CONSTRAINED SPACE-TIME ZERO-FORCING PRE-EQUALIZER FOR THE DOWNLINK CHANNEL OF UMTS-TDD António Morgado1, Atílio Gameiro1 and José Fernandes1 1
Universidade de Aveiro, Instituto de Telecomunicações, Aveiro – Portugal Phone: +351 234 377900, Fax: +351 234 377901, e-mail:
[email protected]
Abstract – The great diversity of services expected to be delivered by third generation mobile radio systems will impose severe operating conditions on the mobile terminal in terms of computational requirements and power consumption. Therefore, in this work we propose to move the most demanding signal processing tasks, usually performed by the mobile unit, to the base station. This technique is developed here for a UMTS-TDD downlink scenario through an equalizer synthesis method based on the redundancy between non-overlapping bands of a Direct Sequence Spread Spectrum (DS-SS) signal, with the design optimised for minimum power transmission under the zeroforcing criterion. Keywords - Equalization, zero forcing, antenna arrays, CDMA. I. INTRODUCTION Driven by the success of second generation success, more and more people are becoming familiar with wireless access availability anywhere, any time [1]. In fact, the mobile and wireless markets have grown exponentially over the last decade and it is likely to continue to grow in the next years, which requires high capacity and spectrum efficient networks. These requirements were the main drivers for the emergence of the so-called third generation mobile radio systems. These systems, called UMTS (Universal Mobile Telecommunications System) in Europe, are also intended to provide higher bit rates, more flexibility, simultaneous multiple services for a user, and services with different quality of service classes [1]. The expected growth demand for system capacity, especially in urban areas, will require the use of advanced techniques to increase the spectrum efficiency achievable with today’s technology. One possible solution is the use of antenna arrays [2]-[4]. This has already been considered by 3GPP (3rd Generation Partnership Project) as a possible performance enhancing feature for Universal Terrestrial Radio Access (UTRA) in both Frequency Division Duplex (FDD) and Time Division Duplex (TDD) modes. In TDD mode the same carrier frequency is used for both uplink and downlink. Thus, assuming that the time between an up and downlink transmission is shorter than the channel coherence time, the TDD transmitter knows the fast fading multipath channel based on the reciprocal channel. This
0-7803-7589-0/02/$17.00 ©2002 IEEE
feature can be explored to provide improved transmission, especially in downlink transmission where the channel is the fundamental source of intra-cell interference [5]. In fact, in a synchronous spread-spectrum downlink transmission using orthogonal codes, as is the case of UMTS TDD systems, the multipath propagation destroys the code orthogonality, causing multiple access interference (MAI) to appear at the receiver. It also introduces inter-symbol interference (ISI) on the signals due to their own delayed replicas. Therefore, if the transmitter knows the multipath channel it could apply, a priori, some kind of equalization to format the transmitted signal so that the multipath channel act as a matched filter to the transmitted signal. A simple receiver would then be sufficient to despread the signal. Such a scheme is particularly suitable for the downlink, because it moves the complexity needed to mitigate the interference from the user equipment (UE) to the base station (BS), where the power and complexity constraints are much less pronounced. This idea is developed in this paper where we consider a downlink pre-equalization scheme, using an array of antennas at the base station as shown in Figure 1, and propose a synthesis method for the filters based on the redundancy between non-overlapping bands of a Direct Sequence Spread Spectrum (DS-SS) signal. In the scheme developed in this paper, the filters of the space-time pre-equalizer were designed using a zero-forcing (ZF) criterion so that each UE receives a signal that after despreading is free from MAI and ISI. Such a design allows the use of a simple low-cost, low power-consuming receiver in the mobile, e.g. a single correlator. The synthesis method we propose for the pre-equalizers explores the frequency redundancy of spread-spectrum signals. A baseband DS-SS signal is in fact a PAM signal where the elementary pulse has a bandwidth much higher than the baud rate, and it is easy to show that if one considers two non-overlapping frequency bands separated by a multiple of the baud rate we get two signals whose baseband equivalents are simply related by a linear filtering operation. This implies that if the bandwidth of the DS-SS signal is L times the baud rate we have in fact L order frequency diversity. The frequency redundancy in conjunction with the KM pre-equalizing filters (where K is the number of users and M the number of antenna elements) implies that as UMTS TDD is concerned, we have enough degrees of freedom to synthesize the filters so that zero MAI
PIMRC 2002
H1,1(f) User 1
QPSK modulator
1
M
M
AWGN
C1,d(f)
H1,M(f)
A1(f)
Matched Filter
QPSK demod.
