Multiple Input - Multiple Output (MIMO) Receiver for Wideband Space ...
Multiple Input - Multiple Output (MIMO) Receiver for Wideband Space-Time Communications Jean-Franc¸ois Frigon and Babak Daneshrad UCLA Electrical Engineering Department LA, CA 90095-1594 fjeanf, [email protected] ABSTRACT C11(z)
I.
INTRODUCTION
Recently, it has been shown that by exploiting the fading properties of the wireless environment, a communication structure with multiple transmit/receive antennas can deliver enormous capacities [1]. Several architectures have been proposed to implement space-time systems for the narrowband flat fading wireless environment [2, 3]. However, in order to support higher data rates, the concept of the space-time communication architecture needs to be extended to the frequency selective wideband channel. Three different space-time structures have been studied for the multiple-input multiple-output (MIMO) dispersive channel. The first architecture uses Orthogonal Frequency Division Multiplexing (OFDM) to mitigate the effect of the frequency selective channel and then applies techniques that have been developed for narrowband space-time system on each OFDM substream [4, 5]. Similarly, the second architecture employs Direct-Sequence Spread Spectrum Code Division Multiple Access (DSSS-CDMA) instead of OFDM to combat the multipath channel [6]. The third approach, and the one that will be explored in this paper, uses equalization techniques to provide communications over frequency selective MIMO channels [7, 8, 9, 10]. In this paper, we propose a new framework for the analysis of a modified MIMO minimum mean squared error (MMSE) decision feedback equalizer (DFE). The modified MIMO DFE structure proposed here assumes that decisions for each data streams are made sequentially and that current decisions of detected data streams are used to compute the current estimate of a data stream. This structure therefore performs an operation similar to interfer-
d1(i)
u1(i)
) (z C 1N
A new receiver structure able to deliver high data rates in a multiple-input multiple-output (MIMO) frequency selective wireless environment is proposed and investigated in this paper. The optimal solution for the finite length MIMO decision feedback equalizer (DFE) with cancellation is derived and used to illustrate the potential of this architecture for space-time communications. LMS and RLS adaptive algorithms are also presented for the MIMO architecture. The convergence and performance of these algorithms is confirmed through simulation results. The proposed adaptive solutions do not require channel identification, are less computationally intensive than the optimal solution, and allow the proposed MIMO receiver to adapt to channel changes.
v1(i)
)
(z
CM
1
CMN (z)
vN(i)
dM(i)
uN(i)
FIGURE 1: MIMO channel model.
ence cancellation and is an extension for the frequency selective fading channel of the MMSE equivalent of the VBLAST algorithm [2]. This type of modified MIMO DFE structure has only been studied in [8]. Furthermore, although the problem of MIMO equalization has been previously studied [7, 8, 9, 10], only [9] has explored a recursive least square (RLS) adaptive method for the linear MIMO equalizer. The new MIMO MMSE DFE problem formulation that we are proposing allows us to determine the performance of a receiver using finite length feedforward and feedback matrix filters under ideal conditions (perfectly estimated channel). This framework can be used in the derivation of novel adaptive algorithms for the modified MIMO DFE based on the stochastic gradient algorithm (e.g., least-mean square (LMS) algorithm) or the least square solution (i.e., RLS algorithm). The paper is organized as follows. In Section II we formulate the MIMO DFE with cancellation problem. In Section III we find the optimal MMSE solution for the finite length proposed receiver. In Section IV and V we propose, respectively, an LMS and an RLS adaptive algorithm for the MIMO DFE with cancellation. Simulation results illustrating the concept proposed in the paper are presented in Section VI. The paper is concluded in Section VII. II.
