The 11th International Symposium on Wireless Personal Multimedia Communications (WPMC’08)

                                  Emiliano Dall’Anese, Antonio Assalini and Silvano Pupolin
                              University of Padua - Department of Information Engineering (DEI)
                                            Via Gradenigo 6/B, 35131, Padova, Italy
                                        e–mail: {edallane, assa, pupolin}

                          A BSTRACT                                nique generalizes to the MIMO case known results obtained for
                                                                   SISO-OFDM systems [2, 4].
In this paper reduced-rank least–squares (rr-LS) channel es-
                                                                      The proposed rr-LS scheme is also compared with LMMSE
timation for MIMO-OFDM systems is presented. Conven-
                                                                   [3] estimation from the point of view of both estimation perfor-
tional LS solution is firstly introduced and a low complexity
                                                                   mance and computational complexity. In particular, we show
per-subcarrier based version derived. Then, estimation mean
                                                                   that for not too long channels, rr-LS and LMMSE perform
square error is shown to improve when a reduced-rank ap-
                                                                   close to each other but with the rr-LS having significantly
proach is adopted. In that case, MIMO channel estimation is
                                                                   lower complexity.
refined in time domain by setting to the real value the length of
the estimated channel. It results that rr-LS represents a conve-      The rest of this paper is organized as follows. In Sect. II
nient choice performing close to the linear MMSE solution.         we introduce the MIMO-OFDM system and the adopted
                                                                   channel model. Sect. III describes LS estimation including
                     I   I NTRODUCTION                             per-subcarrier and reduced-rank approaches. Sect. IV reports
                                                                   the training sequence structure. Simulation results are pre-
Orthogonal frequency division multiplexing (OFDM) is an ef-        sented in Sect. V while Sect. VI concludes this paper.
ficient multicarrier transmission system for high data rate com-
munication over frequency-selective fading channels. To im-           Notation: E[·] represents expectation. δ(·) is the Dirac pulse.
prove the capacity of wireless systems, multiple antennas can      The superscripts (·)T , (·)∗ , (·)H , and (·)−1 represent transpose,
be used at both transmitter and receiver sides jointly with        conjugate, transpose conjugate (Hermitian), and matrix inver-
OFDM. Such a transmission scheme is referred to as multiple-       sion, respectively. The superscript (·)+ represents the Moore-
input multiple-output (MIMO) OFDM system [1].                      Penrose inverse matrix (pseudoinverse matrix), i.e., A+ =
   The achievable performance with a coherent detection of the     (AH A)−1 AH , A ∈ CM ×N , where the columns of A are lin-
transmitted information symbols may be compromised by im-          early independents. x ∼ CN(mx , Rx ) denotes a random vec-
perfect knowledge of the channel state information (CSI) at the    tor x with complex circularly-symmetric Gaussian distributed
receiver. Therefore, accurate channel estimation techniques re-    entries, mean mx and correlation matrix Rx . A = diag(a)
sult of basic importance in order to ensure reliable data recov-   is a diagonal matrix with the elements of a ∈ CM ×1 on its
ery over frequency-selective fading channels. However, since       main diagonal. A = bdiag(a1 , . . . , aN ) ∈ CN M ×N P is a
in MIMO-OFDM systems there is a higher number of propa-            block diagonal matrix with the matrices ai ∈ CM ×P as ele-
gation links to be characterized, channel estimation becomes       ments. · 2 is the squared Euclidean norm. ⊗ denotes the
more complex compared with the single-input single-output          Kronecker product. vec[·] is the matrix linear transformation
(SISO) case. Hence, it is relevant to achieve a good trade-off     known as matrix vectorization which stacks the columns of a
between estimator complexity and global system performance.        matrix A ∈ CM ×N on a M N × 1 vector.
   Channel estimation is often based on least-squares (LS)
[1, 2] and linear minimum mean square error (LMMSE) [2, 3]
techniques. Incidentally, while LS estimator handles the                        II     S YSTEM AND C HANNEL MODEL
channel coefficients as deterministic but unknown constants,        We consider a point-to-point MIMO-OFDM wireless system
MMSE-based solutions require perfect knowledge of the sec-         with Nt transmit and Nr receive antennas as depicted in Fig. 1.
ond order statistical description of the channel, i.e., power      In the following we introduce the system and channel models.
delay profile (PDP), spatial correlations, etc. . Nevertheless,
MIMO channel statistics depends on many random compo-
                                                                   II.A Transmitter
nents which joint estimation may result not sufficiently accu-
                                                                         (t)         (t)   (t)      (t)
rate and then compromise MMSE estimation capabilities.             Let Ak      [Ak,0 , Ak,1 , . . . , Ak,M −1 ]T be the information data
   In this paper, we study a simple and efficient channel estima-   vector modulated at time instant kT in correspondence to the
tor which presents good performance but lower computational        tth transmit antenna, t = 1, 2, . . . , Nt , with M the number
complexity than LMMSE. In particular, we firstly introduce the      of OFDM subcarriers. We assume data symbols with average
well-known (full-rank) frequency domain LS estimator [1, 2].                          2
                                                                   energy equal to σA E[|Ak,m |2 ], ∀ m. The modulated vector
Successively, in order to save computational complexity, we        is extended with a cyclic prefix (CP) of Lcp < M samples.
implement an LS solution working on a per-subcarrier basis.        Therefore, let B be the transmission bandwidth and Tc 1/B
Thereinafter, we derive a reduced-rank (rr) LS estimator that      the channel sampling time, then the modulation rate is equal to
significantly outperforms the full-rank LS approach. That tech-     1/T with T (M + Lcp )Tc .
The 11th International Symposium on Wireless Personal Multimedia Communications (WPMC’08)

