Can someone point me to a webpage or other resource that shows how to analytically solve the beamformer Unconstrained Array Gain expression in Henry Cox's 1987 IEEE paper "Robust Adaptive Beamforming"?
$$ \max_{\mathbf{w}} \frac{|\mathbf{w}^H\mathbf{d}|^2}{\mathbf{w}^H\mathbf{Q}\mathbf{w}} $$
Cox says:
The well-known solution is $\mathbf{w} = \alpha\mathbf{Q}^{-1}\mathbf{d}$
I'd just like to better understand this by learning how to derive this myself.
A common way is to make use of the Schwarz inequality. First note that:
$$\frac{|w^Hd|^2}{w^HQw} = \frac{|w^HQ^{1/2}Q^{-1/2}d|^2}{w^HQw}$$
Using the Schwarz inequality on the numerator: $$\frac{|w^HQ^{1/2}Q^{-1/2}d|^2}{w^HQw} \leq \frac{(w^HQw)(d^HQ^{-1}d)}{w^HQw} = d^HQ^{-1}d$$
Thus, $$\frac{|w^HQ^{1/2}Q^{-1/2}d|^2}{w^HQw} \leq d^HQ^{-1}d$$
From this, it can easily be seen that the beamformer that achieves equality is $w = \alpha Q^{-1}d$.
Sources:
$$\frac{|w^HQ^{1/2}Q^{-1/2}d|^2}{w^HQw} = \frac{(w^HQ^{1/2}Q^{-1/2}d)^H(w^HQ^{1/2}Q^{-1/2}d)}{w^HQw} = \frac{d^Hww^Hd}{w^HQw} $$
Then, substituting $w = Q^{-1}d$: $$ \frac{d^Hww^Hd}{w^HQw} = \frac{d^HQ^{-1}dd^HQ^{-1}d}{d^HQ^{-1}QQ^{-1}d} = \frac{d^HQ^{-1}dd^HQ^{-1}d}{d^HQ^{-1}d} = d^HQ^{-1}d $$
Thus, this definition of $w$ achieves the maximum possible SNR indicated by the Schwartz inequality.
– Gillespie Feb 12 '21 at 18:37You can solve such a problem using the method of Lagrange multipliers. First note that maximizing the expression in your question is equivalent to minimizing the inverse function:
$$\min_{\mathbf{w}}\frac{\mathbf{w}^H\mathbf{Q}\mathbf{w}}{|\mathbf{w}^H\mathbf{d}|^2}\tag{1}$$
Next note that the solution of $(1)$ is invariant to scaling of $\mathbf{w}$, i.e., replacing $\mathbf{w}$ by $c\cdot\mathbf{w}$ in $(1)$ with an arbitrary scalar constant $c$ will not change the value of the function. So we may as well use a scaling such that $\mathbf{w}^H\mathbf{d}=1$ is satisfied. This scaling corresponds to a unity response for the desired signal. With this constraint, problem $(1)$ can be reformulated as
$$\min_{\mathbf{w}}\mathbf{w}^H\mathbf{Q}\mathbf{w}\qquad\textrm{s.t.}\qquad \mathbf{w}^H\mathbf{d}=1\tag{2}$$
We can solve $(2)$ using the method of Lagrange multipliers by minimizing
$$\mathbf{w}^H\mathbf{Q}\mathbf{w}-\lambda(\mathbf{w}^H\mathbf{d}-1)\tag{3}$$
Formally taking the derivative of $(3)$ with respect to $\mathbf{w}^H$ and setting it to zero gives
$$\mathbf{w}=\lambda\mathbf{Q}^{-1}\mathbf{d}\tag{4}$$
The constraint in $(2)$ is satisfied for
$$\lambda=\frac{1}{\mathbf{d}^H\mathbf{Q}^{-1}\mathbf{d}}\tag{5}$$
From $(4)$ and $(5)$ we finally obtain
$$\mathbf{w}=\frac{\mathbf{Q}^{-1}\mathbf{d}}{\mathbf{d}^H\mathbf{Q}^{-1}\mathbf{d}}\tag{6}$$
Note that the scaling in $(6)$ is optional and the general solution is given by $(4)$.
First, a sketch of the solution for the maximum SINR beamformer problem $$ \text{max}_{\mathbf{w}} \frac{|\mathbf{w}^H\mathbf{d}|^2}{\mathbf{w}^H\mathbf{Q}\mathbf{w}} $$ Start with writing down a functional $$ \mathbf{w}^H\mathbf{Q}\mathbf{w} $$ to be minimized, and a set of constraints. Indeed, the weight vectors w and wH are considered the two independent set of variables when taking derivates with respect to these variables; therefore, the output signal energy, typically written as a squared modulus of the weights-signals coproduct, has to be written down as an analytic function, without computing the norm that takes the square root: $$ |\mathbf{w}^H\mathbf{d}|^2 = \mathbf{w}^H\mathbf{d}·\mathbf{d}^H\mathbf{w} $$ The resulting set of linear constraints is $$ \mathbf{w}^H\mathbf{d} = c \\ \mathbf{d}^H\mathbf{w} = c^* $$ and we have to write down a Lagrangian with two Lagrange multipliers, λ and μ: $$ \mathbf{w}^H\mathbf{Q}\mathbf{w}-λ(\mathbf{w}^H\mathbf{d}-c)-μ(\mathbf{d}^H\mathbf{w}-c^*) $$ Taking the two derivatives of the Lagrangian -- the first, with respect to w, and the second, with respect to wH -- we obtain the expressions for λ and μ, and, substituting these to the constraint expressions, finally arrive at the formula for weights: $$ \mathbf{w}=c\frac{\mathbf{Q}^{-1}\mathbf{d}}{\mathbf{d}^H\mathbf{Q}^{-1}\mathbf{d}} $$ To my surprize, searching a web for "a webpage or other resource that shows how to analytically solve the beamformer" per OP's request, I could find only curtailed, flawed versions of this formula's derivation, a typical document being the course notes Optimal Beamforming, a detailed and useful introduction into the subject in all the other aspects. I even suspect that the OP posted the question with the purpose to broadcast this omission of the learning resource (excuse my awkward attempt to joke).
For now, I can only recommend the learning material on general linear constraint quadratic programming to students interested in the optimal beamforming. For example, refs. https://www.math.uh.edu/~rohop/fall_06/Chapter3.pdf and https://www.cis.upenn.edu/~cis515/cis515-20-sl15.pdf . Only real-valued quadratic forms are considered in these documents, but the main results can be generalized to the complex domain.