PROBLEM FORMULATION
Figure 1 shows a discrete-time model for the MIMO channel with M transmit antennas and N receive antennas. A sequence of symbols fdm i g is transmitted from each antenna, where the symbol sequences are assumed to be independent identically distributed (IID) sequences (both in time and space) and drawn from a QAM constellation with power σ2d . The symbols are transmitted over the MIMO dispersive channel consisting of M N finite impulse
()
()
response (FIR) channels, Cm;n z , of order Nc (i.e., each channel consists of Nc multipaths). Each FIR channel Cm;n z connects the transmit antenna m to the receive antenna n. As shown in Figure 1, at any time instant i, the signal received at each of the N antennas consists of a linear combination of the ; : : : ; M, and of the previous symbols current symbols dm i , m of each sequence fdm j ; j < ig. Therefore, the channel introduces inter-symbol interference (ISI) and co-channel interference (CCI). Furthermore, the MIMO channel output at each antenna is corrupted by additive white-noise vn i , n ; : : : ; n, which is assumed to have zero-mean and variance σ2v . The noise is assumed to be uncorrelated with the transmitted sequences. d1 i dM i , the Let us define the data vector transmit data vector t , c , the received data vector u1 i uN i , and the noise vector v1 i vN i . Additionally, let us expand the channel impulse response between transmit antenna m and receive antenna n as follows: Cm;n z cm;n cm;n z,1 cm;n Nc z,Nc . We can then express the MIMO channel in matrix form:
+1
()
u(i)
N
Z-1
F0
( ) =1 ()
Z-1
F1
FNf d(i- )
M
[ ()
2
(0)
c1;1 6 .. 6 .
C(i) =
6 6 cM;1 6 6 .. 6 . 6 6 c1;1 Nc 6 6 .. 4 .
..
.
..
.
..
.
(0)
( ) ( )
cM;1 Nc
c1;N .. .
(0)
3
7 7 7 7 cM;N 7 7 .. 7: . 7 c1;N Nc 7 7 7 .. 5 .
(0)
( )
( )
cM;N Nc
BQ
:
B1
Z-1
B0
Z-1
FIGURE 2: Architecture of the MIMO DFE with cancellation.
ing. We can observe that the MIMO DFE is a matrix counterpart of the scalar DFE where the scalar delay line, the taps and the decision are replaced, respectively, by a vector delay line, matrix taps and a decision vector. Suppose that we have N f feedforward delays and Q feedback delays. The N M feedforward tap matrices are denoted by j , j N f , and the M M feedback tap matrices by j , j Q. To satisfy the “causal” constraint, 0 must be strictly lower triangular (i.e., b0;i j for i j). , c , , Let us introduce the data vector the receive vector , f , the noise vector , f , and the feedback vector f , , , . Let us also define0 the channel matrix0 0 00( 0 Nf where j M j N M(N f , j )N denotes the transpose operation and kl denotes a matrix with k rows and l columns). It can then be verify that:
B 0
F 0
B =0 d = [d(i) d(i N Nf )] u = [u(i) u(i N )] v = [v(i) v(i N )] d = [d(i ) d(i Q)] C = [C C ] C = [0 C(i) 0 ] () () u = dC + v
Then the output of the channel at any instant i is given by:
u(i) = dt(i)C(i)+ v(i)
d(i- )
M
() =1
d(i) = [ ( ) ( )] d (i) = [d(i) d(i N )] u(i) = [ ( ) ( )] v(i) = ( )] ( ) = (0)+ (1) + + ( )
M
(1)
(2)
:
Then, if we define the feedforward coefficient matrix by
F = [F00 FNf 0]0 and the feedback coefficient matrix by B = The purpose of the receiver is to process the received data vec- [B 0 B 0 ]0 , the data estimate vector d ^(i , ) is given by: 1 Q tor u(i), the previous data decisions, and the current available data decisions in order to obtain an estimate of a delayed version of the d^(i , ) = ,df B + uF ^(i , ) = [d^1(i , ) d^M (i , (3) data vector d(i , ) denoted by d = yW )]. The delay is a parameter chosen by the designer. The data decisions are obtained by applying the slicer function to the vector y = [df u] and W = [,B0 F0]0. The MIMO DFE error d^(i , ) that maps each of the data estimates to the closest QAM where vector can also be written as e(i , ) = d(i , ) , yW. constellation point (in order to facilitate the analysis, we will as;
sume that all the decisions are correct). This proposed modified DFE differs from conventional DFE structure by using the current decisions that are available to obtain the data estimate. In the remainder of this text, we will refer to the modified MIMO DFE as a MIMO DFE with cancellation since using current decisions in the feedback in effect tries to cancel the interference caused by these transmitted data. Note that in order to have a “causal” receiver, the data must be detected in a given order to be able to use current data decision for the data estimate. Let assume that higher-indexed data streams are detected first. Then, the current decisions from data streams m, j < m M, are used to obtain the data estimate for stream j. Figure 2 shows the modified MIMO DFE that we are consider-
III.
