Multiplying audio signals

We have been routinely adding audio signals together, and multiplying them by slowly-varying signals (used as amplitude envelopes for example) since chapter 1. In order to complete our understanding of the algebra of audio signals we now consider the situation where we multiply two audio signals neither of which may be assumed to change slowly. The key to understanding what happens is the:

COSINE PRODUCT FORMULA

$$\cos(a) \cos (b) = {1 \over 2} { \left [ \cos (a+b) + \cos(a-b) \right ] }.$$

To see why this is true, recall the formula for the cosine of a sum of two angles:

$$\cos(a+b) = \cos(a)\cos(b) - \sin(a) \sin(b)$$

and evaluate the right hand side of the cosine product formula; it immediately collapses to the left hand side.

We can use this formula to see what happens when we multiply two SINUSOIDS (page ):

$${\cos(\alpha n + \phi) \cos (\beta n + \xi)} =$$

$$= {1 \over 2} { \left [ {\cos \left ( (\alpha + \beta) n +... ...t ( (\alpha - \beta) n + (\phi - \xi) \right ) } \right ] } .$$

In words, multiply two sinusoids and you get a result with two partials, one at the sum of the two original frequencies, and one at their difference. (If the difference $\alpha-\beta$ happens to be negative, simply switch $\alpha$ and $\beta$ in the formula and the difference will then be positive.)

This gives us a very easy to use tool for shifting the component frequencies of a sound, which is called ring modulation, which is shown in its simplest form in Figure 5.2. An oscillator provides the modulating signal, which is simply multiplied by the input. The term ``ring modulation" is used more generally to mean multiplying any two signals together, but here we'll just consider using a sinusoidal modulating signal.

Figure 5.2: Block diagram for ring modulating an input signal with a sinusoid.

Figure 5.3 shows the result of multiplying a sinusoid of angular frequency $\alpha$ and amplitude $a$, by another of angular frequency $\beta$ and amplitude 1:

$$\left [ a \cos (\alpha n) \right ] \cdot \left [ \cos (\beta n) \right ] .$$

(For simplicity we're omitting the phase term here.) The result is shown as a spectrum. The left-hand side sinusoid appears as a single frequency, $\alpha$. The product of the two sinusoids has two component frequencies. Each of the two appears at an amplitude of $a/2$.

Figure 5.3: Sidebands arising from multiplying two sinusoids of frequency $\alpha$ and $\beta$. Part (a) shows the case where $\alpha > \beta > 0$; part (b) shows the case where $\beta > \alpha$ but $\beta < 2\alpha$, so that the lower sideband is reflected about the $f=0$ axis. Part (c) shows the special case $\alpha =\beta$; in this one special case the amplitude of the zero-frequency sideband depends on the phases of the two sinusoids. In part (d), $\alpha$ is zero, so that only one sideband appears.

In the special case where $\alpha =\beta$, the second (difference) sideband has zero frequency and its amplitude depends on the relative phases of the two multiplicands. In this case we replace $\beta$ by $\alpha$ in the product formula to get:

$${\cos(\alpha n + \phi) \cos (\alpha n + \xi)} =$$

$$= {1 \over 2} { \left [ {\cos \left ( 2 \alpha n + (\phi +... ...ight ) } + {\cos \left ( \phi - \xi \right ) } \right ] } .$$

The second term has zero frequency; its amplitude ranges from $+1/2$ to $-1/2$ as the phase difference $\phi - \xi$ varies from $0$ to $\pi$ radians. This situation is shown in part (c) of Figure 5.3.

We can use the distributive rule for multiplication to find out what happens when we multiply signals together which consist of more than one partial each. For example, in the situation above we can replace the signal of frequency $\alpha$ with a sum of several sinusoids:

$${a_1} \cos({\alpha _1} n ) + \cdots + {a_k} \cos({\alpha _1} k ) .$$

Multiplying by the signal of frequency $\beta$ gives partials at frequencies equal to:

$$\alpha_1 + \beta, \alpha_1 - \beta, \cdots, \alpha_n + \beta, \alpha_n - \beta .$$

As before if any frequency is negative we take its absolute value since

$$\cos(-\omega n) = \cos(\omega n) ,$$

so that any signal with a negative frequency is equal to one with a positive frequency.

Figure 5.4 shows the result of multiplying a complex periodic signal (with several components tuned in the ratio 0:1:2:$\cdots$) by a sinusoid. Both the spectral envelope and the component frequencies of the result transform by relatively simple rules.

Figure 5.4: Result of ring modulation of a complex signal by a pure sinusoid: (a) the original signal's spectrum and spectral envelope; (b) modulated by a relatively low modulating frequency (1/3 of the fundamental); (c) modulated by a higher frequency, 10/3 of the fundamental.

The resulting spectrum is essentially the original spectrum combined with its reflection about the vertical axis. This combined spectrum is then shifted to the right by the modulating frequency. Finally, if any components of the shifted spectrum are still left of the vertical axis, they are reflected about it to make positive frequencies again.

In part (b) of the figure, the modulating frequency (the frequency of the sinusoid) is below the fundamental frequency of the complex signal. In this case teh amount of shifting is small, so that re-folding the spectrum at the end almost places the two halves on top of each other. The result is a spectral envelope roughly the same as the original (although half as high) and a spectrum twice as dense.

A special case, not shown, is modulation by a frequency exactly half the fundamental. In this case, pairs of partials will fall on top of each other, and will have the ratios 1/2 : 3/2 : 5/2 :$\cdots$ - an odd-partial-only signal an octave below the original. This is a very simple and effective octave divider for a harmonic signal, asuming you know or can find its fundamental frequency. If you want even partials as well as odd ones (for the octave-down signal), simply mix the original signal with the modulated one.

Part (c) of the figure shows the effect of using a modulating frequency much higher than the fundamental frequency of the complex signal. Here the unfolding effect is much more clearly visible (only one partial, the leftmost one, had to be reflected to make its frequency positive.) The spectral envelope is now widely displaced from the original; this displacement is a more strongly audible effect than the relocation of partials.

In another special case, the modulating frequency may be a multiple of the fundamental of the complex periodic signal; in this case the partials all land back on other partials of the same fundamental, and the only effect is the shift in spectral envelope.