDM channel estimation techniques use interpolation methods on the channels obtained from
the Kpilots to estimate the channel for all Nsubcarriers. Example interpolators include minimum mean
squared error [2] and linear interpolation [3]–[6].
We are interested not in channel estimation but rather in adapting the transmitted signal to the current
channel conditions. Linear precoding [7]–[9] is a practical solution that adapts the transmitted space-time
signal (e.g., spatial multiplexing, orthogonal space-time block coding, etc.) to the channel by multiplying the
signal by a matrix before transmission. The intuition here is that the matrix directs the transmitted space-
time signal towards the “good” directions of the channel while avoiding the “bad”. In TDD narrowband
systems, precoders can be designed at the transmitter for the forward-link channel using the channel estimate
on the reverse-link. The transmitter in an FDD narrowband system, however, does not have knowledge of
the forward-link channel matrix. This can be overcome through the use of feedback from the receiver to the
transmitter [10]–[18].
In this paper, we address the general problem of designing precoding matrices for all the tones given
only the precoding matrices for the pilot tones. Just as in [12], [13], [15], [16], we view the precoders as points
in the Grassmann manifold. The interpolator solves a weighted least squares problem on the Grassmann
manifold. Thus, the interpolation is done on subspaces rather than matrices using a subspace, rather than
Euclidean, distance. Our interpolators can be based on any number of pilots and can be simply implemented
2
using common techniques from linear algebra. Specifically, we show that this interpolation relates to finding
the principle components of a matrix obtained by combining the weighted generator matrices of the precoder
subspaces.
This is a different approach than taken in [19]–[21] where the precoders are designed using a phase
invariance weighting of the nearest two pilots meant to maximize some condition. Our technique can include
any number of pilot tones. In addition, we do not require feedback of a unitary subspace rotation matrix as
in [21]. This reduction in feedback can be significant when the number of pilots is large. This problem has
practical significance particularly with application to next generation wireless local area networks (WLANs)
and wireless metropolitan access.
The organization of this paper is as follows. Section 2 gives a general overview of precoding for MIMO-
OFDM. We propose an interpolation scheme that uses the precoders obtained from pilot subcarriers to design
precoders for all Nsubcarriers in Section 3. Section 4 gives simulation results. We conclude in Section 5.
2 Precoding Systems
Consider an Mttransmit antenna and Mrreceive antenna MIMO-OFDM system transmitting with N
subcarriers. We will assume that the signal experiences an L-tap frequency selective channel in time given
by 1
G[i] =
L−1
X
l=0
Glδ[i−l] (1)
where Gl∈CMr×Mtfor all l. We will assume that Glhas independent and identically CN(0,1) distributed
entries and that Gl1is independent of Gl2for all 0 ≤l1< l2< L.
We will deal with the signal design and analysis in the frequency domain using an N-point inverse fast
Fourier transformation (IFFT) at the transmitter and an N-point fast Fourier transformation (FFT) at the
1We use j=√−1, δ[·] to denote the Kronecker delta function, ∗to denote matrix conjugate transposition,+to
denote the pseudo-inverse, E[·] to denote expectation, Ck×lto denote the vector space of k×lcomplex matrices,
IMto denote the M×Midentity matrix, CN(0,1) to denote a complex Gaussian random variable with zero mean
and unit variance, λi{A}to denote the ith singular value of A,| · | to denote the absolute value, tr(·) to denote the
trace of a matrix, k · k2to denote the matrix two-norm, k·kFto denote the matrix Frobenius norm, U(Mt, M) to
denote the set of Mt×Mmatrices with orthonormal columns, card(·) to denote the cardinality of a set, and the
combinatorial function
n
k
=n!/[k!(n−k)!].
