Time reversal space time block coding with channel estimation and synchronisation errors
Kishore Mehrotra, Ian Vince McLoughlin
Group Research Tait Electronics Limited
PO Box 1645 Christchurch, New Zealand
[email protected], [email protected] Abstract – time reversal (TR) space time block coding
(STBC) is a novel time domain block coding approach developed to mitigate inter-symbol interference (ISI) in dispersive fading channels. This paper discusses a particular implementation of TR-STBC and investigates the effect of both synchronisation errors and channel estimation accuracy on overall system bit error rate. This has particular relevance to the number format employed for the low-level system implementation.
I. INTRODUCTION
The Alamouti scheme [1] is a transmit diversity technique introduced to combat flat fading channels. This approach has been extended [2-4] to time-domain block symbol processing schemes that lend themselves to decoupled and parallel equalisation for frequency selective fading channels. These schemes, known as the time reversal (TR) space time block coding (STBC) approach, are explored in the time domain, then implemented as a simulation and investigated in terms of sensitivity to errors, with a view to real-time system implementation.
Finally, initial results from a real-time hardware implementation are presented to support some of the simulation findings.
II. SYSTEM MODEL
Let us consider the case when there are two transmit antennas and one receive antenna. During the first operational time slot, the symbol block transmitted by antenna-1 at the transmitter is given by
)}
( ),...
1 ( ), 0 (
{ 1 1 1
1 d d d N
S = (2.1)
and the simultaneously transmitted symbol block from antenna-2 is
)}
( ),...
1 ( ), 0 (
{ 2 2 2
2 d d d N
S = (2.2)
It can be seen each antenna transmits (N+1) symbols in the first burst. In the second transmission burst, antenna- 1 transmits a time reversed, complex conjugated and sign inverted version of the block transmitted by antenna-2 in the first burst. Similarly, in the second burst antenna-2 transmits a time reversed and complex conjugated version of the block transmitted by antenna-1 in the first burst.
The time reversal and complex conjugation operation may be represented mathematically as
)}
( , ),...
1 ( ), 0 ( {
)}
0 ( ),...
1 ( ), ( {
1 1
1
* 1
* 1
* 1 1
~
N D D
D
d N
d N d S
=
−
= (2.3)
)}
( , ),...
1 ( ), 0 ( {
)}
0 ( ., ),...
1 ( ), ( {
2 2
2
* 2
* 2
* 2 2
~
N D D
D
d N
d N d S
=
−
−
−
−
=
− (2.4)
where
~
S1 is the time reversed and complex conjugated version of S while 1
~
S2 is the time reversed and complex conjugated version of S . The minus sign is prefixed2 before
~
S2to remember that what is transmitted is
~
S2
− and not
~
S2. The table below summarises the transmitted blocks from the two antennae:
Burst 1 Burst 2
Antenna 1 S1 ~
S2
−
Antenna 2 S2 ~
S1
Let the impulse response from transmit antenna-1 to the receive antenna be denoted by
g
0, g1, g2 andg
3where a 4-tap channel response is assumed. Although this is chosen arbitrarily, even a 4-tap response can illustrate the principle equally well. Similarly, let the channel impulse response from antenna-2 to the receive antenna be denoted by
p
0, p1, p2 andp
3. In this case we assume that the channel is stationary over a block of symbols and during the two bursts. The received signal for the first burst can be expressed as:N to t for
t n t d p t d p
t d p t d p t d g
t d g t d g t d g t r
0
) ( ) 3 ( )
2 (
) 1 ( )
( )
3 (
) 2 ( ) 1 ( ) ( )
(
1 2
3 2
2
2 1 2 0 1
3
1 2 1
1 1 0 1
= +
− +
− +
− +
+
− +
− +
− +
=
(2.5)
where n1(t) is assumed to be white noise with zero mean and variance σ02. Similarly, the received signal for the second burst can be expressed as:
N to t for
t n t D p t D p
t D p t D p t D g
t D g t D g t D g t r
0
) ( ) 3 ( )
2 (
) 1 ( ) ( )
3 (
) 2 ( )
1 ( )
( )
(
2 1
3 1
2
1 1 1 0 2
3
2 2 2
1 2 0 2
= +
− +
− +
− +
+
− +
− +
− +
=
(2.6)
The second burst of received signal is then time reversed and complex conjugated as follows:
N to t for
t N r t r
0 ) ( ) ( 2*
3
=
−
=
(2.7)
Therefore, the time reversed and complex conjugated received signal can be expressed as follows:
N to t for
t N n t N D p
t N D p t N D p
t N D p t N D g t N D g
t N D g t N D g t N r t r
0 ) ( ) 3 (
) 2 ( ) 1 (
) ( ) 3 ( ) 2 (
) 1 ( ) ( ) ( ) (
* 2
* 1
* 3
* 1
* 2
* 1
* 1
* 1
* 0
* 2
* 3
* 2
* 2
* 2
* 1
* 2
* 0
* 2 3
=
− +
−
− +
−
− +
−
− +
− +
−
− +
−
− +
−
− +
−
=
−
=
(2.8)
Since
) ( ) (
) ( ) (
* 1 1
* 2 2
t N d t D
t N d t D
−
=
−
−
= we can derive
) (
)) (
( ) (
2
2
* 2
k t d
k t N N d k t N D
+
−
=
−
−
−
−
=
−
−
(2.9)
) (
)) (
( ) (
1
1
* 1
k t d
k t N N d k t N D
+
=
−
−
−
=
−
−
(2.10) For any arbitrary integer k.
