Multiple Antenna Wireless Communication Systems: Channel models and their Properties Leif Hanlen
[email protected]
Minyue Fu
[email protected]
Department of Electrical and Computer Engineering The University of Newcastle, Callaghan, NSW 2308, Australia
Abstract. We present a general multiple path scattering model for multiple transmit, multiple receive wireless systems. The model is generated using physical modeling of the scatterers surrounding transmit and the receive arrays. The condition of the resulting multiple-input multipleoutput (MIMO) transfer matrix is then examined. A key parameter that determines the channel condition is identi ed. This parameter depends on the local scatterer radii, the separation of arrays, and the wavelength. We show that there exists a critical value for this parameter at which the channel condition changes sharply. The implication is that the promised linear growth in channel capacity may not eventuatate if the separation of the transmit and receive arrays is large ( 10 or 20 at 2GHz) compared with the product of the scattering radii. Monte-Carlo simulations are used to demonstrate the claims.
I Introduction Following the work of [3] and [4] there has been a large amount of literature [1, 5] concerning multiple transmit, multiple receive wireless systems. In [3] and [4] a linear growth in capacity is predicted, proportional to the minimum number of transmit and receive antennas. Fundamental to this work, has been the assumption that the wireless channel may be modeled by an independent random N M matrix, where N and M are the numbers of the transmit and receive elements, respectively. That is, the entries of the transfer matrix are assumed to be independent, complex random variables. These may appear as Rayleigh channel approximations or as independent Gaussian matrices [4]. The assumption of independence guarantees a well-conditioned transfer matrix (statistically) and e ectively hides the loss of capacity due to possible correlations between the channels. Besides [2] very little literature exists relating the channel model and scatterer geometries to the actual MIMO transfer matrix and few authors have addressed the physical conditions necessary for the required well-conditioned transfer matrix.
The work of [2] studies a case where scatterers are placed in uniform arrays. Two classes of channel model are predicted: an \uncorrelated high rank model" - the well known i.i.d. random matrix approximation and an \uncorrelated low rank" or \pin-hole" model where the channels are highly dependent and MIMO capacity is consequently low. These models are disjoint - the transition from one model to the other is predicted (qualitatively) to occur smoothly at large separation of arrays. This paper follows the direction of [2] but aims to provide a somewhat more detailed analysis of channel models for multi-antenna systems. More speci cally, we study a case where scatterers are placed randomly within given circular regions, surrounding the transmit and receive arrays. This corresponds to the well-known local scattering models. This model has the advantage that it provides more scope for analysis of the e ects of separation of the arrays, and includes the results of [2] as a subset, without a noticeable increase in model complexity. We present a channel model using simple physical principles. We then proceed to simplify this model and analyse its condtion. We have identi ed a key parameter that determines the condition of the channel model. This parameter is dependent on the local scatterer radii, the separation between the transmit and receive arrays, and the wavelength. We show that there exists a critical value for this parameter at which the condition of the channel model (and hence the channel capacity) alters sharply. The promised linear growth in channel capacity can not eventuate when the value of this parameter is too low, which corresponds to the case when the separation of transmit and receive arrays is too large compared to the scatterer radii, or when the wavelength is too large. This paper is organized as follows: The con guration of the multi-antenna system is explained in Section II. Section III develops a MIMO transfer matrix for the channel.
scatterer
R
Rx array
βs
R
αs
D
β
ΘT
r Tx array
Rt Figure 2: Circular scatterer.
Figure 1: Physical arrangement
Each scatterer is assumed to be a small, lossless, memoryless re ector. The scatterer receives all transmitted This transfer matrix is simpli ed and its condition is anal- signals isotropically, and re-transmits them isotropically. ysed in Section IV. Some Monte-Carlo simulations are Later we will consider two types of scatterers: circular given in Section V. Section VI provides a summary of our and linear. work.
