Chapter 4. Fourier Transformation and data processing:

Signal: In complex space (Phase sensitive detection): With T2 relaxation: Frequency Decay rate Amplitude  1/2 = 1/  T 2 Determined by 

Zero order: Set  cor = -  First order (Linear phase correction)   : Set  cor = -  t p where  is the offset frequency and t p is the pulse length. Weighting function:  Enhance Signal/Noise ratio (SNR)  Increase linewidth  1/2 = (R LB + R 2 )/  Matched line broadening: R LB = R 2

If we multiply the signal by a weighting function: W(t) = exp(R RE t) where R RE > 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 (-  t 2 ) Gaussian function falling off slower at small t and rapid at large t. If we multiply the signal by W(t) = exp(R RE t)exp(-  t 2 ) R RE is related to the linewidth L by R RE = -  L, we will have W(t) = exp(-  Lt)exp(-  t 2 ) 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/  T 2 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 = 0 o 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: : Square Cos  tSineExponentialGaussian Sinx/X (SINC) Two  functionsLorenzianGaussian Two  functions Questions: 0  1/  --  --  0   +T 2+T2+T Convolution theory: FT(AxB) = FT(A)  FT (B) + FT ( )= G(t) = exp(-a 2 t 2 )

Fourier Transformation: Signal: Fourier transform: Inverse Fourier : Absorption line S y ( ): Dispersion line (S x ( ): A max = A( o ) = T 2 ;  1/2 = 1/  T 2 Cosine FT: Sine FT:  F = F c – iF s  F(e 2  ot ) = (F c – iF s )[cos(2  o t) + isin(2  o t)] = 2  ( - o )