Simplifying complex multiplications by means of polar coordinates

I have a complex matrix A of size and another complex matrix P that has same size as A. But abs(P) = ones(size(M,N)) which indicates that P only contains phase related components and since it only contains phase related components it wont effect abs(A) when A.*P is performed. Coming to my problem. I want to reduce the complex multiplications of A.*P to complex additions angle(A) + angle(B). The pseudo code is as follows.

while above the interface
    A = A.*P;
    Accum(i,:) = sum(A,1);
    i = i+1
end

I am stuck in the accum part. I am not getting how to join both absolute and phase without multiplying them before I sum all the M rows.

My approach so far as been follows:

absA = abs(A);
anA = angle(A);
anP = angle(P)
while above the interface
    anA = anA + anP;
    Accum(i,:) = sum(A,1); % I am stuck in the Accum part
    i = i+1
end

[James Tursa]

Getting the phase angles from complex numbers basically involves atan2( ) calculations. You want to replace a complex multiply with two atan2( ) calls and an addition? For what purpose? Is this just for fun or do you think this is somehow going to be faster? How large are M and N?

[D_coder]

The size of M is 5000 and the size of N is 300. The atan2 calls are only twice and not inside the while loop . So basically I want to add the angles instead of multiplying the whole thing, i.e. converting complex multiplications to real addition. However when I do the sum operation, I dont know if there is way to combine both angle and magnitude without any complex multiplication.

[Walter Roberson]

If the magnitudes are all 1 then the sum of the magnitudes should be the number of items being added. So perhaps you just need to sum angles?

But since the angle of any one P is not changing, Accum should come out with angle sum(angle(original A), 1) + iterations * sum(angle(P), 1) and magnitude iterations * size(A, 1)

[D_coder]

The magnitudes are not 1, the magnitude is 1 only for matrix P. Was wondering if we had some function that would do some operation in polar form and then convert back to rectangular form?

[Walter Roberson]

Ok but you are not changing the magnitudes of any A entry so

Accum should come out with angle sum(angle(original A), 1) + iterations * sum(angle(P), 1) and magnitude sum(magnitude(original A), 1)

I think.

Answer 1

[James Tursa]

It looks like your accumulation is trying to sum polar coordinates. You can't do that. I.e., if you have (r1,theta1) and (r2,theta2), you can't directly sum them in the fashion you are trying to do in polar coordinates. You would have to convert them back to cartesian coordinates, do the sum, and then convert back to polar to get your new r and theta values. Or do some combination of Law of Cosines and Law of Sines to get the new polar result. I don't see how you can get your proposed scheme to work.