Joint Estimation of Channel Parameters forMIMO Communication Systems
Ji Li, Jean Conan, Samuel PierreMobile Computing and Networking Research Group (GRIM), Ecole Polytechnique of Montreal
P.O.Box 6079, Station Centre-ville, Montreal, Que., Canada H3C 3A7
Abstract- In the next generation mobile communication sys-tems, high data rates and high capacity are expected if multipleantennas are used at both receive and transmit sides. Such aradio propagation channel constitutes a Multiple-Input Multiple-Output (MIMO) system. In a wireless MIMO system, it is possibleto estimate channel parameters in a multipath environment byextending the classical parameter estimation methods to the jointspace and time domain. In this paper, we propose a subspace-based approach to jointly estimate the angle-of-arrival (AOA),angle-of-departure (AOD) and delay-of-arrival (DOA) of digitallymodulated multipath signals in MIMO communication systems.The novel approach uses a collection of estimates of a space-time manifold vector of the channel which utilizes a Khatri-Raoproduct to transfer the estimated channel response matrix tothe classical model. Simulation results show that the proposedmethods can achieve high resolution of channel parameters andresolve more multipath components than the number of arrayelements.
In high speed wireless communications, analyzing multipathpropagation parameter is often an important process. Channelparameter estimation is a classical problem encountered inradar, sonar, and wireless position-location systems and futurewireless intelligent networks . Furthermore, to understandthe characteristic of the spatial radio channel is important forthe design of space-time processors.The estimation of channel parameters such as angle-of-
arrival (AOA), angle-of-departure (AOD) and delay-of-Arrival(DOA) for a known signal is the central function for wire-less location systems -. The conventional schemes forestimating the angle and delay of the received signal inwireless communication systems are based on transmitting aknown signal, such as a pulse, and performing correlationor parametric estimations separately , . Unfortunately,in many cases, the received signal is composed of multiplereflections having different AOAs and DOAs, which causesthe signal to overlap in either the time or space domain. Theclassical algorithms for estimating the AOA and DOA are nolonger optimal in such situations -.
Recently, Joint Angle and Delay Estimation (JADE) al-gorithms ,  for smart antenna have been proposedfor more robust estimation. In a multipath environment, thedirection of received signals and associated delays of the pathdo not change quickly, so that it is possible to estimate theseparameters by extending the classical methods to the jointspace and time domain.
In the next generation mobile radio systems, high data ratesand high capacity are expected if multiple antennas are usedat both receive and transmit sides. Such a radio propagationchannel constitutes a Multiple-Input Multiple-Output(MIMO)system which offers a new dimension by exploiting the spatialproperty of the multipath channel , . Therefore, in awireless MIMO system, it is possible to estimate more channelparameters in a multipath environment. Recently, double-directional estimation methods for MIMO channel have beenproposed to estimate both AOD at the transmit site and AOA atthe receive site simultaneously . However, the estimationof channel features has to include some other parameters suchas the delay of arrival.
Therefore, in this paper, we propose a subspace-basedapproach to jointly estimate the AOA, AOD and DOA ofdigitally modulated multipath signals in MIMO communica-tion systems. This approach uses a collection of estimates ofa space-time manifold vector of the channel, then exploitsthe eigenstructure of the input covariance matrix. Finally, bysearching the peaks of the MUSIC spatial spectrum to estimatethe parameters of the multiple incident signals.
This paper is organized as follows, Section 2 introducesthe system model of the MIMO channel, which assumesthe propagation path to be specular rather than dispersed. InSection 3, we define the concept of space-time manifold vectorwhich utilizes a Khatri-Rao product to transfer the estimatedchannel response matrix to the classical model. The proposedsubspace-based joint parameter estimation method for MIMOmultipath channel is proposed in Section 4, whereas CRB isalso derived. In Section 5, simulation results are presented
0-7803-9206-X/05/$20.00 2005 IEEE
to demonstrate the proposed approach. Section 6 providesconcluding remarks to summarize the paper.
II. THE SPECULAR CHANNEL MODELA. Assumptions
In order to jointly estimate multipath channel parameters,we shall consider the following conditions on the mobile radiopropagation scenario.
. The MIMO multipath environment is modelled by adiscrete number of rays, each parameterized by a delay,complex amplitude (path gain), angle of arrival and angleof departure.
. The source signals are digital sequences that are linearlymodulated by known pulse shape functions.
. The parameters such as AOD, AOA, and DOA are notchanging significantly from each time slot to the next.
. The data transmitted by the antennas is sampled at orabove the Nyquist rate.
The MIMO multipath propagation channel is illustrated asFig. 1.
where T is the symbol period.The received baseband signal is given by:
x(t) = Ho(t) * s(t) + n(t) (2)where * is convolution operator. s(t) is the vector form ofSm(t), and n(t) is assumed to be Gaussian white noise.
