Zero Forcing Equalizing Filter for MIMO Channels ... - Semantic Scholar
ZERO FORCING EQUALIZING FILTER FOR MIMO CHANNELS WITH INTERSYMBOL INTERFERENCE Volker Pohl, Volker Jungnickel, Eduard Jorswieck, Clemens von Helmolt Heinrich-Hertz-Institut für Nachrichtentechnik Berlin GmbH, Einsteinufer 37, 10587 Berlin, Germany, e-mail: [email protected], [email protected], [email protected], [email protected]
Abstract – Equalization in the time domain is a wellknown technique for combating intersymbol interference (ISI) in frequency selective channels. In this paper we derive the general structure of the linear zero forcing (ZF) equalizing filters for multiple input multiple output (MIMO) communication systems with more outputs than inputs. These filters are finite impulse response (FIR) filters in general and their structure is determined by the number of inputs and outputs. The relation between the spatial diversity of the system and the necessary filter length is worked out and it is shown, that an optimal time lag in the filter will improve the performance considerably. Keywords – MIMO, equalization, spatial multiplexing I. INTRODUCTION MIMO communication systems operating on channels with ISI arise in many different applications. By using the same frequency- and time slot for different data streams, these systems promise an enormous capacity [1,2]. One method to combat the delay spread is to use Orthogonal Frequency Division Multiplexing (OFDM) [3,4]. In single carrier systems the receiver has not only to separate the different data streams, transmitted over the multiple transmit antennas, but also to suppress the ISI. There are two major equalizing techniques. The first one are sequence estimation algorithms, which process a whole block of consecutive data [5,6]. The drawback of these equalizers is, that the computational complexity grows rapidly with the length of the data blocks. The second kind of equalizers, use filters which operate on the received data stream symbol by symbol [7-9]. Fig. 1 shows the general structure of such equalizing filters. To determine the coefficient matrices Gi normally an arbitrary length L and time-lag τ is chosen. The matrices are then determined by the wellknown Wiener-Hopf equation. But it is not clear at the outset which filter-length and time lag will be optimal. In this paper we solve the system of difference equations, which describe the transmission over the frequency selective MIMO channel. In this way we obtain the general structure of the ZF inverse filters and so we are able to specify the minimal- and the optimal filter length. Moreover, the optimal time lag can be determined. The paper is organised as follows: Section II describes the channel and signal model used in this paper. In Section III the structure of the filters is derived and in Section IV an improved filter structure is proposed. Section V investigates the performance of the filters by means of bit error rat
0-7803-7589-0/02/$17.00 ©2002 IEEE
y(k)
y(k−1) z-1
G1
y(k−L+2) z-1
L L
G2
L
+
z-1
y(k−L+1) z-1
GL-1
GL
+
+
xˆ (k−τ)
Fig. 1: General structure of a linear equalizer of length L and with a time lag τ.
(BER) simulations, followed by a short summary in Section VI. II. CHANNEL AND SIGNAL MODEL We consider a single-user MIMO communication system with n antennas at the transmitter (Tx) and m antennas at the receiver (Rx). A discrete-time complex baseband signal model is used, where the received signals are sampled at symbol rate. The channel is assumed to be time invariant and so it can be described by the following linear difference equation with constant coefficients: y(k) =
N
Ki x(k−i) + ν(k) Σ i=0
(1)
where N is the order of the equation, k is the time index and the difference between the number of outputs and inputs d=m−n is the antenna excess of the system. Ki are complex valued, constant matrices with n columns and m rows. It is always assumed that the entries in these matrices are statistically independent random numbers. The entries in the vector x(k)=[x1(k) , ... , xn(k)]T and y(k)=[y1(k) , ... , ym(k)]T are the n symbols and m signals simultaneously transmitted and received, respectively at time instant k. The noise ν(k)=[ν1(k), ... , νm(k)]T is assumed to be statistically independent but equally distributed for different time instances k. It has zero mean (E[ν ν]=0) and the covariance matrix is denoted by Ν=E[νν ννH]. The single entries xi(k) and yi(k) in the above vectors are functions Z→ →C, mapping the set of all integers Z into the field of complex numbers C, and so the whole vector x(k) is a function Z→ →C n where C n denotes a n-dimensional vector field over the complex numbers C. If we denote the set of all functions Z→ →C n by FC n then the transmission over the channel is described by a linear mapping Φ : FC n→FC m y(k) = (Φ Φ x)(k) + ν(k)
(2)
PIMRC 2002
where x∈FC n, y∈FC m, ν∈FC m and Φ has the form (Φ Φ x)(k) = Ko x(k) + ... + KN x(k−N).