Thus, the received time reversed and complex conjugated signal can be written as
N to t for
t N n t d p
t d p t d p
t d p t d g t d g
t d g t d g t N r t r
0
) ( ) 3 (
) 2 ( ) 1 (
) ( ) 3 ( )
2 (
) 1 ( ) ( )
( ) (
* 2 1
* 3
1
* 2 1
* 1
1
* 0 2
* 3 2
* 2
2
* 1 2
* 0
* 2 3
=
− + + +
+ +
+ +
+ +
− +
−
+
−
−
=
−
=
(2.11)
III. LINEAR COMBINING
To illustrate time domain linear combining we apply a slight simplification, but without compromising the elegance of proof or generality of application. We assume that the channel impulse response is only 2 taps long, rather than 4 taps, so that the number of terms to be handled is reduced. When the number of taps is increased,
this will only result in extra terms of a similar nature, whilst the concept will not change.
The original derivations in [2-4] utilise the q-operator to avoid carrying over too many terms. However, the idea here is to formulate all the equations in discrete time in order to aid implementation. However, if we wished to compare the received sample expressions in the q-domain we can simply re-write the expressions for r1(t) and r3(t) as the following
− +
= − −
) (
) ( ) (
) ( ) ( ) (
) ( ) ( ) (
) (
3 1 2
1
*
*
1 1
3 1
t n
t n t d
t d q g q p
q p q g t r
t r
(3.1) where
3 3 2 2 1 1 0 1)
(q− =g +gq− +g q− +g q− g
3 3 2 2 1 1 0 1)
(q− = p +pq− + p q− + pq− p
3
* 3 2
* 2
* 1
* 0
*(q) g g q g q g q
g = + + +
3
* 3 2
* 2
* 1
* 0
*(q) p p q p q p q
p = + + +
(3.2)
Now applying the simplification of assuming that there are only 2 taps in the channel impulse response, i.e.
1 1 0 1)
(q− =g +g q− g
1 1 0 1)
(q− = p +pq− p
q g g q
g*( )= 0*+ 1* q p p q
p*( )= *0+ 1*
(3.3)
To perform linear combination, the received data samples are filtered in the following manner:
To generate antenna-1 measurements z1(t), pass r1(t) through a non-causal filter with coefficients g0* and g1*, pass r3(t) through a causal filter with coefficients p0 and
p1, and sum the results.
To generate antenna-2 measurements z2(t), pass r1(t) through a non-causal filter with coefficients p*0 and p1*, pass r3(t) through a causal filter with coefficients −g0 and −g1, and sum up the results.
) 1 ( ) ( ) 1 ( ) ( )
( *0 1 1*1 0 3 13
1 t =g r t +g r t+ +p r t +p r t−
z (3.4)
) 1 ( ) ( ) 1 ( ) ( )
( *01 1*1 0 3 13
2 t = p r t +p r t+ −g r t −g r t−
z (3.5)
Substituting the expressions for r1(t) and r3(t) and performing algebraic simplifications results in the following expressions:
) 1 ( ) 1 ( ) ( )
( 0 1 1 1 1* 1
1 t = d t + d t− + d t+
z γ γ γ (3.6)
) 1 ( ) 1 ( ) ( )
( 0 2 1 2 1* 2
2 t = d t + d t− + d t+
z γ γ γ (3.7)
As shown in [2-4] for a general nr-tap channel the complex conjugate coefficients are
) ( ) ( ) ( ) (
...
..
...
) , (
1 2
* 2 1 1
* 1
) 1 ( 1 0
* 1 1
* 1
−
−
−
−
− −
− −
−
+
≡
+ +
+ +
+ +
=
q h q h q h q h
q q
q q
q q
n n n n
n n n n
γ γ γ γ
γ γ γ γ
γ γ
γ
γ γ
γ
(3.8)
It is clear that for a 4 tap channel response we have
=3 nγ
2 3 2 2 2 1 2 0
2 3 2 2 2 1 2 0 0
g g g g
p p p p
+ + + +
+ + + γ =
(3.9)
3
* 2 2
* 1
1
* 0 3
* 2 2
* 1 1
* 0 1
g g g g
g g p p p p p p
+ +
+ + + γ =
(3.10)
3
* 1 2
* 0 3
* 1 2
* 0
2 = p p + p p +g g +g g
γ (3.11)
3
* 0 3
* 0
3 = p p +g g
γ (3.12)
Fig. 2 Fig. 1
Fig. 2
IV. SIMULATION RESULTS
The dual transmit antenna and single receive antenna scenario was configured into a number of system simulations. In the simulations, each antenna transmits a block of 55 bits in the forward and reverse data slots. The simulations were performed for two different cases:
1: Perfect channel knowledge and perfect synchronisation.
