Algorithm for Detecting the Number of Transmit
Antennas in MIMO-OFDM Systems: Receiver
Integration
Eckhard Ohlmer, Ting-Jung Liang and Gerhard Fettweis
Vodafone Chair Mobile Communications Systems, Technische Universität Dresden, Germany
email: {eckhard.ohlmer, liang, fettweis}@ifn.et.tu-dresden.de
Abstract—Knowledge of the channel between all transmitreceive antenna pairs is essential for enabling the decoupling of
spatial streams and coherent data detection in wireless MIMOOFDM systems. The design of the preamble structure for MIMO
channel estimation in an ad-hoc system depends on the number
of transmit antennas in the system. A challenge at the receiver is
therefore to accurately detect the number of transmit antennas in
order to perform MIMO channel estimation. In this contribution
we extend our previous algorithm for detecting the number of
transmit antennas as presented in [1] and show how to integrate
this algorithm in a typical receiver flow. Results show that our
algorithm can be successfully applied in a realistic system if the
unavoidable synchronization inaccuracies are carefully taken into
account.
I. I NTRODUCTION
This work focuses on an ad-hoc MIMO-OFDM system
operating on a burst transmission basis. In such a system the
initial acquisition of incoming packets is of critical importance.
Typically, a receiver would first perform packet detection,
followed by timing synchronization and subsequent frequency
synchronization before estimating the MIMO channel and
detecting the received data [2].
Synchronization, as well as channel estimation, in MIMOOFDM systems is usually achieved by employing a preamble
composed of a short training field (STF), primarily intended
for synchronization, followed by a long training field (LTF),
primarily intended for channel estimation. Synchronization can
be performed without knowledge of the actual number of
transmit antennas if the STF is carefully designed. The LTF
structure, as well as the pilot sequences transmitted within the
LTF, typically depend on the number of transmit antennas,
which means that the number of transmit antennas has to
be known before the MIMO channel estimation. A simple
solution, adopted in [3], is the transmission of additional
signaling information before the MIMO-LTF, but this approach
has the disadvantage of increased overhead and transmission
latency.
In [1] we proposed a novel algorithm to detect the number of
transmit antennas before MIMO channel estimation based only
on the MIMO-LTF and hypothesis testing. We then evaluated
its performance under perfect synchronization conditions. In
this work we extend the work in [1] by integrating it into
a real world communication system. This is not a straight
forward extension because the algorithm is applied after
the first, coarse synchronization stage and therefore has to
cope with unavoidable frequency and timing synchronization
mismatches. It turns out that for our previously proposed
algorithm imperfect timing synchronization leads to severe
performance degradation, whereas imperfect frequency synchronization has little impact on the algorithm performance. In
this work we propose a method to measure the residual timing
synchronization inaccuracies and by applying this method
show that the algorithm for detecting the number of transmit
antennas retains its robust performance even under imperfect
timing synchronization conditions. The algorithm performance
will be evaluated using a MIMO-OFDM system based on
IEEE802.11n parameters.
The remainder of the paper is organized as follows: In
section II we introduce the system model. In section III we discuss receiver functions required for synchronization, channel
estimation and detection of the number of transmit antennas. In
section IV we show how to integrate our algorithm in a typical
receiver flow. The performance of the integrated system will
be evaluated in section V. Finally, conclusions are drawn in
section VI.
II. S IGNAL M ODEL AND P REAMBLE D ESIGN
In this work, we consider the OFDM multicarrier modulation technique, where (in general complex valued) transmit symbols are modulated onto multiple subcarriers in the
frequency domain. Transformation to the time domain is
achieved by means of a discrete Fourier transform (DFT).
Each time domain OFDM symbol is protected against inter
symbol interference by a cyclic prefix (CP). At the receiver,
after deleting the CP, the signal is converted back into the
frequency domain by an inverse DFT operation.
