simple fourier transform example

Fourier Transform

P. J. Grandinetti

time

*

loudersofter

frequency

amp

litu

de time

*

louder softer

frequency

amp

litu

de

time

frequency

amp

litu

de

Fourier Transform

P. J. Grandinetti

What is the mathematicalrelationship between two

signal domains

time

frequency

amp

litu

de

Fourier Transform

P. J. Grandinetti

time

frequency

amp

litu

de

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

FT

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.