User d
HK,1(f) User K
QPSK modulator
CM,d(f)
M
M
HK,M(f)
AK(f)
Ts
Ad*(f)
M
Channel Estimates
Figure 1 - Downlink transmission using pre-equalization at the base station and ISI is guaranteed even with maximum number of resource units (RU) in a time-slot. As M increases we get additional degrees of freedom that will allow to further optimize the space-time pre-equalizer performance or make the implementation of the several filters easier. In this communication we present some refinements to the unconstrained ZF criterion presented in [6] intending to improve the power efficiency in downlink operation. Hence, this communication proposes using the available degrees of freedom to impose additional constrains on the ZF criterion in order to reduce the transmitted power while still guarantying ISI and MAI values close to zero in the UE.
M K Yd ( f ) = Ak ( f ) ⋅ H k , m ( f ) ⋅ C m , d ( f ) ⋅ Ad* ( f ) (2) m =1 k =1 d=1,…,K
∑
where Ak(f) are the Fourier transforms of the signature waveforms, while Hk,m(f) and Cm,d(f) represent the preequalizer filters and channel frequency responses, respectively. In order to have the desired signal Ydd(f) and the interference signal Ykd(f) components explicit specified, we rearrange (2) obtaining M 2 Yd ( f ) = Ad ( f ) ⋅ ∑ H d , m ( f ) ⋅ Cm, d ( f ) + m =1
K M + ∑ Ak ( f )∑ H k , m ( f )C m , d ( f ) Ad* ( f ) k =1 m =1 k≠d
II. PRE-EQUALIZATION FILTER DESIGN Figure 1 intends to represent all the relevant signals involved in a downlink transmission of K pre-equalized resource units (RU) by a base station (BS) comprising M antennas spaced several wavelengths apart. Basically, after QPSK modulation the several RUs are spread using the signature waveforms defined in (1), where p(t) designates the root-raised cosine filter impulse response and F{.} the Fourier transform operation.
Ak ( f ) = F {a k (t )* p(t )} = F {a k (t )}× F {p(t )}
= P( f ) ⋅
Q −1
∑a
k ,n
⋅e
− j 2πfnTC
(1)
;
k=1,…,K
n =0
∑
= Ydd ( f ) +
(3)
K
∑Y ( f ) kd
, d=1,…,K
k =1 k ≠d
To obtain a signal at the dth UE without inter-symbol interference (ISI) and MAI, after sampling with a rate 1/T, the Fourier transforms given by (3) must verify
n f − =1 T n Ykd f − = 0 T
∑Y n
∑ n
dd
,
k=1,…,K ; k ≠ d
(4)
After spreading, the signals enter the pre-equalizer unit where the pre-equalization filters Hk,m(f), obtained using the uplink channel estimates, will format the signals in such a way that, after being transmitted by the several antennas, combined at the user equipment (UE) antenna, despread by a filter matched to the dth spreading code Ad(f) and sampled, will result in a signal without ISI and MAI.
Now, consider that each direct-sequence-spread-spectrum (DS-SS) signal component Yij(f) before sampling is subdivided in N non-overlapping frequency bands, so that we can write
Observing Figure 1 it is easy to see that the global transfer function between the base station spreaders and the dth UE code matched filter up to the point immediately before the sampler, is given by
where Yij(n)(f) are bandwidth limited functions with support
Yij ( f ) =
N
∑Y
(n) ij
n=1
n N +1 f − + T 2T
(5)
1 1 in − , ∀ i, k , n . Because we are dealing with
2T 2T
signals sampled with rate 1/T, the left-hand side terms in (4)
are periodic with period equal to 1/T. Consequently it is sufficient to test the conditions of (4) in the interval 1 1 , − . Therefore, making use of (5), the processing 2T 2T for the dth UE should be such as to verify
N (n) Ydd ( f ) = 1 n=1 N Ykd( n ) ( f ) = 0 n=1
∑
;
∑
k=1,…,K ; k ≠ d
(6)
Consequently, (9) can be written as S = NS
2
(n) d
n=1
( n) m ,d
m=1
∑ A ( f )A ( f )⋅ ∑ H ( f )⋅ C ( f ) = 0 M