3
receiver. In addition, we will assume that the length of the cyclic prefix is greater than the intersymbol
interference (ISI) length introduced by the channel. After removing the cyclic prefix at the receiver the
MIMO frequency-domain channel for the nth subcarrier (where n= 0,1, . . . , N −1) is given by
Hn=1
√N
L−1
X
l=0
Glexp µ−j2πnl
N¶.(2)
This allows us to write the input-output relationship for the nth tone as
Yn=HnXn+Nn(3)
where Nnis an Mr×Tmatrix with CN(0,1) entries, Xnis an Mt×Ttransmitted matrix with tr (E[X∗
nXn]) =
T ρ, ρ is the signal-to-noise ratio (SNR), and Tis the number of channel uses involved in transmitting one
space-time signal. Examples of Tinclude T= 1 for spatial multiplexing [22], [23], T= 2 for a two-antenna
Alamouti code [24], and T= 1 for beamforming (i.e. one-dimensional precoding) [22].
The transmitted signal Xncan be classified into two cases: open-loop and closed-loop. In an open-loop
system, the signal Xnis designed independently of the channel Hn. In a closed-loop system, the signal
Xnis designed as a function of Hn.We will assume that the MIMO-OFDM system uses a special kind of
closed-loop signaling known as linear precoding [7]–[9]. In linear precoding, the transmitted matrix can be
decomposed as
Xn=FnSn(4)
where Fnis an Mt×M(with 1 ≤M≤Mt) precoding matrix and Snis an M×Tspace-time signal. The
space-time signal could be generated with an M-dimensional spatial multiplexing encoder (Sn=sn∈CM×1)
[23], a single-dimensional modulator (Sn=sn∈C1×1) [22], or an Mdimensional orthogonal space-time block
code [25] (Sn∈CM×T).To constrain the peak transmitted power, we will assume that Fn∈ U(Mt, M ) for all
subcarriers . This approach is a natural extension of the work in antenna subset selection space-time signaling
(see for example [26]–[28]). This kind of channel orthogonalization precoding has been used for beamforming
[10], [12], [13], precoded orthogonal space-time block codes [15], and precoded spatial multiplexing [7], [16].
4
The optimal unquantized precoders for the above space-time signaling schemes can be determined from
the selection criteria being considered. Precoding criteria can generally be divided into two categories i)
minimizing the error rate based on the receiver for the space-time signaling scheme ii) maximizing the
mutual information for the precoded channel.
i) Minimizing Error Rate
a) Beamforming: An equivalent objective to minimizing the average probability of symbol error is
to maximize the post processed SNR [29]. For beamforming systems the SNR γnseen at the
receiver for each subcarrier after combining is
γn=ρ|F∗
nH∗
nHnFn|2
kF∗
nH∗
nk2
2
.(5)
In this case the precoder vector Fopt that maximizes (5) is the right singular vector of Hn
corresponding to the largest singular value of Hn[12]. Note that ej φFopt for an arbitrary φalso
maximizes (5) implying that the optimal precoder is not unique. In fact, all vectors with the
same column space provide the same performance.
b) Spatial Multiplexing: For spatial multiplexing systems, commonly used receiver architectures do
either i) maximum likelihood (ML) decoding, ii) minimize the minimum mean square error
(MMSE) or iii) zero-forcing (ZF) . The ML receiver while providing the best performance has the
highest computational complexity. MMSE and ZF decoders are linear receivers whereby a linear
transformation C, an M×Mrmatrix, is first applied to the received signal Yn. This is followed
by an operation Q(·) which does single dimension ML decoding on each substream to get the
estimate of the transmitted space-time vector signal i.e., b
Sn=Q(CYn). For MMSE decoding,
C= [F∗
nH∗
nHnFn+ (MN0/Es)IM]−1F∗
nH∗
nwhile the ZF receiver has the form C= (HnFn)+.
We now review the precoding selection criterion for ZF systems. Precoding for ML and MMSE
receivers can be found in [16] and references therein.
