Goals For This Class

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Goals For This Class

• Quickly review of the main results from last
  class
• Convolution and Cross-correlation
• Discrete Fourier Analysis Important
  Considerations
• Some examples How to do Fourier Analysis (IDL,
  MATLAB)
• Windowed Fourier Transforms and Wavelets
• Tapering
• Coherency

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From Last Class..
Fourier Transform (Spectral Analysis)
Time (Space) Domain
Frequency Domain
Fourier Transform
Inverse Fourier Transform
The Fourier transform decomposes a function into
a continuous spectrum of its frequency components
(using sine and cosine functions), and the
inverse transform synthesizes a function from its
spectrum of frequency components
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Frequency Vs Time (Space Domain)
A time domain graph shows how a signal changes
over time.
A frequency domain graph shows how much of the
signal lies within each given frequency band over
a range of frequencies. A frequency domain
representation can also include information on
the phase shift that must be applied to each
sinusoid in order to be able to recombine the
frequency components to recover the original time
signal.
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Important Properties to remember
Fourier Transform is a special case of Integral
Transforms
Kernel Function
An integral transform "maps" an equation from its
original "domain" to a different one.
Parsevals Theorem
Fourier Transform conserves variance!!
Spectral Estimation
Complex Conjugate
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Convolution
is defined as
The convolution of two functions
Books also use
Convolution expresses the amount of overlap of
one function as it is shifted over another
function.
A convolution is a kind of very general moving
average (weighted).
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Convolution Properties
Derivation
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Cross-Correlation
The cross-correlation of two functions
is defined as
Relationship Between Convolution And
Cross-Correlation
In General
if
Spectral density
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Discrete Fourier Transform
In this case we do not have a continuous function
but a time series.
Time series Sequence of data points, measured
typically at successive times, separated by time
intervals (often uniform).
Sampling Interval
DFT
IDFT
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Discrete Fourier Transform Properties
Fourier Transform of a real sequence of numbers
results in a sequence of complex numbers of the
same length.
is real
is real
and
If
Parsevals Theorem
Nyquist Frequency
In order to recover all Fourier components of a
periodic waveform (band-limited), it is necessary
to use a sampling rate at least twice the highest
waveform frequency. This implies that the Nyquist
frequency is the highest frequency that can be
resolved at a given sampling rate in a DFT
Nyquist Freq.
Sampling rate
Similarly Lowest Frequency?
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Aliasing
Aliasing is an effect that causes different
continuous signals to become indistinguishable
when sampled.
Good example to do in MatLab or IDL!!
Classic Example Wagon wheels in old western
movies
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Leakage
Allow frequency components that are not present
in the original waveform to leak into the DFT.
Spectral leakage appears due to the finite length
of the time series (non integer number of
periods, discontinuities, sampling is not and
integer multiple of the period).
How to handle this? Tapering
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How-To (see the code)
Matlab?
IDL?