Click here to purchase the entire book in PDF format. Phase vs. Addition If I take any two sinusoidal waves that have the same frequency, but they have different amplitudes, and they're offset in phase, and I add them, the result will be a sinusoidal wave with the same frequency with another amplitude and phase. For example, take a look at Figure 1.10. The top plot shows one period of a 1 kHz sinusoidal wave starting with a phase of 0 radians and a peak amplitude of 0.5. The second plot shows one period of a 1 kHz a sinusoidal wave starting with a phase of $\frac{\pi}{4}$ radians ($45^{\circ}$) and a peak amplitude of 0.8. If these two waves are added together, the result, shown on the bottom, is one period of a 1 kHz sinusoidal wave with a different peak amplitude and starting phase. The important thing that I'm trying to get across here is that the frequency and wave shape stay the same - only the amplitude and phase change. Figure 1.10: Adding two sinusiods with the same frequency and different phases. The sum of the top two waveforms is the bottom waveform. So what? Well, most recording engineers talk about phase. They'll say things like ``a sine wave, $135^{\circ }$ late'' which looks like the curve shown in Figure 1.11. Figure 1.11: A sine wave, starting at a phase of $135^{\circ }$. If we wanted to be a little geeky about this, we could use the equation below to say the same thing: $$y(\alpha) = A \sin(\alpha + \phi)$$ (2.13) which means the value of $y$ at a given value of $\alpha$ is equal to $A$ multiplied by the sine of the sum of the values $\alpha$ and $\phi$. In other words, the amplitude $y$ at angle $\alpha$ equals the sine of the angle$\alpha$ added to a constant value $\phi$ and the peak value will be $A$. In the above example,$y(\alpha)$ would be equal to $1 * \sin(\alpha + 135^{\circ})$ where $\alpha$ can be any value. Unfortunately, we have to be even more geeky than that. We have to talk about cosine waves instead of sine waves. We've already seen that these are really the same thing, just $90^\circ$ apart, so we can already figure out that a sine wave that's starting $135^{\circ }$ late is the same as a cosine wave that's starting $45^{\circ}$ late. Now that we've made that transition, there is another way to describe a wave. If we scale the sine and cosine components correctly and add them together, the result will be a sinusoidal wave at any phase and amplitude we want. Take a look at the equation below: $$A \cos(\alpha + \phi) = a \cos(\alpha) - b \sin(\alpha)$$ (2.14) where A is the amplitude $\phi$ is the phase angle $\alpha$ is any angle $a = A \cos(\phi)$ $b = A \sin(\phi)$ What does this mean? Well, all it means is that we can now specify values for a and b and, using this equation, wind up with a sinusoidal waveform of any amplitude and phase that we want. Essentially, we just have an alternate way of describing the waveform. For example, where you used to say ``A cosine wave with a peak amplitude of 0.93 and $\frac{\pi}{3}$ radians ($60^\circ$) late'' you can now say: $A = 0.93$ $\phi = \frac{\pi}{3}$ $a = 0.93 * \cos( \frac{\pi}{3}) = 0.93 * 0.5 = 0.4650$ $b = 0.93 * \sin( \frac{\pi}{3}) = 0.93 * 0.8660 = 0.8054$ Therefore $$A \cos(\alpha + 27^\circ) = 0.4650 \cos(\alpha) - 0.8054 \sin(\alpha)$$ (2.15) So we could say that it's the combination of an upside-down sine wave (upside-down because the sine component is subtracted in the equation) with a peak amplitude of 0.8054 and a cosine wave with a peak amplitude of 0.4650. We'll see in Chapter 1.6 how to write this a little more easily. Remember that, if you're used to thinking in terms of a peak amplitude and a fixed phase offset, then this might seem less intuitive. However, if your job is to build a synthesizer that makes a sinusoidal wave with a given phase offset, you'd much rather just mix (in other words, add) an appropriately scaled cosine and sine rather than having to build a delay.