MMSE SOLUTION
Finding an optimal solution for an architecture is important in several key aspects. First it provides a means to quickly analyze its performance in different environments and configurations as will be shown in Section VI. Second, it provides the optimal solution against which the mean squared error (MSE) performance of the adaptive algorithms can be compared. The objective of the MMSE solution is to select the unknown entries of that minimize the covariance matrix of . In order to solve this problem we will use the innovations of the vector [11]. Let us denote the innovation vector by x1 ; : : : ; xK and the receiver input vector by y1 ; : : : ; yK , where K
W
y
e
y = [d u ] = [
x=[
]
]
=
( +1)+ ( +1)
N NF M Q . We want the innovation vector to be white (i.e., we want to have x , where denotes the Hermitian transpose operation) and x j to be a linear combination , where is lower triangular of fy j ; : : : ; yK g. Next, let y ,1 (i.e., is the Cholesky decomposition of y ). Then, if the innovation vector has the desired properties.
R = E[x x] = I R =L L
L
x
()
L
R
x = yL
^
The MIMO receiver finds the estimates dm using a linear combination of fym+1 ; : : : ; yK g. Since fxm+1 ; : : : ; xK g spans the same space as fym+1 ; : : : ; yK g [11], dm i , can also be written as:
m is the mth column of , and j j2 . We see where m from this equation that Jm does not depend on the other i, 6 m indepencolumns . Therefore, if each column dently converges to its optimal value, its associated cost function m will converge to its minimum and then the global cost Jm function will attain its minimum. We can thus apply the steepest descent algorithm to each of the vectors m . Recall that the first m elements of m are equal to zero. We can then find that
W W i=m (W )
^ ( ) = ∑K x j wxjm j =m+1
m
=1
;::: ;
M:
(4)
E[ ( ( ) ^ ( ))] = 0 x
Wxlm = 0Rd(i,)x lm ml ml K
I.
INTRODUCTION
Recently, it has been shown that by exploiting the fading properties of the wireless environment, a communication structure with multiple transmit/receive antennas can deliver enormous capacities [1]. Several architectures have been proposed to implement space-time systems for the narrowband flat fading wireless environment [2, 3]. However, in order to support higher data rates, the concept of the space-time communication architecture needs to be extended to the frequency selective wideband channel. Three different space-time structures have been studied for the multiple-input multiple-output (MIMO) dispersive channel. The first architecture uses Orthogonal Frequency Division Multiplexing (OFDM) to mitigate the effect of the frequency selective channel and then applies techniques that have been developed for narrowband space-time system on each OFDM substream [4, 5]. Similarly, the second architecture employs Direct-Sequence Spread Spectrum Code Division Multiple Access (DSSS-CDMA) instead of OFDM to combat the multipath channel [6]. The third approach, and the one that will be explored in this paper, uses equalization techniques to provide communications over frequency selective MIMO channels [7, 8, 9, 10]. In this paper, we propose a new framework for the analysis of a modified MIMO minimum mean squared error (MMSE) decision feedback equalizer (DFE). The modified MIMO DFE structure proposed here assumes that decisions for each data streams are made sequentially and that current decisions of detected data streams are used to compute the current estimate of a data stream. This structure therefore performs an operation similar to interfer-
d1(i)
u1(i)
) (z C 1N
A new receiver structure able to deliver high data rates in a multiple-input multiple-output (MIMO) frequency selective wireless environment is proposed and investigated in this paper. The optimal solution for the finite length MIMO decision feedback equalizer (DFE) with cancellation is derived and used to illustrate the potential of this architecture for space-time communications. LMS and RLS adaptive algorithms are also presented for the MIMO architecture. The convergence and performance of these algorithms is confirmed through simulation results. The proposed adaptive solutions do not require channel identification, are less computationally intensive than the optimal solution, and allow the proposed MIMO receiver to adapt to channel changes.