In [30], it is shown that the union bound of the vector symbol error performance for a ZF system
5
depends on the substream with minimum SNR. For this case
SN R(ZF )
min ≥λ2
min{HnFn}Es
MN0
(6)
where λmin{HnFn}is the minimum singular value of HnFn. The optimum unquantized precoder
maximizes (6) which in turn minimizes the vector symbol error probability. Let the singular value
decomposition of Hnbe given by
Hn=VLΣV∗
R(7)
where VL∈ U(Mr, Mr), VR∈ U(Mt, Mt) and Σis an Mr×Mtdiagonal matrix containing the
singular values of Hn. In [16], it is shown that the optimal precoder matrix is Fopt =VRwhere
VRis the matrix constructed from the first Mcolumns of VR. The optimal precoder matrix is
not unique as FoptU,U∈ U(M , M) provides performance identical to that of Fopt. This can be
easily seen because [31]
λmin{HnFopt U}=λM{VLΣV∗
RFoptU} ≤ λM(Hn) (8)
where λM(Hn) is the Mth singular value of Hn. In (8) the upper bound is achieved when
Fopt =VR. The minimum singular value is invariant to right multiplication by a unitary
matrix. Therefore, all matrices that generate the same column space give the same minimum
singular value performance.
c) Precoded Orthogonal Space-Time Block Coding (OSTBC): For precoded OSTBC, the probability
of symbol error (SER) given channel knowledge Hnwhen using an ML detector can be written
as [32]
P r(ERROR|Hn)≤exp(−γkHnFnk2
F) (9)
where γis a function that depends on M,ρand Sn. If γis fixed, minimizing SER corresponds to
maximizing kHnFnk2
F. In [15] it is shown that the optimal precoder corresponds to Fopt =VR.
6
Again, precoder performance is invariant to right multiplication by a unitary matrix as
kHnFnUk2
F=tr(U∗F∗
nH∗
nHnFnU) = tr(F∗
nH∗
nHnFn) = kHnFnk2
F.(10)
ii) Capacity
As opposed to minimizing symbol error rates, an alternative criteria for precoding is to determine Fn
that maximizes the capacity of the channel [27],[33]. The uninformed transmitter capacity or mutual
information I(Fn) assuming an uncorrelated complex Gaussian source given Hnand a fixed Fnis
I(Fn) = log2det µIM+Es
MN0
F∗
nH∗
nHnFn¶.(11)
The precoder maximizing the above is also Fopt =VR[16]. This precoding criteria applies to all the
above space-time signaling schemes. Finally, it is worth noting that precoder performance depends
only on its column space as maximizing I(FnU) for U∈ U(M, M ) is equivalent to maximizing
detÃU∗³IM+Es
MN0F∗
nH∗
nHnFn´U!which is equivalent to maximizing det ³IM+Es
MN0F∗
nH∗
nHnFn´.
For MIMO-OFDM systems the linear precoding matrix Fnmust be designed as a function of Hnfor every
subcarrier. In this paper, we will study the use of pilots that are interspersed throughout the OFDM symbol.
We will assume that there are Kpilots transmitting known data. The kth pilot (with k= 0,1, ..., K −1) is
located at subcarrier kN
K.This is a comb-type tone arrangement and has been studied in [1].
Mathematically, the pilots correspond to
Y(k)=H(k)Ξ(k)+N(k)(12)
for the kth pilot. The matrix Ξ(k)is some sort of training space-time signal that is known to both the
transmitter and the receiver and allows the receiver to solve for H(k)for 0 ≤k≤K−1.
In TDD, perfect channel reciprocity would allow the transmitter to also know H(k)for 0 ≤k≤K−1.
In FDD, the transmitter has no knowledge of the channels corresponding to the forward-link pilots. Thus,
some form of feedback must be used. We will consider the idea of using the narrowband feedback techniques
developed in [10]–[13], [15]–[18] for each pilot subcarrier in the OFDM symbol. The receiver will use its
7
knowledge of H(k)for 0 ≤k≤K−1 to design precoders F(k)for each pilot. The idea is to restrict each
F(k)to lie in a codebook F={F1,F2, . . . , F2B}.The codebook is known to both the transmitter and the
receiver. Because of this, a total of K·Bbits of feedback can be sent from the receiver to the transmitter
to convey the Kprecoders for the pilots that were chosen from the codebook.
