Click here to purchase the entire book in PDF format. Complex notation or... Who cares? This is probably the most important question for us. Imaginary numbers are great for mathematicians who like wrapping up loose ends that are incurred when a student asks ``what's the square root of -1?'' but what use are complex numbers for people in audio? Well, it turns out that they're used all the time, by the people doing analog electronics as well as the people working on digital signal processing. We'll get into how they apply to each specific field in a little more detail once we know what we're talking about, but let's do a little right now to get a taste. In the chapter that introduces the trigonometric functions sine and cosine, we looked at how both functions are just one-dimensional representations of a two-dimensional rotation of a wheel. Essentially, the cosine is the horizontal displacement of a point on the wheel as it rotates. The sine is the vertical displacement of the same point at the same time. Also, if we know one of these two components, we know the diameter of the wheel and how fast it's rotating but we need to know both components to know the direction of rotation. At any given moment in time, if we froze the wheel, we'd have some contribution of these two components - a cosine component and a sine component for a given angle of rotation. Since these two components are effectively identical functions that are $90^{\circ}$ apart (for example, a sine wave is the same as a cosine that's been delayed by $90^{\circ}$) and since we're thinking of the real and imaginary components in a complex number as being $90^{\circ}$ apart, then we can use complex math to describe the contributions of the sine and cosine components to a signal. Figure 1.13: A signal consisting only of a cosine wave Huh? Let's look at an example. If the signal we wanted to look at a signal that consisted only of a cosine wave as is shown in Figure 1.13, then we'd know that the signal had 100$\%$ cosine and 0$\%$ sine. So, if we express the cosine component as the real component and the sine as the imaginary, then what we have is: $$1 + j0$$ (2.56) If the signal was an upside-down cosine, then the complex notation for it would be $(-1 + 0j)$ because it would essentially be a cosine * -1 and no sine component. Similarly, if the signal was a sine wave, it would be notated as $(0 - 1j)$. This last statement should raise at least one eyebrow... Why is the complex notation for a positive sine wave $(0 - 1j)$? In other words, why is there a negative sign there to represent a positive sine component? Well... Actually there is no good explanation for this at this point in the book, but it should become clear when we discuss a concept known as the Fourier Transform in Section 9.2. For now, you'll just have to trust me. This is fine, but what if the signal looks like a sinusoidal wave that's been delayed a little like the one in Figure 1.14? Figure 1.14: A signal (the black one) consisting of a combination of attenuated cosine and sine waves with the same frequency. This signal was created by a specific combination of a sine and cosine wave. In fact, it's 70.7$\%$ sine and 70.7$\%$ cosine. (If you don't know how I arrived that those numbers, check out Equation 1.13.) How would you express this using complex notation? Well, you just look at the relative contributions of the two components as before: $$0.707 - j 0.707$$ (2.57) It's interesting to notice that, although Figure 1.14 is actually a combination of a cosine and a sine with a specific ratio of amplitudes (in this case, both at 0.707 of ``normal''), the result looks like a sine wave that's been shifted in phase by $-45^{\circ}$ or $-\frac{\pi}{4}$ radians (or a cosine that's been phase-shifted by $45^{\circ}$ or $\frac{\pi}{4}$ radians). In fact, this is the case - any phase-shifted sine wave can be expressed as the combination of its sine and cosine components with a specific amplitude relationship. Therefore, any sinusoidal waveform with any phase can be simplified into its two elemental components, the cosine (or real) and the sine (or imaginary). Once the signal is broken into these two constituent components, it cannot be further simplified. If we look at the example at the end of Section 1.5, we calculated using the equation $$A \cos(\alpha + \phi) = a \cos(\alpha) - b \sin(\alpha)$$ (2.58) that a cosine wave with a peak amplitude of 0.93 and a delay of $\frac{\pi}{3}$ radians was equivalent to the combination of a cosine wave with a peak amplitude of 0.4650 and an upside-down sine wave with a peak amplitude of 0.8054. Since the cosine is the real component and the sine is the imaginary component, this can be expressed using the complex number as follows: $$0.93 \cos(\alpha + \frac{\pi}{3}) = 0.4650 \cos(\alpha) - 0.8054 \sin(\alpha)$$ (2.59) which is represented as 0.4650 + j 0.8054 which is a much simpler way of doing things. (Notice that I flipped the ``-'' sign to a ``+.'') For more information on this, check out The Scientist and Engineer's Guide to Digital Signal Processing available at http://www.dspguide.comwww.dspguide.com Figure 1.15: A quick guide to what things mean when you see a cosine wave expressed as this type of equation. Note that both angles ($\alpha$ and $\phi$) can be expressed in either degrees or radians - just make sure that they're both the same.