                                                                                 Now, we define
                                                                                                              (1,1)                  (1,N )
                                                                                                     Hk,m                    ...    Hk,m t
                                                                                                       .                    ..         .     
                                                                                          H k,m    
                                                                                                  =    .                       .       .          (2)
                                                                                                        .                               .     
                                                                                                      (Nr ,1)                        (Nr ,N )
                                                                                                     Hk,m                    ...    Hk,m t
                                                                               as the Nr × Nt matrix with elements given by the sam-
                                                                               ples of the channel frequency response (CFR) of the
                                                                               MIMO channel in correspondence to the mth OFDM tone,
                                                                               m = 0, . . . , M − 1. Moreover, we also introduce the fol-
         Figure 1: MIMO-OFDM transmission chain.
                                                                               lowing Nr Nt × 1 column vectors H k,m and hk,l obtained
                                                                               by stacking the columns of H k,m and the columns of hk,l ,
II.B Channel Model                                                             respectively, i.e., H k,m = vec[H k,m ], and hk,l = vec[hk,l ].
                                                                               Finally, let us define H k = [H T , . . . , H T −1 ]T and
                                                                                                                        k,0        k,M
The adopted channel model has Lc distinct Tc -spaced taps. In
particular, at time instant kT , the lth component of the MIMO                 hk = [hT , . . . , hT c −1 ]T , then the coefficients of the MIMO
                                                                                        k,0        k,L
channel is characterized by: i) an Nr × Nt Gaussian random                     CFR and channel impulse response (CIR) are related as
matrix Gk,l , ii) a random angle-of-departure (AoD) θk,l , iii) a                                  H k = (F Lc ⊗ I Nr Nt ) hk ,                     (3)
random angle-of-arrival (AoA) θk,l and iv) a propagation delay
τk,l . The Lc paths are assumed to be mutually uncorrelated [5].                                  hk =        F +c
                                                                                                                L    ⊗ I Nr Nt H k ,                (4)
   Performance of MIMO systems strongly depends on the                         where [F Lc ]m,l = exp(−j2πlm/M ), m = 0, 1, . . . M − 1,
spatial correlation that arises when the separation among ei-                  l = 0, 1, . . . Lc − 1, is a sub-block of the discrete Fourier trans-
ther transmit or receive antennas is insufficient. In this paper,               form (DFT) matrix F = [F ]m,l , m, l = 0, 1, . . . M − 1. F +c    L
we assume that both transmit and receive sides are equipped                    represents the Lc × M Moore-Penrose inverse matrix of F Lc .
with uniform antenna arrays having omnidirectional elements
spaced of dt and dr meters, respectively. Therefore, with the                  II.C Receiver
model given in [6], the spatial correlation matrix for the lth tap,            At the receiver, we assume perfect time and frequency syn-
Rl ∈ CNr Nt ×Nr Nt , is given by the Kronecker product of the                  chronization. After CP removal and OFDM demodulation, the
spatial correlation matrices at the receiver RR,l ∈ CNr ×Nr and                resulting signal at the rth receive antenna is given by the super-
at the transmitter RT,l ∈ CNt ×Nt as Rl = RR,l ⊗ RT,l . In                     imposition of Nt components. We can write the received data
particular, for small angle spreads the spatial correlation coef-              sample, at time kT for the mth OFDM tone, as
ficients can be approximated as ([5])                                                                         Nt
                                                                                                 (r)                 (r,t)    (t)
                                                                                               Yk,m     =          Hk,m Ak,m + nm ,                 (5)
                 −j2π∆t (m−n) cos θk,l − 1 (2π∆t (m−n)δt sin θk,l )
                                   D                          D
[RT,l ]m,n ≈ e                         e 2                                 ,                                 t=1

