Chapter 4. Fourier Transformation and data processing:

Signal:

In complex space (Phase sensitive detection):

With T2 relaxation:

2/Tte

Frequency Decay rate

Amplitude

1/2 = 1/T2

Determined by

Zero order:

Set cor = -

First order (Linear phase correction) :Set cor = - tp where is the offset frequency and tp is the pulse length.

Weighting function:

Enhance Signal/Noise ratio (SNR) Increase linewidth

1/2 = (RLB + R2)/

Matched line broadening: RLB = R2

If we multiply the signal by a weighting function:W(t) = exp(RREt) where RRE > 0 then the resonance will be narrowed. However, the S/N ratio will decrease (Increasing noise).

To compensate for that we can multiply the signal by another Gaussian function of the form: W(t) = exp (- t2)

Gaussian function falling off slower at small t and rapid at large t.

If we multiply the signal by W(t) = exp(RREt)exp(- t2) RRE is related to the linewidth L by RRE = - L, we will have W(t) = exp(- Lt)exp(- t2)Where L is the line width. In this notation L > 0 causes line Broadening and L < 0 leads to line narrowing.

Lorentzian lineshape (liquid state):

f() = f()max when = o; 1/2 = 1/T2

Gaussian lineshape (Solid state):

g() = g()max when = o; 1/2 = 2(ln2)1/2/a

222

22

)(41

2)(

oT

Tf

])(exp[)( 20

2 ag

Sine bell: First 1/2 of the sine function to fit into the acquisition region

Phase shift = 0o

Phase shift =

Sine bell square: First 1/2 of the sine square function to fit into the acquisition region (Faster rising and falling)

Only need to adjust one parameter !

Add points of amplitude zero tothe end of FID to increase resolution(Get more points in a given spectrumwithout adding noise).

Discard points at the end of a FID Reduce resolution Reduce noise Cause “ringing” or “wiggle”. Linear prediction, maximum entropy etc

Fourier Transformation:

Signal:

Fourier transform:

Inverse Fourier :

Fourier pairs:

t:

:

tiTtiTt eeetS o

)21(

2/ 022)(

dtetSdteetSdtetSSti

Ttiti

Tti oo )](21[

2)2

1(

2 22 )()()()(

dteStS ti 2)()(

Square Cost Sine Exponential Gaussian

Sinx/X (SINC) Two functions Lorenzian GaussianTwo functions

Questions:

0

1/

-

-

0 +T2+T

Convolution theory: FT(AxB) = FT(A) FT (B)

+FT ( ) = FT ( ) FT ( )

G(t) = exp(-a2t2)

])(exp[)( 2

aaF

Fourier Transformation:Signal:

Fourier transform:

Inverse Fourier :

Absorption line Sy():

Dispersion line (Sx():

Amax = A(o) = T2 ; 1/2 = 1/T2

Cosine FT:

Sine FT: F = Fc – iFs

F(e2ot) = (Fc – iFs)[cos(2ot) + isin(2ot)] = 2( - o)

tiTtiTt eeetS o

)21(

2/ 022)(

dtetSdteetSdtetSSti

Ttiti

Tti oo )](21[

2)2

1(

2 22 )()()()(

dteStS ti 2)()(

)()()](2/1[(

1)()(

2

)](21[2

iDA

iTdtetS�S

o

tiT o

222

2

220

)(41

)(2)(

oT

TD

222

22

)(41)(

oT

TA

dtttStSFc )2cos()()(

dtttStSFs )2sin()()(