Calculus

Fourier Series

How much of a given frequency is inside a signal? Multiply the two together and add up the area. If the wave fits, the product sits mostly above the axis and the area is large; if it does not, the positives and negatives cancel and the answer is exactly zero. Every coefficient of a Fourier series is that one measurement.

the signalthe sum of what has been measuredarea above the axisarea below it

Drag on the top panel to draw your own signal · click a bar to test that frequency

How much fits—

—

Signal

Testing with

Frequencies tested 1

b_k, how much sin kx fits—

a_k, how much cos kx fits—

of the signal accounted for—

What to observe

Test the square wave at frequency 2: the product has as much area below the axis as above, so b₂ is exactly zero. A square wave has no even harmonics.
Switch to cos kx on the square wave and every coefficient is zero. The signal is odd and cosines are even, so their product always cancels.
Press Run: each frequency is tested in turn, its bar joins the spectrum, and the teal curve gains that term. The reconstruction is the running total of the measurements.
On two tones only bars 2 and 5 stand up. The spectrum is a much shorter description of the signal than the signal itself.

Shortcuts:space run/pause · s step · r reset · f fullscreen