Creating and recovering a frequency shift in time domain

I am trying to simulate and process Doppler signal. 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 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 wihtout 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

%% Task1:

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


%% Task 2
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.