(3)
In what follows we will denote such mappings by upper case Greek letters, matrices by bold upper case Latin letters and vectors and vector valued functions by bold lower case letters. AT will denote the transposed of the matrix A and AH the transposed conjugate complex of A. The n×n identity matrix is denoted by In. III. FILTER STRUCTURE The structure of the ZF filters will follow from the solution of the system of m difference equations for n unknown functions (1). To solve (1) we neglect the noise term ν(k) and write this system of Nth order equations as system of first order equations as follows: w(k) = (Ψ u)(k) = A u(k) − B u(k−1) ∼
(4)
∼
with a linear mapping Ψ : FC n→FC m, where A and B are ~ = (N−1) n + m rows and ~ matrices with m n = N n columns:
0M A= 0 K
0
∼
In M 0 K1
L 0 O M L In L KN-1
0 0 IM L O M M ;B=0 L I 0 0 L 0 -K n
n
~ ~~ ~ (k) = P w y(k) and x(k) = Q u (k) (7) ~ ~ respectively. Where the m×m matrix P consists of the last m ~ ~ matrix Q columns of the matrix P, and the n×n consist of the first n rows of the matrix Q. ~ ~ The concrete form of the matrices A and B is determined by the elementary divisors and minimal indices of the matrix pencil Aλ−Bµ. For m=n it follows from the assumption of regular, independent random matrices Ki that the pencil Aλ−Bµ has (with probability one) only linear elementary divisors (αiλ−βiµ , i=1... ~ n) and no minimal indices. From there it can be shown, that the ZF filter of such a system is an IIR filter, in general. We will concentrate here on the case where the system of difference equations is over-determined (m>n). Therefore we have m equations for only n unknown functions x1(k), ... , xn(k). So (2) will have a solution only if y(k) lies in the range of Φ, but since the received signals y(k) are corrupted by the noise ν(k), this requirement will in general not be met. Our aim is to find a pseudo-inverse mapping Φ-1 : FC m→FC n which satisfies the following ZF condition (Φ Φ-1 Φ x)(k) = x(k)
(4a)
N
∼
and u∈FC n , w∈FC m are given by: u(k) = [ xT(k) , xT(k−1) , ... , xT(k−N+1) ] T w(k) = [ 0 , 0 , ... , yT(k) ]
of (6) back to the desired solution x(k) of (1) can thus be written as
T
(4b)
Note, that the difference in the number of equations and unknowns is unchanged by this transformation from (1) to ~ −n ~. (4), i.e. we have d = m−n = m To solve (4) we consider the matrix pencil Aλ−Bµ (A comprehensive introduction into matrix pencils including an extensive reference list can be found in [10]). A well known result from Weierstrass and Kronecker [11,12] states, that the pencil is completely determined by its elementary divisors and its minimal indices, and so these numbers determine the structure of the ZF filter completely. From the same result follows also (see [10]), that (4) can always be transformed into a canonical form by changing the co-ordinates according to ~(k) ; w(k) = P-1 w ~ (k) u(k) = Q u (5) ~ ~ ~ ~ where P and Q are regular m×m and n×n matrices respectively. So (4) is written in canonical form as: ~ ~ ~~ ~~ ~ (k) = (Ψ w u)(k) = A u (k) − B u (k-1) (6) ~ ~ with A=PAQ and B=PBQ. One method how the transformation matrices P and Q can be determined from the given matrices A and B is described in [10, chapter 12.3]. The transformation from the given y(k) in (1) to the inhomogen~ (k) in (6) and the transformation from the solution u ~(k) ity w
for all x ∈ FC n .
(8)
It can be shown that, for m>n and under the assumption of regular, independent random matrices Ki, the matrix pencil Aλ−Bµ has (with probability one) no elementary divisor, l1 minimal indices ε1 and l2 minimal indices ε2 of rows, such ~ ×n ~ matrices in the canonical form (6) have the that the two m following quasi-diagonal form:
~ A=
Aε L 0 1 M
l1
0
M
O 0 L Aε 1
Aε L 2
0
M 0
l2
O
0 M
L Aε2
; B~=
Bε L 0 1
l
1 M O 0 L Bε 1
M
0 Bε L 0 2
0
l
2 M O 0 L Bε 2
M
(9)
where the diagonal elements are l1 matrices Aε1 and l2 matrices Aε2 (resp. l1 matrices Bε1 and l2 matrices Bε2). The matrices Aε and Bε (ε = ε1 or ε2) with ε+1 rows and ε columns are given by:
1 0 0 1 = M 0 O 0 L 0 0
Aε
L 0 M 0 0 0 1 0 0 1
;B
0 1 0 0 O = M 0 0 L 1 0
ε
L 0 M 0 1 0 0 1 0 0
(10)
The numbers ε1, ε2, l1, l2 are determined only by the dimen~ ~ sions of the matrices A and B: • ε1 ist the smallest integer for which (ε1 + 1) d > N n • l1 = (ε1 + 1) d − N n (11) • ε2 = ε1 + 1 • l2 = d − l1
m
~ P
z-1
~ (k− w −1)
z-1
L
~ m
L
M2
M1
z-1
~ (k− w −ε2+1)
ΓM
~ (k− w −ε2)
Mε2-1
Mε2
+
+
L
+
z-1
~ n
n
~ Q
m
~ (k− w −1)
~ (k) w
~ P
z-1
z-1
z-1
L
~ m -Lε2
~ (k− w −ε2+1)
~ (k− w −ε2+2)
L
-Lε2-1
L
+
-L2
-L1
+
+
MIMO : 8 Tx- / 8+d Rx antennas Channel of order N=4 (5 taps)
0
5
10
~ n
n
~ Q
15
20
xˆL(k− − ε2 )
and not by the concrete realisation of the channel matrices Ki. So only the transformation matrices P and Q will change if the channel varies. Because of the quasi-diagonal structure of the canonical form the whole equation (6) separates into l1 + l2 = d elementary equation of rank ε1 (respectively ε2) of the form: ~ ~ ~ )(k) ; i=1 ... d (12) ~ (k) = A u Θεu w i ε i(k) − Bε ui(k−1) = (Θ i where Θε: FC ε→FC ε+1 is called elementary mapping of rank ε. It remains to solve these elementary equations on the diagonal. This can be done by one of the following two elementary pseudo-inverse filters of rank ε : ΓMε and ΓLε : FC ε+1→FC ε :
It is easily verified that both filters will satisfy the following ZF condition: ~ (k) = (Γ ~ )(k) u ΓMε Θε u i i ~ ∈FC ε ; for all u i ~ ~ ΓL Θ u )(k) u (k) = (Γ i
ε
ε
So the solution of (12) can be obtained by ~ )(k) ; I = M or L ~ ΓεI w ui(k) = (Γ i
Each of the elementary equations on the diagonal of the canonical form (6) can be solved by one of the elementary filters ΓMε or ΓLε , to form the overall pseudo-inverse solution. We consider two possible solutions of (6): the first one ΓM is formed by solving all the separate elementary equations by the corresponding filters ΓMε and the second ΓL is formed by all filters ΓLε . So after the transformation back to the original -1 -1 co-ordinates using (7) we get two ZF filters ΦM and Φ L which can be used to calculate an estimate of the transmitted data x(k):
i=1
j=1
(13)
ε
~ ~ )(k) = − ∑ Lε w ~ ui(k) = (Γ ΓLε w i j i(k+j)
ε
~ ~ ~ 2 ~ -1 x$ L(k) = (Q ΓL P y)(k) = − Q·∑Li P·y(k+i) = (Φ Φ L y)(k)
ε
where L i and M i are matrices with ε rows and ε+1 columns defined by
00 10 01 0 L 0M 00 00 01 1 L 0M 00 L0 ε ε ε O O M = ... Mε = M1 = M M 0 0 1 0 2 M 0 0 0 1 0 L 0 0 0 0 L 0 L 0 0 1
Lε = 2
0 0 0 L0 1 0 00 M O M 10 0 0 0L 1 0 0
0 0
Mi = Li = M ε
ε
L 0 O M L 0
... Lε = ε
0 01 00 O 0 M 000
0 0 0 L0 M M 0 O 00 00 1 0 0 L0
for i>ε.