Thus, the channel impulse response can be expressed as:R
Ho(t) = E aR(Or)br(g(t - rr) aT(Or))r=1
where denotes Kronnecker product.The vector aR(Or) is the receive array manifold vector for a
ULA array at direction Or, while aT(O,) is the correspondingarray manifold vector for a signal emitting from the directionOr. Here aR( ) = [aR1(Or), -- ,aRn(0r)fT, and aT(Or) =[aTl(Or),--- aT.(9r)]T. It is reasonable to assume that thepulse shape function g(t) has finite support, i.e., it is nonzeroonly for t e [0, LgT]. Then (2) implies that the (integer)channel length is L, where LT := L,T + AT seconds.
Suppose the channel has a finite duration with channellength L measured in terms of symbol periods, We collectdata over K symbol periods, and we sample received basebandsignal x(t) from r1, then the discrete-time baseband model ofthe received signal can be expressed as:
x(k) = E H(l)s(k - 1) + n(k)1=0
Let the oversampling factor be P, i.e., at each symbol periodwe take P samples of x(t). The MIMO transfer function Hrelates the transmit array excitation S to the observed receivearray. We write X, H and S into following matrices:
X=HS+NFig. 1. A specular MIMO multipath propagation channel model
B. Data ModelIn the MIMO multipath model shown in Fig. 1, we as-
sume a MIMO channel employing ULAs (uniform lineararray) at both end with M transmit elements and N receiveelements. There are R different paths, each parameterizedby 0r,rr,b. The parameter 9tr and r are the departureand arrival angles of the rth path respectively, while eachassociated path has a complex path gain b, and delay rr.The transmitted signal sm (t) at antenna m can be repre-
sented as a convolution of the data bits sim, and the pulseshaping function g(t):
H(T) ...H((1 + p)T) ...
H((2 -)T) ...
S1 ... SK-1
S_L+2 ... SK-L
Sm(t) = E simg(t - iT) (1)
III. THE CHANNEL ESTIMATIONA. Space-time Manifold Vector
In order to estimate the angle of departure, angle of arrival,and the delay of arrival with knowledge only of the transfermatrix H, we can first vectorize the matrix by stacking itscolumns. Using the vec operator, we get from (6)
h = vec(H) = A(O, b) o G(r)b (8)
where o denotes Khatri-Rao product. G(r) =[g(-r1)T ...g(R)T]T, and b = [b1 ... bR]T. The matrixA(O, q) here is a MN x R matrix given by
IV. PROPOSED ALGORITHM AND CRBA. The proposed Subspace-based Algorithm
The multiple signal classification(MUSIC) method proposedin  is a spectral-based algorithm but relies on the propertiesof the eigenvalue decomposition of the covariance matrix. Thedivided signal and noise subspace are orthogonal. For MIMOchannel, the noise eigenvectors An are perpendicular to thespace-time response matrix W(O, 0, r) or the signal subspacespanned by As. Then we have the following orthogonalitycondition
An W(O, X, T) = 0 (12)
The idea of the algorithm is to find the R vectors W(O, 0, r)A(O,=q ) A[~ec(O) A~(q)a Q1H) vc(~($)a(9)H]which are the most orthogonal to the estimate of An.- A()OA(Q)0(9) Let A1, A2, ... , AMNLP be the eigenvectors of the esti-
where A(0) and A(Q) represents array manifold vector for mated covariance matrix R0 arranged in the descending ordertransmit and receive side respectively, of the corresponding eigenvalues. The eigenvectors spanning
With noise, we can rewrite (8) into the signal subspace corresponding to the R largest eigen valueswill be
u = h+ n = W(O,q5,T)b+ n (10) (13)where w(0r, , Tr) gg(Tr) 0 aT(Or) 0 aR(q$r) is defined asthe MIMO space-time manifold vector.
This MNPL-dimensional vector is the spatial-temporal re-sponse to the antenna array to a signal path with AOD (0),AOA (h), and delay (r).The space-time response matrix for R paths can also be
W(0, -,T) = G(r) o A(0) o A($) (1)
Since the array and time manifold are all assumed functions,this will enable to extract the desired parameters AOD, AOA,and DOA from knowledge of the column span of W(O, (t, r).
B. Transfer Function EstimationThe goal we are pursuing is: given estimates of the MIMO
channel pulse response, estimate the channel parameters suchas AOD, AOA, and DOA. In the first step, we will estimatethe channel impulse response for the MIMO channel. Thiscan be done by using training sequence or blindly . Theestimation of H can be obtained by collecting Q consecutivetime slots.The second step of this approach consists of estimating the
angle of departure and arrival, as well as delay of arrival.There're various ways to estimate MIMO channel parameters,we focus on an improved Music algorithms for joint parameterestimation in this work.