(n) k
*( n ) d
(n) k ,m
(n) m ,d
M
∑H
(7)
m=1
d=1,…,K ; k=1,…,K ; k ≠ d
m =1
A(1)H(1)C(1)A (1)H+...+A(N)H(N)C(N)A(N)H = IK
S=
N
∑S
(n)
=N S
(8)
Therefore, considering all the K transmitted RUs, we will have NMK functions or degrees of freedom to fulfill K2 conditions. III. PRE-EQUALIZER OPTIMIZATION In [6] the pre-equalizer was designed to meet (8) without optimization. As we usually have more degrees of freedom than constraints, frequency bands were grouped to allow a desired solution. In this paper we derive an optimized design that minimizes the overall transmitted power or, more correctly, the power spectral density (PSD), given by S = E {dAHH H A H d H }
(9)
where d is the 1×K data vector containing the Fourier transforms of the transmitted RUs, A is the K×K signature waveform diagonal matrix and H represents the K×M pre-equalization matrix. In the expressions bellow, C is the M×K channel matrix defining the channels between the M antennas and the K mobile UE. It can be shown that because the Ns transmitted RU data symbols (bi) are uncorrelated, then
}= 0
; ∀i ≠ k
(10)
and N −1 N −1 E{Di D } = E bi , k e − j 2πfkT bi*,m e j 2πfmT = N S m=0 k =0
∑ S
* i
∑ tr (A N
n =1
(12)
( n)
H (n ) H (n ) H A ( n) H )
(13)
The objective of the proposed optimization is the calculation of the NMK degrees of freedom H k( n, m) which minimize the transmitted PSD (S) under de K2 constrains of (8). This can be accomplished resorting to the variational calculus’ Lagrange Multipliers technique [7].
then the (i,j) condition of (8) is N
∑x
where, with the purpose of notation simplicity, we dropped the variable f.
E{D k D
= N S ⋅ tr (AHH H A H )
Designating by xi(,nj) the (i,j) element of A(n)H(n)C(n)A(n)H
or, equivalently, in matrix form
* i
2 k ,m
Because at the mobile UE the received DS-SS signal is going to be sampled at rate 1/T, then designating by N the number of frequency bands with bandwidth 1/T, we are going to have the overlapping of N DS-SS signal spectra replicas, spaced by 1/T. Therefore, instead of minimizing the PSD in each frequency point, we should minimize the PSD of the sum of these replicas.
M
(n) d ,m
2
n =1
∑ A ( f ) ⋅ ∑ H ( f )⋅ C ( f ) = 1 N
∑
Ak
k =1
Then, using (3) the ZF design criteria for the pre-equalizer is expressed by N n=1
K
∑ S
(11)
(n) i, j
= δ i, j
(14)
n =1
And the function that we should minimize is N
N
K
K
n =1
n =1
i =1
j =1
∑ S ( n ) − ∑ ∑∑ λi , j x i(,nj) = f (H )
(15)
Next we must derive the Lagrange multipliers which minimize the function above. To achieve that, we impose that the NKM derivatives of f in order to H m( n,l) were identical to 0. ∂f (H ) = 0 ; ∀l , m, n ; m=1,…,K; l=1,…,M; n=1,…,N (16) ∂H m( n, l)
After straightforward but tedious manipulations, making B(n)=C(n)A(n)H and solving (16) in order to H m( n,l) we have K
H
( n )* m,l
=
∑λ
m, j
(n) m
A B
j =1
2⋅ A
(n) m
2
( n) l, j
⇔ H m( n,l)
=
K
∑ j =1
λm , j A B 2 2 ⋅ Am( n ) (n) m
( n) l, j
*
(17)
Using matrices (17) can be expressed as H= ½ A-1Λ*ACH
(18)
Therefore substituting (18) in (8), and solving for Λ we obtain N Λ∗ = 2 ∑ A ( n ) C ( n ) H C ( n ) A ( n ) H n =1
−1
(19)
Substituting (19) in (18) and the result in (13), we finally obtain the expression for minimum PSD (Smin) and for the filters which meet the ZF criterion constrained to the minimum transmit PSD. N 1 S min = N S ∑ tr Λ* A ( n ) C ( n ) H C ( n ) A ( n ) H ΛT n =1 4
(20)
V. CONCLUSIONS
(21)
In this communication we considered a frequency domain pre-equalizer that explores the frequency redundancy present in DS-SS signals and the space diversity provided by the use of multiple antennas at the BS.