                                                                               where nm represents thermal noise with nm ∼ CN(0, σn ) for
[RR,l ]m,n ≈ e
               −j2π∆r (m−n) cos θk,l
                                         2   (                     A
                                                 2π∆r (m−n)δr sin θk,l) ,      any m and r. Here, we define the average signal-to-noise ratio
                                                                                                                     2       2
                                                                               (SNR) per subcarrier as SN R = σA /σn and we collect the
where δt and δr are the angle spreads, while ∆t = dt /λc and                                             (1)    (2)            (Nr )
                                                                               samples (5) as Y k,m = [Yk,m , Yk,m , . . . , Yk,m ]T
∆r = dr /λc are the antenna spacing normalized to the wave-
length λc corresponding to the transmission carrier frequency.                                 Y k,m = H k,m Ak,m + nk,m ,                          (6)
   Therefore, the lth path, l = 0, . . . , Lc − 1, of the MIMO                                      (1)       (2)              (N )
                                                                               where Ak,m = [Ak,m , Ak,m , . . . , Ak,m ]T .

channel at time kT can be represented by the matrix hk,l ∈
CNr ×Nt defined as                                                                                 III       C HANNEL E STIMATION
                                1/2       1/2 H                                In this section, we first introduce the well-known full-rank fre-
                  hk,l =   σhl RR,l Gk,l RT,l          ,              (1)
                                                                               quency domain LS estimator [1, 2]. Then, in order to cut the
where the Nr × Nt matrix Gk,l has independent circularly-                      required computational complexity, we give an LS formula-
symmetric zero-mean complex Gaussian distributed entries                       tion defined on a per-subcarrier basis. Thereafter, we derive a
with unitary variance. The value of the average power σhl is                   “reduced-rank” (rr) LS estimator which is analytically shown
obtained by properly sampling the power delay profile (PDP),                    to significantly outperform the full-rank LS approach. More-
                          Lc −1 2
which is normalized as l=0 σhl = 1 in order to fix the chan-                    over, for completeness, performance comparison also includes
nel energy despite of the number of present scatterer clusters.                LMMSE estimation [3].
The number of channel taps is given by Lc = ⌊Bτrms ⌋ ≤ Lcp ,                      In this paper, performances are measured through the esti-
with τrms the channel root-mean square time delay spread.                      mation mean square error (MSE) defined as
  We also assume that the channel is static over at least Nt                                               1                                2
consecutive OFDM symbols.                                                                  M SE                 E             Hk − Hk           ,   (7)
                                                                                                        Nt Nr M
The 11th International Symposium on Wireless Personal Multimedia Communications (WPMC’08)