(13a)
(13b) (13c)
The elementary filters of rank 0 are defined as 0 ≡ (Γ ΓM0 w)(k) = (Γ ΓL0 w)(k)
(17)
i=1
j=1
ε
(16)
ε
~ ~ )(k) = ∑ Mε w ~ ui(k) = (Γ ΓMε w i j i(k−j+1)
L1 =
(15)
i
~ ~ ~ 2 ~ -1 x$ M(k) = (Q ΓM P y)(k) = Q·∑ Mi P·y(k−i+1) = (Φ ΦM y)(k)
ε
1 0 0 L0 01 00 M O M 0100 0L 0 1 0
30
Fig. 3: Minimal filter length Lmin of the MIMO equalizer as function of the antenna excess for a system with 8 Tx antennas and a channel with 5 filter taps (N=4)
Fig. 2: The structure of the two ZF equalizing filter Φ M-1 and ΦL-1 according to (17). Note the time lag of ε2 in Φ L-1.
ε
25
antenna excess d
ΓL
~L(k− u −ε2)
m=(N+1)n
m=2n
~ (k− w − ε2 )
z-1
z-1
35 30 25 20 15 10 5
xˆM(k)
~ M(k) u
y(k)
min. filter length
~ (k) w
y(k)
for all w∈FC.
(14)
with the matrices
M= i
Mεi1 L 0 l
1 M O 0 L Mεi1
M
0
0
Mεi2 L 0 M
l2
M O ε 0 L M i2
=
Li
Lεi1 L 0 l
1 M O ε 0 L L i1
M
0
0
Lεi2 L 0 l
2 M O ε 0 L L i2
M
(18)
whereas the matrices Lεi and Mεi are defined in (13). These filters are sketched in Fig. 2. We see that both filters have the same structure as shown in Fig. 1, where the ~ ~ ~ ~ matrices Gi are equal to PMiQ and PLL+1−iQ, respectively. But now we can explicitly specify the necessary filter length: Lmin = ε2 ≤
Nn + 1. d
(19)
~ P
Ξ ~ m
L
10
ΞM
0
MIMO: 7 Rx, 5 Tx Channel: 5 Taps (N=4) min. equalizer length: Lmin=10 Taps optimal time lag: τ=L/2
-1
10
~ n ~M(k− u − ε 2) ~L(k− − ε 2) u
~ Q
FM
n +
~ Q
FL
xˆ(k− − ε 2)
n
Fig. 4: Structure of the equalizer which uses both elementary ZF-filter to improve the performance in the presence of noise.
This is the minimum length of the filter which is necessary to remove all ISI and CCI. We see, that the necessary filter length is reduced considerably by increasing the antenna excess d. At this point the close relation between the spatialand time dimension of the channel becomes obvious, as illustrated in Fig. 3: If the system has no antenna excess (d=0) the ZF filter is an IIR filter with a infinite number of filter taps. With every additional receive antenna the length of the filter decreases until the number of receive antennas is larger as or equal to the channel length times the number of transmit antennas: m ≥ (N + 1) n
resp.
d≥Nn
(20)
In this case Lmin becomes 1 and so the channel can be equalized by a matrix multiplication only e.g. by space-only processing alone (see [5]). Furthermore we see, that if the number of receive antennas is equal or larger than twice the number of transmit antennas (m≥2n) the minimal filter length becomes smaller than the channel length. The knowledge of the minimal filter length is important, since the complexity of the calculation of the filter coefficients increases rapidly with the filter length. So a filter with the length Lmin will be the filter with the least complexity among all filters which can cancel all CCI and ISI. -1 Furthermore we see, that the first filter in Fig. 2, ΦM has no time lag, i.e. this filter uses the actual and the past received symbols y(k), y(k−1), ... to calculate an estimate of the -1 transmitted signals x(k). The second filter Φ L on the other hand has a time lag of ε2, that means, that it uses the future receive symbols y(k+1), ..., y(k+ε2) to calculate the actual transmitted signals x(k). IV. IMPROVED FILTER In fact, there are two different ZF filters. This might be used to improve the performance of the equalizer in presence of noise at the receiver. We can calculate two different estimates of the transmitted symbols x(k) by applying the -1 and Φ-1L on the received signal two pseudo-inverse filters ΦM y(k). Φ-1i y)(k) = x(k) + (Φ Φ-1i ν)(k) ; i = M or L (21) x$ i(k) = (Φ Both estimates are unbiased and since x$ M(k) is effected by the noise at the time instances k, k−1, ... but x$ L(k) by the noise at k+1, k+2, ... both estimates are statistically independent. To improve our estimation we combine x$ M(k) and
aver. uncoded BER
m
~ (k− w −ε2)
~ (k) w
y(k)
-2
10
-3
10
equalizer length L: 4 Taps 6 Taps 8 Taps 10 Taps 14 Taps 18 Taps
-4
10
-5
10
-6
10
0
5
10 15 20 25 aver. SNR at one Rx antenna [dB]
30
Fig. 5: Influence of the filter length on the performance.