−1
H
( n)
=A
−1( n )
N (i ) (i ) H (i ) (i ) H ∑ A C C A A (n) C (n)H i =1
IV. NUMERICAL RESULTS After deriving the optimization criteria we wanted to evaluate the impact of this optimization on the filters shape. So, we have considered a UMTS-TDD single cell downlink scenario where 4 RU are being transmitted using a two-element array antenna at the base station. As an example of the numerical results achieved we show in Figure 2 and Figure 3 the frequency response of the pre-equalizer filters considering respectively an unconstrained and a constrained design. We considered a spreading factor of 16 and the following deterministic 2–path channels c1,1 (t ) = δ (t )
c1, 2 (t ) = 1 − 0.5 2 ⋅ δ (t ) + 0.5e
π
−j
4
⋅ δ (t − 4.50Tc )
π
c1, 3 (t ) = 1 − 0.9 2 ⋅ δ (t ) + 0.9e 6 ⋅ δ (t − 2.75Tc ) j
c1, 4 (t ) = 1 − 0.2 ⋅ δ (t ) + 0.2e
−j
c2 ,1 (t ) = 1 − 0.8 2 ⋅ δ (t ) + 0.8e
−j
c2 , 2 (t ) = 1 − 0.12 ⋅ δ (t ) + 0.1e
j
2
π 2
π 12
3π 5
c2 , 3 (t ) = 1 − 0.4 2 ⋅ δ (t ) + 0.4e
−j
c2 , 4 (t ) = 1 − 0.7 2 ⋅ δ (t ) + 0.7e
j
π 5
2π 5
constrained filters have a not so sharper frequency behavior when compared with the unconstrained ones. However we think this design can still be improved through the inclusion of practical implementation constraints in the design process.
⋅ δ (t − 8.25Tc ) ⋅ δ (t − 2.75Tc ) ⋅ δ (t − 7.75Tc ) ⋅ δ (t − 5.50Tc ) ⋅ δ (t − 3.50Tc )
The communication presented the design technique for the case of a ZF criterion constrained to the minimization of the transmitted power, a useful feature in TDD. Numerical results have illustrated the technique, showing that additional benefits can be achieved, beyond the decrease of the UE computational requirements, when moving the most demanding UE signal processing tasks to the base station. ACKNOWLEDGEMENTS The first author would like to acknowledge the financial support of this work by the Portuguese Foundation for Science and Technology (FCT) through grant PRAXIS XXI/BD/18393/98, and by the European Commission through the project IST-1999-10741 (ASILUM). REFERENCES [1] Tero Ojanperä, Ramjee Prasad, Wideband CDMA for third generation mobile communications, Artech House Publishers, 1998. [2] A. Paulraj, C. Papadias, “Space-Time Processing for Wireless Communications”, IEEE Signal Processing Magazine, November 1997. [3] J. Litva, T. Kwok-Yeung Lo, Digital Beamforming in Wireless Communications, Artech House, 1996. [4] J. C. Liberti, T. S. Rappaport, Smart Antennas for Wireless Communications: IS-95 and Third Generation CDMA Applications, Prentice Hall, 1999.
where cm,k(t) represents the propagation channel between the mth BS antenna and the kth UE. Figure 2 and Figure 3 show the change in the pre-equalizing filters frequency response that occurs when going from an unconstrained design to the one involving transmitted power optimization.
[5] Markku Heikkilä, Petri Komulainen, Jorma Lilleberg, “Interference Suppression in CDMA Downlink Through Adaptive Channel Equalization”, Proceedings of VTC’99 Fall – 50th IEEE Vehicular Technology Conference, Amsterdam, September 1999.
As can be seen in Figure 2a-d, in this scenario we obtained a ZF unconstrained solution by considering groups of 1/T frequency bands, thus making the available degrees of freedom equal to the ZF constraints.
[6] A. Morgado, P. Pinho, A. Gameiro, J. Fernandes, “Preequalization Technique for Interference Cancellation in the UMTS-TDD Downlink Channel”, Proceedings of VTC’01 Fall – 54th IEEE Vehicular Technology Conference, Atlantic City, October 2001
Besides the transmitted power minimization, which can be observed by the lower magnitude in the frequency response of the filters in Figure 3, the results also show that the
[7] Monson H. Hayes, Statistical Digital Signal Processing and Modeling, John Wiley & Sons, 1996.
-3
6
-4
Pre-eq. filters - RU 1 - Antenna 1
x 10
3
5
2 |H(f)|
|H(f)|
4 3
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1
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x 10
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x 10
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0.8 0.6
|H(f)|
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3 2
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x 10
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1
2
0.5 1 0
0
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c)
d) Figure 2 - Unconstrained results a) RU1 b) RU2 c) RU3 d) RU4 -4
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x 10
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Pre-eq. filters - RU 2 - Antenna 1
x 10
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|H(f)|
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Figure 3 – Transmitted power constrained results a) RU1 b) RU2 c) RU3 d) RU4
