I think it's interesting we can manipulate functions like this by simply adding more functions. The same applies in music. Since notes can be created from pure sine waves, we can manipulate the sounds by adding more notes or more functions. Some examples of this are square, triangle, and sawtooth waves as variations of the sine wave.
keenan
@bebert Yes, superposition of different sounds is a great way to think about adding functions—especially the sinusoidal bases we talk about later in this lecture, which sort of correspond to different tones. (Real physical sound waves don't quite add linearly, but it is often a reasonable approximation.)
Huster
This makes me think of some special waves that we can see on the oscilloscope. For example, the square wave. Square wave can be represented as an infinite summation of sinusoidal waves. We cannot construct an ideal square wave with finite functions. As we zoom in to see the detail of the square wave on the screen of an oscilloscope, we always see some sawtooth part. On the other hand, some complex functions can be decomposed into some simple functions. A typical example is Fourier Series.
I think it's interesting we can manipulate functions like this by simply adding more functions. The same applies in music. Since notes can be created from pure sine waves, we can manipulate the sounds by adding more notes or more functions. Some examples of this are square, triangle, and sawtooth waves as variations of the sine wave.
@bebert Yes, superposition of different sounds is a great way to think about adding functions—especially the sinusoidal bases we talk about later in this lecture, which sort of correspond to different tones. (Real physical sound waves don't quite add linearly, but it is often a reasonable approximation.)
This makes me think of some special waves that we can see on the oscilloscope. For example, the square wave. Square wave can be represented as an infinite summation of sinusoidal waves. We cannot construct an ideal square wave with finite functions. As we zoom in to see the detail of the square wave on the screen of an oscilloscope, we always see some sawtooth part. On the other hand, some complex functions can be decomposed into some simple functions. A typical example is Fourier Series.