A. Signal Model
The received time domain signal can be written as
#
"
Nt NX
CIR −1
X
r
t
rt
yk =
xk−b hb + vrk ,
t=1
b=0
(1)
where ykr denotes the signal at receive antenna r, xtk is the
signal transmitted from transmit antenna t, vkr denotes complex
valued, zero mean additive white Gaussian noise (AWGN) and
k is the sample index 1 . The channel impulse response (CIR)
hrt
b between the receive-transmit antenna pair rt has a length
of NCIR taps with power σh2 k at tap k and sum power σh2 .
The DFT (or inverse DFT) of some arbitrary vector b (B) of
dimension [N × 1] in the time (frequency) domain is defined
by
1
B = Fb, b = FH B and FFH = N IN ,
(2)
N
where F represents the [N × N ] Fourier matrix with elements
{F}n,k = exp(−j2πnk/N ), (·)H is the hermitian operator
and IN is the [N ×N ] identity matrix. Under the assumption of
perfect receiver synchronization, the N received time domain
samples, belonging to the m-th OFDM symbol, are stacked
r
r
into a column vector ym
. Left-sided multiplication of ym
with
F yields the frequency domain representation of the received
signal in vector-matrix notation
 r
  1 T  r1   r

Xm
Hm
Ym,0
Vm,1
 .
  .
  .
  .

 ..
 =  ..
  ..
 +  ..
.

 
 
 

r
Nt
rNt
r
Ym,N
V
X
H
m
m
−1
m,NR
|
{z
} |
{z
}| {z } |
{z
}
r
Ym
Xm
Hrm
r
Vm
(3)
r
Similar to (1), Ym
denotes the signal at receive antenna
r, Xm is the signal transmitted from all transmit antennas
t = [1 · · · Nt ], Hrm denotes the channel between transmitr
receive antenna pair rt and Vm
denotes AWGN. The transt
t
. . . Xm,N
mitted signal is defined by Xtm = diag{Xm,0
−1 },
where diag{·} creates a diagonal matrix from the comt
plex valued symbols Xm,n
, modulated onto subcarriers
n = [0 . . . N − 1]. Finally, a single channel vector is given
¤T
£ rt
P Nt 2
2
rt
. Given
by Hrt
Hm,0 . . . Hm,N
m =
−1
t=1 σX t , σV ,
P
N −1 2
2
2
σH = k=0 σhk = σh = 1 as the received signal, noise and
channel power per subcarrier,
ratio (SNR)
PNt the2signal-to-noise
σX t /σV2 .
at the receiver is SNR = t=1
B. Preamble Design
In packet based OFDM systems the transmitted data is typically preceded by a bipartite preamble. The generic structure
for this preamble that we will consider in this work is depicted
in Fig. 1. The upper and lower indices in Fig. 1, such as LTFtm ,
denote transmit antenna t and OFDM symbol m.
STF
LTF
a1
Tx NT a Nt
...
a Nt
...
...
1
LTFNt
Data1
Fig. 1.
LTF1Nt
Nt
LTFNt
DataNt
...
...
...
a1
Tx 1
LTF11
...
MIMO preamble structure
1 Lowercase and uppercase letters denote the time and frequency domain
signal representation. Normal and boldface letters denote the sample and
matrix (vector) signal representation.
The first part, the STF, has a length of NSTF samples and
comprises L identical sequences a of length Na = NSTF /L.
The STF is transmitted in an orthogonal manner, e.g. from only
one antenna or cyclically shifted from all antennas. The second
part, the LTF comprises Nt OFDM symbols, each guarded by
a cyclic prefix, as adopted in [3]. Every LTF-OFDM symbol
contains pilot sequences for channel estimation. In order to
minimize the channel estimation error, pilot sequences transmitted from different antennas are designed to be mutually
orthogonal [4]. Orthogonality can be achieved by transmitting
sequences separated in the time (TO), frequency (FO) or code
(CO) domain. In [1] we discussed all three designs, although
in this work we will restrict ourselves to the TO design without
loss of generality. In a TO design, antenna t transmits pilot
sequences only during the t(th) LTF symbol, i.e. only within
LTFtm=t .
