NMR Part IV, Apodization and Zero Filling

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Quentin Murphy
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1 NMR Part IV, Apodization and Zro Filling This worksht is an introduction to how apodization and zro filling nhanc S/N and rsolution It is intractiv, so you can chang th variabls and s how thy ffct th signal Signals Gnratd: This sction dfins th obsrvd signals You can chang th amplitud, frquncy and rlaxation constants for ach nuclus Nuclus a Nuclus b Amplitud A a A b Frquncy ν a Hz ν b Hz Rlaxation T a sc T b sc Sampling Paramtrs: This sction dfins th spctromtr sampling paramtrs Ths will ffct th rsolution, spctral window, and acquisition tim Numbr of Data Points Sampld (binary) N 8 N = 56 Dwll Tim DW sc Nois Lvl: Nois nois Calculatd Paramtrs: Th paramtrs abov dtrmin th following variabls Acquisition Tim AT DW N AT = 56 sc Spctral Window Digital Rsolution Angular Frquncy Wav a Wav b SW DW Rsolution AT SW = 5 Hz Rsolution = 9 Hz ω a π ν a ω a = 68 rad sc ω b π ν b ω b = 566 rad sc Indx: Ths Indxs ar usd for various calculations i a, ( N ) t i a i a DW j a, N frquncy aj a j a N DW

2 Calculat Obsrvd Signal Wavform: Calculat th FID from th abov information Signal Wavform W Signali A a a cos t i a ω a T a A b cos t i a ω b T b Random Distribution NORM σ n σ n ln( rnd( ) ) cos( π rnd( ) ) Random Nois Nois i a NORM( nois) Obsrvd Signal Wavform W Obsrv Nois W Signal Obsrvd Signal Wavform W Obsrv i a Fourir Transforms: Fourir transform of FID to gnrat a frquncy domain signal (spctrum) F Obsrv fft W Obsrv Frquncy Spctrum of Obsrvd Signal Amplitud F Obsrv j a 4 5 frquncy a j a Frquncy (Hz)

3 Zro Fill: Incras th digital rsolution by adding null data to th FID, incrasing th siz of th data array Numbr of zro fills: ZF Crat zro array: k, ( N) ZF zro k Rst indxs and data array: N N ( N) ZF i b, ( N ) j, N i b DW frquncy j j N DW W ZF stack W Obsrv, zro F ZF fft W ZF Obsrvd Signal Wavform W ZF i b Frquncy Spctrum of Obsrvd Signal F Obsrv j a 4 5 frquncy j, frquncy a j a (Hz)

4 Exponntial Multiplication for S/N nhancmnt: Lin Broadning Factor LB Calculat Apodization Function: Apodiz EMi a LB π sc Calculat Apodizd Wavform: W EMi a W Obsrvi a Apodiz EMi a W EM stack W EM, zro Apodiz EM stack Apodiz EM, zro Fourir Transform: F EM fft W EM Apodization Function Apodiz EM i b W EM i b Comparison of Exponntial Multiplication 5 F EMj frquncy j

5 Gaussian Multiplication for S/N nhancmnt: Lin Broadning: LB Calculat Apodization Function: Apodiz GMi a π sc LB Calculat Apodizd Wavform: W GMi a W Obsrvi a Apodiz GMi a W GM stack W GM, zro Apodiz GM stack Apodiz GM, zro Fourir Transform: F GM fft W GM Apodization Function Apodiz GM i b W GM i b Comparison of Gaussian Multiplication 5 F GMj frquncy j

6 Doubl Exponntial Multiplication for Rsolution nhancmnt: Lin Broadning Factor: LB Gaussian Multiplication Factor: GM Calculat Apodization Function: Apodiz DEi a π sc LB GM AT π sc Calculat Apodizd Wavform: W DEi a W Obsrvi a Apodiz DEi a W DE stack W DE, zro Apodiz DE stack Apodiz DE, zro Fourir Transform: F DE fft W DE Apodization Function Apodiz DE i b W DE i b Comparison of Doubl Exponntial 5 F DEj frquncy j

7 TRAF Function for Rsolution Enhancmnt: Lin Broadning Factor: LB 4 Calculat Apodization Function: Apodiz TRAFi a t i a LB π sc t i a LB π sc AT π sc LB Calculat Apodizd Wavform: W TRAFi a W Obsrvi a Apodiz TRAFi a W TRAF stack W TRAF, zro Apodiz TRAF stack Apodiz TRAF, zro Fourir Transform: F TRAF fft W TRAF 4 Apodization Function Apodiz TRAF i b W TRAF i b Comparison of TRAF Rsolution Function F TRAFj 4 5 frquncy j

8 6 Comparison of Apodization Functions F EMj F GMj 5 F DEj F TRAFj frquncy j

9 Qustions Mak th following changs and obsrv thir ffct on th FID and th spctra a Chang th numbr of zro fills b Chang th lin broadning factor for Exponntial Multiplication c Chang th lin broadning factor for Gaussian Multiplication d Chang th lin broadning factor and th GM factor for Doubl Exponntial Multiplication Chang th lin broadning factor for th TRAF Function Chang th Rlaxation constant for on of th Nucli and rpat qustion How dos th natural linwidth ffct th spctrum Chang th numbr of data points sampld and rpat qustion and How dos th acquisition tim chang, and how dos this ffct th S/N? This documnt was dvlopd by: Scott E Van Bramr Dpartmnt of Chmistry Widnr Univrsity Chstr, PA 9 svanbram@scincwidnrdu

Introduction to Medical Imaging. Signal Processing Basics. Strange Effects. Ever tried to reduce the size of an image and you got this?

Introduction to Medical Imaging. Signal Processing Basics. Strange Effects. Ever tried to reduce the size of an image and you got this? Strang Effcts Introduction to Mdical Imaging Evr trid to rduc th siz of an imag and you got this? Signal Procssing Basics Klaus Mullr Computr Scinc Dpartmnt Stony Brook Univrsity W call this ffct aliasing