v1(i)
)
(z
CM
1
CMN (z)
vN(i)
dM(i)
uN(i)
FIGURE 1: MIMO channel model.
ence cancellation and is an extension for the frequency selective fading channel of the MMSE equivalent of the VBLAST algorithm [2]. This type of modified MIMO DFE structure has only been studied in [8]. Furthermore, although the problem of MIMO equalization has been previously studied [7, 8, 9, 10], only [9] has explored a recursive least square (RLS) adaptive method for the linear MIMO equalizer. The new MIMO MMSE DFE problem formulation that we are proposing allows us to determine the performance of a receiver using finite length feedforward and feedback matrix filters under ideal conditions (perfectly estimated channel). This framework can be used in the derivation of novel adaptive algorithms for the modified MIMO DFE based on the stochastic gradient algorithm (e.g., least-mean square (LMS) algorithm) or the least square solution (i.e., RLS algorithm). The paper is organized as follows. In Section II we formulate the MIMO DFE with cancellation problem. In Section III we find the optimal MMSE solution for the finite length proposed receiver. In Section IV and V we propose, respectively, an LMS and an RLS adaptive algorithm for the MIMO DFE with cancellation. Simulation results illustrating the concept proposed in the paper are presented in Section VI. The paper is concluded in Section VII. II.
PROBLEM FORMULATION
Figure 1 shows a discrete-time model for the MIMO channel with M transmit antennas and N receive antennas. A sequence of symbols fdm i g is transmitted from each antenna, where the symbol sequences are assumed to be independent identically distributed (IID) sequences (both in time and space) and drawn from a QAM constellation with power σ2d . The symbols are transmitted over the MIMO dispersive channel consisting of M N finite impulse
()
()
response (FIR) channels, Cm;n z , of order Nc (i.e., each channel consists of Nc multipaths). Each FIR channel Cm;n z connects the transmit antenna m to the receive antenna n. As shown in Figure 1, at any time instant i, the signal received at each of the N antennas consists of a linear combination of the ; : : : ; M, and of the previous symbols current symbols dm i , m of each sequence fdm j ; j < ig. Therefore, the channel introduces inter-symbol interference (ISI) and co-channel interference (CCI). Furthermore, the MIMO channel output at each antenna is corrupted by additive white-noise vn i , n ; : : : ; n, which is assumed to have zero-mean and variance σ2v . The noise is assumed to be uncorrelated with the transmitted sequences. d1 i dM i , the Let us define the data vector transmit data vector t , c , the received data vector u1 i uN i , and the noise vector v1 i vN i . Additionally, let us expand the channel impulse response between transmit antenna m and receive antenna n as follows: Cm;n z cm;n cm;n z,1 cm;n Nc z,Nc . We can then express the MIMO channel in matrix form:
+1
()
u(i)
N
Z-1
F0
( ) =1 ()
Z-1
F1
FNf d(i- )
M
[ ()
2
(0)
c1;1 6 .. 6 .
C(i) =
6 6 cM;1 6 6 .. 6 . 6 6 c1;1 Nc 6 6 .. 4 .
..
.
..
.
..
.
(0)
( ) ( )
cM;1 Nc
c1;N .. .
(0)
3
7 7 7 7 cM;N 7 7 .. 7: . 7 c1;N Nc 7 7 7 .. 5 .
(0)
( )
( )
cM;N Nc
BQ
:
B1
Z-1
B0
Z-1
FIGURE 2: Architecture of the MIMO DFE with cancellation.
