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Chapter 4. Fourier Transformation and data processing:. Signal:. In complex space (Phase sensitive detection):. With T2 relaxation:. Amplitude. Frequency. Decay rate. 1/2 = 1/T 2. Determined by . Zero order: Set cor = - . First order (Linear phase correction) :
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Signal: In complex space (Phase sensitive detection): With T2 relaxation: Amplitude Frequency Decay rate 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
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 to the end of FID to increase resolution (Get more points in a given spectrum without 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: : G(t) = exp(-a2t2) Cost Sine Exponential Gaussian Square 0 1/ - - Two functions Lorenzian Gaussian Sinx/X (SINC) Two functions Questions: 0 +T 2+T Convolution theory: FT(AxB) = FT(A) FT (B) + FT ( ) = FT ( ) FT ( )
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)
