How can I generate sinusoidal signal with a time-varying frequency?

Hi. I want to generate time-varying frequency signal.

I have frequency data per every time sample.

For example, for a 1 second time duration signal with a sampling frequency of 100 Hz, I have 100 frequency values.

The picture above is showing the frequency data per time sample that I actually have. It is a signal of 0.85 seconds sampled at 100 kHz.

I want to get a spectrogram by making sound data from this value, but something goes wrong.

Down below is the code I made and the result. I'm not sure where I went wrong in the code.

clc;clear all;close all;

fs = 100e3; % 100kHz sampling frequency
ts = 1/fs;

Time = ts:ts:84986*ts;   % 0.84986 sec 
load('Freq.mat');  % 84.986 frequency value

Signal = cos(2*pi*Freq.*Time) + 1j*sin(2*pi*Freq.*Time);   % generating signal

[s,ff,ttt] = spectrogram(y,1024,800,1024,fs);
s = abs(s);

%% ploting spectrogram of Signal

X = figure;
axes1 = axes('Parent',X); 

imagesc(ttt,ff,s);
set(axes1,'YDir','normal');
axis([ttt(1) ttt(end) 4e3 17e3]);xlabel('sec');ylabel('Hz');

The above spectrogram figure is the result of the generated spectrogram. As you can see, the signal was generated a lot different from the frequency data I had.

Please help. Thanks.

Answer 1

[Chunru]

fs = 100e3; % 100kHz sampling frequency
ts = 1/fs;
Time = ts:ts:84986*ts;   % 0.84986 sec 
load('Freq.mat');  % 84.986 frequency value

figure;
plot(Time, Freq);

You need to integrate the instantaneous fre quency over time:

where is the time-varying phase and is the instantaneous frequency. The signal can be then expressed as:

phase = 2*pi*cumsum(Freq)/fs;
% Signal = cos(2*pi*Freq.*Time) + 1j*sin(2*pi*Freq.*Time);   % generating signal
Signal = exp(1j*phase);

[s,ff,ttt] = spectrogram(Signal,1024,800,1024,fs);
s = abs(s);
%% ploting spectrogram of Signal
X = figure;
axes1 = axes('Parent',X); 
imagesc(ttt,ff,s);
set(axes1,'YDir','normal');
axis([ttt(1) ttt(end) 4e3 17e3]);xlabel('sec');ylabel('Hz');

[인성 이]

Thank you so much for your kind reply. It really helped me a lot.