The structured gradient algorithm exploits the structure of the array correlation matrix.
However, it does not make use of the previous samples when estimating the gradient at the nth iteration. In this section, a method is presented that exploits the structure of the array correlation matrix and uses previous samples. The method is referred to as the improved method [God90a, God93].
An estimate of the gradient using the improved method is given by
(3.8.1) where
(3.8.2) with ˜R(n) given by (3.6.6).
It can easily be shown that the gradient estimate is unbiased, that is,
(3.8.3)
V n
n n R n R
R
H
g (w( ) )=( +41)2w ( ) ( )w
V n n R n R
S
H
g (w( ) )=4w ( ) ( )w
g wI( ( )n)=2R nˆ˜( +1) ( )w n
ˆ˜ ˆ˜ ˜
R n nR n R n
+ n
( )= ( )+ ( + )
1 + 1
1
E[g wI( ( )n)w( )n]=2Rw( )n
The performance and the signal sensitivity of the above algorithm is now compared with a RLS algorithm that makes use of the previous samples and requires the same order of computation for computing the weights. The following form of the RLS algorithm is used for the comparison:
(3.8.4) where ˆR–1(n) is updated using the Matrix Inversion Lemma and is given by (3.1.6) and (3.1.7). Note that in the absence of errors, n →∞, ˆR–1(n) → R–1, and w(n) → ˆw.
Figure 3.9 to Figure 3.12 compare the mean output noise power PN(w(n)) vs. the iteration number for various look direction signal powers when the weights w(n) are adjusted using the two algorithms. The mean output noise power is calculated using
(3.8.5) A linear array of ten elements with half-wavelength spacing is assumed for these examples. The variance of uncorrelated noise present on each element is assumed to be equal to 0.1. Two interference sources are assumed to be present. The first interference falls in the main lobe of the conventional array pattern and makes an angle of 98° with the line of the array. The power of this interference is taken to be 10 dB more than the uncorrelated noise power. The second interference makes an angle of 72° with the line of the array and falls in the first side-lobe of the conventional pattern. The power of this interference is 30 dB more than the uncorrelated noise power. The look direction is broadside to the array. The signal power for the four plots is varied from −10 dB below the uncor- related power to 30 dB above the uncorrelated noise power. The gradient algorithm is initialized with the conventional weights.
FIGURE 3.9
PN(w(n)) vs. the iteration number for a 10-element linear array with one-half wavelength spacing. Two interfer- ences: θ1 = 98°, p1 = 1, θ2 = 72°, p2 = 100, σ2n= 0.1, look direction angle θ0 = 90°. (From Godara, L.C., IEEE Trans.
Antennas Propagat., 38, 1631–1635, 1990. ©IEEE. With permission.)
w S
S S
n R n
R n
( )= H ( ) ( )
−
−
ˆ ˆ
1 0 0
1 0
PN(w( )n)=wH( )n RNw( )n
For the improved LMS algorithm the gradient step size à is taken to be equal to 0.00005 and for the RLS algorithm ε0 is taken to be 0.0001. According to these figures, for a weak signal the RLS algorithm performs better than the improved algorithm. However, as the input signal power increases the output noise power of the processor using the RLS
FIGURE 3.10
PN(w(n)) vs. the iteration number for a 10-element linear array with one-half wavelength spacing. Two interfer- ences: θ1 = 98°, p1 = 1, θ2 = 72°, p2 = 100, σ2n= 0.1, look direction angle θ0 = 90°. (From Godara, L.C., IEEE Trans.
Antennas Propagat., 38, 1631–1635, 1990. ©IEEE. With permission.)
FIGURE 3.11
PN(w(n)) vs. the iteration number for a 10-element linear array with one-half wavelength spacing. Two interfer- ences: θ1 = 98°, p1 = 1, θ2 = 72°, p2 = 100, σ2n= 0.1, look direction angle θ0 = 90°. (From Godara, L.C., IEEE Trans.
Antennas Propagat., 38, 1631–1635, 1990. ©IEEE. With permission.)
algorithm increases. Thus, the RLS algorithm used in the present form is sensitive to the look direction signal. On the other hand, this is not the case for the improved LMS algorithm. Performance of the improved LMS algorithm improves as the signal power is increased, and in the presence of a strong signal it performs much better than the RLS algorithm, both in terms of convergence and the output SNR. See for example, the plots in Figure 3.12 where the input signal power is 30 dB more than the uncorrelated noise power.
Figure 3.13 compares the performance of the standard LMS algorithm, recursive LMS algorithm, and improved LMS algorithm. The noise field and array geometry used for this example are the same as those used in previous examples. The input signal power is 30 dB more than the uncorrelated noise power and the gradient step size is 0.00005. It is clear from Figure 3.13 that the output noise power of the processor at each iteration is less when the recursive algorithm and the improved algorithm are used in comparison to the output noise power using the standard algorithm. A large fluctuation in the output of the processor using the standard algorithm in comparison to the other two algorithms indi- cates the sensitivity of this algorithm to the look direction signal. A comparison of the recursive LMS and improved LMS show that the latter performs better, both in terms of the amount of the noise and its variation as a function of iteration number.