Friday, May 10, 2013

What goes into a sign?

Sine that is... a sinusoid... That is where we left off, correct? (Looks back at Up, Down, Up, Down...)

Let's go back a little bit first. What are sine (and cosine)? They represent a function where the output is the ratio of one of the two legs of a right triangle over the hypotenuse. You say, "This sounds a LOT like trigonometry," Well that's where a LOT of our math in engineering comes from. Sine is the ratio of the leg opposite the measured angle and the hypotenuse, which, when the angle is zero, the length of that side is zero. In this instance you really don't have a triangle, you have a line segment the length of the hypotenuse, but that's not a big deal. Maybe a picture is needed for this.
This is called the Unit Circle. It's made by changing the angle that the hypotenuse of a right triangle make with the x-axis, while keeping the radius one (hence unit). Looking at sine (the vertical component of the triangle we see that as we increase the angle, the ratio gets bigger until the angle measured reaches a right angle, 90 degrees, or pi/2 radian (we'll probably stick to radians a lot, since the math later will be much easier that way). At this point, sine begins to shrink again until and angle of pi, where it becomes zero. 

Now, on the lower half of the circle, we see sine grow negative, to a maximum of 1 at 3pi/2, and then shrink back to zero at 2pi, or 0 radians (same thing). If we were to plot each of these points as we went around the unit circle, we would get a plot of what we call a sinusoid. Wikipedia has a great graphic for this:

Now because the function's parameter, theta, is locked in to radians, we can adjust the scale of sine across other variables. This parameter is called the phase of the function. Commonly we use space and time variables (x,y,z, and t) to scale sinusoids across those domains. This is shown below:
The higher the frequency, the more cycles per duration of the domain the sinusoid goes through. In terms of acoustics, the higher the frequency, the higher the pitch of the sound we're hearing. For those music types, Middle C is about 264Hz. We'll talk about frequency more later. Because the shape of the sinusoid is being stretched over the domain we're working over (x,y,z, or t) we want to be able to measure that as well. If we measure the distance between consecutive peaks or troughs, we get the wavelength, often stated with the variable lambda:
Wavelength can bring on many discussions of speed of a wave in a medium (Speed of sound or Speed of Light), as well as a whole other realm of mathematics. Needless to say, we're going to hold off on that. Speed of sound will come soon enough.

Shifting a sinusoid can be done by adding a number to the phase, this then called a phase shift. Here's an example of that:
This will become very important later when we want to compare to signals, and when we want to talk about physical phenomena like resonance and electrical phenomenon like filtering.

I'm going to save talking about the amplitude of a sinusoid until next time, since there are a ton of ways to describe it, and getting the all straight is something that I know many people have a hard time with.

Sorry about the delay in getting this out, real life caught up with me this week, and I was super busy at work, which made me crazy tired when I got home, and therefore crashed early, when I should have been writing this...

Anyway, if this weekend isn't too nice, I'll try to get back to amplitude on Sunday! Cheers!

-CMW

Thanks to Wikipedia for use of all of these images. Without you, I would have had to generate them in MATLab. I believe that all of them are linked properly, so you can check out the various Wikipedia sites where they reside.