II System Con guration We consider a non-line-of-sight (NLOS) channel, where fading results from scatterers at both the transmit and receive ends of the link. We consider only scatterers close to either the transmitter or receiver arrays since distant scatterers tend to su er large attenuation in signal strength, and have a correspondingly small e ect on transmission. All signals are assumed to be narrow-band, and the channel is assumed to exhibit frequency- at fading. Consider the multiple antenna system with N transmit elements (denoted by tx) and M receive elements (denoted by rx), both placed in linear arrays as shown in Figure 1. The arrays are separated by a distance D. Without loss of generality, we may place the arrays parallel to each other. This removes the need to introduce orientation angles of one array relative to the other. This simpli cation is accurate if the waves transmitted may be approximated as plane waves and the signals are narrow-band. In this case, rotation of arrays is equivalent to adding extra phase o sets at each element. The arrays are not necessarily uniform. The vertical position for transmit (resp. receive) element i is denoted by t;i (resp. r;i . For simplicity it is assumed that t;0 = r;0 = 0. Surrounding each array is a disc of local scatterers, denoted tx scatterers and rx scatterers respectively. These scatterers are placed on an annulus of maximum radius R. The discs (containing the scatterers) will be referred to as local scatter-rings. The placement is assumed to be random. It is assumed that R is large compared with the array length, and that D is large compared with R. Using spherical coordinates, each tx scatterer is placed at (Rts ; s ) where ts 2 (0; 1] and s 2 [0; 2) chosen at random. Similarly, each rx scatterer is placed at (Rrs ; s ) with the appropriate translation of origin.
III Modeling
A Scatterer Model
We consider two types of scatterers: circular and linear, as depicted in Figures 2 and 3, respectively. We assume that the scatterers receive signals from transmitters and re-transmit signals omni-directionally without any loss or delay. The amount of power received by a scatterer depends only on the surface area exposed toward the transmitter. For a circular scatterer (Figure 2), let the lines from the surface of the scatterer s, drawn to the transmitter tx, subtend an angle T . radius of the scatterer is rscat . The scatterer is placed at distance R0 from the transmitter. The power received at scatterer s, will be a fraction, f , of the power transmitted by tx. We have 1 r def T scat (1) f = 2 = arcsin R 0 We will denote f by f (rscat =R0 ) to show its explicit dependence. Using a similar analysis, we can consider the case where a scatterer s is a line segment (Figure 3). The scatterer is placed at coordinates (R0 ; s ) from the transmitter tx. It has a length 2rscat and an angle s . The proportion of the power fs received at scatterer s will depend on the length presented to the transmitter, i.e., f = 1 arcsin rscat k cos( s ? s )k (2)
R0
The term rscat k cos( s ? s )k can be viewed as the e ective radius of a circular scatterer. Therefore, in our future discussions, we will consider all scatterers to be circular, for which (1) applies.
It should be noted that f is the power attenuation where rr is the e ective radius of the receive element r and p factor. Therefore, the signal attenuation factor is proporp R3 (r; s) = R2 rs2 ? 2Rrs r sin s tional to f . To determine Hs , we consider the st th scatterer at the B Channel Model transmitter side and the sr th scatterer at the receiver side. Consider the channel in Figure 1. De ne x and y to De ne R2 (st ; sr ) as the path length between the two scatthe vectors of transmitted signals and received signals, re- terers. Then, we have v spetively, i.e., u 2 uD + 2DR(ts cos s ? rs cos s ) t R ? (6) T T def def 2 (st ; sr ) x = x0 : : : xN ?1 ; y = y0 : : : yM ?1 + R2 t2s + rs2 ? 2ts rs cos( s ? s ) Then, the transfer matrix H from tx to rx consists of three Therefore, the (sr ; st )th element of Hs is given by factors: a transfer matrix Ht from tx to the tx scatterers, s a transfer matrix Hr from the rx scatterers to rx, and a 2 R (s ; s ) (7) rS;r def H ( s ; s ) = exp i f s r t transfer matrix Hs between the two sets of local scatterers. R2 (sr ; st ) 2 t r That is, we have where rS;r is the e ective radius of the scatterer sr . y = Hx = aHr Hs Ht x (3)
IV Conditioning
where a is a constant attenuation factor which is independent of N and M . To determine Ht , we consider the tth transmit element It is well-known [4] that the capacity of the channel is and the sth local scatterer surrounding the transmit ar- given by ray. >From Figure 1, we see that the distance between the transmit element and the scatterer is given by C = log2 det(IM + N HH ) (8) p
R1 (s; t) = R2 t2s ? 2Rtst sin s Hence, the (s; t)th element of Ht is given by s
Ht (s; t) = f R r(T;s exp i 2 R1 (t; s) t; s ) 1 def
(4)
where rT;s is the e ective radius of the scatterer. In a similar fashion, we may compute Hr by considering the sth local scatterer surrounding the receive array and the rth receive element. The (r; s)th element of Hr is given by s
Hr (r; s) def = f R (rr;r s) exp i 2 R3 (r; s) 3
φ
Rt
α Tx
Figure 3: Linear scatterer.
