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()()(
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