I don't think anyone here has really answered your question. In this case, the sound is "focused" using phased arrays. The face of the audio spotlight has multiple transducers:
Flickr
The same signal is output from each of them, but delayed slightly by different amounts, so that the wavefronts all reach the same point in front of the device at the same time. This "virtual focus" is called beamforming.
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This is how modern radars focus their beams, too. Instead of spinning a satellite dish around, they have lots of little elements that don't move, but the signals are delayed to produce different beam shapes.
A single measurement like this has a lot of noise on it - and random signal is always going to have some random correlation. You should definitely not pay too much attention to the stuff that is in the tail of the correlation distribution - it's all noise.
The fact that the built in function does not produce negative values is related to you only looking at the first 10 bins (which is the default). If you plotted the full range, it would go negative just like the others - use
plt.acorr(f, maxlags=None)
Incidentally, it does return all the values in four variables - if you do
result = plt.acorr(f, maxlags=None)
you can examine result[0] and result[1] to give you the values the function calculated - you will find it's similar (identical) to the one you get using other functions. The reason the plot is different, as I said, is that you are not looking at the same range of lags.
Let me demonstrate a few principles. I will be generating some white noise, filtering it with the response of an RC circuit, and plotting the autocorrelation multiple times. We are going to learn how to create white noise (and how to filter it); that the autocorrelation will not be the same twice; that it can go negative, even if you take the mean of 20; and that the measured time constant will vary by quite a bit. Bottom line - you will come away with a better feeling for this. There is much more to learn...
Here is the code:
# white noise
import numpy as np
import matplotlib.pyplot as plt
import math
from scipy import stats
def autocorr(f):
temp = np.correlate(f, f, mode='full')
mid = temp.size/2
return temp[mid:]
plt.close('all')
# simulate a simple RC circuit
# V = 1/jwC/(R+1/jwC) = 1/(jwRC+1)
C = 1e-6 # 1 uF
R = 1.0e3 # 1 kOhm
RC = R*C
fs = 100000 # sampling frequency
fc = 1.0 / RC # cutoff frequency
repeats = 100 # number of times experiment is repeated
ns = 10000 # number of samples per repeat (1/10th of a second)
nplot = 500 # number of samples in autocorrelation to plot
t = np.arange(ns)/(1.0*fs) # time corresponding to each sample
fr = fs / (2.0 * ns) # resolution per bin in the FFT
freq = np.arange(ns) * fr # frequency bins in FFT
response = np.divide(1, 1j * 2 * math.pi * freq * RC + 1) # complex response per bin
# plot response function:
fn = 4* np.floor(fc / freq[1]) # two octaves above 3dB point
plt.figure()
plt.subplot(2,2,1)
plt.plot(freq[:fn], np.abs(response[:fn]))
plt.xlabel('frequency')
plt.title('Filter response: amplitude')
plt.subplot(2,2,2)
plt.plot(freq[:fn], np.angle(response[:fn], deg=1))
plt.title('Filter response: phase')
plt.ylabel('angle (deg)')
plt.xlabel('frequency')
fa = [] # array where we will store the filtered results
plt.subplot(2,2,3)
for j in range(repeats):
e = np.random.normal(size=ns)
# take Fourier transform:
ff = np.fft.fft(e)
# roll off the frequencies
fr = ff * response
# take inverse FFT:
filtered = np.fft.ifft(fr)
ac = autocorr(filtered)
acNorm = np.abs(ac / ac.max()) # <<< using abs function here: get amplitude
fa.append(acNorm)
plt.plot(acNorm[:nplot])
plt.xlabel('correlation distance')
plt.title('Autocorrelation: result of %d samples'%repeats)
plt.show()
# calculate the mean of the repeated samples:
repeatedMean = np.mean(fa, axis=0)
plt.subplot(2,2,4)
plt.semilogy(repeatedMean[:nplot])
plt.title('mean of %d repeats'%repeats)
plt.xlabel('correlation distance')
plt.show()
# fit an exponential to the data
# anything beyond the first 20 points does not make much sense - contains no useful information
# fit an exponential to the data
# anything beyond the first 20 points does not make much sense - contains no useful information
nfit = fs*RC # <<<< added
slopes = []
plt.figure()
for j in range(repeats):
slope, intercept, r_value, p_value, std_err = stats.linregress(t[:nfit], np.log(fa[j][:nfit]))
plt.plot(t[:nfit], fa[j][:nfit])
slopes.append(slope)
plt.title('autocorrelation used for fitting') # <<< fixed typo
plt.xlabel('correlation time')
plt.show()
print 'mean time constant is %.2f ms'%np.mean(np.divide(-1000,slopes)) # <<< changed
print 'standard deviation is %.3f ms'%np.std(np.divide(1000, slopes)) # <<< changed
And here are the plots it produces:
The time constant derived from these fits (inverse of slope) averaged over 100 fits was 1.01 ± 0.15 ms - consistent with the time constant of 1.0 ms that was implied in the choice of components in my code (1 kOhm and 1 uF).
Play around with the code - and let me know if this clears things up for you.
Best Answer
I think a simpler explanation than some kind of sound filtering as a result of the speed of rotation is that when you hit the coin with more force (resulting in a faster speed of rotation), it increases the amplitude of the coin's vibrations, enabling higher modes of vibration, and thus changing the spectrum. I'm skeptical about your claim that the spectrum is more uniform unless you've done a spectral analysis of the sound. It's difficult to draw specific relations between the timbre of a sound and the specific qualities of the sound's spectrum, but generally a "mellow" sound is one with a lower amount of high overtones relative to its low ones, whereas brighter, more piercing, or grittier sounds have a higher amount of the high overtones. When I flip my coin, I hear a fairly bright sound, so I would predict more of the high overtones.
I'd expect the coin would vibrate in modes similar but not identical to those pictured on this Wikipedia page.
Another note is that if you did a spectrum analysis on the coin's sound, it would be non-harmonic e.g the important frequencies in the spectrum would be non-integer multiples of the lowest frequency. This is a property of most percussion instruments, because their different modes of vibration don't vibrate in integer multiples of the lowest overtone.
I hope I've answered the question sufficiently.