The method described below is a version of the CDEML algorithm presented in [15]. Let the overall channel model be represented by the vector FIR model in (2.18)
y(t) =Bd(t) +n(t) (2.53) where d(t) = [d(t) d(t;1) ::: d(t;nb)]T and B is the channel matrix (2.19) (2.20) of sizeM(nb+ 1). The signald(t) is the transmitted discrete symbol sequence. The noise plus interference, n(t), is here assumed to be circularly symmetric6, zero-mean and Gaussian with second order moments given by
E[n(t)nH(s)] =Qt;s E[n(t)nT(s)] =0: (2.54) Consider the DOA's for the incoming signals7
= [01 ::: 0;k1::: nb;1::: nb;knb]T (2.55) and the corresponding gains
= [01 ::: 0;k1::: nb;1::: nb;knb]T: (2.56) The indices in the expressions above for and denote tap number (delay) and path number within each tap, in that order. Each tap is assumed to be aected bykcsignal paths arriving from signicantly dierent directionsc1
to ckc. The matrix B can now be parametrized in terms of these DOA's and gains
B(;) =A();() (2.57) where
A() = [a(01) ::: a(0;k1) ::: a(nb;1) ::: a(nb;knb)] (2.58)
6A complex random variable is said to be circularly symmetric if its probability density function is circularly symmetric around its zero mean. This will be the case if its real and imaginary parts are uncorrelated, equally scaled and have zero mean.
7The rst number in the subscript of the DOAij refers to the tap number and the second number refers to the number of the DOA within the tap.
and
;() =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
01 0 0
... ... ...
0k1
0 11 ...
1...k2
... ... 0
nb1
0 nbk...nb
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
: (2.59)
The vectors, a(ij)), in (2.58) constitute the array response vectors for a signals arriving from the anglesij.
The parameter vectors and, as seen above, are partitioned according to which delay inB they correspond to:
= [T0 T1 ::: Tnb] where c= [c1 ::: ckc]T (2.60)
= [T0 T1 ::: Tnb], where c= [c1 ::: ckc]T: (2.61) It can be shown (see [15]) that a large sample maximum likelihood estimate of and can be obtained by minimizing
F(;) =tr[Rdd(B(;);B^LS)HQ^;1(B(;);B^LS)] (2.62) where
R
dd= limN
!1
N ;1nb
N
X
t=nb+1
d(t)dH(t): (2.63) Above, ^BLS is the least squares channel matrix estimate (2.52) and
^
Q= 1N ;nb
N
X
t=nb+1(y(t);B^LSd(t))(y(t);B^LSd(t))H (2.64)
2.5. Spatial Parametrization 65 is an estimate of the noise plus interference covariance matrix.
In digital communications the symbol sequence d(t) is assumed to be white.
The covariance matrix, Rdd, will then be diagonal. In this case, the maxi- mum likelihood estimates of the angles,, and the gains,, can be found by considering the following minimization for eachcolumn, ^bc, in ^B separately:
f^c;^cg= arg min
c;c[A(c)c;^bc]HQ^;1[A(c)c;b^c] (2.65) where ^bc is the relevant column in ^B which corresponds to a vector tap in the FIR channel b(q;1).
The fact that the minimization can be decoupled greatly reduces the com- putational complexity. From (2.65) we can see that we are looking for c's andc's that minimize the weighted squared norm of the dierence between the least squares estimated column ^bcof ^BLSand the parametrized estimate
A(c)c.
Returning to (2.62), we see that when Rdd is diagonal, the overall criterion we want to minimize, is simply the weighted squared Frobenius norm of the dierence between the parametrized channel matrix, B(;) and the least squares estimated channel matrix ^BLS, i.e. kB(;);B^LSk2
Q
;1.
For a given set of angles ^c, the minimizing gain ^cin (2.65) is given by the weighted least squares solution
^
c(^c) =fAH(^c) ^Q;1A(^c)g;1AH(^c) ^Q;1^bc: (2.66) By substituting (2.66) into (2.65) we obtain an expression for the estimated directions of arrival [15]
^
c = arg min
c f^bHc [ ^Q;1;Q^;1A(c)
(AH(c) ^Q;1A(c));1AH(c) ^Q;1]^bcg: (2.67) In order to assure convergence to the global minimum, a good initial value is required for c. In [15], it is proposed to initializec with the kc lowest
local minima of the function f() = ^bHc
"
^
Q
;1
;
^
Q
;1
a()aH() ^Q;1
aH() ^Q;1a()
#
^
bc: (2.68) This is exactly the cost function to be minimized if there was only one signal arriving per symbol delay. As the rst term is independent of, we can instead look for localmaximas to the function
f0;c() = aH()^Q;1^bc^bHc Q^;1a()
aH()^Q;1a() : (2.69) In the simulations performed in this study, this initialization procedure has been found to have some problems. It could, for example, have diculties in estimating DOA's of signals that were close to a strong co-channel interferer.
The presence of a strong co-channel interferer in the noise plus interference covariance matrix,Q, can cause a dip in the functionf0;c().
In an attempt to alleviate this problem,Q;1 can be removed from f0;c().
The result is a simple \beamformer" DOA estimator. The initial values of the components ofccan thus be chosen as the local maximas of the function
f1;c() = aH()^bc^bHc a()
aH()a() : (2.70)
However, both of these methods have been found to have considerable dif- culties in estimating initial values for the DOA's when coherent sources are present. The peaks for the two functions in (2.69) and (2.70) can then have peaks in directions not corresponding to a DOA. This is because side lobes of the \beamformers" involved may pick up the signals and combine them constructively depending on the particular relative phases of the sig- nals involved. These combined signals can have a stronger amplitude then the signals caught by the main lobes of the \beamformers". A DOA will then be indicated at the wrong direction. A solution to this problem is to constrain the antenna array to a uniform linear array. In this case the non- linear minimization of (2.67) can be replaced with a polynomial root-nding technique similar to the one proposed in [94] and [95].
Once the directions of arrival inc have been estimated the gains inccan be computed using (2.66). The estimated ^ and ^ are used to form the
2.5. Spatial Parametrization 67 improved parametrized channel matrix estimate
^
BCDEML =A(^);(^): (2.71)