Fourier Transform. louder softer. louder. softer. amplitude. time. amplitude. time. frequency. frequency. P. J. Grandinetti

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1 Fourier Transform * * amplitude louder softer amplitude louder softer frequency frequency

2 Fourier Transform amplitude What is the mathematical relationship between two signal domains frequency

3 Fourier Transform amplitude frequency

4 Inverse Fourier Transform amplitude frequency

5 Simple Fourier Transform Example

6 Simple Fourier Transform Example

7 Simple Fourier Transform Example FT

8 Simple Fourier Transform Example FT Ω 0 Ω frequency

9 Simple Fourier Transform Example FT Ω 0 What is the meaning of negative frequency? Ω frequency

10 Circular (Counter Clockwise) Motion in Complex Plane y r x -r r x r y -r FT Ω 0 Ω frequency

11 Circular (Clockwise) Motion in Complex Plane y r x -r r x r y -r FT Ω 0 Ω frequency

12 Exponential Decay : Lorentzian Lineshape X Y Lorentzian

13 Exponential Decay : Lorentzian Lineshape FT Ω Ω Real Imaginary Absorption Mode 2/T 2 Dispersion Mode 2/T 2

14 In a perfect world... Spectral Phase Correction path of tip of magnetization vector as it precesses x detector y detector Real Imaginary

15 Spectral Phase Correction : Zeroth Order First problem is a minor one... φ 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. x detector y detector Real Imaginary Absorption and Dispersion mode lineshapes become mixed in real and imaginary parts.

16 Spectral Phase Correction : Zeroth Order Solution is simple... Real Imaginary Absorption and Dispersion mode lineshapes mixed in real and imaginary parts. Real Imaginary Absorption and Dispersion mode lineshapes cleanly separated into real and imaginary parts.

17 Spectral Phase Correction : First Order Ω 2 Ω 1 X Real Imaginary Ω 1 Ω 2 y at t=0, when receiver is turned on, the two magnetization vectors are aligned along x axis.

18 Spectral Phase Correction : First Order Ω 2 Ω 1 X Real Imaginary Ω 1 Ω 2 y 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?

19 Spectral Phase Correction : First Order Receiver is turn on at t 0 after pulse. Ω 1 Ω 1 X Real Ω 2 Imaginary Ω 2 y 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 define phase correction as linearly dependent on frequency: that we were late in starting the detector

20 Spectral Phase Correction : First Order Ω 1 Real Ω 2 Imaginary Ω 1 Ω 2 Real Imaginary

21 Spectral Phase Correction : First Order Ω 1 Real Ω 2 Imaginary Ω 1 Ω 2 Real Imaginary Somes see baseline roll

22 Spectral Phase Correction : First Order S 1 (t) F. T. S 1 (ν) S 2 (t) 1 0 X = (Multiplication) F. T. S 2 (ν) * = (Convolution) S T (t) F. T. S T (ν)

23 Spectral Phase Correction : Algorithm ν

24 Spectral Phase Correction : Algorithm ν Ω 1 Apply zeroth order phase correction until one peak is completely absorption mode lineshape. one peak "phased" ν

25 Spectral Phase Correction : Algorithm ν No further phase correction should affect this peak Ω 1 Apply zeroth order phase correction until one peak is completely absorption mode lineshape. one peak "phased" ν

26 Spectral Phase Correction : Algorithm ν No further phase correction should affect this peak Ω 1 Apply zeroth order phase correction until one peak is completely absorption mode lineshape. one peak "phased" ν Pivot Frequency

27 Spectral Phase Correction : Algorithm ν No further phase correction should affect this peak Ω 1 Apply zeroth order phase correction until one peak is completely absorption mode lineshape. one peak "phased" ν Pivot Frequency Adjust t 0 until spectrum is phased. ν

NMR Basics. Lecture 2

NMR Basics. Lecture 2 NMR Basics Lecture 2 Continuous wave (CW) vs. FT NMR There are two ways of tuning a piano: - key by key and recording each sound (or frequency). - or, kind of brutal, is to hit with a sledgehammer and