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Playing around with FM synthesis

Preliminaries

var('t')
t 
# Fourier transform plotting function:
# tmin and tmax give the range of values to sample.
# size is the number of sample points:
#     According to the Nyquist–Shannon sampling theorem,
#     this must be at least twice the highest frequency component we wish to include.
# barlimit is the largest frequency plotted.
def plotFT(func, tmin, tmax, size, barlimit):
    delta = (tmax - tmin)/size
    # Create an empty Fast Fourier Transform object
    fft_data = FFT(size)
    # Populate it with data sampled from func.
    for t in range(size):
        fft_data[t] = func(tmin + t * delta)
    # Fourier transform to convert to frequency domain
    # (Modifies fft_data in place.)
    fft_data.forward_transform()
    # Create array for amplitudes
    fft_freq = [0 for k in range(1/delta)]
    # Populate fft_freq with the computed amplitudes
    # (For technical reasons, we only have to compute this for half of the sampled frequencies)
    for f in range(size/2):
        fft_freq[int(f/(size * delta))] = abs(vector(fft_data[f]))
    # Plot results
    return bar_chart(fft_freq[0:barlimit])

The formula

The formula is y=Asin⁡(2πfct+Bsin⁡(2π(fmt)))y = A\sin\left( 2\pi f_{c} t + B\sin(2\pi(f_{m}t)) \right) where fcf_{c} is the carrier frequency, fmf_{m} is the modulating frequency, and BB is the modulation index.

Here's an example with fc=20 Hzf_{c} = 20 \text{ Hz}, fm=2 Hzf_{m}= 2 \text{ Hz}, and B=1B = 1:

f1(t) = sin(2*pi*(20*t) + sin(2*pi*(2*t)))
plot(f1, (t, 0, 1))

Here's the Fourier transform of that:

plotFT(f1, tmin = 0, tmax = 1, size = 100, barlimit = 40)

One can see the "side bands" around 20 Hz. These side bands spread as the modulation index increases.

f2(t) = sin(2*pi*(20*t) + 2*sin(2*pi*(2*t)))
plot(f2, (t, 0, 1))

plotFT(f2, tmin = 0, tmax = 1, size = 100, barlimit = 40)

f3(t) = sin(2*pi*(20*t) + 3*sin(2*pi*(2*t)))
plot(f3, (t, 0, 1))

plotFT(f3, tmin = 0, tmax = 1, size = 100, barlimit = 40)

TODO: Check that the amplitudes are scaled correctly. (Bessel functions.)

In the typical application, the modulating frequency is the same or close to the carrier frequency. If they are the same:

f4(t) = sin(2*pi*(20*t) + sin(2*pi*(20*t)))
plot(f4, (t, 0, 0.2))

plotFT(f4, tmin = 0, tmax = 0.2, size = 500, barlimit = 200)

f5(t) = sin(2*pi*(20*t) + 5*sin(2*pi*(20*t)))
plot(f5, (t, 0, 0.2))

plotFT(f5, tmin = 0, tmax = 0.2, size = 500, barlimit = 200)

What about a modulation frequency that is a perfect fifth away from the carrier frequency?

f6(t) = sin(2*pi*(20*t) + sin(2*pi*(30*t)))
plot(f6, (t, 0, 0.2))

plotFT(f6, tmin = 0, tmax = 0.2, size = 500, barlimit = 200)

f7(t) = sin(2*pi*(20*t) + 5*sin(2*pi*(30*t)))
plot(f7, (t, 0, 0.2))

plotFT(f7, tmin = 0, tmax = 0.2, size = 600, barlimit = 300)