III. I NDEPENDENT R ECEIVER F UNCTIONS
A. Packet Detection and Coarse Timing
For packet detection, using the STF presented in the previous section, we employ a scheme presented in [5] which
exploits the repetitive structure of the STF. At each time instant
k the received signal ykr , within a window [k . . . k + NSTF − 1],
is assumed to be the received STF. Subsequently, a metric
Ωk is calculated so that the correlation between all possible
combinations of received sequences a within that window,
averaged over all receive antennas, is calculated and normalized by the power of the received sequences a. Normalization
ensures that Ωk can only take values between zero and
one. Expressing a certain received sequence a by the vector
r
r
yir = [yk+N
. . . yk+N
]T , Ωk can be calculated
a ×i
a ×(i+1)−1
according to:
PNa −1 PNr ¯¯PNa −l−1 r H r ¯¯
(yi ) yi+l ¯
l=1
r=1 ¯
i=0
´.
´ ³P
Ωk = ³ P
P
Nr
Na
Na −1
1
r )H y r
(y
i
×
i
r=1
i=1 i
i=1
Na
(4)
Once Ωk exceeds a certain threshold Tc at time index k̂PD ,
the receiver assumes a packet has arrived and searches for the
maximum Ωk in a window of length NSTF +NCP starting from
0
k̂P D . The sample index k̂S,CT
at which the maximum Ωk is
detected, is regarded as the start of the packet. In multipath
0
fading environments k̂S,CT
will typically have a positive delay
w.r.t. the exact packet start. Therefore, we propose to shift the
estimated packet start by NCT samples backwards to obtain the
0
final estimate as k̂S,CT = k̂S,CT
− NS,CT . We define a packet to
be detected correctly if k̂S,CT lies within a window of ±NCP
around the exact packet start kS [6].
B. Frequency synchronization
If the carrier frequencies fc,T x at the transmitter and fc,Rx
at the receiver are unsynchronized then the received signal ykr
experiences a linear phase rotation φk = 2πεk/N in terms of
parameter k, where ε = (fc,Tx −fc,Rx )/fsc represents the carrier
frequency offset (CFO) relative to the subcarrier spacing fsc .
In the presence of CFO the received signal on subcarrier n
can be expressed as:
N
−1
X
Ynr = Ψn,n Tn +
Ψn,s Ts + Vnr
| {z } s=0,s6=n
useful signal
|
{z
}
(5)
incorrect hypothetical numbers of transmit antennas can be
distinguished by employing the metric Θu
io
PNCP −1 n h
H
E
ĥ
ĥ
u u
k=0
n h
io k,k
(10)
Θu = P
N −1
H
E
ĥ
ĥ
u
u
k=0
k,k
ICI
where Ti =
PN t
Ψn,s =
t=1
Hir,t Xit ,
1 jπ(1− N1 )(s−n+ε) sin (π(s − n + ε))
¢.
¡π
e
N
(s − n + ε)
sin N
(6)
and we have dropped the OFDM symbol index m. Note that
ICI is the intercarrier interference and Ψm,n,s represents the
frequency domain influence of CFO.
In this contribution we are only interested in how the
residual CFO εres after an initial estimation and compensation
(again a linear phase rotation of the received time domain
signal) might degrade the performance of our algorithm for
detecting the number of transmit antennas. We therefore
assume that an estimator based on the STF as proposed in
[7] is employed. Additionally, we can model εres as a zero
mean Gaussian random variable. The variance of εres is given
by [7]:
£ ¤
3N 2 SN R−1
E ε2res =
(7)
2 − N 2) .
2π 2 NSTF (NSTF
a
where ĥu is the estimated channel impulse response belonging
to a certain hypothesis u. Note that ĥu is the discrete Fourier
transform of (9). If u, the number of transmit antennas
assumed in MIMO channel estimation, is incorrect, Θu yields
a constant value which equals the ratio NCP /N according to
the length of the summation intervals in (10). If u is correct
Θu only depends on the SN R and is always greater than the
same metric Θu for an incorrect assumption on Nt . Therefore,
the hypothesis that maximizes Θu is assumed to be correct,
i.e.