where H k is the CFR vector given in (3) while its estimation          The last sign of equality holds being data and noise mutually
is denoted by H k . In (7) expectation is taken over time. The         independent and, as addressed in Sect. IV, Ak is designed
following proposition states as the MSE can equivalently be            such as training sequences are orthogonal and then AH Ak is
evaluated by considering either the CFR or the CIR [3, 7].             diagonal
                                                                                       AH Ak = diag(αk )I Nr Nt M ,
                                                                                         k                                      (11)
Proposition 1 : Let M be the number of subcarriers of the
                                                                       where the elements of αk ∈ CNr Nt M ×1 can be found from the
OFDM modulators and let hk and H k be the estimated coeffi-             specific realization of Ak .
cients of the MIMO CIR and CFR, respectively, then
                                                                       III.B Least-Squares per-subcarrier estimator (LS-PS)
         1                       2                    2
           E      Hk − Hk            =E    hk − hk        .     (8)    With the objective of reducing computational complexity of LS
                                                                       estimator, in the following, we describe a criterion that works
Proof: Let ∆H k = H k − H k and ∆hk = hk − hk be the                   on a per-subcarrier (PS) basis. In principle, complexity is re-
Nr Nt M ×1 and Nr Nt Lc ×1 vectors representing the CFR and            duced by dividing the channel coefficient vector H k into a set
CIR estimation errors, respectively. From the basic relation           of blocks of smaller size and then by performing CFR estima-
∆H k = (F Lc ⊗ I Nr Nt )∆hk , it follows that                          tion independently on each sub-block.
                                                                          Therefore, let Ak,m = [Ak−Nt +1,m , . . . , Ak,m ] denote the
E ∆H H ∆H k =                                                          sequence of symbols transmitted on the mth subcarrier at time
                                                                       instants (k − Nt + 1)T, . . . , kT . We can write from (6)
     = E ∆hH (F Lc ⊗ I Nr Nt ) (F Lc ⊗ I Nr Nt ) ∆hk
           k                                                                             Yk,m = H k,m Ak,m + Nk,m ,                            (12)
     = E     ∆hH
               k     F Hc F Lc
                       L         ⊗ I Nr Nt ∆hk                         where Ym,k = [Y k,m , . . . , Y k−Nt +1,m ] is the Nr × Nt
                                                                       matrix containing the received samples on the mth subcar-
     = E ∆hH M I Nr Nt Lc ∆hk = M E ∆hH ∆hk .
           k                          k                                rier at time instants (k − Nt + 1)T, . . . , kT , and Nk,m =
                                                                       [nk,m , . . . , nk−Nt +1,m ]. In this case the minimization of the
III.A Least Squares estimator (LS)                                                               LS−P S              2