(5)
assuming the receiver has the full knowledge of the channel, where is the signal to noise ratio at the receiver elements. Using singular value decomposition, we may rewrite
N HH = U U
where U is a unitary matrix and is a diagonal matrix containing all the signular values i of =NHH . a diagonal matrix of singular values i . Without loss of generality, we may assume that 1 2 :::. Therefore,
C = log2
M Y i=1
(1 + i ) =
M X i=1
log2 (1 + i )
X i=1
log2 (1 + i )
where is the number of singular values which are not negligible. Consequently, if H is poorly conditioned (i.e., is small), then the channel will have reduced capacity. We will show that the channel model (and hence its capacity) is largely determined by the parameter 2 = 2R
D
We rst analyse the conditioning of Ht and Hr . For
Ht , we assume that
N ?1 Rts
It is clear that D1 ; D2 ; D3 are well-conditioned, profor every scatterer s near the transmit array. With this vided that the rx scatterers are of similar sizes. Hence, assumption, we have we need to analyse M in detail. To this end, we approxs r 2 2 imate M using th Taylor expansion, where is to be T;s Ht (t; s) f Rt exp(i Rts ) exp(?i t sin s ) determined later: s
Therefore,
(s
)
r T;s Ht D11 H12 D11 = diag f Rt exp(i 2 Rts ) s 2 H12 (x; y) = exp(?i x sin y ) It is obvious that D11 is well-conditioned, provided that the ratio of rT;s and Rts is similar for di erent scatterers. The matrix H12 is also well-conditioned provided that the tranmit elements are spaced at least =2 apart and that
M (x; y) 1 + itx ry sin x sin y + + (its rs sin s sin s ) De ning = V (t1 sin 1 ; t2 sin 2 ; ) V1 def def V2 = V (r1 sin 1 ; r2 sin 2 ; )
(11)
where V (x1 ; x2 ; xn ) denotes the n Vandermonde matrix, i.e., the matrix with the (i; j )th element equal to xij . Then, M can be written as
the angles of the scatterers are well-separated. M = V2 ?V1 (12) A similar analysis applies to Hr , which is omitted. ? def = diagf1; i; ; (i) g (13) The transfer matrix Hs , on the other hand, is highly dependent on the separation of the antenna arrays and the Giving radii of the scatterer rings. To analyse the conditioning of Hs , we assume H = Hr D3 V10 ?V2 D2 D1 D11 H12 = HR ?HT (14) 2 D R def = D 1 with d = i and = ?id"2 (9) It can be seen from (14) that the magnitude of the singular values of H will be exponentially decreasing. Each With the above assumption, we have the following ap- term will have an expected magnitude of ? 1 4 . 2 proximation: The analysis above shows clearly the role the param eter plays in the condition of the channel model. Since S;r f R (rsS;r; s ) f rD and j ts rs sin s sin s j 1, the contribution of the th term 2 t r diminishes exponentially as a function of when < 1. 2 Small implies that the channel model is poorly condiR2 (st ; sr ) D 1 + a(st ; sr ) + 2 b(st ; sr ) tioned. That is, the \e ective rank" of the channel model is small. In this case, increasing the number of transmit where or receive elements has little e ect on the e ective rank of the channel once the number of elements reaches certain a(st ; sr ) = tst cos st ? rsr cos sr b(st ; sr ) = t2s + rs2 ? 2ts rs cos( s ? s ) ? a2 (st ; sr ) (10) limit. Similarly, increasing the number of scatterers has little e ect on the e ective rank either. = (ts sin s ? rs sin s )2 On the other hand, the transfer matrix H can be wellconditioned if > 100. Only in this case, it is possible = t2s sin2 s + rs2 sin2 s ? 2ts rs sin s sin s to assume that the transfer from the tth transmit element Subsequently, we have to the rth receive element to be independent, leading to linear growth in channel capacity as promised by [3] and Hs D3 MD2 D1 [4].