Because subcarrier numbers can possibly number in the hundreds, it is impractical for an FDD system
to feedback a total of N·Bbits of feedback to design Fnfor 0 ≤n≤N−1.For this reason, it is of utmost
importance to find efficient methods for determining all Fnprecoders from F(1) ,F(2),··· ,F(K).
3 Interpolation Precoding
We will discuss the interpretation of precoders as points in the Grassmann manifold before presenting the
interpolator.
3.1 Grassmannian Precoding
As described in the previous section, the precoder Fnis usually chosen with respect to some criterion
such as i) maximizing the minimum singular vector of HnFn,ii) maximizing the capacity of the effective
channel HnFn,iii) minimizing the mean squared error (with a minimum mean squared error receiver) when
transmitting over HnFn,or iv) maximizing the Frobenius norm of HnFn.All of these criteria are invariant
to multiplication of Fnby an M×Munitary matrix U∈ U(M , M).This means that, from the perspective of
these criteria, FnUprovides performance identical to Fn.Note that this means the performance is dependent
only on the column space of Fn. This kind of invariance was used in [12], [13], [15], [16] to significantly reduce
the quantization problem of designing Ffor use in FDD systems.
Because of this invariance, we will view each precoder Fnas a subspace rather than a matrix. The set
of all M-dimensional subspaces of CMtis known as the Grassmann manifold and denoted by G(Mt, M).
Distances can be defined on the Grassmann manifold just as on standard Euclidean spaces [34]. The chordal
distance between subspace P1and P2with corresponding orthogonal basis matrices FP1and FP2(i.e.,
8
FP1,FP2∈ U(Mt, M )) is given by
d(FP1,FP2) = 1
√2°
°FP1F∗
P1−FP2F∗
P2°
°F(13)
where k·kFdenotes the Frobenius norm. While this distance is written as a function of the basis matrix, the
distance is only a function of the subspaces since FUU∗F∗=FF∗when F∈ U(Mt, M ) and U∈ U(M , M).
Two subspaces in G(Mt, M) can be oriented in relation to each other using the Mprincipal angles
φ1, . . . , φMbetween the subspaces. In the M= 1 case, there is only one principal angle and it is given by
arccos ¡¯¯f∗
P1fP2¯¯¢where fPiis the unit vector that generates the one-dimensional subspace (or line) Pi.More
generally,
M
X
i=1
cos2(φi) = °
°F∗
P1FP2°
°2
F.(14)
Multiplying out (13) and using (14), we can see that
d(FP1,FP2) = qM−°
°F∗
P1FP2°
°2
F=°
°sin(φ)°
°2(15)
where sin(φ) = [sin(φ1)···sin(φM)]T.
3.2 Grassmannian Interpolator
This different view will make the problem one of obtaining reliable subspace knowledge at the trans-
mitter. Thus, we must design Fnfor all subcarriers assuming that the transmitter has knowledge of
F(1),F(2) , . . . , F(K)for the pilot precoders.
It has been shown in [15], [35] that precoder performance is dependent on the subspace distance
d2(Fn,Fn,opt) where Fn,opt is the optimal precoder for subcarrier nassuming that Hnis perfectly known to
the transmitter. Thus, we would like the precoder subspace used for signal transmission at each subcarrier to
be “close” to the optimal subspace assuming perfect transmitter knowledge of all subcarrier channels. There-
fore, we would like to reconstruct the subspaces (and corresponding precoder matrices) for all subcarriers
using the pilot precoder matrices.
To solve this subspace precoding problem, we can formulate a weighted least squares problem with the
9
cost function
Fn= argmin
F∈U(Mt,M )
K−1
X
k=0
α2
n(k)d2(F,F(k)) (16)
where αn(1),··· , αn(K) are real valued weights for the estimation at subcarrier n. This weighted least
squares solution will provide a good approximation to the optimal precoder without requiring any channel
knowledge besides the pilot precoder matrices.
The solution we will develop for interpolating will be assuming that a weight set is determined offline.