x$ L(k) linearly to form an overall estimate x$ (k) with minimum covariance. x$ (k) = FM x$ M(k) + FL x$ L(k)
(22)
FM = CL (CM + CL)-1 ; FL = CM (CM + CL)-1
(23)
with where CM and CL are the covariance matrices of x$ M(k) and x$ L(k) respectively: ε
~ 2 ~ ~ H ~H CM = Q ∑ (Mi P) Ν (Mi P) Q
i=1 ε2 ~ ~ ~ H ~H CL = Q ∑ (Li P) Ν (Li P) Q i=1
(24)
and where Ν is the covariance matrix of the noise ν(k). Let C be the covariance matrix of the so formed estimate x$ (k). The matrices FM and FL are chosen such that any other − − choice will result in a covariance matrix C such that C−C is negative semidefinite. The structure of this improved (optimal) linear filter is shown in Fig. 4. Again, the filter has the general structure of Fig.1, but now the filter length is Lopt=2*Lmin, and the filter has a time lag of τopt=Lmin. As it become obvious from the above calculation, this filter is optimal in the sense, that it make use of all information it can get from previous and future received signals to eliminate all ISI and CCI. V. SIMULATION RESULTS To illustrated the above results we performed some BER simulations. During this simulations the entries in the channel matrices Ki were complex-valued, independently normally distributed random numbers with an exponential power delay profile, i.e. the variance of the random numbers in the tap matrices Ki was set according to σ2i ∝ exp(−i/τT). The normalised descending time τT= τ/T was chosen to one (were 1/T denotes the sampling rate). We assumed always perfect channel information at the Rx. During every simulation 104 QPSK symbols were transmitted over each transmit antenna and it was averaged over 104 random channels. The
0
10
MIMO: 7 Rx, 5 Tx Channel: 5 Taps (N=4) Equalizer: L=10 Taps
-1
aver. uncoded BER
10
-2
10
-3
10
time lag τ 0 symbol 1 symbol 2 symbols 3 symbols 5 symbols
-4
10
-5
10
-6
10
0
5
10
sity. Furthermore it was shown, that a time lag will improve the performance of the filters considerably and that the optimal time lag is equal to half the filter length. Bit error rat simulations indicated, that even filters with the minimal number of filter taps will perform nearly as good as filters with the optimal length, provided that the optimal time lag is used. We remark, that if the system has more inputs than outputs, the necessary length for pre-equalizing filters can be determined in a similar way as it was shown here. REFERENCES
15
20
25
30
aver. SNR at one Rx antenna [dB]
Fig. 6: Influence of the time lag on the performance.
equalizer were determined by choosing a filter length L and a time lag τ and then using the Wiener-Hopf equation to calculate the filter coefficients (Of course it would be possible to calculate the coefficients according to the calculation ~ ~ above using the transformation matrices P and Q. But since the Wiener-Hopf solution is optimal in the MMSE sense, this way would give no better performance.). We consider a MIMO system with seven antennas at the Rx (m=7), five antennas at the Tx (n=5) and a channel with five taps (N=4). From (11) resp. (19) we can determine the minimal necessary filter length to Lmin=10 taps. Fig. 5 shows the BER curves for different filter lengths L but always optimal time lag (τ=L/2). If the length of the filter is smaller than Lmin the equalizer is unable to fully cancel the ISI and CCI, as is evident from the error floors in the graphs. If the filter is made longer than Lmin the performance is improved slightly. If the length is increased above Lopt=2*Lmin no further improvement is observed. The influence of the time lag τ is investigated in Fig. 6. We used a filter with Lmin=10 filter taps. The graphs show, that the filter performs best with the optimal time lag τopt=L/2. Using a filter without any time lag will result in a considerably loss of more than 10 dB. If the time lag is increased above τopt the BER performance decreases again. So a filter with a time lag of 9 symbols performs equal to a filter with time lag of 1 symbol. VI. SUMMERY A multiple input multiple output communication channel can be described by a system of coupled difference equations. The general structure of the linear zero forcing equalizing filters was obtained by solving these equations. In this way we were able to determine the minimal necessary length and the optimal length of the equalizing filters. This lengths are determined by the channel length and the number of antennas at the receiver and at the transmitter. It was shown that the length of these filters (and so the computational complexity) is reduced by an increasing space diver-
[1] G.J. Foschini, M.J. Gans, “On limits of wireless communication in a fading enviroment when using multiple antenna”, Wireless Personal Communications, vol 6, no. 3, pp. 311-335, March 1998 [2] G.G. Raleigh, J.M. Cioffi, “Spatio-temporal coding for wireless communication”, IEEE Trans. Comm., vol. 46, no. 2, pp. 357-366, 1998 [3] H. Bölcskei, D. Gesbert, A.J. Paulraj, “On the capacity of wireless systems employing OFDM-based spatial multiplexing”, IEEE Trans. Comm., vol. 50, no. 2, pp. 225-234, Feb. 2002. [4] G. Wunder, H. Boche, “Performance Bounds and Optimal Pilot Signals in OFDM-MIMO Systems,” in Proc. of 10th Achen Symposium on Signal Theory, pp. 319-324, Sept. 20-21, 2001 [5] A.J. Paulraj, C.B. Papadias, “Space-Time Processing for Wireless Communications”, IEEE Signal Processing Magazine, pp. 49-83, Nov 1997 [6] E. Jorswieck, T. Haustein, V. Pohl, C. von Helmolt, “On the Performance of Signal Detection Algorithms in Flat-Fading and Frequency-Selective MIMO Channels”, in Proc. of IEEE VTC Spring, Birmingham, USA, May 6-9, 2002 [7] J. Salz, “Digital Transmission over cross-coupled linear channels”, AT&T Tech. J.,vol 64, no. 6, pp 1147-1159, July/Aug. 1985 [8] A. Duel-Hallen, “Equalizers for Multiple Input/Multiple Output Channels and PAM Systems with Cycloststionary Input Sequences,” IEEE Journal on Selected Areas in Communications, vol. 10, no. 3, pp. 630-639, Apr. 1992 [9] H. Sampath, H. Bölcskei, A.J. Paulraj, “PreEqualization for MIMO Wireless Channels with Delay Spread”, in Proc. of IEEE VTC Fall, Boston, USA, Sept. 24-28, 2000 [10] Ф.