D1 = diag exp(dts cos s + d2 t2s sin2 s ) D2 = diag exp(drs cos s + d2 rs2 sin2 s ) r 2 D r S;r D3 = diag f D exp i and M is a matrix with M (st ; sr ) = exp(d2 ts rs sin s sin s ) = exp(its rs sin s sin s )
0
V Simulation We use Monte-Carlo simulations to analyze the model predictions. A wavelength of = 0:03m, corresponding to approximately 10GHz, is used. We set M = N = 5, and positioned array elements in uniform linear arrays, with inter-element spacings of 2 . Scatterers were placed uniformly at random, within circular regions surrounding
Condition number of H as arrays are separated
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Figure 4: Condition number of H vs. η
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Figure 6: Channel capacity vs.
Singular values of H as arrays separate
η
rameter, , which plays a crucial role in determining the condition of the channel and thus the channel capacity of the system. This analysis gives a quantative measure of the system con guration which can possibly achieve high channel capacity. In particular, large separation between the transmit array and receive array leads to poor condition for the channel transfer matrix and thus the poor channel capacity. The simulation results clearly indicate the signi cance of the parameter . The model presented covers NLOS transmission only. It is reasonable to ask how much more capacity can be Figure 5: Singular values of H vs. expected if LOS transmission is present, or transmission paths involving a single scatterer are included. It has been each array. The regions had a xed radius of 10m. Scat- found that a similar parameter dominates the condition of terers were given random gains selected uniformly over an the channel transfer matrix. Having the additional paths interval G 2 [0; 1]. These gains mimic the sizes of the does not signi cantly alter the channel capacity. scatterers. It is found from the simulations that the expected rollo in capacity is independent of the physical characteristics of the scatterers. Increasing the number of scatterers and reducing variation in losses and placement of individ- [1] N. Seshadri A. F. Naguib and A.R. Calderbank, Increasing data rate over wireless channels, IEEE Signal ual scatterers have little e ect on the channel condition Processing Magazine 17 (2000), no. 3, 76{92. and capacity as expected. Figure 4 shows the condition number of the transfer [2] D. Gore D. Gesbert, H. Bolcskei and A. Paulraj, matrix H as the function of . Note the sharp transition MIMO Wireless channels: Capacity and performance from small condition number to large condition number. prediction, IEEE Proc. Globecom 2000, 2000. Figure 5 shows that the poor conditioning is not con ned to a few extreme singular values. Rather, the sin- [3] G. J. Foschini and M. J. Gans, On limits of wireless gular values of H decrease geometrically. Plotted is the communications in a fading environment when using singular value decomposition at di erent values of from multiple antennas, Wireless Personal Communications a Monte-Carlo simulation. the arrays increases. 6 (1998), 311{335. Figure 6 shows the channel capacity as a function of , using (8). It is assumed in the simulation that = 10dB . [4] I. E. Telatar, Capacity of multi-antenna gaussian channels, European Transactions on Telecommunications 10 (1999), no. 6, 585{595. [5] I. E. Telatar and D. N. C. Tse, Capacity and mutual information of wideband multipath fading chanIn this paper, we have developed a channel model for nels, IEEE Transactions on Information Theory 46 multi-antenna wireless systems which involve local scat(2000), no. 4, 1384{1400. terers at both the transmit and receive ends. This model has been analysed in detail. We have identi ed a key pa2
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References
VI Conclusion
