Properties Of The Fourier Transform; Fast Fourier Transform (Fft) - HP 50g User Manual

Properties of the Fourier transform

Linearity: If a and b are constants, and f and g functions, then F{a f + b g} = a
F{f }+ b F{g}.
Transformation of partial derivatives. Let u = u(x,t). If the Fourier transform
transforms the variable x, then
Convolution: For Fourier transform applications, the operation of convolution is
defined as
The following property holds for convolution:

Fast Fourier Transform (FFT)

The Fast Fourier Transform is a computer algorithm by which one can calculate
very efficiently a discrete Fourier transform (DFT). This algorithm has
applications in the analysis of different types of time-dependent signals, from
turbulence measurements to communication signals.
The discrete Fourier transform of a sequence of data values {x
n-1, is a new finite sequence {X
X
k
The direct calculation of the sequence X
involve enormous amounts of computer (or calculator) time particularly for large
values of n. The Fast Fourier Transform reduces the number of operations to the
order of n log n.
2
operations, while the direct calculation would require 10,000 operations. Thus,
F{ u/ x} = i F{u}, F{
F{ u/ t} = F{u}/ t, F{
( * )( )
f g x
F{f*g} = F{f} F{g}.
}, defined as
k
1 n 1
∑
x exp(
j
n
j 0
For example, for n = 100, the FFT requires about 664
2 2
u/ x } = -
2 2
u/ t } =
1
∫
(
f x
2
i 2 kj / n ),
involves n
k
F{u},
2 2
F{u}/ t
) ( ) .
g d
}, j = 0, 1, 2, ...,
j
k 0 1 , 2 , ,..., n
2
products, which would
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