The MUSIC estimation of the channel parameter is to findthe R minima of the following cost function:
F(O, X, T) = WH(0, X, T)[I_-AAS]W(0,O, T) (14)B. The Cramer-Rao Bound
The conventional CRB provides a lower bound for thevariance of any unbiased parameter estimator and is typicallyused for a stationary gaussian signal in the presence ofstationary gaussian noise . The derived CRB is
CRB(J)-1 = 2real(BHDwHP' DwB) (15)
where P = I-W(WHW)-WH is the projector onto thenoise space, Dw = [DO, Do, DT], and B = 13x3 X diag(b).DO= [d0,... d0R Do = [do,,... , doR, and DT =[drl , ddrR], where do, is the derivative of the rth columnofW and dOr 9(Tr) 0) aR(q$r) 09 daT(Or)/d0r. The similardefinition holds for d Or and d r
V. SIMULATION RESULTS AND DISCUSSIONSIn this simulation, we assume a single user, three multipaths,
and a two-element ULA at transmitter and a three-elementULA at receiver.The AOD's are [-18, 540, 360] relative to the array broad-
side, the corresponding AOA are [36, 00, -54] while thepath delay are [0.2,0.6, 1.6]T seconds, where T is normalizedto one. The collected data are corrupted by noise with inversevariance 1/a = 20dB. The modulation waveform is a raised
As= [A,,A2,''- . AR]
cosine pulse with excess bandwidth 0.35, assumed to be zerooutside the interval [-3, 3). We sampled at rate T/2. Data iscollected over 30 time slots, and at each time slot the channelis estimated via Least Square method using 30 training bits.The experiment variance of the angle and delay estimation iscomputed from 100 runs.The following figures show our simulation results for the
proposed spectral based approach.
Fig. 4. Plot of the comparison of estimated results with true value
Fig. 2. Plot of the Music Spatial Spectrum for Multipath(AOD vs. DOA)
Fig. 2 illustrates the computer simulations of the proposedjoint estimation algorithm for MIMO channel that using atwo uniformly spaced linear array with half wavelength inter-element spacing at both site. This method is able to distinguishbetween the three signals of equal power at an SNR of20 dB arriving at -18,54 and 60 degrees and with delay[0.2,0.6, 1.6]T seconds, respectively.
Fig. 3 is the plot of the double direction estimation for theMIMO channel. Here we illustrate three digitally modulatedsignals of different frequencies, transmitting at -18,54 and60 degrees and arriving at angles of [36,0, -54] degreesrespectively.
Po: th SpaI Spet- for D-Mt*-i-no .. Agb Eti-t.,(AOD AOA
Fig. 3. Plot of the Spatial Spectrum for Double-directional Angle Estima-tion(AOD vs. AOA)
Fig. 4 shows the comparison of estimated results with truevalue for five multipaths.
Fig. 5 shows that the proposed methods can achieve highresolution of channel parameters and resolve more multipathcomponents than the number of array elements.
Pbt of th. Sp.t1.1 SpWatur I., D..bW-dwretiol.. IMg Et1-i.,,go(A0D AsOA)
"i. d A-1va -tOO -t00 ArVb d Depa-ur
Fig. 5. Plot of the resolution of AOA and AOD estimation for ten multipaths
If we consider multiple users in this communication system,we can use each user's unique training sequence which couldbe orthogonal to each other to independently estimate thechannel matrices H.
The proposed joint estimation of MIMO channel parameteralgorithm exploits the spatial time properties of the multipathchannel. It has the following advantages:
. If received signals arrive in same direction, the con-ventional estimation algorithm can not distinguish theirdifference. But with joint estimation, the two signals withsame direction can be separated by their delays.
. Unlike traditional subspace-based algorithms, the jointestimation algorithm can work in cases when the numberof paths exceeds the number of antennas or array ele-ments. Thus, it can resolve more multipath signals thanthe number of array elements.
. By assuming the first arrived signal corresponds to theline-of-sight path, the LOS signal will be separated fromother multipath signals.
. Since it's a spectral based algorithm, high-resolution ofchannel parameters can be achieved even under low SNR.
1.6- V T- -k-E.b.,W.d .W-IA- 0
I I- 0toZ- IsA0.6-
0.4 40 00.2.0
A,V. A,,i..l -60 -60 A.g.. Dp....
However, the proposed spectral based approach requiresa three-dimension extrema search of the cost function. Thissearch can be performed using dynamic programming oralternate projection methods. But this technique does give usa practical insight into the resolving power of the subspacebased approach for MIMO communication systems.
VI. CONCLUSIONSIn this research, we have developed a novel approach to
jointly estimate channel parameters for MIMO communicationsystems. By using the fact that the AOA, AOD and DOA arealmost constant over several time slots, this approach collectsestimates of space-time manifold vector through several timeslots, then analyze the eigen structure of covariance of MIMOchannel transfer function. A high-resolution estimation ofchannel parameters can be achieved through subspace basedmethods. Cramer-Rao Bound for this approach is also derived.
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