time What is the mathematical relationship between two signal domains
frequency P. J. Grandinetti
Fourier Transform
amplitude
time
frequency P. J. Grandinetti
Inverse Fourier Transform
amplitude
time
frequency P. J. Grandinetti
Simple Fourier Transform Example
time
P. J. Grandinetti
Simple Fourier Transform Example
time
P. J. Grandinetti
Simple Fourier Transform Example
time
FT
P. J. Grandinetti
Simple Fourier Transform Example
time
FT
−Ω
P. J. Grandinetti
0
Ω
frequency
Simple Fourier Transform Example
time
FT
−Ω What is the meaning of negative frequency? P. J. Grandinetti
0
Ω
frequency
Circular (Counter Clockwise) Motion in Complex Plane
y
r x -r r x
time
r y -r
time
FT −Ω P. J. Grandinetti
0
Ω
frequency
Circular (Clockwise) Motion in Complex Plane y
r x -r r x
time
r y -r
time
FT −Ω P. J. Grandinetti
0
Ω
frequency
Exponential Decay : Lorentzian Lineshape
X
Y
time
P. J. Grandinetti
Lorentzian
time
Exponential Decay : Lorentzian Lineshape
FT
Ω
Ω
Real Absorption Mode
Imaginary 2/T2
Dispersion Mode
2/T2 P. J. Grandinetti
Spectral Phase Correction In a perfect world... path of tip of magnetization vector as it precesses
x detector
time
y detector
Real time P. J. Grandinetti
Imaginary
Spectral Phase Correction : Zeroth Order First problem is a minor one...
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.
φ
x detector
y detector
Real time
P. J. Grandinetti
time
Imaginary
Absorption and Dispersion mode lineshapes become mixed in real and imaginary parts.
Spectral Phase Correction : Zeroth Order Solution is simple...
Real
Imaginary
Absorption and Dispersion mode lineshapes mixed in real and imaginary parts.
Real
Imaginary
Absorption and Dispersion mode lineshapes cleanly separated into real and imaginary parts. P. J. Grandinetti
Spectral Phase Correction : First Order
Ω2 Ω1
X
Real
Imaginary
Ω1 Ω2
y
P. J. Grandinetti
at t=0, when receiver is turned on, the two magnetization vectors are aligned along x axis.
Spectral Phase Correction : First Order
Ω2 Ω1
X
Real
Imaginary
Ω1 Ω2
y
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? P. J. Grandinetti
Spectral Phase Correction : First Order Receiver is turn on at time t0 after pulse. Ω1 Ω1 X
Real
Ω2
Imaginary
Ω2 y
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 define phase correction as linearly dependent on frequency: time that we were late in starting the detector P. J. Grandinetti
Spectral Phase Correction : First Order Ω1 Real
Ω2
Imaginary
Ω1 Ω2 Real
P. J. Grandinetti
Imaginary
Spectral Phase Correction : First Order Ω1 Real
Ω2
Imaginary
Ω1 Ω2 Real
Sometimes see baseline roll P. J. Grandinetti
Imaginary
Spectral Phase Correction : First Order
F. T.
S1(t)
S1(ν)
*
X
S2(t)
1 0
(Multiplication)
S2(ν)
F. T.
ST(ν)
=
=
F. T.
ST(t)
P. J. Grandinetti
(Convolution)
Spectral Phase Correction : Algorithm
ν
P. J. Grandinetti
Spectral Phase Correction : Algorithm
ν Ω1
one peak "phased"
P. J. Grandinetti
Apply zeroth order phase correction until one peak is completely absorption mode lineshape.
ν
Spectral Phase Correction : Algorithm
ν No further phase correction should affect this peak
Ω1
one peak "phased"
P. J. Grandinetti
Apply zeroth order phase correction until one peak is completely absorption mode lineshape.
ν
Spectral Phase Correction : Algorithm
ν No further phase correction should affect this peak
Ω1
one peak "phased"
Pivot Frequency
P. J. Grandinetti
Apply zeroth order phase correction until one peak is completely absorption mode lineshape.
ν
Spectral Phase Correction : Algorithm
ν No further phase correction should affect this peak
Ω1
one peak "phased"
Apply zeroth order phase correction until one peak is completely absorption mode lineshape.
ν
Adjust t0 until spectrum is phased. Pivot Frequency