Notice that an arbitrary subcarrier number ncan always be uniquely written as n=kN
K+lwhere l=
mod(n, N/K ). Using this notation, the weights for a traditional linear interpolator can be written as [1], [6]
αn(k) =
N−Kmod(n,N/K)
Nif k=bnK/Nc
Kmod(n,N/K)
Nif k=bnK/Nc+ 1
0 otherwise.
Alternatively, second-order interpolation could be used [36]
αn(k) =
β(β−1)
2if k=bnK/Nc − 1
−(β−1)(β+ 1) if k=bnK/Nc
β(β+1)
2if k=bnK/Nc+ 1
0 otherwise
where β=Kmod(n,N/K)
N.
Given {αn(k)}for all subcarriers, we will now solve (16). Note that
K−1
X
k=0
α2
n(k)d2(F,F(k)) =
K−1
X
k=0
α2
n(k)µM−°
°
°F(k)∗F°
°
°
2
F¶
=ÃM
K−1
X
k=0
α2
n(k)!−
K−1
X
k=0
α2
n(k)°
°
°F(k)∗F°
°
°
2
F.(17)
The first term in (17) does not have any effect on the optimization in (16). Therefore, the problem can be
reformulated as
Fn= argmax
F∈U(Mt,M )
K−1
X
k=0
α2
n(k)°
°
°F(k)∗F°
°
°
2
F.(18)
Let
Wn=£αn(0)F(0) αn(1)F(1) ··· αn(K−1)F(K−1) ¤.(19)
This weighted matrix allows (18) to be rewritten as
Fn= argmax
F∈U(Mt,M )kW∗
nFk2
F.(20)
10
Consider the singular value decomposition (SVD) of Wndenoted by
Wn=ULΣU∗
R
where UL∈ U(Mt, Mt),UR∈ U(K M, KM ),and Σis an Mt×KM diagonal matrix with the diagonal
element at position (i, i) equal to the ith largest singular value. It follows that [31]
kW∗
nFk2
F=kΣTU∗
LFk2
F
≤
M
X
i=1
λi{F∗ULΣΣTU∗
LF}
=kΣk2
F
(21)
where λiis the ith largest singular value and Σis the matrix consisting of the first Mcolumns of Σ. The
upper bound is achieved when F=ULwhere ULis the matrix formed from the Mleft singular vectors of
Wncorresponding to the Mlargest singular values.
This solution is equivalent to signaling on the Mlargest principal components in the weighted pilot
precoder matrix Wn.The idea is that the weights {αn(k)}adjust the effect of the subspace directionality
in F(k)on the precoder at subcarrier n. In the case when L= 1 (i.e., flat fading), the optimization in (20)
yields that all tones will signal using the same precoder as would be expected.
4 Numerical Performance Analysis
In this section, we compare the performance of the proposed Grassmannian interpolator using linear inter-
polation weights with the modified spherical (MS) interpolators in [19]–[21] and “brick-wall” type precoding
as in [37]. For the beamforming scenario, the MS interpolator is
Fn=αn(k)F(k)+αn(k+ 1)ejθkF(k+1)
kαn(k)F(k)+αn(k+ 1)ejθkF(k+1) k2
(22)
where θkis a phase parameter that maximizes the effective channel gain of the subcarrier furthest from the
interpolating precoders. It is chosen from a finite sized codebook Θ and is calculated by
θk= argmax
ΘkHnFnk2
2n= (k+ 1/2)N/K. (23)
11
The weights αn(k) are for those of linear interpolation. The modified spherical interpolator can be thought
of as a variant of a normalized linear interpolator with an extra parameter ej θk. For the spatial multiplexing
case, a similar expression to (22) is obtained in [21] where the ejθkparameter is replaced by a rotation matrix
Qfollowed by an appropriate orthonormalization. Alternatively, a simpler precoding scheme is to reuse the
pilot precoders for the neighboring subcarriers resulting in a “brick-wall” type arrangement of the precoding
matrices with
Fn=F(k)n= (k−1/2)N/K ···(k+ 1/2)N/K. (24)
This is computationally the simplest scheme possible.
