Hence, we can properly choose the modulation sequences to improve the performance of estimation.
We now discuss how to design the modulation sequences to combat the effect of noise on DOA estimation.
The final feature of the proposed method is that the performance of the estimation algorithm depends on the choice of the modulation sequences to resist the noise effects.
Since q is formed from the modulation sequences, we need to design the modulation sequences to minimize ∥q∥2 and the effect of the resulting perturbation term.
The third feature of the proposed method is that it provides a design of the modulation sequences to minimize the effect of noise on DOA estimation and thus improves the accuracy of the solution.
Particularly in [9, 10], the signal sources transmitted are first multiplied at a symbol rate by known amplitude-variation sequences, called modulation sequences, to aid symbol recovery or channel estimation at the receiver.
Use the autocorrelation matrices in (3.27) to form V 0, V 1, ⋯, V M−1, and use the designed modulation sequences (3.23) to form W. 4.
Use the autocorrelation matrices in (3.27) to form V 0, V 1, ⋯, V M−1, and use the designed modulation sequences (3.23) to form W. Obtain the matrices X 0, X 1, ⋯, X M−1 using the least squares solutions in (3.7) with the aid of the matrices V 0, V 1, ⋯, V M−1, and W obtained from the previous step.
where w i = c i ( 1 ) 2 c i ( 2 ) 2 ⋯ c i ( P ) 2 T ∈ R P is the i th column of W, ∀i = 1, 2, ⋯, N. If the modulation sequences can be selected to meet the orthogonality condition (3.19), the effect of noise is completely eliminated, but this is impossible since the entries of w 1, w 2, ⋯, w N and 1 P are positive.
Therefore, we seek to choose the modulation sequences such that 1 P is as close to being orthogonal to w i as possible, for i = 1, 2, ⋯, N. To this end, we define the following correlation coefficients: γ i = w i T 1 P ∥ w i ∥ 2 ∥ 1 P ∥ 2, i = 1, 2, ⋯, N, (3.20).
However, to make the matrix W be full column rank such that the least squares solutions (3.7) can be computed, the peak locations of the N modulation sequences in one period need to be distinct with one another, i.e., m k ≠ m l for all k ≠ l.
