Yesterday I was looking into calculating fluctuation strength and playing around with some examples. Along the way I discovered how to create files that sound like police sirens. These are sounds with high fluctuation strength.

The Python code below starts with a carrier wave at *f*_{c} = 1500 Hz. Not surprisingly, this frequency is near where hearing is most sensitive. Then this signal is modulated with a signal with frequency *f*_{m}. This frequency determines the frequency of the fluctuations.

The slower example produced by the code below sounds like a police siren. The faster example makes me think more of an ambulance or fire truck. Next time I hear an emergency vehicle I’ll pay more attention.

If you use a larger value of the modulation index β and a smaller value of the modulation frequency *f*_{m} you can make a sound like someone tuning a radio, which is no coincidence.

Here are the output audio files in .wav format:

from scipy.io.wavfile import write from numpy import arange, pi, sin, int16 def f(t, f_c, f_m, beta): # t = time # f_c = carrier frequency # f_m = modulation frequency # beta = modulation index return sin(2*pi*f_c*t - beta*sin(2*f_m*pi*t)) def to_integer(signal): # Take samples in [-1, 1] and scale to 16-bit integers, # values between -2^15 and 2^15 - 1. return int16(signal*(2**15 - 1)) N = 48000 # samples per second x = arange(3*N) # three seconds of audio data = f(x/N, 1500, 2, 100) write("slow.wav", N, to_integer(data)) data = f(x/N, 1500, 8, 100) write("fast.wav", N, to_integer(data))

## More frequency modulation posts

Note that this code works for smooth modulating functions. You are smoothly distorting the time axis and calculating cos(2pi f*distorted_time).

If, for example, you want a 2-tone siren, then the modulating function would be a square wave, and the code above will return a wave with phase discontinuities at each tone change.

In general, to produce a frequency modulated wave, you need to integrate the instantaneous frequency in order to get a continuous phase, and then calculate cos(phase).

Very cool. How close/distant is this to pulse code modulation ?

I think you might be mistaken on the “most sensitive” part. I’ve heard 4000 before, and a quick googling suggests 2000-5000.

The difference between those numbers isn’t so much on a log scale, and perception is kinda flat in that range.

The second one sounds like the small UFOs in Asteroids

Fredrik,

I too generated soundless .wav files. Examination showed that it was because all zeroes were getting written to them. I fixed this by making sure that the data generated was using floats:

x = arange(3*N, dtype=float32)

I’m using Python 2, and numpy.arange() produces integers by default.

This is very cool! The fast one sounds like a little toy :) we implemented a similar code in Matlab (analyzing the Fourier Transforms and spectra), in case someone is interested:http://www.behindthesciences.com/signal-processing/frequency-modulation-fm-matlab-tutorial-part2

Thank you!