Through-Zero Pulse-Width Modulations

Through-Zero Pulse-Width Modulations

Demonstration 2: Positive and Negative Relative Frequency Representations

Comparisons and other operations involving relatively periodic waveforms at different frequencies are quite common in communications and signal processing systems. Here a view in terms of "positive and negative relative frequency" is presented which NRI finds useful in the design and analysis of modulation spectral, aliasing, and symbolic dynamics.

Consider two time-varying sine waves, which to begin are assumed to be of unit amplitude and zero phase but with two different (temporal) frequencies of oscillation f₁ and f₂:

a(t) = sin(2πf₁t)
b(t) = sin(2πf₂t)

If both sine waves are sounded simultaneous in an acoustic or linear audio mixing environment, these are signals are simply added together.

If each frequency is well within the audio range between 16Hz and 16kHz and the two frequencies are sufficiently far apart, the human ear perceives these as two separate tones, especially if the ratio of the frequencies is not an integer. (If the ratio is a small integer, the ear may perceive these as harmonics of a more complex tone; the ear may also do this for ratios of frequencies that are ratios of small integers.[1])

If the frequencies are within 15-20 Hz of one another, the ear perceives these as one tone with a time-varying amplitude. The frequency of the time-varying amplitude is called the "beat frequency" well known to musicians in fine-tuning their instruments. Mathematically the trigonometric sum and product relations ("Prosthaphaeresis formula") [2])



naturally provides the relevant equivalent representation. (Note since cos(-x) = cos(x), A and B can be freely exchanged.) Applying one obtains



The first term is a sine wave oscillatory at the average of f1 and f2. The second term is the amplitude variotion which oscillates at half the difference of the two frequencies.

Because the human ear doesn't perceive the low frequency reversal of sign as the cosine term crosses zero, the amplitude variation is perceived as



which repeats its cycle twice as fast as



This is shown in the plot below.



Since cos(x) = cos(-x), one also has cos (x) = cos|x|. Thus the perceived beat frequency is



or the simple size of the difference in the frequencies of the two sine waves.

The animation below shows two sine waves of differing frequencies; here f1 = 3.0 (red) and f2= 2.5 (blue)



The animation below compares these to a sine wave (purple) whose frequency is the average of these, in this case with a frequency of (3+2.5)/2= 2.75. Using this waveform as a reference, the waveform with higher frequency (red) is seen to change more frequently, and the waveform with the lowest frequency (blue) is seen to change less frequency in time.



Quantitatively, using the average frequency as a reference, one finds





Thus f₁ is a positive frequency increment from the reference frequency and f₂ is a negative frequency decrement (of the same size) from the reference frequency.

Thus, one can say qualitatively that the higher-frequency waveform have f₁=3.0 (red, has a "positive relative frequency" with respect to the average frequency (purple), while the lower-frequency waveform, here f₂ =2.5 (blue), has a "negative relative frequency" with respect to the average frequency (green). Note that sin [2π(-f)t]= sin [2π(-t)f] so that negative frequency can be represented as a time reversal. Thus, if one treats the average frequency as the observation point graphically one may obtain the following view.



This is exactly what is observed when applying a strobscope to rotating fan blades. If the stroboscope flashes at slightly high frequency than the rotational frequency of the fan blades, the strobed blades appear to rotate slowly in one direction, while if the strobe light flashes at a slighty lower speed than the rotational speed of the fan blades, the strobed fan blades appear to rotate slowly in the opposite direction.

For subsequent discussions, there is no need for the fixed reference (purple) waveform, so the following direct representation (with reference to the implicit average frequency) is used.



REFERENCES

[1] Helmholtz, Herman L. F., Sensations of Tone as a Physiological Basis for the Theory of Music, Dover Publications, Inc., New York, 1954; ISBN 0-486-60753-4.

[2] Beyer, William H, CRC Standard Mathematical Tables, CRC press, Inc, Boca Raton, 1987 ISBN: 0-8493-0628-0 (see also http://mathworld.wolfram.com/ProsthaphaeresisFormulas.html).