The cyclic nature of FFT

Summary

FFT is by definition cyclic. The transition from end to start is part of the signal.

The time domain samples may represent samples of a cyclic, continuous-time waveform.
The bandwidth is limited, leading to the “smoothest possible” waveform through the points.
But the “smoothness” is also enforced across the transition over the end of the period!

This can lead to unwanted consequences, when a signal has a sharp discontinuity that isn't actually part of the signal.

Example

Consider the “sawtooth” wave in Figure 1.
The blue line is not the equivalent continous-time signal, but wishful thinking:

Figure 1: Sawtooth generator? Not!

Figure 2 shows the sequence of samples for one single period:

Figure 2: Samples

Now there is a sharp discontinuity at the end, when one period leads into the next.

An ideal lowpass filter (FFT) reconstructs the continuous-time signal, as shown in Figure 3:

Figure 3: Continuous-time signal, reconstructed by ideal LP filter

This result is correct - the waveform passes through all the samples.
And looking at several periods of the cyclic signal confirms that the signal is the “smoothest possible signal” that can follow the transition into the next cycle!

Figure 4

It does not matter, whether the discontinuity occurs between start- and endpoint, or somewhere else:
The first (or last) point have no different role than all other points.
A time-shifted version of the signal shows exactly the same “ringing”:

Figure 5

The blue line sketched out in Figure 1 is simply not possible, and requires change. For example,

increase the sample rate
use a longer ramp cycle
change the waveform to a shape that contains less high-frequency content (triangular)

© Markus Nentwig 2007-2008
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