# Fast Fourier Transform (fft) - HP 49g+ User Manual

Graphing calculator.

convolution: For Fourier transform applications, the operation of convolution
is defined as
(
*
f
g
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
2, ..., n-1, is a new finite sequence {X
1
n
1
X
x
k
n
j
=
0
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
requires about 664 operations, while the direct calculation would require
10,000 operations. Thus, the number of operations using the FFT is reduced
by a factor of 10000/664 ≈ 15.
The FFT operates on the sequence {x
shorter sequences. The DFT's of the shorter sequences are calculated and
later combined together in a highly efficient manner. For details on the
algorithm refer, for example, to Chapter 12 in Newland, D.E., 1993, "An
1
)(
)
(
x
f
x
2
π
F{f*g} = F{f}⋅F{g}.
}, defined as
k
exp(
i
2
π
kj
/
n
),
j
involves n
k
n.
For example, for n = 100, the FFT
2
} by partitioning it into a number of
j
ξ
)
(
ξ
)
ξ
.
g
d
}, j = 0, 1,
j
k
0
1 ,
2 ,
,...,
n
1
2
products, which would
Page 16-49  