Simulations are done for a MIMO-OFDM system using QPSK with 128 subcarriers (N= 128) and 16
pilot tones (K= 16). We assume that the discrete-time channel impulse response has eight taps between
each transmit and receive antennas pair with a uniform power delay profile and i.i.d. complex Gaussian
distribution as in (1). We also assume that the feedback channel has no delay and no transmission error
and that the receiver has perfect channel knowledge. This will allow us to isolate the effect of subspace
interpolation.
Fig. 1 shows the performance of a 2 ×4 (Mt×Mr) system with beamforming at the transmitter and
maximum ratio combining at the receiver. For the MS interpolators, θis uniformly quantized to 2 bits.
Therefore, with MS interpolation, using a codebook size card(F) = 2 or card(F) = 4 for the beamforming
vectors, a total of 48 bits (K+ 2K) or 64 bits (2K+ 2K) of feedback information are required per OFDM
symbol. The Grassmannian interpolator used a codebook size card(F) = 8 which requires 48 bits (3K) of
feedback information. The benchmark for comparison is where the indices from a codebook size card(F) = 16
are fed back for all beamforming vectors. This requires a total of 512 bits of feedback information. The
Grassmann interpolator using 48 bits of feedback has the same performance as that of the MS interpolator
with 64 bits and “brick-wall” scheme with 80 bits. The performance gains of the Grassmann interpolator
over the MS interpolator are also presented in a coded system. Fig. 2 shows the bit error rate (BER) curves
when using a rate 1/2 convolutional code in the above system with generator polynomials g0= 1338and
12
g1= 1718. The interleaver specified in the industry standard [38] is used on a per MIMO-OFDM symbol
basis and soft decision decoding is done with the Viterbi algorithm.
It is also of interest to compare the performance of the above system in a multi-user scenario. In this
model, up to Pusers are allowed to transmit in each of the Nsubcarriers. The received signal in the nth
subcarrier for the pth user can be written as
Yn,p =Hn,pXn,p +
P
X
µ=1,µ6=p
Hn,µXn,µ
+Nn,p.(25)
If the channel model for each of the users follows (1) then the interference term in the brackets above for
the pth user is Gaussian. In Fig. 3 we plot the interpolator performance with the signal to interferer power
fixed at 5dB. The Grassmann interpolator using 48 bits of feedback has a performance comparable to the
MS interpolator using 64 bits of feedback.
In Fig. 4 the performance of the different interpolators are compared for a two substream precoded
spatial multiplexing 4 ×2 system. We plot the vector symbol error probability when using zero-forcing (ZF)
receivers. The precoding matrices are found using the minimum singular value criterion [16] and are selected
from codebooks generated using methods in [39]. For the MS interpolator, the unitary rotation matrix Q
is chosen from a codebook size card(Q) = 4. With the precoding matrices chosen from a codebook size
card(F) = 4, the MS scheme requires 64 bits (2K+ 2K) of feedback. The corresponding Grassmannian
interpolator with card(F) = 16 also uses 64 bits (4K) of feedback. The ideal case for comparison is where
the indices of the precoder matrices for all subcarriers are sent back from a codebook size of card(F) = 64
corresponding to 768 bits (6N) of feedback. At a vector symbol error rate (VSER) of 10−1the Grassmannian
interpolator outperforms the MS interpolator by 0.55dB and is 1.25dB away from the ideal case.
In Fig. 5 we plot the performance of the precoding schemes when there is channel estimation error in
the above spatial multiplexing system. The receiver is assumed to estimate the channel matrix for the nth
subcarrier as being
Hn,est =αHn+p1−α2Hn,error (26)
13
where the entries of Hn,error are independent and distributed according to CN (0,1). The precoders for the
pilots are designed as a function of Hn,est. As can be seen, the degradation in performance due to increasing
channel estimation error is less with the Grassmann interpolator. When α2= 0.9, the performance of the
Grassmann interpolator is as good as that of the MS interpolator with perfect channel knowledge.
