I want to know the frequency domain spectrum of an exponential which is modulated with a sine function that is changing with time.
The time-domain form is,
$$ s(t) = e^{j \frac{4\pi}{\lambda} \mu \frac{\sin(\Omega t)}{\Omega}} $$
Here, $ \mu $, $ \Omega $ and $ \lambda $ are constants.
A quick implementation in MATLAB gives me the following in the frequency/velocity domain.
The code is given below.
clear;
close all;
lambda = 0.03;
mu = 4 * 2/lambda; % Mean Doppler frequency
Omega_rpm = 60; % in RPM
Omega = 2*pi/60 * Omega_rpm; % In rad/s
BW_deg = 1.8; % beam width in degree
BW = BW_deg * pi/180; % beam width in radian
v_amb = 7.5;
PRT = 1e-3;
f_amb = 1/(2 .* PRT);
p0 = 0*pi/180; % start angle
p1 = 360*pi/180; % end angle
N_BW = 1; % Number of beam widths to integrate
M = round((p1 - p0)/(BW * N_BW)); % Number of azimuth points
hs = N_BW * round(BW/Omega/PRT); % hits per scan -> Sweeps in one beamwidth
N = hs * M; % Total number of points in the time axis
th = linspace(p0, p1, N); % All the angles
phi = linspace(th(1), th(end), M); % Angle of the sectors
t1 = 0:PRT:(N - 1)*PRT; % Time axis
ph_ = (2 * pi * mu .* t1);
s_ = (exp(1j .* ph_ .* (sin(eps + Omega .* t1)./(eps + Omega .* t1))));
vel_axis = linspace(-f_amb, f_amb, N); % frequency axis for the entire rotation
s_f = fftshift(fft(s_));
figure; plot(vel_axis*lambda/2, (abs(s_f)), 'LineWidth', 2); grid on;
xlabel('Doppler velocity [ms^{-1}]', 'FontSize', 12, 'FontWeight', 'bold');
ylabel('Spectrum', 'FontSize', 12, 'FontWeight', 'bold');
title('Spectrum function'); grid on;
I want to know what this function should look like in the spectrum domain analytically.