Checking Fourier Coefficients
Step 1: Draw the function for a few complete cycles. This enables you to see what the repeat
period T is, and hence also the fundamental frequency f0.
Step 2: Look at the function to decide whether it is odd or even, in which case you can delete all
the an terms or all the bn terms respectively. If the function f(t) is neither odd nor even,
you need both an and bn terms.
Step 3: It is almost always possible to draw a very good approximation to the constant a0/2 and
to the lowest frequency term of the series. This enables you to check the signs and the
approximate magnitudes of these coefficients in your calculations.
Step 4: In some cases you can also see from your diagram what the next term must be in order
to correct for the deficiencies of the leading term. For example, you may well be able to
check its frequency. This in general will be 2f0, but for some functions, this term will be
missing and you will have only a 3f0 term as the next one. If this is so, it would not be
too surprising if all higher terms of the type 2nf0 were zero as well.
You might also hope to check the sign of the second term. Again it will often be the
case that if the second term has the same sign as the first, then all the terms have this
sign, while if they are opposite, then the signs along the complete series may alternate.
Step 5: Next look at the n dependence of your coefficients, and see if they agree with your
expectations based on the function's behavior.
Step 6: As a final test you can substitute some specific value t' for the variable t (e.g. t' = π or
π/2), such that all the sine and cosine terms take on simple values. You should then be
able to use a few terms of your series to check numerically that the sum gives a plausible
approximation to f(t'). (You need to sum a few terms, since if you use only the first
term, you are doing no more than the check of Step 3). It is important to remember that,
if the function displays a discontinuity at t = t', the Fourier series should give the
average of the two values on either side of the discontinuity, e.g. for a unit boxcar, it
yields an answer of ½ at the discontinuity (the edge of the boxcar).
If your solution passes all these tests, it may well even be correct.
Adapted from: Lyons, L., 1998, All you wanted to know about mathematics but were afraid to ask, Vol. 2.,
Cambridge University Press, Cambridge, p. 161-162.
Geos 469/569 Seismic Data Processing