Р. Гантмахер, Теория матриц, Москва, 1966 F. R. Gantmacher, Matrizentheorie, translation into German by H. Boseck, D. Soyka and K. Stengert, Berlin-Heidelberg-New York-Tokyo, Springer, 1986 [11] K. Weierstrass, “Zur Theorie der bilinearen und quadratischen Formen”, Monatsberichte der Königl. Preuss. Akademie der Wissenschaften zu Berlin, pp. 310-338, 18. Mai 1868 [12] L. Kronecker, “Algebraische Reduction der Schaaren bilinearer Formen”, Sitzungsberichte der Königl. Preuss. Akademie der Wissenschaften zu Berlin, pp. 12225-1237, 27. Nov. 1890
Abstract – Equalization in the time domain is a wellknown technique for combating intersymbol interference (ISI) in frequency selective channels. In this paper we derive the general structure of the linear zero forcing (ZF) equalizing filters for multiple input multiple output (MIMO) communication systems with more outputs than inputs. These filters are finite impulse response (FIR) filters in general and their structure is determined by the number of inputs and outputs. The relation between the spatial diversity of the system and the necessary filter length is worked out and it is shown, that an optimal time lag in the filter will improve the performance considerably. Keywords – MIMO, equalization, spatial multiplexing I. INTRODUCTION MIMO communication systems operating on channels with ISI arise in many different applications. By using the same frequency- and time slot for different data streams, these systems promise an enormous capacity [1,2]. One method to combat the delay spread is to use Orthogonal Frequency Division Multiplexing (OFDM) [3,4]. In single carrier systems the receiver has not only to separate the different data streams, transmitted over the multiple transmit antennas, but also to suppress the ISI. There are two major equalizing techniques. The first one are sequence estimation algorithms, which process a whole block of consecutive data [5,6]. The drawback of these equalizers is, that the computational complexity grows rapidly with the length of the data blocks. The second kind of equalizers, use filters which operate on the received data stream symbol by symbol [7-9]. Fig. 1 shows the general structure of such equalizing filters. To determine the coefficient matrices Gi normally an arbitrary length L and time-lag τ is chosen. The matrices are then determined by the wellknown Wiener-Hopf equation. But it is not clear at the outset which filter-length and time lag will be optimal. In this paper we solve the system of difference equations, which describe the transmission over the frequency selective MIMO channel. In this way we obtain the general structure of the ZF inverse filters and so we are able to specify the minimal- and the optimal filter length. Moreover, the optimal time lag can be determined. The paper is organised as follows: Section II describes the channel and signal model used in this paper. In Section III the structure of the filters is derived and in Section IV an improved filter structure is proposed. Section V investigates the performance of the filters by means of bit error rat
0-7803-7589-0/02/$17.00 ©2002 IEEE
y(k)
y(k−1) z-1
G1
y(k−L+2) z-1
L L
G2
L
+
z-1
y(k−L+1) z-1
GL-1
GL
+
+
xˆ (k−τ)
Fig. 1: General structure of a linear equalizer of length L and with a time lag τ.
(BER) simulations, followed by a short summary in Section VI. II. CHANNEL AND SIGNAL MODEL We consider a single-user MIMO communication system with n antennas at the transmitter (Tx) and m antennas at the receiver (Rx). A discrete-time complex baseband signal model is used, where the received signals are sampled at symbol rate. The channel is assumed to be time invariant and so it can be described by the following linear difference equation with constant coefficients: y(k) =
N
Ki x(k−i) + ν(k) Σ i=0
(1)
where N is the order of the equation, k is the time index and the difference between the number of outputs and inputs d=m−n is the antenna excess of the system. Ki are complex valued, constant matrices with n columns and m rows. It is always assumed that the entries in these matrices are statistically independent random numbers. The entries in the vector x(k)=[x1(k) , ... , xn(k)]T and y(k)=[y1(k) , ... , ym(k)]T are the n symbols and m signals simultaneously transmitted and received, respectively at time instant k. The noise ν(k)=[ν1(k), ... , νm(k)]T is assumed to be statistically independent but equally distributed for different time instances k. It has zero mean (E[ν ν]=0) and the covariance matrix is denoted by Ν=E[νν ννH]. The single entries xi(k) and yi(k) in the above vectors are functions Z→ →C, mapping the set of all integers Z into the field of complex numbers C, and so the whole vector x(k) is a function Z→ →C n where C n denotes a n-dimensional vector field over the complex numbers C. If we denote the set of all functions Z→ →C n by FC n then the transmission over the channel is described by a linear mapping Φ : FC n→FC m y(k) = (Φ Φ x)(k) + ν(k)
(2)
PIMRC 2002
where x∈FC n, y∈FC m, ν∈FC m and Φ has the form (Φ Φ x)(k) = Ko x(k) + ... + KN x(k−N).
