Problem with Matlab solver

I have the following expression (transfer function):

H(w) = 0.1/(1 - 0.9exp(-jw))

where w is a symbolic variable (omega).

I'm trying to solve the following expression for w:

|H(w)| == 1/sqrt(2)

Solving this by hand I believe the answer should be 0.105, but I cannot get this answer. I've tried adding assumptions for w is real and w>0

I've tried the following commands:

solve(abs(H)==1/sqrt(2),w)

and

solve(H^2==1/2,w)

With no luck. Any assistance would be greatly appreciated.

Thanks!

Answer 1

[Carlos Felipe Rengifo]

Hi, You can solve the equation without using symbolic math. The answer can be find in a numerical way:

H  = @(w) 0.1/(1 - 0.9*exp(-1i*w));
F  = @(w) (abs(H(w))-1/sqrt(2))^2;
Wo = fminbnd(F,0,pi);

As the function H has a period of two times pi, there are infinite solutions: Wo+2*pi, Wo+4*pi, ....

If you really need to use symbolic math, the following sentences give the right solution in Matlab '9.4.0.885841 (R2018a) Update 3'

syms w real;
H   = 0.1/(1 - 0.9*exp(-1i*w));
Sol = double(solve(H'*H == 1/2,w));
Sol(Sol < 0) = [];