Python- FM Modulation

Question

I am trying to Frequency modulate a sine signal using Python. I have written the following code that should do it:

def generateSignalFM(time_vector,data):
    TWO_PI = 2 * np.pi
    fc = 100
    b = 15
    data = np.sin(TWO_PI * 1 * time_vector)
    fm = np.sin(TWO_PI * (fc + b * data) * time_vector)

    plot_graph2d(time_vector,data,time_vector,fm)

def plot_graph2d(t1,data1,t2,data2):

    plt.plot(t2,data2)
    plt.xlabel("Time(s)")
    plt.ylabel("Amplitude")
    plt.plot(t1,data1)
    plt.legend(['FM Signal', 'Original Signal'])
    plt.show()

However this is the result I'm getting back:

As you can see it does work, but it is not synchronized, the lowest frequency of the sine wave does not appear below the lowest frequency of the FM signal.

Can anyone explain why this is happening?

Tying to FM an audio file:

Answer 1

1. Input to sine should be phase, not frequency
2. The * t is only to apply to fc per 1

Corrected:

Code

import numpy as np
import matplotlib.pyplot as plt

#%% Generate #################################################################
t = np.linspace(0, 1, 2048, 0)
fc = 200
b = 15
data = -np.cos(2*np.pi * 1 * t)
phi = fc*t + b * data
fm = np.sin(2*np.pi * phi)
    
#%% Plot #####################################################################
plt.plot(t, fm)
plt.xlabel("Time(s)")
plt.ylabel("Amplitude")
plt.plot(t, data)
intended_freq_approx = np.hstack([0, *np.diff([data])])
intended_freq_approx *= np.abs(data).max() / np.abs(intended_freq_approx).max()
plt.plot(t, intended_freq_approx)
plt.legend(['FM Signal', 'Original Signal', '~Intended Freq'])
plt.show()

#%% Bonus ####################################################################
from ssqueezepy import ssq_stft
from ssqueezepy.visuals import imshow

Tx = ssq_stft(fm)[0][::-1]
imshow(Tx, abs=1, title="abs(SSQ_STFT)", ylabel="frequency", xlabel="time", 
       ticks=0)

Answer 2

The answer given by OverLordGoldDragon was close but not quite right. As can be seen from https://en.wikipedia.org/wiki/Frequency_modulation, the phase requires the time integral of the signal, not just the signal itself. I believe the correct implementation looks like,

import numpy as np
import matplotlib.pyplot as plt

#%% signal generation
fn = 500 # Nyquist Frequency
fs = 2*fn # sampling frequency
t = np.arange(0, 10, 1/fs) # time axis

f_sig = 0.1 # base signal frequency
sig = np.cos(2*np.pi*f_sig*t) # base signal

#%% modulation
fc = 10 # carrier frequency
k = 0.05 # sensitivity
phi = 2*np.pi*fc*t + k*np.cumsum(sig) # phase

sig_mod = np.cos(phi) # modulated signal

#%% plotting
fig, ax = plt.subplots(2, 1, num=0, clear=True, sharex=True)

ax[0].set_title('Signal')
ax[0].plot(t, sig)

ax[1].set_title('Modulated Signal')
ax[1].plot(t, sig_mod)
ax[1].set_xlabel('Time')

Answer 3

This should really be a comment on the accepted answer, but I lack the reputation to do so.

@OverLordGoldDragon, what is the motivation behind multiplying the whole phi by $2\pi$, as opposed to only multiplying its first half (fc*t) by $2\pi$? It seems that this just boosts the modulation index b by a relatively arbitrary amount. This isn't too bad when b is just an uninterpretable control parameter for artistic applications (e.g. sound synthesis), but if your intention includes demodulation (e.g. FM radio transmission), the modulation index is not a free parameter, but depends on the modulator frequency and amplitude.