Click here to purchase the entire book in PDF format. Negative Frequency Back in Section 1.5 we looked at how two wheels rotating at the same speed (or frequency) but in opposite directions will look exactly the same if we look at them from only one angle. This was our big excuse for getting into the whole concept of complex numbers - without both the sine and cosine components, we can only know the speed of rotation (frequency) and diameter (amplitude) of the sine wave. In other words, we'll never know the direction of rotation. As we walk through the world listening to sinusoidal waves, we only get one signal for each sine wave - we don't get a sine and cosine component, just a pressure wave that changes in time. We can measure the frequency and the amplitude, but not the direction of rotation. In other words, the frequency that we're looking at might be either positive or negative, depending on which direction the imaginary wheel is turning. Here's another way to think of this. Take a Slinky, stretch it out, and look at it from the side. If you didn't have the benefit of perspective, you wouldn't be able to tell if the Slinky was coiled clockwise or counterclockwise from left to right. One is positive frequency, the other is the negative equivalent. In the real world, this doesn't really matter too much, but as we'll see later on, when you're doing things like digital filtering, you need to worry about such things.