The Fourier Series
Some time ago, Fourier, doing heat transfer work, demonstrated that any periodic signal can be viewed as a linear composition of sine waves. Lets look at a periodic wave. Here is an example plot of a signal that repeats every second.
Clearly this signal is not a sinusoid - and it looks as though it has no relationship to sinusoidal signals. However, over a century ago, Fourier showed that a periodic signal can always be represented as a sum of sinusoids (sines and cosines, or sines with angles). That representation is now called a Fourier Series in his honor.
Fourier not only showed that it was possible to represent a periodic signal with sinusoids, he showed how to do it. Assuming this signal repeats every T seconds, then we can describe it as a sum of sinusoids. Here is the form of the sum. Fourier gave an explicit way to get the coefficients in a Fourier Series and we need to look at that in a while. First we are going to look at how a signal can be built from a sum of sinusoids.
Here's that signal again. Is this signal a sum of sinusoids? We will examine that question here now, starting with a single sine signal.
Here is a single sine signal.
The expression for this signal is just:
Sig(t) = 1 * sin(2pt/T) and T = 1 second.
Now, we are going to add one other sine to our original sine signal. The sine we add will be at three times the frequency of the original and it will be one third as large.
Sig(t) = 1 * sin(2pt/T) + (1/3) * sin(6pt/T)
This looks a little different. Continue by adding one more sine signal - at five times the original frequency and one-fifth of the original size.
Sig(t) = 1 * sin(2pt/T) + (1/3) * sin(6pt/T) + (1/5) * sin(10pt/T)
This is getting interesting. We are just adding in terms at odd multiples of the original frequency. Here's what the signal looks like with the terms up to the 11th multiple.
This looks like a fairly lousy square wave. Let's add a lot more terms and see what happens.
Here is the signal with terms up to the 49th multiple.
At this point is seems that this process is giving us a signal that is getting closer and closer to a square wave signal. However, this looks like a fairly lousy square wave. Let's add a lot more terms and see what happens.
Here is the signal with odd terms up to the 79th multiple. Now we're getting a pretty clear indication of a square wave with an amplitude a little under 0.8. In fact, the way we are building this signal we are using Fourier's results. We know the formula for the series that converges to a square wave.
In fact, the way we are building this signal we are using Fourier's results. We know the formula for the series that converges to a square wave. Here's the formula. For a perfectly accurate representation, let N go to infinity.
Some time ago, Fourier, doing heat transfer work, demonstrated that any periodic signal can be viewed as a linear composition of sine waves. Lets look at a periodic wave. Here is an example plot of a signal that repeats every second.
Fourier not only showed that it was possible to represent a periodic signal with sinusoids, he showed how to do it. Assuming this signal repeats every T seconds, then we can describe it as a sum of sinusoids. Here is the form of the sum. Fourier gave an explicit way to get the coefficients in a Fourier Series and we need to look at that in a while. First we are going to look at how a signal can be built from a sum of sinusoids.
Here is the signal with terms up to the 49th multiple.
Here is the signal with odd terms up to the 79th multiple. Now we're getting a pretty clear indication of a square wave with an amplitude a little under 0.8. In fact, the way we are building this signal we are using Fourier's results. We know the formula for the series that converges to a square wave.
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