Creating and recovering a frequency shift in time domain in MATLAB

• MATLAB
I am trying to simulate and process the Doppler signals. My main problem is a little more complex so I am only posting a simple version of it. Task1: I have a time-domain signal with the velocity of the target as mu. I need to change the velocity to mu cos(theta) where theta is a vector from 0 to 2 pi with the same number of points that I have in the time domain signal. So, every point in the original time-domain signal should have this change. This means that the Doppler frequency (velocity) is not constant with time. Task2: Then, after I get this signal, I want to recover the original spectrum (with constant velocity) but without knowing mu. I just need to know the phase and then modify it somehow to get the real signal.

 clear;
close all;

lambda = 0.03;
M = 61;
hs = 128;

N = hs * M; % Number of points in the time axis

mu = 4; % Doppler velocity

PRT = 1e-3;  % Time step

t1 = 0:PRT:(N - 1)*PRT; % Time axis

s_ = exp(1j .* 4 * pi ./ lambda .* mu .* t1); % Original Time domain signal

s_f = fftshift(fft(s_)); % FT of the original time domain signal

v_amb = 7.5;

v_axis = linspace(-N/2, N/2-1, N)/N .* 2 * v_amb; % Velocity axis

figure; plot(v_axis, db(abs(s_f))); title('Original spectrum') % Original spectrum
BW = 1*pi/180;
p0 = 0*pi/180;
p1 = 360*pi/180;
th = linspace(p0, p1, N); % Angle axis for cos(theta)

s_man_ = abs(s_) .* exp(1j .* (angle(s_)) + 1j * 4*pi/lambda * mu * (cos(th) - 1) .* t1); % Modified time domain signal with cos(theta)

s_man_f_full = fftshift(fft(s_man_)); % FT of the modified time domain signal

s_man_comp = abs(s_man_) .* exp(1j .* (angle(s_man_)) ./ cos(th)); % Reconstruction of the original signal in time domain

s_man_comp_f = fftshift(fft(s_man_comp)); % FT of the reconstructed signal

figure(1); hold on; plot(v_axis, db(abs(s_man_f_full)));

hold on; plot(v_axis, db(abs(s_man_comp_f)));

legend({'HD spectrum', 'Manipulated with cosine Spectrum', 'Compensated from Manipulated spectrum'});  grid on;

The figure is shown below. The Yellow plot should resemble the blue blot but it doesn't. Any idea what I do wrong here? The red plot also should show multiple spikes in the plot and not like a noise signal like this I suppose.

Delta2