In this section, we introduce the channel estimation criterion         functional Yk,m − H k,m              Ak,m |       gives
known as frequency domain LS estimator [1, 2].                                                 LS−P S
   Let Y k,m = [Y T , Y T
                               k−1,m , . . . , Y k−Nt +1,m ]
                                                              be the                       H k,m           = Yk,m A+ .
                                                                                                                   k,m                         (13)
signal samples received at time instants (k − Nt + 1)T, . . . , kT
                                                                       The complete set of M Nr Nt channel fading coefficients is
in correspondence to the mth OFDM tone for all re-                       LS−P S            T       T            T           T
ceive antennas and let Yk = [Y T Y T , . . . , Y T −1 ]T be
                                       k,1 k,2            k,M          Hk           = H k,0 , H k,1 , . . . , H k,M −1          , where H k,m =
the Nr Nt M × 1 column vector holding all received sam-                      LS−P S
ples.    Moreover, we also define Ak,m = [Ak,m ⊗                        vec[H k,m    ],  m = 0, . . . , M − 1.
                                                                          We note (13) leads to the same MSE as for the LS estimator
I Nr , Ak−1,m ⊗ I Nr , . . . , Ak−Nt +1,m ⊗ I Nr ]T , where Ak,m
                                                                       (9). In particular, an error component Nk,m A+ is present on
is given in Sect. II.C. Finally, we introduce the follow-
                                                                       a generic mth subcarrier.
ing Nt Nr M × Nt Nr M block diagonal matrix Ak =
bdiag Ak,0 , Ak,1 , . . . , Ak,M −1 . Note that the number of          III.C Reduced-rank LS estimator (rr-LS)
consecutive OFDM symbols used for channel estimation is
                                                                       With a PS approach the LS estimator presents a significantly
fixed to Nt .
                                                                       lower complexity, but the MSE is still high. Therefore, to
   With the above notation the received symbol column vec-
                                                                       achieve a better estimation accuracy we extend the reduced-
tor reads Yk = Ak H k + nk . Therefore, frequency domain
                   LS                                                  rank strategy for single-antenna systems [2] to the MIMO case.
LS estimation H k       is obtained by minimizing the functional          Let us suppose the receiver knows the length of the CIR of
             LS 2
 Yk −   Ak H k       yielding [7]                                      the MIMO channel. Then, “reduced-rank” estimation of the
                                                                       CIR, hrr−LS ∈ CNr Nt Lc ×1 , is obtained as
                            LS                                                                                                           2
                         H k = A+ Yk ,
                                k                               (9)      rr−LS
                                                                        hk     = arg min H k
                                                                                                   LS−P S
                                                                                                             − (F Lc ⊗ I Nr Nt ) h            . (14)
where  A+
        k  =   (AH Ak )−1 AH
                 k         k  is the pseudo-inverse of Ak .            The solution is
   LS technique presents an high MSE. In fact, directly from
                                                                                                                           LS−P S
(7) and by (9), we find that LS estimation is affected by a                            rr−LS
                                                                                     hk     = F +c ⊗ I Nr Nt H k
                                                                                                L                                    .         (15)
noise component A+ nk which leads to the following estima-
                   k                                                                                   +
tion MSE per subcarrier                                                Note that (F Lc ⊗ I Nr Nt ) = F +c ⊗ I Nr Nt .
                                1                    2                   Therefore, the estimated CFR H k                 becomes
            M SELS      =             E A+ nkk
                             Nr Nt M                                        rr−LS                                                    LS−P S
                                                                        Hk          = (F Lc ⊗ I Nr Nt ) F +c ⊗ I Nr Nt H k
                        .    σn        1                                                                                                       (16)
                        =       E              .               (10)                                                  LS−P S
                             Nt     |Ak,m |2                                        = F Lc F +c ⊗ I Nr Nt H k
                                                                                             L                                   .
The 11th International Symposium on Wireless Personal Multimedia Communications (WPMC’08)

                         Table 1: Complex operations required by the considered estimation techniques.
   Estimator                      (×)                                            (+)                                inv(·)

   LS                        3 3       2 2
                         2Nr Nt M + Nr Nt M                                      2 2
                                                                    Nt Nr M (2Nt Nr − Nt Nr − 1)                 inv(Nr Nt M )
   LS-PS                          3    2
                            M (2Nt + Nt Nr )                             3    2          2
                                                                  M (2Nt + Nt Nr − 2Nt − Nt Nr )                  M · inv(Nt )
   rr-LS            3     2
               M (2Nt + Nt Nr ) + M (M Lc + 2L2 + Nr Nt )
                                                                   3     2
                                                             M (2Nt + Nt Nr ) + M (M Lc + 2L2 + Nr Nt )
                                                                                               c            M · inv(Nt ) + inv(Lc )
                                                                     −(M 2 + L2 + Lc M + Nt Nr )
   LMMSE                   3 3
                        M Nt Nr (1 + Lc + Lc M )                                      2N 2) + N 3N 3M 2L
                                                            (Nt Nr − 1)(1 + Lc )(M Nt r                        2 · inv(Nr Nt Lc )
                                                                                               t r      c
                                                                     2 2             2 2
                                                                 +Nt Nr M 2 − M (Nt Nr Lc + Nt Nr )

