Frequency Modulation

Pre-view

I’m glad that I still have time to write this blog in spite of so much things i need to do. The pressure of my grade is overweighted. But I feel lucky to have a little spare time to do something meaningful rather just playing games and wasting it. So back to the topic. In this blog, i will introduce a more useful modulation method called Frequency Modulation.

Definition

Unlike AM modulation which use the amplitude of the carrier signal to maintain the amplitude of modulation signal, which enable to transmit the infomation of modulation signal in a high frequency range(carrier signal’s frequency), FM modulation add the modulation signal to carrier signal’s frequency. So although the signal still transmit in high frequency range, the signal frequency is changed by the value of modulation signal. In this way, the signal carries the infomation of modulation signal in its frequency.

Math Theory

So, what is the official definition of FM?

Let’s think about a modulation signal and a carrier signal. Its formula is :

$$ Modulation-Signal = Am * \cos(2\pi f_{m}t) \tag{1}$$

$$ Carrier-Signal = Ac * \cos(2\pi f_{c}t) \tag{2}$$

And the modulation process is add modulation signal to the frequency of carrier signal.

$$ FM(t) = A \cos \left[ \omega_c t + A_c \int_0^t (cos2\pi f_c\tau )d\tau \right] = A\cos \left[ (\omega_c + Ac*cos(2\pi f_ct)t + \theta) \right] \tag{3}$$

If you look more close to it, you will find that the integeration of frequency is the phase

$$ \frac{d \left[ A_c \int_0^t (cos2\pi f_c\tau )d\tau \right]}{dt} = A_c cos2\pi f_ct \tag{4}$$

which means if you add a phase varable $ \theta(t^2) $ to the phase of carrier signal, it is not only a FM but a PM also.

$$\beta = \frac{A_c |cos2\pi f_ct|_{max}}{W} \tag{5}$$

And the $ A_c $ represents the frequency modulation sensitivity. The $W$ is the bandwidth (or the highest frequency) of the baseband signal (modulating signal).

MATLAB Simulation

The Simulation code of MATLAB is as followed :

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% FM调制解调过程
clc;clear;
% 基本参数
fm = 1e2;                   % 基带信号频率 -> modulation signal
T = 5;                      % 信号时长
fs = 2e4;                   % 采样频率 奈奎斯特采样定理为最大频率的两倍，这里取20倍为了绘制更多的细节，让时域信号更平滑
dt=1/fs;                    % 时间采样间隔，采样频率的倒数
N=T/dt;                     % 采样点个数，总时长除以采样间隔
t = (0 : N-1) * dt;         % 采样点的时间序列，作为横坐标

% *********Modulated Signal in Time domain*******
Am = 1;                             % 基带信号幅度 
modu_s = Am*cos(2*pi*fm*t);         % 基带信号
figure;                             % 绘制第一幅图
plot(t,modu_s);                     % 时间t为横坐标，基带信号mt为纵坐标绘图，线宽为2
xlabel('Time');                     % 横坐标标注
ylabel('Amp');                      % 纵坐标标注
title('Base signal');               % 图标题标注
axis([0 0.1 -1.1 1.1]);             % 横纵坐标范围设置
% 绘制一条从(0,0)到(0.1,0)的蓝色实线，线宽为2
line([0,0.1],[0,0],'color','b','Linewidth',2);

% *****Modulated Signal in Frequency domain*****
[mf, msf] = T2F(t, modu_s);
figure;
plot(mf, abs(msf));
title('Modu_s in Freq domain');
xlabel('Frequency Hz');
ylabel('Amp H');
axis([-250 250 0 3]);

% *****Carrier Signal in Time domain*******
fc = 1e3;
A = 1;
carrier_wave = A * cos(2 * pi * fc * t);
figure;
plot(t, carrier_wave);
title('carrier wave');
xlabel('time');
ylabel('Amp');
axis([0 0.01 -1.1 1.1]);
line([0,0.01],[0,0],'color','b','Linewidth',2);

% ******carrier signal in Frequency domain******
[cf, csf] = T2F(t, carrier_wave);
figure;
plot(cf, abs(csf));
title('carrier signal in freq');
xlabel('f Hz');
ylabel('Amp H');
axis([-1200 1200 0 3]);

% FM
Ksfm = 4e3;     % influence 
SFM = A * cos(2*pi*fc*t + (Ksfm*Am)/(2*pi*fm).*sin(2*pi*fm*t) + pi);
figure;
plot(t, SFM, 'LineWidth', 1, 'color', 'r');
xlabel('Time');
ylabel('Amp');
title('Frequency Modulation');
axis([0 0.1 -1.2 1.2])

% Linear Frequency modulate
Ksfm1 = 1e5;
SFM1 = A * cos(2*pi*fc*t + pi*Ksfm1*t.^2);
figure;
plot(t, real(SFM1), 'LineWidth', 1, 'color', 'r');
xlabel('Time');
ylabel('Amp');
title('Frequency Modulation');
axis([0.03 0.08 -1.2 1.2])

for i=1:N-1
    diff_SFM(i) = (SFM(i + 1) - SFM(i)) / dt;
end
diff_SFM = abs(hilbert(diff_SFM));
figure;
plot((1:N-1) * dt, diff_SFM);
title('envelope with DC');
xlabel('Time');
ylabel('Amp');
axis([0 0.1 1000 11000]);
line([0, 0.1], [0, 0], 'color', 'b', 'linewidth', 2);

demodu_s = (diff_SFM / A - 2*pi*fc)/Ksfm;
figure;
plot((1:N-1) * dt, demodu_s);
hold on
plot(t, modu_s, 'r-', 'color', 'b');
title('envelope and raw');
xlabel('Time');
ylabel('Amp');
axis([0 0.1 -1.2 1.2]);
line([0, 0.1], [0, 0], 'color', 'b', 'linewidth', 2);

Above all is the whole content of our blog. I hope it will be of great help to you. See you in the next blog.