(3)
In what follows we will denote such mappings by upper case Greek letters, matrices by bold upper case Latin letters and vectors and vector valued functions by bold lower case letters. AT will denote the transposed of the matrix A and AH the transposed conjugate complex of A. The n×n identity matrix is denoted by In. III. FILTER STRUCTURE The structure of the ZF filters will follow from the solution of the system of m difference equations for n unknown functions (1). To solve (1) we neglect the noise term ν(k) and write this system of Nth order equations as system of first order equations as follows: w(k) = (Ψ u)(k) = A u(k) − B u(k−1) ∼
(4)
∼
with a linear mapping Ψ : FC n→FC m, where A and B are ~ = (N−1) n + m rows and ~ matrices with m n = N n columns:
0M A= 0 K
0
∼
In M 0 K1
L 0 O M L In L KN-1
0 0 IM L O M M ;B=0 L I 0 0 L 0 -K n
n
~ ~~ ~ (k) = P w y(k) and x(k) = Q u (k) (7) ~ ~ respectively. Where the m×m matrix P consists of the last m ~ ~ matrix Q columns of the matrix P, and the n×n consist of the first n rows of the matrix Q. ~ ~ The concrete form of the matrices A and B is determined by the elementary divisors and minimal indices of the matrix pencil Aλ−Bµ. For m=n it follows from the assumption of regular, independent random matrices Ki that the pencil Aλ−Bµ has (with probability one) only linear elementary divisors (αiλ−βiµ , i=1... ~ n) and no minimal indices. From there it can be shown, that the ZF filter of such a system is an IIR filter, in general. We will concentrate here on the case where the system of difference equations is over-determined (m>n). Therefore we have m equations for only n unknown functions x1(k), ... , xn(k). So (2) will have a solution only if y(k) lies in the range of Φ, but since the received signals y(k) are corrupted by the noise ν(k), this requirement will in general not be met. Our aim is to find a pseudo-inverse mapping Φ-1 : FC m→FC n which satisfies the following ZF condition (Φ Φ-1 Φ x)(k) = x(k)
(4a)
N
∼
and u∈FC n , w∈FC m are given by: u(k) = [ xT(k) , xT(k−1) , ... , xT(k−N+1) ] T w(k) = [ 0 , 0 , ... , yT(k) ]
of (6) back to the desired solution x(k) of (1) can thus be written as
T
(4b)
Note, that the difference in the number of equations and unknowns is unchanged by this transformation from (1) to ~ −n ~. (4), i.e. we have d = m−n = m To solve (4) we consider the matrix pencil Aλ−Bµ (A comprehensive introduction into matrix pencils including an extensive reference list can be found in [10]). A well known result from Weierstrass and Kronecker [11,12] states, that the pencil is completely determined by its elementary divisors and its minimal indices, and so these numbers determine the structure of the ZF filter completely. From the same result follows also (see [10]), that (4) can always be transformed into a canonical form by changing the co-ordinates according to ~(k) ; w(k) = P-1 w ~ (k) u(k) = Q u (5) ~ ~ ~ ~ where P and Q are regular m×m and n×n matrices respectively. So (4) is written in canonical form as: ~ ~ ~~ ~~ ~ (k) = (Ψ w u)(k) = A u (k) − B u (k-1) (6) ~ ~ with A=PAQ and B=PBQ. One method how the transformation matrices P and Q can be determined from the given matrices A and B is described in [10, chapter 12.3]. The transformation from the given y(k) in (1) to the inhomogen~ (k) in (6) and the transformation from the solution u ~(k) ity w
for all x ∈ FC n .
(8)
It can be shown that, for m>n and under the assumption of regular, independent random matrices Ki, the matrix pencil Aλ−Bµ has (with probability one) no elementary divisor, l1 minimal indices ε1 and l2 minimal indices ε2 of rows, such ~ ×n ~ matrices in the canonical form (6) have the that the two m following quasi-diagonal form:
~ A=
Aε L 0 1 M
l1
0
M
O 0 L Aε 1
Aε L 2
0
M 0
l2
O
0 M
L Aε2
; B~=
Bε L 0 1
l
1 M O 0 L Bε 1
M
0 Bε L 0 2
0
l
2 M O 0 L Bε 2
M
(9)
where the diagonal elements are l1 matrices Aε1 and l2 matrices Aε2 (resp. l1 matrices Bε1 and l2 matrices Bε2). The matrices Aε and Bε (ε = ε1 or ε2) with ε+1 rows and ε columns are given by:
1 0 0 1 = M 0 O 0 L 0 0
Aε
L 0 M 0 0 0 1 0 0 1
;B
0 1 0 0 O = M 0 0 L 1 0
ε
L 0 M 0 1 0 0 1 0 0
(10)
The numbers ε1, ε2, l1, l2 are determined only by the dimen~ ~ sions of the matrices A and B: • ε1 ist the smallest integer for which (ε1 + 1) d > N n • l1 = (ε1 + 1) d − N n (11) • ε2 = ε1 + 1 • l2 = d − l1
m
~ P
z-1
~ (k− w −1)
z-1
L
~ m
