Simple Fourier Transform Example
Transcript of Simple Fourier Transform Example
Fourier Transform
P. J. Grandinetti
time
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loudersofter
frequency
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louder softer
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time
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Fourier Transform
P. J. Grandinetti
What is the mathematicalrelationship between two
signal domains
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Fourier Transform
P. J. Grandinetti
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Inverse Fourier Transform
P. J. Grandinetti
Simple Fourier Transform Example
P. J. Grandinetti
time
Simple Fourier Transform Example
P. J. Grandinetti
time
Simple Fourier Transform Example
P. J. Grandinetti
time
FT
Simple Fourier Transform Example
P. J. Grandinetti
time
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frequencyΩ−Ω 0
Simple Fourier Transform Example
P. J. Grandinetti
time
FT
frequencyΩ−Ω 0
What is the meaningof negative frequency?
Circular (Counter Clockwise) Motion in Complex Plane
P. J. Grandinetti
time
rx
y
x
timey
r
-r
r
-r
FT
frequencyΩ−Ω 0
Circular (Clockwise) Motion in Complex Plane
P. J. Grandinetti
time
rx
y
x
timey
r
-r
r
-r
FT
frequencyΩ−Ω 0
Exponential Decay : Lorentzian Lineshape
P. J. Grandinetti
X Y
time time
Lorentzian
Exponential Decay : Lorentzian Lineshape
P. J. Grandinetti
Real
AbsorptionMode
Imaginary
DispersionMode
2/T2
Ω Ω
2/T2
FT
Spectral Phase Correction
P. J. Grandinetti
x detector
y detector
path of tip of magnetization vectoras it precesses
time
time
In a perfect world...
Real Imaginary
Spectral Phase Correction : Zeroth Order
P. J. Grandinetti
First problem is a minor one...
x detector
y detector
time
time
φ
Real Imaginary
Receiver phase of zero does not correspond to zero phase from x in rotating frame.
Depends on cable lengths and probe tuning. Otherwise should remain constant.
Absorption and Dispersion mode lineshapesbecome mixed in real and imaginary parts.
Spectral Phase Correction : Zeroth Order
P. J. Grandinetti
Solution is simple...
Real Imaginary
Real Imaginary
Absorption and Dispersion mode lineshapesmixed in real and imaginary parts.
Absorption and Dispersion mode lineshapescleanly separated into real and imaginary parts.
Spectral Phase Correction : First Order
P. J. Grandinetti
Ω1 Ω2
y
X
Ω2
Ω1
Real Imaginary
at t=0, when receiver is turned on, the two magnetization vectors are aligned along x axis.
Spectral Phase Correction : First Order
P. J. Grandinetti
Ω1 Ω2
y
X
Ω2
Ω1
Real Imaginary
at t=0, when receiver is turned on, the two magnetization vectors are aligned along x axis.
What happens if we were late in turning on the receiver?
Spectral Phase Correction : First Order
P. J. Grandinetti
y
X
Ω2
Ω1
Ω1
Ω2Real Imaginary
Receiver is turn on at time t0 after pulse.
Phase needed to make site 1 have a pure absorption mode spectrum in real part is not the same as the phase needed for site 2.
The phase correction needed can be calculated from the frequency of each site.We de�ne phase correction as linearly dependent on frequency:
time that we were latein starting the detector
Spectral Phase Correction : First Order
P. J. Grandinetti
Ω1
Ω2Real Imaginary
Ω1 Ω2
Real Imaginary
Spectral Phase Correction : First Order
P. J. Grandinetti
Ω1
Ω2Real Imaginary
Ω1 Ω2
Real Imaginary
Sometimes seebaseline roll
Spectral Phase Correction : First Order
P. J. Grandinetti
F. T.
F. T.
F. T.
S2(t) S2(ν)
S1(t)
ST(t)
S1(ν)
ST(ν)
0
1
X=
*=
(Convolution)
(Multiplication)
Spectral Phase Correction : Algorithm
P. J. Grandinetti
ν
Spectral Phase Correction : Algorithm
P. J. Grandinetti
ν
ν
Ω1
one peak"phased"
Apply zeroth order phase correction until one peak is completely absorption mode lineshape.
Spectral Phase Correction : Algorithm
P. J. Grandinetti
ν
ν
Ω1
one peak"phased"
Apply zeroth order phase correction until one peak is completely absorption mode lineshape.
No further phasecorrection should
a�ect this peak
Spectral Phase Correction : Algorithm
P. J. Grandinetti
ν
ν
Ω1
one peak"phased"
Apply zeroth order phase correction until one peak is completely absorption mode lineshape.
No further phasecorrection should
a�ect this peak
PivotFrequency
Spectral Phase Correction : Algorithm
P. J. Grandinetti
ν
ν
Ω1
one peak"phased"
ν
Apply zeroth order phase correction until one peak is completely absorption mode lineshape.
No further phasecorrection should
a�ect this peak
PivotFrequency
Adjust t0 until spectrum is phased.