Fourier Series Intro

Today I introduced Fourier Series.  The fundamental reason that we study Fourier Series is that it allows us to represent almost any periodic time-domain signals (with a few degenerate exceptions that are of more interest to mathematicians then engineers -- i.e., they cannot generally be produced by a physical system) by a sum of sinusoidal signals.  This is useful if we know the frequency domain response (Bode Plot) of a system.  If the input is a sum of sinusoids at different frequencies, the output is also a sum of sinusoids whose magnitude and phase are changed by the magnitude and phase of the transfer function at that frequency (i.e., the Bode Plot).  We have not yet touched upon this, but will do so in later weeks.

Given a periodic function x(t) with period T, where

The quantity is the fundamental frequency, or frequency of the first harmonic.

We can write x(t) in several forms with a trigonometric series

or with an exponential series.

The value of the coefficients can be derived by using orthogonality.  In class I derived the expression for an.  The other coefficients can be found in a similar way and are given by (the equation for an is not valid for n=0: that expression is given in the next paragraph):

The exponential series is generally more useful.   The two are, unsurprisingly related.  For n≠0, 2Re(cn)=an, -2Im(cn)=bn.  For n=0, a0=c0.  Each coefficient of the Fourier Series represnet the amplitude of the sinusoidal signal at the frequency nw0.  The zero frequency (non-oscillating) term from the Fourier series is just the average value of the function and can often be determine by inspection

You can see some examples of the results of this integration here (this page is similar to the demo I did in the last class).

I also quickly demonstrated that you can make simplifications if x(t) is even or odd.  We usedthe facts that

• an odd function multiplied by an odd function yields an even function
• an even function multiplied by an even function yields an even function
• an odd function multiplied by an even function yield an odd function
• the "sine" function is odd
• the "cosine" function is even
• The integral of an odd function from -A to A is zero.
• The integral of an even function for -A to A is twice the integral from 0 to A.

If we wan to find the Fourier coefficients for an even function, e(t), then it is often easiest to represent the function with a Fourier cosine series (i.e., the bn=0).

In a similar way, if we start with an odd function, o(t), and using the fact that the product of two odd functions is an even function, we get:

In class I derived the Fourier coefficients for an odd square wave using both the exponential series, and the sine series.  Because the function was odd, the integration for the sine series coefficients was easier.

Convergence of the Fourier Series

At the end of class I briefly mentioned convergence criteria for Fourier Series, which we have not considered up to this point.  The Fourier Series Converges as long as

• ,
• x(t) has a finite number of maxima and minima in one period, and
• x(t) has a finite number of discontinuities in one period

The Fourier Series Representation of x(t) converges to x(t) except at discontinuities.  At a discontinuity the Fourier Series converges to the midpoint of the discontinuity.  On either side of the discontinuity there is always an overshoot of about 9% (Gibb's phenomenon).  However this discontinuity becomes vanishingly narrow as we sum up more terms of the series.

If x(t) is given by the function shown below (a rectangular pulse with amplitude=1, width= Tp/2 and period=T).

The Fourier Series coefficients are given by

These can be plotted vs. n    They are represented by the blue circles in the graph shown below; the horizontal axis is n).

The dotted red line shows the values of cn as n is varied continuously from -25 to 25.   Note that non-integer values of n have no meaning (yet).

Note several things.

• This type of function (sin(x)/x) will come about frequently.  We will call these types of functions "sinc" functions (though strictly speaking the definition is sinc(x)=sin(π·x)/(π·x)).
• The width of the main lobe (between n=-9 and n=+9) is 2T/Tp.  In other words the red line goes through zero when n=T/Tp (in this case it is not an integer value).  As Tp decreases (the function becomes more localized in the time domain) the width of the main lobe increases (the Fourier series becomes less localized in the frequency domain).   We will come back to this with the Fourier Transform in the next class, where it will be (hopefully) clearer.
• Each value of an corresponds to an sinusoidal signal at a frequency of

• The spacing between consecutive frequencies is

As T increases, the spacing decreases.  As T goes to infinity the spacing goes to zero.  This leads to the Fourier Transform.  (See notes from next lecture).