A Few Key Ideas

The distinguished pulsar radio astronomer,
Jim Cordes at Cornell University, recently wrote to me:

"There is the general dictum, if it fluctuates, Fourier transform it."

The Fourier Transform, FT is an analog tool used to analyze the frequency content of continuous signals.

The Discrete Fourier Transform, DFT is a digital tool used to analyze the frequency content of discrete signals.

The Fast Fourier Transform, FFT is an algorithm to rapidly compute the DFT.

The analog signal must be frequently digitized to faithfully reproduce it. Observance of the Nyquist criterion will ensure this:

Sampling frequency > 2 maximum fluctuating frequency

(Higher is advisable; e.g., FFT in image processing may use a factor of 2.57 for optimal brightness and contrast).

Aliasing (fold-over or mixing) occurs if Nyquist sampling is violated.

Typical examples of aliasing are listed below:

(1) Analog electronics: heterodyning is used for tuning; anti-aliasing filters (low pass) filter unwanted signals before the A/D conversion.

(2) Engine timing: slow sampling by a strobe light can arrest the motion of a rotating engine.

(3) Movie making: frames per second may be too slow and "wagon wheels" will appear to stop or rotate backwards.

(4) Moire patterns: slight motion of one of two overlapping (semitransparent) repetitive patterns creates large scale changes in patterns.

Conjugate variable pairs such as position and velocity or time and frequency, are related by Fourier Transforms.

In some applications, f(t) may be periodic, as in harmonic analysis of tidal action, musical instruments tones, resonances of rotating machinery, and AC circuits. Such sinusoids are periodic. But, in many other applications, f(t) may be aperiodic. Nonetheless, since the sine and cosine functions form an orthogonal basis set (viz. functions in vector space), then any well-behaved function on a finite interval can be expanded in terms of such a basis set. This is one way of expressing Parseval's Theorem. Therefore, f(t), when expanded in terms of sines and cosines (or equivalently, as complex exponentials), forms a Fourier series (a composition of sinusoids of various discrete frequencies or harmonics).

Some important examples are depicted below:

courtesy of Agilent Technologies

See standard references or these found on the internet:

"The Fundamentals of Signal Analysis," Application Note 243, Agilent Technologies.

"Introduction to Communication Systems," 2nd ed., Ferrel Stremler, Addison-Wesley Publishing Co., Reading, MA, 1982.

"The Fundamentals of FFT-Based Signal Analysis and Measurement," Michael Cerna and Audrey F. Harvey, Application Note 041, National Instruments, July 2000.

"Fourier Transforms, DFTs, and FFTs", rev. 2/13/02
http://www.me.psu.edu/me82/Learning/FFT/FFT.html

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