| Phys 198 | March 17, 1997 |
Time Domain: x(t) |
Frequency Domain: X(f) |
| Real | Real Part Even, Imaginary Part Odd |
| Imaginary | Real Part Odd, Imaginary Part Even |
| Real even, Imaginary Odd | Real |
| Real Odd, Imaginary Even | Imaginary |
| Real and Even | Real and Even |
| Real and Odd | Imaginary and Odd |
| Imaginary and Even | Real and Odd |
| Complex and Even | Complex and Even |
| Complex and Odd | Complex and Odd |
(Table taken from Brigham, E., The Fast Fourier Transform,
p. 45.)
Some of these properties impact heavily on the way the FFT is used. For instance, the fact that real data input produces a transform with real part even and imaginary part odd means that the transform is symmetric and only half the output produces new information. Thus the result of entering only real values (only half of the input the transform expects since all the imaginary components are zero) is that only half the number of harmonics are useful. Thus a data series of 512 real values will give a transform where only 256 harmonics are useful. Indeed, with a little mathematical manipulation, 512 real values can be transformed with a 256 entry FFT by inserting half of the input in the otherwise unused imaginary parts.
Another result of this is that if only symmetric filtering is performed in transform space, then real data is obtained when the data is inversely transformed back to real space. On the other hand, if non-symmetric filtering is performed, the resulting data in real space will be a set of complex numbers, rather than a real set corresponding to the original real data.
If x(k) and y(k), consisting of N samples, have discrete Fourier transforms X(n) and Y(n), respectively, then the following properties hold:
| Linearity | x(k) + y(k) <=> X(n) + Y(n) |
| Symmetry | (1/N)X(k) <=> x(-n) |
| Time Shifting | x(k - m) <=> X(n) exp(-i2pmn/N) |
| Frequency Shifting | x(k)exp(i2pmk/N) <=> X(n - m) |
The frequency shifting property is widely used to produce a display of the Fourier transform which has the (0,0) frequency located in the center of the image, rather than in the lower left corner. Another common change applied to the data to get a good display of the transform is to plot the logarithm of the transformed data, rather than the data itself:
F'(x,y) = log10(1 + F(x,y))
where F' is the displayed data, and F is the original transformed data. This gives a much wider dynamic range.
These properties of the Fourier transform are rather simple to prove by simple substitution.
Email comments or questions to: matthysd@vms.csd.mu.edu
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