Simple Phase Shift for Removing Instrument Response - matlab

Question

So I have a simple program where I want to shift a simple sine function pi/2. Now I knwo this would be extremely easy by just inserting a bias (i.e. A*sin(2*pi*frequency + bias). But this program is a simple way to test a theory. I need to shift complicated magnetic data, but the shift is frequency dependent. So to figure out how to do that I just want to shift this sin wave by a set shift, but I want to do it in the frequency domain. While the code below doesn't show any errors, it does not shift the data properly and effects magnitude. Code is below. Thank you!
clear all
time = 1:0.01:2*pi; %Create time vector
mag = sin(2*pi*time);
Y = fft(mag); %transform
Pnewr = pi/2;
%t = angle(mag);
t = imag(Y);
%t_fin = t-Pnewr; %Subtract the pahse delay from the original phase vector
R=real(Y);
I=t;
k = I./R
Phi = tan(k);
PhiFinal = Phi-Pnewr;
PhiFinal = PhiFinal'
IFinal = R * atan(PhiFinal);
spec=complex(R,IFinal);
Finalspec = ifft(spec); %Invert the transform
Final = Finalspec;
plot(time,mag);
hold on
plot(time,Final,'r')
grid on

Answer 1

For one thing you are not recombining imaginary and real components properly, since
PhiFinal = PhiFinal'
IFinal = R * atan(PhiFinal);
is effectively a dot product, not an element by element product. In general it doesn't look like you are making correct use of the complex relationships. The following generates a clean phase shift:
time = 1:0.01:2*pi;
mag = sin(2*pi*time);
Y = fft(mag); %transform
Pnewr = pi/2;
R = real(Y);
I = imag(Y);
It = abs(Y);
% It = sqrt(I.^2 + R.^2);
Phi= angle(Y); % <-- using matlab function, equivalent to `Phi = atan2(I, R);`
k = I./R;
Phi0 = atan(k);
figure, subplot(121), plot(Phi0,Phi,'.'), xlabel('Phi from tan'), ylabel('Phi from ''angle'''), grid on, axis('tight')
PhiFinal = Phi-Pnewr; % <-- phase shift
IFinal = It .* sin(PhiFinal);
RFinal = It .* cos(PhiFinal);
spec= RFinal + 1i*IFinal;
Final = ifft(spec); %Invert the transform
subplot(122)
plot(time,mag);
hold on
plot(time,real(Final),'r')
plot(time,imag(Final),'r:')
grid on
axis('tight')
legend('Initial','Final'),xlabel('t'), ylabel('I')
mag*mag' % <-- check that total power is conserved
Final*Final'
These figures show the phase as computed with tan vs matlab's angle (which uses atan2), and the results of the phase shift (right panel):