We emphasize that the entries of the pseudo-inverse matrix         by counting the number of operations on complex numbers re-
F +c can be stored for some values of Lc of interest.
   L                                                               quired to realize the different criteria.
   Nevertheless, we note that the CIR length supposed at the          In particular, (+) and (×) denote respectively the number
receiver may be incorrect. In [4] the robustness of the reduced-   of sums and products, while inv(C) represents inversion of a
rank estimator to a wrong setting of the CIR length was inves-     non-singular C × C matrix. In the analysis, we neglect the cost
tigated for the single antenna case.                               of transpose-conjugate (Hermitian) operation and Kronecker
   We now compute the MSE of rr-LS estimator. For the              products involving identity matrices.
sake of simplicity, we suppose CFR estimation is performed            In general, matrix inversion has a heavy impact on total com-
by using (9). Therefore, from (9) and (15), we have that           plexity. However, we note that under (11) inv(Nr Nt M ) and
hk S = hk + F + ⊗ I Nr Nt A+ nk , and then the MSE is
                      L              k                             inv(Nt ) in Tab. 1 for (9) and (13) become straightforward.
given by                                                              The analysis shows that LS-PS estimator requires a lower
                                                                   number of complex operations than full-rank LS. The imple-
                        1                            2
  M SErr−LS      =          E    F + ⊗ I Nr Nt A+ nk
                                    L           k
                                                                   mentation of rr-LS needs additional complexity than for LS-PS
                      Nr Nt                                        as emphasized by (16). Tab. 1 also shows the higher complex-
                 .    σn       1      Lc                           ity of LMMSE with respect to rr-LS.
                 =       E         2 M
                                          .           (17)
                      Nt    |Ak,m |
                                                                            IV    O RTHOGONAL T RAINING S EQUENCES
The last sign of equality holds under hypothesis (11).
                                                                   In this section we introduce the space-time structure of the
III.D Remarks                                                      training symbols transmitted on each subcarrier for channel
  • In OFDM-based systems the channel length is much               estimation purposes. In particular, sequences simultaneously
    shorter than M . In fact, usually Lc ≤ Lcp . From (10)         transmitted from the Nt antennas are orthogonal in both space
    and (17), we note rr-LS significantly outperforms full-         and time and are Nt OFDM symbols long [1, 8, 9].
    rank LS. In details, we obtain                                    We focus on symbols transmitted on a given subcarrier
                                                                   m. Let us consider the Nt × Nt symbol matrix Ak,m =
       M SELS,dB − M SErr−LS,dB = 10 log10 (M/Lc ) ,               [Ak−Nt +1,m , . . . , Ak,m ] introduced in Sect. III.B. The inner
                                                          (18)     product of any two distinct rows (space dimension) or columns
     hence, the gain in estimation accuracy increases with the     (time dimension) of Ak,m is zero if they are designed as in
     decreasing of the channel length Lc .                         [10], where orthogonal space-time block codes were proposed.
                                                                   In particular, with a BPSK mapping orthogonal sequences can
  • The estimation MSE of both estimators does not depend          be easily generated from an Hadamard matrix for any value of
    on the second order statistics of the channel, e.g., PDP and   Nt [8, 9]. The use of orthogonal sequences is also convenient
    spatial correlation. In fact, LS technique treats the chan-    because it simplifies the computation of the pseudo-inverse ma-
    nel coefficients as deterministic but unknown constants.        trices in (9) and (13).
    Other MIMO channel estimation criteria, such as LMMSE
    [3], exploit the spatial correlations to improve channel es-                       V   N UMERICAL R ESULTS
    timation accuracy. However, the performance of such es-
    timators may severely be compromised if channel statistic      We consider a MIMO-OFDM system with M = 64 and
    is not perfectly available.                                    Lcp = 16. Channel estimation is performed in the frame
                                                                   preamble. Training sequences are construed by Hadamard or-
  • LS and rr-LS can be used for both pilot-aided and              thogonal bases of length M [9].
    decision-directed channel estimation. However, in both            The normalized antenna spacing is fixed to ∆t = ∆r = 1/2.
    cases pilot and data sequences should be generated as un-      The AoDs and the AoAs are obtained from a Gaussian random
    derlined in Sect. IV in order to comply with (11).             variable with zero mean and standard deviation 90◦ and 360◦ ,
                                                                   respectively. The angle spread for all taps at both transmitter
III.E Computational Complexity                                     and receiver are the same and fixed to 4◦ (2π/9 rad). The chan-
In Tab. 1 we consider the computational complexity required        nel has an exponentially decaying PDP truncated to its first Lc
for channel estimation. Specifically, complexity is measured        taps and normalized to one.
The 11th International Symposium on Wireless Personal Multimedia Communications (WPMC’08)