L
M2
M1
z-1
~ (k− w −ε2+1)
ΓM
~ (k− w −ε2)
Mε2-1
Mε2
+
+
L
+
z-1
~ n
n
~ Q
m
~ (k− w −1)
~ (k) w
~ P
z-1
z-1
z-1
L
~ m -Lε2
~ (k− w −ε2+1)
~ (k− w −ε2+2)
L
-Lε2-1
L
+
-L2
-L1
+
+
MIMO : 8 Tx- / 8+d Rx antennas Channel of order N=4 (5 taps)
0
5
10
~ n
n
~ Q
15
20
xˆL(k− − ε2 )
and not by the concrete realisation of the channel matrices Ki. So only the transformation matrices P and Q will change if the channel varies. Because of the quasi-diagonal structure of the canonical form the whole equation (6) separates into l1 + l2 = d elementary equation of rank ε1 (respectively ε2) of the form: ~ ~ ~ )(k) ; i=1 ... d (12) ~ (k) = A u Θεu w i ε i(k) − Bε ui(k−1) = (Θ i where Θε: FC ε→FC ε+1 is called elementary mapping of rank ε. It remains to solve these elementary equations on the diagonal. This can be done by one of the following two elementary pseudo-inverse filters of rank ε : ΓMε and ΓLε : FC ε+1→FC ε :
It is easily verified that both filters will satisfy the following ZF condition: ~ (k) = (Γ ~ )(k) u ΓMε Θε u i i ~ ∈FC ε ; for all u i ~ ~ ΓL Θ u )(k) u (k) = (Γ i
ε
ε
So the solution of (12) can be obtained by ~ )(k) ; I = M or L ~ ΓεI w ui(k) = (Γ i
Each of the elementary equations on the diagonal of the canonical form (6) can be solved by one of the elementary filters ΓMε or ΓLε , to form the overall pseudo-inverse solution. We consider two possible solutions of (6): the first one ΓM is formed by solving all the separate elementary equations by the corresponding filters ΓMε and the second ΓL is formed by all filters ΓLε . So after the transformation back to the original -1 -1 co-ordinates using (7) we get two ZF filters ΦM and Φ L which can be used to calculate an estimate of the transmitted data x(k):
i=1
j=1
(13)
ε
~ ~ )(k) = − ∑ Lε w ~ ui(k) = (Γ ΓLε w i j i(k+j)
ε
~ ~ ~ 2 ~ -1 x$ L(k) = (Q ΓL P y)(k) = − Q·∑Li P·y(k+i) = (Φ Φ L y)(k)
ε
where L i and M i are matrices with ε rows and ε+1 columns defined by
00 10 01 0 L 0M 00 00 01 1 L 0M 00 L0 ε ε ε O O M = ... Mε = M1 = M M 0 0 1 0 2 M 0 0 0 1 0 L 0 0 0 0 L 0 L 0 0 1
Lε = 2
0 0 0 L0 1 0 00 M O M 10 0 0 0L 1 0 0
0 0
Mi = Li = M ε
ε
L 0 O M L 0
... Lε = ε
0 01 00 O 0 M 000
0 0 0 L0 M M 0 O 00 00 1 0 0 L0
for i>ε.
(13a)
(13b) (13c)
The elementary filters of rank 0 are defined as 0 ≡ (Γ ΓM0 w)(k) = (Γ ΓL0 w)(k)
(17)
i=1
j=1
ε
(16)
ε
~ ~ )(k) = ∑ Mε w ~ ui(k) = (Γ ΓMε w i j i(k−j+1)
L1 =
(15)
i
~ ~ ~ 2 ~ -1 x$ M(k) = (Q ΓM P y)(k) = Q·∑ Mi P·y(k−i+1) = (Φ ΦM y)(k)
ε
1 0 0 L0 01 00 M O M 0100 0L 0 1 0
30
Fig. 3: Minimal filter length Lmin of the MIMO equalizer as function of the antenna excess for a system with 8 Tx antennas and a channel with 5 filter taps (N=4)
Fig. 2: The structure of the two ZF equalizing filter Φ M-1 and ΦL-1 according to (17). Note the time lag of ε2 in Φ L-1.
ε
25
antenna excess d
ΓL
~L(k− u −ε2)
m=(N+1)n
m=2n
~ (k− w − ε2 )
z-1
z-1
35 30 25 20 15 10 5
xˆM(k)
~ M(k) u
y(k)
min. filter length
~ (k) w
y(k)
for all w∈FC.
(14)
with the matrices
M= i
Mεi1 L 0 l
1 M O 0 L Mεi1
M
0
0
Mεi2 L 0 M
l2
M O ε 0 L M i2
=
Li
Lεi1 L 0 l
1 M O ε 0 L L i1
M
0
0
Lεi2 L 0 l
2 M O ε 0 L L i2
M
(18)
whereas the matrices Lεi and Mεi are defined in (13). These filters are sketched in Fig. 2. We see that both filters have the same structure as shown in Fig. 1, where the ~ ~ ~ ~ matrices Gi are equal to PMiQ and PLL+1−iQ, respectively. But now we can explicitly specify the necessary filter length: Lmin = ε2 ≤
Nn + 1. d
(19)
~ P
Ξ ~ m
L
10
ΞM
0
MIMO: 7 Rx, 5 Tx Channel: 5 Taps (N=4) min. equalizer length: Lmin=10 Taps optimal time lag: τ=L/2
-1
10
~ n ~M(k− u − ε 2) ~L(k− − ε 2) u
~ Q
FM
n +
~ Q
FL
xˆ(k− − ε 2)
n
Fig. 4: Structure of the equalizer which uses both elementary ZF-filter to improve the performance in the presence of noise.
This is the minimum length of the filter which is necessary to remove all ISI and CCI. We see, that the necessary filter length is reduced considerably by increasing the antenna excess d. At this point the close relation between the spatialand time dimension of the channel becomes obvious, as illustrated in Fig. 3: If the system has no antenna excess (d=0) the ZF filter is an IIR filter with a infinite number of filter taps. With every additional receive antenna the length of the filter decreases until the number of receive antennas is larger as or equal to the channel length times the number of transmit antennas: m ≥ (N + 1) n
resp.