N̂t = arg max {Θu } .
(11)
u
Note that we assumed for each number of transmit antennas
allowed in the system a unique LTF is transmitted.
IV. R ECEIVER I NTEGRATION
A MIMO-OFDM receiver incorporating the algorithm for
detecting the number of transmit antennas is depicted in Fig. 2
C. Preamble Based Channel Estimation
Stacking the Nt LTF-OFDM symbols, received at antenna
r, into a column vector and using (3), the received signal can
be rewritten as
 r  
 r1   r 
H
Y1
X1
V1
 .
  .
  .

 .
 ..
 =  ..
 +  ..
. (8)
 ..

 
 


r
rNt
r
YN t
VN t
XN t
H
| {z } | {z }| {z } | {z }
r
YLTF
XLTF
Hr
r
VLTF
Note that in (8) the channel is assumed to be constant during
the LTF. If XLTF is of full rank, which can be ensured by
an appropriate preamble design, the least squares channel
estimate is calculated as
r
Ĥr = X†LTF YLTF
,
(9)
−1 H
where X†LTF = (XH
XLTF denotes the pseudo inverse
LTF XLTF )
of the pilot sequences, transmitted within the LTF. A comprehensive overview on preamble based channel estimation in
MIMO-OFDM can be found in [8].
D. Detection of the Number of Transmit Antennas
In [1] we derived an algorithm which employs the LTF in
order to detect the number of transmit antennas in a MIMOOFDM system. We now highlight the important results of
this work. Specifically, the derived algorithm is based on
carrying out channel estimation for each possible number of
transmit antennas u belonging to the set {1 . . . Nt,max } allowed
in the system, where Nt,max is typically small. Correct and
PD,
Timing
Sync .
Freq.
Sync .
MIMO
Detector
DFT
DFT Window
Adjust
Fig. 2.
N
t
Nˆ
Detect.
Receiver flow
t
Channel
Est.
.
Firstly, the receiver performs packet detection (PD) and
coarse timing synchronization according to sec. III-A. Note
that a potential CFO does not degrade the coarse timing
synchronization. This results from the fact that the differential
r
in the numerator of (4),
phase rotation between yir and yi+l
due to the CFO, is the same ∀ i and a certain l.
Secondly, the CFO is estimated and compensated. The
residual CFO after compensation can be characterized as
discussed in sec. III-B. After conversion of the coarsely
synchronized signal to the frequency domain, the number of
transmit antennas has to be detected (accomplished by the
Nt detection block) before MIMO channel estimation. The
Nt detection block essentially implements (10) and needs to
compute an estimate ĥ of the CIR. Before we show how to
compute (10) the effects of imperfect synchronization on the
estimated CIR need to be discussed.
Fig. 3 illustrates the effect of imperfect timing synchronization on ĥ. The grey shaded areas represent the part of
the estimated CIR which contributes to the sum Σnum in
the numerator of (10). In the case of perfect coarse timing
synchronization (Fig. 3b)) the nonzero part of the estimated
hˆk
hˆk
Σnum
hˆk
Σnum
kˆS ,CT > kS
kˆS ,CT < kS
0
Fig. 3.
NCP-1
a)
N-1
Σnum
k
0
NCP-1
b)
N-1
k
0
N CP-1
c)
N-1
k
Effect of imperfect timing synchronization on the estimated CIR
CIR is located exactly within Σnum as long as NCIR ≤ NCP
holds; a typical assumption in OFDM systems. If the estimate
for the packet start k̂S,CT is smaller (Fig. 3a) ) or bigger
(Fig. 3c) ) than the actual packet start kS , the estimated CIR
experiences a left or right cyclic shift, respectively. Thus, in the
case of imperfect coarse timing synchronization, Σnum does not
cover the complete CIR. This results in a severe degradation
in the performance of the proposed algorithm.
Considering imperfect frequency synchronization we evaluate the ICI term in (5). The ICI term is assumed to be
an additive, zero mean complex Gaussian random variable.
