Fourier Transforms and Properties
Outline
Fourier transforms (FT)
Forward and inverse
Discrete (DFT)
Fourier series
Properties of FT:
Symmetry and reciprocity
Scaling in time and space
Resolution in time (space) and frequency
FT’s of derivatives and time/space-shifted functions
The Dirac’s delta function
Sin/cos() or exp() forms
of Fourier series
Note that the
cos()
and
sin()
basis in Fourier series can be replaced
with a basis of
complex exponential functions of positive and
negative frequencies
:
where:
The
e
inx
functions also form an orthogonal basis for all
n
The Fourier series becomes simply:
Time- (or space-) frequency uncertainty
relation
If we have a signal localized in time (space) within interval
T
,
then its frequency bandwidth
(
f
) is limited by:
This is known as the Heisenberg uncertainty relation in
quantum mechanics
or
For example, for a boxcar function
B
(
t
)
of length
T
in time,
the spectrum equals:
The width of its main lobe is:
Dirac’s delta function
The “generalized function”
plays the role of identity matrix
in integral transforms:
(
x
)
can be viewed as an infinite spike of zero width at
x
= 0,
so that:
Another useful way to look at
(
x
):
(
x
)
is the Heavyside
function:
(
x
)
= 0 for
x
< 0
and
(
x
)
= 1 for
x
> 0
Dirac’s delta function
Recalling the formula we had for the forward and inverse
Fourier transforms:
…we also see another useful form for
(
y
) (
y
=
x
-
x
here)
:
Fourier transforms play a crucial role in signal processing by transforming signals between time and frequency domains. This outline covers the basics of Fourier transforms, discrete Fourier transforms, Fourier series, properties like symmetry and reciprocity, resolution in time and frequency, the Dirac delta function, sin/cos or exp forms, and the time-frequency uncertainty relation.
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Presentation Transcript
Outline Fourier transforms (FT) Forward and inverse Discrete (DFT) Fourier series Properties of FT: Symmetry and reciprocity Scaling in time and space Resolution in time (space) and frequency FT s of derivatives and time/space-shifted functions The Dirac s delta function
Sin/cos() or exp() forms of Fourier series Note that the cos() and sin() basis in Fourier series can be replaced with a basis of complex exponential functions of positive and negative frequencies: ( ) ( ) + + inx inx cos sin a nx b nx e e n n n n + a ib a a ib , , and 0 n n n n where: 0 n n 2 2 2 The einx functions also form an orthogonal basis for all n The Fourier series becomes simply: = ( ) f x inx e n = n
Time- (or space-) frequency uncertainty relation If we have a signal localized in time (space) within interval T, then its frequency bandwidth ( f) is limited by: 2 T or f T 1 For example, for a boxcar function B(t) of length T in time, the spectrum equals: ( ) B ( ) 2sin 2 T = 2 = = The width of its main lobe is: 2 T T This is known as the Heisenberg uncertainty relation in quantum mechanics
Diracs delta function The generalized function plays the role of identity matrix in integral transforms: ( ) ( f x x ) ( ) x f x dx = (x) can be viewed as an infinite spike of zero width at x = 0, so that: ( ) x dx Another useful way to look at (x): = 1 (x) is the Heavyside function: (x) = 0 for x < 0 and (x) = 1 for x > 0 d dx ( ) x ( ) x
Diracs delta function Recalling the formula we had for the forward and inverse Fourier transforms: 1 ( ) f x ( ) f x dx dk = = ikx ikx e e 2 1 ( ) ( ) ik x x = e dk f x dx 2 we also see another useful form for (y) (y = x - x here): 1 ( ) y = iky e dk 2