Finally, we compare the computational complexities for the Grassmann and MS interpolators for pre-
coded spatial multiplexing systems. For the Grassmann interpolator an O(K·2B) search is first required at
the receiver to determine the best precoder matrices for the pilots. The computational cost for the transmit-
ter to compute the precoder matrices for the non-pilot subcarriers is (N−K)·O(M3
t). In the MS interpolator
the receiver has to do an O(K·(2B+ 2B1)) search where B1= log2(card(Q)) to determine the precoder
matrices for the pilots. The computational cost at the transmitter to compute the precoder matrices for the
non-pilot subcarriers is (N−K)·O(M3).
5 Conclusions
In this correspondence, we proposed a method for performing precoding with pilot-based feedback. The
idea is to formulate the per subcarrier precoder design problem using pilot tone feedback as a weighted least
squared subspace distance problem. With this formulation, we show the problem can be easily solved using
principle component techniques. The algorithm can use any interpolation weighting scheme.
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−4 −2 0 2 4 6
10−4
10−3
10−2
10−1
100
Es/No (dB)
Average Probability of Symbol Error
Comparison of Precoder Interpolating Schemes ina 2x4 MIMO−OFDM System using QPSK
Ideal Precoding
All Subcarriers w/512bits
Grassmann w/48bits
Modified Spherical w/64bits
Brickwall w/80bits
Modified Spherical w/48bits
Normalized Linear w/48bits
Figure 1: Probability of symbol error comparison for different precoder interpolation schemes in a
2×4 beamforming MIMO-OFDM system using QPSK.
16
−7 −6.5 −6 −5.5 −5 −4.5 −4 −3.5 −3
10−5
10−4
10−3
10−2
10−1
100
Eb/No (dB)
Probability of Bit Error
Comparison of Precoder Interpolating Schemes in a 2x4 MIMO−OFDM System using QPSK with coding
Ideal Precoding
All Subcarriers w/384 bits feedback
Grassmann w/48 bits feedback
Modified Spherical w/48 bits feedback
Figure 2: Bit error rate comparison for different precoder interpolation schemes in a 2 ×4 beam-
forming MIMO-OFDM system using QPSK with coding.
0 2 4 6 8 10 12 14 16 18 20 22
10−4
10−3
10−2
10−1
Es/No (dB)
Probability of Symbol Error
Comparison of Precoder Interpolating Schemes in a 2 x 4 MIMO−OFDM System with 5dB Interference per Subcarrier
Ideal beamforming w/512 bits feedback
Grassmann w/64 bits feedback
Grassmann w/48 bits feedback
Modified Spherical w/64 bits feedback
Modified Spherical w/48 bits feedback
Figure 3: Probability of symbol error comparison for different precoder interpolation schemes in a
2×4 beamforming MIMO-OFDM system using QPSK. The signal-to-interference power was fixed
at 5dB.
17
0 2 4 6 8 10 12
10−2
10−1
100
Es/No (dB)
Probability of Vector Symbol Error
Comparison of Precoder Interpolating Schemes in a 2 Substream 4x2 MIMO−OFDM system using QPSK
Ideal Precoding
All Subcarriers w/ 768 bits feedback
Grassmann w/ 64 bits feedback
Modified Spherical w/ 64 bits feedback
Figure 4: Probability of symbol vector error for different precoder interpolation schemes in a two
substream 4 ×2 MIMO-OFDM system using QPSK with a zero-forcing receiver.
0 2 4 6 8 10 12
10−2
10−1
Es/No (dB)
Probability of Vector Symbol Error
Comparison of Precoder Interpolating Schemes in a 2 Substream 4 x 2 MIMO−OFDM System
Grassmann w/64 bits feedback α2=1
Grassmann w/64 bits feedback α2=0.9
Grassmann w/64 bits feedback α2=0.75
Modified Spherical w/64 bits feedback α2=1
Modified Spherical w/64 bits feedback α2=0.9
Modified Spherical w/64 bits feedback α2=0.75
Figure 5: Probability of vector symbol error for different precoder interpolation schemes with
channel estimation error in a two substream 4 ×2 MIMO-OFDM system using QPSK with a
zero-forcing receiver.
18