d≥Nn
(20)
In this case Lmin becomes 1 and so the channel can be equalized by a matrix multiplication only e.g. by space-only processing alone (see [5]). Furthermore we see, that if the number of receive antennas is equal or larger than twice the number of transmit antennas (m≥2n) the minimal filter length becomes smaller than the channel length. The knowledge of the minimal filter length is important, since the complexity of the calculation of the filter coefficients increases rapidly with the filter length. So a filter with the length Lmin will be the filter with the least complexity among all filters which can cancel all CCI and ISI. -1 Furthermore we see, that the first filter in Fig. 2, ΦM has no time lag, i.e. this filter uses the actual and the past received symbols y(k), y(k−1), ... to calculate an estimate of the -1 transmitted signals x(k). The second filter Φ L on the other hand has a time lag of ε2, that means, that it uses the future receive symbols y(k+1), ..., y(k+ε2) to calculate the actual transmitted signals x(k). IV. IMPROVED FILTER In fact, there are two different ZF filters. This might be used to improve the performance of the equalizer in presence of noise at the receiver. We can calculate two different estimates of the transmitted symbols x(k) by applying the -1 and Φ-1L on the received signal two pseudo-inverse filters ΦM y(k). Φ-1i y)(k) = x(k) + (Φ Φ-1i ν)(k) ; i = M or L (21) x$ i(k) = (Φ Both estimates are unbiased and since x$ M(k) is effected by the noise at the time instances k, k−1, ... but x$ L(k) by the noise at k+1, k+2, ... both estimates are statistically independent. To improve our estimation we combine x$ M(k) and
aver. uncoded BER
m
~ (k− w −ε2)
~ (k) w
y(k)
-2
10
-3
10
equalizer length L: 4 Taps 6 Taps 8 Taps 10 Taps 14 Taps 18 Taps
-4
10
-5
10
-6
10
0
5
10 15 20 25 aver. SNR at one Rx antenna [dB]
30
Fig. 5: Influence of the filter length on the performance.
x$ L(k) linearly to form an overall estimate x$ (k) with minimum covariance. x$ (k) = FM x$ M(k) + FL x$ L(k)
(22)
FM = CL (CM + CL)-1 ; FL = CM (CM + CL)-1
(23)
with where CM and CL are the covariance matrices of x$ M(k) and x$ L(k) respectively: ε
~ 2 ~ ~ H ~H CM = Q ∑ (Mi P) Ν (Mi P) Q
i=1 ε2 ~ ~ ~ H ~H CL = Q ∑ (Li P) Ν (Li P) Q i=1
(24)
and where Ν is the covariance matrix of the noise ν(k). Let C be the covariance matrix of the so formed estimate x$ (k). The matrices FM and FL are chosen such that any other − − choice will result in a covariance matrix C such that C−C is negative semidefinite. The structure of this improved (optimal) linear filter is shown in Fig. 4. Again, the filter has the general structure of Fig.1, but now the filter length is Lopt=2*Lmin, and the filter has a time lag of τopt=Lmin. As it become obvious from the above calculation, this filter is optimal in the sense, that it make use of all information it can get from previous and future received signals to eliminate all ISI and CCI. V. SIMULATION RESULTS To illustrated the above results we performed some BER simulations. During this simulations the entries in the channel matrices Ki were complex-valued, independently normally distributed random numbers with an exponential power delay profile, i.e. the variance of the random numbers in the tap matrices Ki was set according to σ2i ∝ exp(−i/τT). The normalised descending time τT= τ/T was chosen to one (were 1/T denotes the sampling rate). We assumed always perfect channel information at the Rx. During every simulation 104 QPSK symbols were transmitted over each transmit antenna and it was averaged over 104 random channels. The
0
10
MIMO: 7 Rx, 5 Tx Channel: 5 Taps (N=4) Equalizer: L=10 Taps
-1
aver. uncoded BER
10
-2
10
-3
10
time lag τ 0 symbol 1 symbol 2 symbols 3 symbols 5 symbols
-4
10
-5
10
-6
10
0
5
10
sity. Furthermore it was shown, that a time lag will improve the performance of the filters considerably and that the optimal time lag is equal to half the filter length. Bit error rat simulations indicated, that even filters with the minimal number of filter taps will perform nearly as good as filters with the optimal length, provided that the optimal time lag is used. We remark, that if the system has more inputs than outputs, the necessary length for pre-equalizing filters can be determined in a similar way as it was shown here. REFERENCES
15
20
25
30
aver. SNR at one Rx antenna [dB]
Fig. 6: Influence of the time lag on the performance.
equalizer were determined by choosing a filter length L and a time lag τ and then using the Wiener-Hopf equation to calculate the filter coefficients (Of course it would be possible to calculate the coefficients according to the calculation ~ ~ above using the transformation matrices P and Q. But since the Wiener-Hopf solution is optimal in the MMSE sense, this way would give no better performance.). We consider a MIMO system with seven antennas at the Rx (m=7), five antennas at the Tx (n=5) and a channel with five taps (N=4). From (11) resp. (19) we can determine the minimal necessary filter length to Lmin=10 taps. Fig. 5 shows the BER curves for different filter lengths L but always optimal time lag (τ=L/2). If the length of the filter is smaller than Lmin the equalizer is unable to fully cancel the ISI and CCI, as is evident from the error floors in the graphs. If the filter is made longer than Lmin the performance is improved slightly. If the length is increased above Lopt=2*Lmin no further improvement is observed. The influence of the time lag τ is investigated in Fig. 6. We used a filter with Lmin=10 filter taps. The graphs show, that the filter performs best with the optimal time lag τopt=L/2. Using a filter without any time lag will result in a considerably loss of more than 10 dB. If the time lag is increased above τopt the BER performance decreases again. So a filter with a time lag of 9 symbols performs equal to a filter with time lag of 1 symbol. VI. SUMMERY A multiple input multiple output communication channel can be described by a system of coupled difference equations. The general structure of the linear zero forcing equalizing filters was obtained by solving these equations. In this way we were able to determine the minimal necessary length and the optimal length of the equalizing filters. This lengths are determined by the channel length and the number of antennas at the receiver and at the transmitter. It was shown that the length of these filters (and so the computational complexity) is reduced by an increasing space diver-
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