1/25/2010
1
Odd-Even Symmetry
•
If x(n) is real and even
, that is: x(n)=x*(n)
and x(-n)=x(n), then:
• X(m) is
real
and
even
X(m) is
and
.
•
If x(n) is real and odd
, that is: x(n)=x*(n)
and x(-n)=-x(n), then
•
X(m) is
imaginary
and
odd
.
Discrete Fourier Transform
•
There are three possible phenomena that
result in errors between the computed and
the desired transform. These three
phenomena are
– (a)
aliasing
(Lyons 2.1)
– (b)
leakage
(Lyons 3.8)
– (c)
scalloping loss
(Lyons 3.10)
http://www.cage.curtin.edu.au/mechanical/info/vibrations/tut4.htm
Leakage and the Discrete Fourier
Transform
•
The DFT gives exact frequency results for
sinusoidal signals if the N-samples
sequence contains an
integer number of
complete cycles
of the sampled continuous
signal x(t).
•
If it does not, the DFT produces a spread
spectrum whose values are not exactly
correct. We will discuss frequency leakage
in detail later.
DFT Leakage
DFT Leakage
Discrete Fourier Transform
• (b)
Leakage
. This problem arises because we must
limit observation of the signal to a finite interval.
The process of terminating the signal after a finite
number of terms is equivalent to multiplying the
number of terms is equivalent to multiplying the
signal by a
window function
.
•
The net effect is a distortion of the spectrum.
There is a spreading or leakage of the spectral
components away from the correct frequency,
resulting in an undesirable modification of the
total spectrum.
http://www.cage.curtin.edu.au/mechanical/info/vibrations/tut4.htm

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