The Fourier Transform

Michael Zeineh has already given you extensive and rather complete notes on this subjects. You can find still more data on the web tutorial. For folks who really want to know this material well, the probable very best reference is "The Fourier Transform" by Bracewell. I have a copy in my office for anyone who want to have a look at it.

Notes

Consider the example of a piano. Because the notes on a piano are systematically laid out according to pitch (frequency), by listening to the notes, we can infer the location that a pianist has hit. We do this by detecting the FREQUENCY of the signal that we hear. In this case, our ears perform a Fourier transform of the signal.

If more than one note is struck at a time, the vibrations of the air and eardrum are quite complex, but we perceive a simple stimulus: two notes that sound independent. In fact, the Fourier transform of the sound is also quite simple, having energy at the frequencies of each of the notes.

An important fact about the FT, is that it is a complete (no loss) description of the time domain signal. In some cases it is more compact, and in some cases less compact, than the time domain signal.

The Fourier series for a periodic signal (for example a square wave) is the series of sinusoidal signals which, when added together, will exactly equal the original periodic signal. For example, the sum of

sin(t) + sin(3t)/3 + sin(5t)/5 + sin(7t)/7 + sin(9t)/9 looks very much like a square wave.

As shown in the graph above, the heavy line shows the sum of the signals of the first five elements of the Fourier series. As higher and higher frequencies are added, you can probably see that the oscillations become smaller and more closely spaced. In general, the waveforms that we used in MRI are non-ideal: in many cases we must use truncated waveforms like this, and their Fourier transforms will produce this kind of oscillation in the signal. We will show many examples.

Some properties of the transform that you should know:

1. Shorter pulses (in time) result in wider frequencies.
2. The FT is a "complex" transform: with a "real" input signal, the FT may produce both real and imaginary outputs.
3. The FT is invertable. If we start with a function f(t), and create its Fourier transform F(s), applying the Fourier transform again, will recover our original signal, (though time reversed): f(-t).
4. The mathematical formula for forward and inverse transformation is essentially identical. As a result, the signals come in mirror pairs. For example, if we have a rectangular time domain function, we will have a sinc (sin(s)/s ) frequency domain function. If we have a sinc time domain function, we will have a rectangular frequency domain function.