For εres ¿ 1, using an approximation similar to [9], the ICI
variance can be stated as
µ
¶2
π 2 ε2res
2
σICI ≈ 1 − 1 −
.
(12)
6
2
Since the STF is designed to ensure σICI
¿ σV2 in the
SNR regime of interest, the effect of imperfect frequency
synchronization on the proposed algorithm can be neglected.
2
Another way to view the condition σICI
¿ σV2 is that the
variance of εres in (6) is sufficiently small.
Firstly, to calculate (10) we exploit that in a MIMO system
estimates of the CIR between all receive-transmit antenna
pairs rt are available and approximate {E[ĥu ĥH
u ]}k,k by
PNt PNr
rt 2
2 . Secondly, to account for the
|
ĥ
|
=
|
ĥ
|
k,u
k,u
t=1
r=1
imperfect coarse timing synchronization, Σnum needs to be
adjusted to completely 0cover the estimated CIR. We therefore
PN
PNCP
by k=k̂FP in (10), where k̂FP is an estimate
replace k=0
of the position of the first path for the estimated CIR. Finally,
N 0 = (k̂FP + NCP − 1) mod N is the (potentially) cyclic upper
limit of the summation window. Employing those changes (10)
can now be approximated by:
PN 0
Θu ≈
2
k=k̂FP |ĥk,u |
.
PN −1
2
k=0 |ĥk,u |
(13)
To obtain an estimate k̂FP for the position of the first path
kFP of the estimated CIR, we employ a method originally
proposed for fine timing in [10]. The first path of the CIR
is found by the threshold decision:
o
n
(14)
k̂FP = min |ĥk,u |2 > Γ and |ĥk,u |2 > |ĥk+1,u |2
k
where k = [0 . . . N −1]. The first criterion ensures that the first
path is not chosen from a noise only sample while the second
criterion selects only local maxima of the estimated CIR. The
threshold Γ depends on the instantaneous realization of |ĥk,u |
and is computed according to
o
n
Γ = max |ĥk,u |2 max × 10−Γ1 /10 , |ĥk,u |2 min × 10Γ2 /10 .
(15)
In [10] the authors propose to choose Γ1 = 10 dB and Γ2 =
8 dB. To ensure that kFP can be found in case of k̂S,CT > kS
(see Fig. 3c)), we additionally introduce a cyclic right shift
of NSFT samples to |ĥk,u |2 before the estimation of kFP . Note
that k̂FP , which has been estimated in conjunction with the
true number of transmit antennas, can be reused to adjust the
DFT window as depicted in Fig. 2. This allows for a reduction
in computational overhead.
V. P ERFORMANCE E VALUATION
A. Simulation Setup
An OFDM system with IEEE802.11n physical layer parameters [3] is considered. The DFT size and cyclic prefix
length were N = 128 and NCP = 32. A multipath Rayleigh
fading channel with a power delay profile decaying with
exp(−k × 0.36) and k = [0 . . . 15] has been chosen. The
system comprises Nt ∈ {1, 2, 4} transmit and Nr ∈ {1, 2, 4}
receive antennas. The STF has a length of NSTF = 320 samples
and is composed of L = 10 identical sequences a of length
Na = 32 as proposed in [3]. Additionally, the LTF consists of
Nt OFDM symbols (see section II-B) and all antennas transmit
the same random, BPSK modulated base sequences. Different
base sequences have been used, depending only on the number
of transmit antennas. For packet detection (see section III-A)
a threshold Tc = 0.2 has been found to maximize the packet
detection probability. Shifts of NSCT = 8 and NSFT = 10
samples have been chosen, based on the distribution of coarse
timing estimates at SNR = 0 dB. To model the situation when
a packet is detected in the noise before the actual packet start
(referred to as a false alarm) we prepend three empty OFDM
symbols, which, at the receiver, will appear only as noise, to
each packet. If a packet is detected in the noise, the receiver
is assumed to be blocked and the packet is lost [11]. Finally,
the spectral mask according to [3] has been adopted resulting
in the outermost and DC subcarriers being set to be zero.