ing. We can observe that the MIMO DFE is a matrix counterpart of the scalar DFE where the scalar delay line, the taps and the decision are replaced, respectively, by a vector delay line, matrix taps and a decision vector. Suppose that we have N f feedforward delays and Q feedback delays. The N M feedforward tap matrices are denoted by j , j N f , and the M M feedback tap matrices by j , j Q. To satisfy the “causal” constraint, 0 must be strictly lower triangular (i.e., b0;i j for i j). , c , , Let us introduce the data vector the receive vector , f , the noise vector , f , and the feedback vector f , , , . Let us also define0 the channel matrix0 0 00( 0 Nf where j M j N M(N f , j )N denotes the transpose operation and kl denotes a matrix with k rows and l columns). It can then be verify that:
B 0
F 0
B =0 d = [d(i) d(i N Nf )] u = [u(i) u(i N )] v = [v(i) v(i N )] d = [d(i ) d(i Q)] C = [C C ] C = [0 C(i) 0 ] () () u = dC + v
Then the output of the channel at any instant i is given by:
u(i) = dt(i)C(i)+ v(i)
d(i- )
M
() =1
d(i) = [ ( ) ( )] d (i) = [d(i) d(i N )] u(i) = [ ( ) ( )] v(i) = ( )] ( ) = (0)+ (1) + + ( )
M
(1)
(2)
:
Then, if we define the feedforward coefficient matrix by
F = [F00 FNf 0]0 and the feedback coefficient matrix by B = The purpose of the receiver is to process the received data vec- [B 0 B 0 ]0 , the data estimate vector d ^(i , ) is given by: 1 Q tor u(i), the previous data decisions, and the current available data decisions in order to obtain an estimate of a delayed version of the d^(i , ) = ,df B + uF ^(i , ) = [d^1(i , ) d^M (i , (3) data vector d(i , ) denoted by d = yW )]. The delay is a parameter chosen by the designer. The data decisions are obtained by applying the slicer function to the vector y = [df u] and W = [,B0 F0]0. The MIMO DFE error d^(i , ) that maps each of the data estimates to the closest QAM where vector can also be written as e(i , ) = d(i , ) , yW. constellation point (in order to facilitate the analysis, we will as;
sume that all the decisions are correct). This proposed modified DFE differs from conventional DFE structure by using the current decisions that are available to obtain the data estimate. In the remainder of this text, we will refer to the modified MIMO DFE as a MIMO DFE with cancellation since using current decisions in the feedback in effect tries to cancel the interference caused by these transmitted data. Note that in order to have a “causal” receiver, the data must be detected in a given order to be able to use current data decision for the data estimate. Let assume that higher-indexed data streams are detected first. Then, the current decisions from data streams m, j < m M, are used to obtain the data estimate for stream j. Figure 2 shows the modified MIMO DFE that we are consider-
III.
MMSE SOLUTION
Finding an optimal solution for an architecture is important in several key aspects. First it provides a means to quickly analyze its performance in different environments and configurations as will be shown in Section VI. Second, it provides the optimal solution against which the mean squared error (MSE) performance of the adaptive algorithms can be compared. The objective of the MMSE solution is to select the unknown entries of that minimize the covariance matrix of . In order to solve this problem we will use the innovations of the vector [11]. Let us denote the innovation vector by x1 ; : : : ; xK and the receiver input vector by y1 ; : : : ; yK , where K
W
y
e
y = [d u ] = [
x=[
]
]
=
( +1)+ ( +1)
N NF M Q . We want the innovation vector to be white (i.e., we want to have x , where denotes the Hermitian transpose operation) and x j to be a linear combination , where is lower triangular of fy j ; : : : ; yK g. Next, let y ,1 (i.e., is the Cholesky decomposition of y ). Then, if the innovation vector has the desired properties.
R = E[x x] = I R =L L
L
x
()
L
R
x = yL
^
The MIMO receiver finds the estimates dm using a linear combination of fym+1 ; : : : ; yK g. Since fxm+1 ; : : : ; xK g spans the same space as fym+1 ; : : : ; yK g [11], dm i , can also be written as:
m is the mth column of , and j j2 . We see where m from this equation that Jm does not depend on the other i, 6 m indepencolumns . Therefore, if each column dently converges to its optimal value, its associated cost function m will converge to its minimum and then the global cost Jm function will attain its minimum. We can thus apply the steepest descent algorithm to each of the vectors m . Recall that the first m elements of m are equal to zero. We can then find that
W W i=m (W )
^ ( ) = ∑K x j wxjm j =m+1
m
=1
;::: ;
M:
(4)
E[ ( ( ) ^ ( ))] = 0 x
Wxlm = 0Rd(i,)x lm ml ml K
