A MIMO WLAN BASED ON LINEAR CHANNEL INVERSION Volker Jungnickel, Thomas Haustein, Eduard Jorswieck and Clemens von Helmolt1
Abstract—Multiple-input multiple-output (MIMO) techniques may increase the capacity of wireless local area networks (WLANs) since they profit from rich scattering of radio waves in indoor environments. On the other hand, much signal processing must be performed in a very short time to reach practical data rates. This paper reports on elementary steps towards MIMO transmission when the channel is known to the transmitter. In most cases, channel information can be obtained at the transmitter by means of reciprocity. Linear channel inversion (LCI) may then be employed allowing simple processing suitable for real-time applications. In contrast to signal processing at the receiver, LCI offers isotropic noise gain which simplifies coding. On the other hand, power distribution at the transmitter depends on the statistical properties of the MIMO channel. Results in the case of Rayleigh fading indicate that at least one additional antenna is needed at the transmitter for reliable operation. A simple formula for the mean transmitter power is reported and confirmed by simulation. Performance can now be expressed in terms of transmitter power and a fair comparison with transmission based on signal processing at the receiver is possible. Introduction Recently it has been shown that the capacity of wireless systems can be significantly improved with multiple antennas both at the transmitter (Tx) and at the receiver (Rx) provided that the Rx has proper knowledge about the channel [1]. In the case of flat fading, where H is independent on frequency, the multiple-input multipleoutput (MIMO) transmission between the various antennas can be written as
y = H⋅x +n
(1)
where x is a vector containing m Tx signals, y is the vector with the n Rx signals and n is the noise vector at the multi-antenna Rx. The Matrix H contains the complex-valued channel coefficients from each Tx antenna to each Rx antenna. These coefficients can be estimated, for instance, by sending orthogonal training sequences from the Tx to the Rx prior to the data transmission and using a bank of m correlation circuits at each Rx [2]. The most simple method of signal reconstruction at the receiver is zero forcing (ZF), i.e. the received signal is multiplied with the Moore-Penrose (MP) pseudo-inverse matrix H-1
x' = H −1y = x + H −1n
(2)
which reconstructs the transmitted signals. But the noise is also multiplied with H-1 in (2) and the bit error performance is significantly degraded. Even if the channel coefficients in H are identically and independently distributed, the entries in H-1 are not, and noise gain in the data paths is not isotropic after the signal processing at the Rx. The V-BLAST approach takes smart advantage of this fact [3]. The particular data stream having the smallest noise gain is detected at first. The detected symbols in this stream are then multiplied with H and subtracted from the received vector y so that the effective number of transmitters is reduced by one. Due to the increased diversity, anisotropy of noise gain is reduced for those data streams detected in the later iterations of BLAST. Step by step, all data signals are detected, and the overall bit error performance is improved compared to zero forcing. 1
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α β
base station
Fig. 1
mobile terminal
On the reciprocity of radio channels.
A digital signal processor (DSP) can be used to implement the BLAST algorithm. But it is difficult to implement BLAST with present DSP technology for high data rates. It was already mentioned in [1] that performance can be improved when the channel is known also at the transmitter. Maximising the capacity is then a problem of variation calculus which can be solved by waterfilling like in the classical theory of parallel data links [4]. But like for ZF and BLAST, signal powers in the data paths differ from each other, and an individual combination of channel coding and modulation is required in each path to approach the optimum capacity. Thus, water-filling multiplies the complexity of coding. A natural question arising from this point is whether there are sub-optimum techniques with signal processing at the Tx which are more practical. In the present paper, LCI is investigated being such a sub-optimum technique. LCI has a number of striking advantages and it might be interesting for WLAN applications. Reciprocity in MIMO channels Reciprocity is a basic property of radio channels and it can be utilised to reduce the signalling effort in MIMO systems. Reciprocity means that the channel coefficients from a particular antenna at the mobile station to a particular antenna at the base station are identical when the direction of transmission is reversed at the same carrier frequency. Channel estimation can thus be performed in arbitrary direction of the link, and it gives the same results except of errors due interference. Signal processing for the down-link from the base station to the mobile can be based on a channel estimation in the up-link and vice versa. For instance, the mobile may transmit the orthogonal training sequences in the up-link period and the base station Rx estimates the channel coefficients in order to detect the up-link data, afterwards. The same information on the channel coefficients is also useful for the signal processing of down-link data at the base station transmitter. No additional effort for channel estimation is therefore required due to reciprocity. Reciprocity can be confirmed experimentally for closely located antennas with a network analyser by alternating measurements of S21 and S12. For instance, for a lab in our institute, reciprocity was fully confirmed over the frequency range 0.9...1 GHz, provided that time variations of the channel due to moving objects near the antennas were avoided during a measurement cycle. An rather intuitive approach to explain reciprocity is sketched in Fig. 1. It is easy to show that the line-of-sight path between base station and mobile terminal is reciprocal. When reflections are involved, one must take care that the reflectivity ρ of materials is reciprocal by itself
ρ ( a , β ) = ρ ( β ,α )
(3)
where α and β are the angles of the incident and of the reflected wave with respect to the normal of the reflection plane (see Fig. 1). Once (3) is given, the entire radio channel is reciprocal when it is considered as a sum of the line-of-sight and the echo signals due to reflections. Of course, (3) is fulfilled for typical materials due to Fresnel's equations. It is also clear that diffuse reflections according to Lambert's law are reciprocal. But there is one example for non-reciprocal reflection known in optical wireless communications. Sometimes, reflections from carpets are shiny from one side but diffuse from the other side, and (3) is not fulfilled [5]. Thus, exceptions are possible, at least in principle. Experimental data concerning reciprocity for larger distances between antennas are rather scarce, presently. Even if it is expected that measurements will confirm reciprocity in most cases, more reliable data on the reciprocity of radio channels should be collected in the future. A second point is that Tx and Rx front-ends must be linear and properly calibrated [6].
Tx signal processing and noise gain Linear channel inversion (LCI) creates isotropic noise gain in the parallel data paths. For this reason, coding can be done for the serial data stream before it is transmitted in parallel over the MIMO channel. The same is true for the decoder which can be placed after parallel-to-serial conversion in the receiver. Common coding may significantly reduce costs. In the most simple form of LCI, the data vector d with the n parallel data streams is multiplied at the transmitter with a right-handed generalised inverse Hr-1 creating mixed base-band signals for the m transmit antennas
x = H r−1d
(3)
This kind of Tx signal processing has been considered also in [7]. The received signals read
y = HH −1 r d+n =d+n .
(4)
Obviously, data are already separated at the receiver and no further processing is needed, which sounds interesting for down-link applications. In addition, the noise in (4) is not affected by signal processing and it remains fully isotropic. But, of course, there is an effective noise gain for LCI even if it is not directly visible in (4). Note that the transmitter power PTx is controlled by the channel coefficients which can be described by
PTx = x H x = d H (H −r 1 ) H H −r 1d ,
(5)
where xH is the conjugate transpose of x. We obtain Hr-1=VΣ Σ-1UH from the singular value decomposition (SVD) of H=UΣ ΣVH. With m Txs and n