2: With channel estimation errors and synchronisation error of 2 symbols.
Fig-1 and Fig-2 illustrate the BER vs SNR curves for these two cases. These plots represent the averaged results from 10 runs of a 55-bit packet transmission block. At low bit error rates, the small sample size in terms of bit errors reduces the accuracy of the measurement and therefore adversely affects the smothness of the plotted results.
Though the results for the case of perfect channel knowledge and imperfect synchronisation are not presented here it was found that the channel estimation errors dominated over the synchronisation errors. A Viterbi equaliser was used for decoding the bit stream transmitted from each antenna. It can be noticed from the curves (fig-2) that at 1% BER upto 7dB penalty is incurred due to channel estimation and synchronisation errors. A preliminary analysis of the results indicates that channel estimation errors to the degree investigated are more critical to overall performance than synchronisation errors. In fact synchronisation errors of up to three symbols produce almost indistinguishable performance results from the perfect synchronisation case.
V. EVALUATION
Throughout the understanding of the simulations, the TR- STBC system has been implemented as a hardware system operating from a remote dual antenna transmitter system to a single antenna receiving system at 2.45GHz an with a 2Mbit/s bitrate. The system implementation was primarily in VHDL (VHSIC hardware description language, where VHSIC is very high speed integrated circuits) running on high speed FPGA (field programmable gate array) hardware [5].
The synchroniser used in the implementation operates on data received from a single transmit antenna using a matched-filter approach to a given synchronisation sequence. Although the synchroniser performs relatively poorly, the equaliser is capable of coping with this.
Figures 3 and 4 show amplitude versus time plots for the estimated responses for each of the transmit channels under the same channel conditions, sampled approximately 2 seconds apart. Although we can assume that the channel has remained fairly static, and more so, that since the distance between transmitters and receiver has not changed, the peak lobes of the channel responses should be nearly identical. In fact it can be seen that the peak lobe in figure 4 has drifted by one or two symbols indicating a mis-synchronisation (or lost synchronisation).
Despite the obvious mis-syncronisation, the channel estimator and Viterbi equaliser in the TR-STBC system has maintained its performance and has not suffered any burst error in this instance. Empirical evidence collected to date indicates that mis-synchronisation of more than about 5 symbols will, typically introduce a burst error
until synchronisation can be re-established.
IV. CONCLUSION
Previous research introduced the TR (time reversal) concept to the space time processing research field. The method has proven particularly suitable for the mitigation of inter-symbol interference. This paper firstly re-formulates the published TR equations into a time-domain representation suitable for implementation.
The implementation equations have then been simulated to check the BER performance of the system, both in a perfect environment and in the presence of channel estimation and synchronisation errors. It was found by simulation that the time reversal space time block coding scheme with a Viterbi equaliser is relatively immune to synchronisation errors but requires accurate channel estimates. The system was implemented in real-time hardware and evaluated to yield results that are consistent with the simulations in terms of immunity to minor synchronisation error.
Fig. 3: magnitude versus time plot of estimated channel response from each transmit antenna to the single receive antenna for a single transmission block in the implemented TR- STBC system.
Fig. 4: magnitude versus time plot as in fig. 3 but captured from the same system approximately 2 seconds afterwards. This data has been captured after the TR-STBC system has experienced a minor mis-synchronisation error.
VI. REFERENCES
[1] S. Alamouti, "A Simple Diversity Technique for Wireless Communications," IEEE JSAC vol-16, No-8, Oct 1998
[2] E. Lindskog, A Paulraj, "A Transmit Diversity Scheme for Channels with Intersymbol Interference", IEEE Int. Conf. On Comms., Vol. 1, pp.18-22, June 2000.
[3] P. Stoica, E. Lindskog, "Space Time Block Coding for Channels with Intersymbol Interference", Conf. Record of 35th Asilomar Conf. On Signals, Systems and Computers, Vol. 1, pp253-256, Nov. 2001.
[4] E.G. Larsson, P.Stoica, E.Lindskog, J. Li, "Space Time Block Coding for Frequency Selective Channels", ICASSP 2002, Vol. 3, pp2405-2408, May 2002.
[5] S.H.Kuo, K.Mehrotra, I.V.McLoughlin,
“Reconfigurable processing framework for Space Time Block Codes”, accepted for ATNAC2003