           0                                                                            0
          10                                                                           10
                                                                LS                                                                     LS
                                                                rr−LS                                                                  rr−LS, L =1
                                                                LMMSE                                                                           c
                                                                Lc = 2                                                                 rr−LS, Lc=2
           −1                                                                           −1
          10                                                                           10
                                                                L =4                                                                   rr−LS, Lc=4
                                                                L = 16                                                                 rr−LS, Lc=6
                                                                                                                                       rr−LS, L =8


           −2                                                                           −2
          10                                                                           10

           −3                                                                           −3
          10                                                                           10

           −4                                                                           −4
          10                                                                           10
                0   2   4    6     8     10     12    14   16   18       20                  0   2   4   6    8     10    12    14   16    18        20
                                   Average SNR [dB]                                                          Average SNR [dB]

Figure 2: MSE vs Average SNR per subcarrier, Nt = Nr = 2,                     Figure 3: MSE vs Average SNR per subcarrier, Nt = Nr = 4,
Lc = 2, 4. Random AoDs and AoAs, 4◦ angle spreads.                            Lc = 1, 2, 4, 6, 8. Random AoDs and AoAs, 4◦ angle spreads.

   In Fig. 2 we set Nt = Nr = 2 and compare the MSE of full-                   [2] J. J. van de Beek, O. Edfors, M. Sandell, S. K. Wilson, and P. O.
                                                                                   B¨rjesson, “On Channel Estimation in OFDM Systems,” in Proc. IEEE
rank LS, rr-LS and LMMSE for different channel lengths. As                         Int. Veh. Technol. Conf., Chicago, IL, Jul. 1995, pp. 815-819.
expected, full-rank LS presents the worst performances that are
                                                                               [3] H. Zhang, Y. (G.) Li, A. Reid, and J. Terry, “Channel Estimation for
indeed independent of the channel length as predicted by (10).                     MIMO OFDM in Correlated Fading Channels,” in Proc. ICC 2005, in
Viceversa, as shown in (18), the MSE for rr-LS is much better                      Proc. IEEE Int. Conf. Commun. (ICC), Seoul, South Korea, May 2005,
than for full-rank LS, especially for short channels. Finally, rr-                 pp. 2626-2630.
LS performs very close to LMMSE which is, however, slightly                    [4] E. Dall’Anese, A. Assalini, and S. Pupolin, “On Reduced-rank Channel
better in the low SNR regime.                                                      Estimation and Prediction for OFDM-based Systems,” in Proc. WPMC
                                                                                   ’07, Jaipur, India, Dec. 2007.
   Fig. 3 reports the MSE performance of rr-LS for a system
with Nt = Nr = 4 varying the number of channel taps. Also in                            o
                                                                               [5] H. B¨lcskei, D. Gesbert, and A. J. Paulraj, “On the Capacity of OFDM-
                                                                                   Based Spatial Multiplexing Systems,” IEEE Trans. Commun., vol. 50,
this case the behavior predicted by (17) is corroborated. MSE                      pp. 225-234, Oct. 2001.
decreases linearly with SNR and it depends on the ratio be-
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LS solution was recalled and a reduced-complexity per-                             2007.
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