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Junior Physics Laboratory
Notes on Fourier Transforms
The Fourier transform is a generalization of the Fourier series representation of functions. The Fourier series is limited to periodic functions, while the Fourier transform can be used for a larger class of functions which are not necessarily periodic. Since the transform is essential to the understanding of several exercises, we briefly explain some basic Fourier transform concepts.
A. Fourier series Recall the definition of the Fourier series expansion for a periodic function u ( t ). If u ( t + T ) = u ( t ), then
u ( t ) = c (! m m ="#
$ ) exp( i^! m t )^ (1)
where !m = 2 " m / T and
c (! m ) = 1 T u ( t ) " T / 2
T / 2
exp(" i^! m t )^ dt^ (2)
If u ( t ) is a function of time it is natural to call! m an angular frequency and to refer to c (! m ) as the amplitude of u ( t ) at frequency! m. Together, the c (! m ) comprise the frequency
spectrum of u ( t ). Computing the frequency spectrum for a sine or square wave of period T is fairly simple. For the sine wave, there are only two non-zero coefficients
c (! m ) = ± 1 2 m = ± 1 (3)
as one might expect. Note, though, the odd fact that the formalism requires both positive and negative frequencies. The spectrum of the square wave is slightly more complicated
c (! m ) = 2 " m m odd (4)
and again contains both positive and negative frequencies. The interesting feature here is that the rapid rise and fall of the square wave leads to spectral components at high frequency, as demonstrated in Fig. 1. This is a general effect, and can help one understand the qualitative connection between time and frequency descriptions. S (^) 1 S (^) 2 S^ 3 S 4
Fig. 1 Building up a square wave from Fourier series components. Each graph is a plot of Sn ,
the sum of the first n terms in the series expansion, for increasing values of n.
B. Fourier transforms For many purposes a knowledge of the c (! m ) may be more useful than knowing u ( t ) itself. It is therefore desirable to extend the idea of a frequency spectrum to non-periodic functions, for example single pulses or finite wave trains. This can be done by replacing the sum of Eq. 1 by an integral, giving
u ( t ) = 1 2! c ( ") exp( i " t ) #$
+$ % d^ " (5)
and
c ( !) = u ( t ) "#
+# $ exp(" i^! t )^ dt^ (6)
where! is now a continuous variable. The complex function c ( !) is called the Fourier transform of u ( t ) and is again the amplitude of u ( t ) at frequency !. Some examples may make the ideas clearer. The transform of exp( i! 0 t ) is a delta function, !(! -! 0 ). This says that the spectrum of exp( i! 0 t ) has a large amplitude only at! =! 0 , which seems reasonable since the function is periodic with angular frequency! 0. The Fourier transform of a pulse centered at t = 0,
u ( t )
= 1 t !" / = 0 t > " /
&%^
u ( x , y ) = 1 2!
c ( " x ," y )
#$
$
#$
$
% exp^ [ i 2!^ ( x^ " x +^ y^ " y )] d " x d " y
c (! x ,! y ) = u ( x , y )
"#
"#
$ exp^ [" i^2 %^ ( x^! x +^ y^! y )] dxdy
where !x and !y are called spatial frequencies, with units of (length)-1. Just as ordinary
frequency measures how fast a signal changes in time, a spatial frequency measures how rapidly a signal varies in space. Since the formalism is the same, all of our previous observations about the properties of the transforms hold for two-variable functions.
C. Sampled data considerations So far we have tacitly assumed that the function u ( t ) is known at all values of t , but it is more common to sample the input with an analog to digital converter at regular intervals! t so that the required computations can be done digitally. Also, the signal can only be sampled for a finite time period, say from -T /2 to T /2. When transforming such sampled data, we compute a discrete approximation to the ideal Fourier transform. The discrete transform of an N -point array u (^) n is defined to be
c (^) m = u (^) n n = 0
N! 1
" exp^! i^2 #^
nm N
%^ &^
(^ )^
for m = 0,1,..., N -1. For a real input function un , c * N-m = c (^) m , so we obtain N /2 distinct points in the frequency spectrum, uniformly spaced between f = 0 and the Nyquist frequency f (^) N = 1/(2! t ). The spacing between points, or the frequency resolution, is just 1/ N! t = 1/ T , where T is the total length of the input record. Except as noted below, the discrete transform is, for our purposes, equivalent to the continuous Fourier transform. A very efficient algorithm, called the Fast Fourier Transform or FFT, exists to compute the sum in Eq. 13, provided that N is exactly a power of two.
The first limitation of the sampling process is the possibility of aliasing, as illustrated in Fig. 2. Evidently input frequencies from zero to 1/(2! t ) are all distinguishable, but if higher frequencies are present they will be confused with some lower frequency. An example is shown in Fig. 3, where the higher harmonics of the square wave signal are aliased to lower frequencies, producing a misleading spectrum. The Nyquist frequency, fN = 1/(2! t ), is the highest frequency that can be measured with a sample spacing of! t. Experimentally, it is important to avoid aliasing by inserting an analog low-pass filter in the input before sampling and then choosing! t small enough that the Nyquist frequency is well above the pass limit of the analog filter. A more subtle problem arises from the finite length of the sampled record. In general, the signal will not be exactly zero at the start and end of the record, so it appears to change very
Fig. 2 Sampling of sinusoidal waves. Note that there is no way to distinguish between the two sine waves on the basis of the regularly-spaced samples.
(^10 0 100 200 300 400 )
10 0
10 5
(^10 0 100 200 300 400 )
10 0
10 5
Frequency (Hz) Frequency
Power
(Hz)
Power
Fig. 3 Power spectra of square wave signals with a fundamental of 157 Hz. (Left) Without filtering, there are numerous spurious peaks due to aliasing. (Right) With proper low-pass filtering, only the first and third harmonic appear below the Nyquist frequency.