B. Performance Measures
Four measures are employed to characterize the performance of our algorithm. The first, 1 − PA , is the probability that Nt has not been correctly detected, given perfect
packet detection. The second, 1 − PD , is the probability of
incorrectly detecting a packet, i.e. a false alarm, or the case
where kS − NCP ≤ k̂S,CT ≤ kS + NCP does not hold. The
third performance measure, 1 − PA/D , is the probability of
incorrectly detecting Nt , given a correct packet detection.
Finally, 1 − PD0 = 1 − (PD × PA/D ) measures the over all detection performance including packet detection and subsequent
detection of Nt .
C. Results
1) Perfect Synchronization: The figure of merit to benchmark the proposed metric is the probability of detecting the
incorrect number of transmit antennas, denoted by 1 − PA .
10
10
0
10
perf. sync.
real freq ., perf . timing
perf freq., real timing
real freq ., real timing
-1
10
P
10
,A
P
−1
10
-1
PD
−
1, 10-2
1x1
-2
1 − PD
1 − PD
'
D
/A
−1
0
1x1
PD
−1
2x2
-3
4x4
-4
10
-20
Fig. 4.
10
-15
2x2
-3
4x4
-4
-10
-5
SNR [dB]
0
5
Probability of detecting the incorrect number of transmit antennas
Results are plotted in Fig. 4 (solid black lines). It can be seen
that under perfect synchronization conditions 1−PA is already
less than 10−4 at an SNR of 0 dB. Note that this is the worst
case scenario (i.e. a SISO transmission). Increasing the number
of transmit and receive antennas, results in performance gains
of roughly 5 dB in a 2x2 MIMO system and 8 dB in a 4x4
MIMO system, respectively, for 1 − PA = 10−4 , as compared
to a SISO system.
2) Effect of Timing Synchronization: Under real packet
detection and coarse timing synchronization conditions (Fig.
4, 1 − PA/D , solid grey lines) a performance loss of roughly
3 dB compared to the former case of perfect synchronization
appears. The reason for this is due to the inaccuracy of both
the coarse timing synchronization and the estimation of the
first path of the estimated CIR (see section IV) in the low
SNR regime.
3) Effect of Frequency Synchronization: The effect of imperfect frequency synchronization on our algorithm is shown
in Fig. 4 for perfect timing synchronization (black crosses)
and real timing synchronization (grey crosses). It can be seen
that the algorithm performance is, as expected, not affected by
imperfect frequency synchronization.
4) Overall Detection Performance: We now show that the
proposed algorithm is suitable for practical implementation.
Specifically, we compare the overall missed packet detection
0
probability 1−PD
with the missed packet detection probability
0
1 − PD . Results are given in Fig. 5. Note that 1 − PD
includes
imperfect frequency synchronization. It can be seen that in
the worst case (i.e. a SISO system) the performance loss is
as low as 0.75 dB at 1 − PD = 10−4 . In a 2x2 and 4x4
MIMO system almost no performance loss is visible because
1 − PA/D is smaller than 1 − PD (compare to the grey curves
in Fig. 4).
VI. C ONCLUSION
In this work we showed how our previously presented
algorithm for detecting the number of transmit antennas in
MIMO-OFDM systems [1] can be integrated into a typical MIMO-OFDM receiver. In particular, we addressed the
crucial issue of detecting the number of transmit antennas
10
-10
-8
Fig. 5.
-6
-4
-2
SNR [dB]
0
2
4
Missed packet detection probability
under imperfect synchronization conditions. It was found that
imperfect timing synchronization has a significant impact on
the algorithmic performance, while the effects of imperfect
frequency synchronization are negligible. It was also shown
that the overall packet detection performance achieved by
applying our algorithm is almost the same as compared to the
case when the receiver has perfect knowledge of the number of
transmit antennas. Consequently, we believe that our algorithm
can be considered for practical application in next generation
ad-hoc mobile wireless standards in order to reduce signaling
overhead and transmission latency.
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