%%% HW #2

%%% Due: October 9, 2008


%% 1. Boxcars and Sincs

% The boxcar function (or rect function) is

% rect(t) = 0 if t < -1
%         = 1 if -1 < t < 1
%         = 0 if t > 1

% a. Sample the rect function on an interval -10 to 10 and
%    its Fourier transform with 1024 points. Plot both functions.

% HINT: Be sure to plot the reciprocal domain (say m^-1 or s^-1) correctly
%       with the correct axis.

% b. Sample the rect function on an interval -100 to 100 and its Fourier
%    transform with 1024 points. Plot both functions.

% c. The full width at half maximum (FWHM) is a measure of linewidth and is
% often defined by the maximum height divided by the width at half of the
% maximum. Measure the FWHM for both plots in (a) and (b). Discuss the
% results. What is Fourier transform of a rect function called?

%% 2. Exponential Decay and Lorentzians

% (a) Plot one second of an exponential decay with T2 = 0.1 s oscillating
%     at a frequency 100 Hz. Take the Fourier transform and plot this also
%     with the correct axis.

% (b) Over the same period, plot an exponential decay with T2=0.01 s
%     oscillating with a frequency 100 Hz. Take the Fourier transform and
%     plot this with the correct axis.

% (c) Again, measure the FWHM of the lines in (a) and (b) and discuss the
%     results. What is the Fourier transform of an exponential decay
%     called?

% (d) In this case, how is the spectral line width (FWHM) related to T2?
%     Give an expression for this.

%% 3. Parts-Per-Million and Alcohols

% Parts-Per-Million (ppm) is a useful independent variable for plotting NMR
% spectra because it is independent of field strength.

% ppm = (frequency - reference frequency) / reference frequency

% For example, with respect to the reference TMS (tetramethysilane), 1H in
% water resonates at 4.7 ppm.

% For example, to calculate the water shift from TMS at 9.4 T...

4.7                 %%% ppm
42.57               %%% gamma MHz/T
9.4                 %%% T
4.7 * 42.57 * 9.4   %%% Hz


% a. Simulate the time domain signal and 1H NMR spectra of 2-phenylethanol
%    C6H5-CH2-CH2-OH.

% %%% HINT:: 2-phenylethanol has 4 groups of protons which resonate at
% different frequencies. For this problem, assume that there is no
% interaction between the protons, so that each group resonates at only one
% frequency (no J-coupling). Also, neglect relaxation.

% %%% Phenyl 1H ~ 7.5 ppm
% %%% OH        ~ 4.5 ppm
% %%% CH2       ~ 4.0 ppm
% %%% CH2       ~ 3.1 ppm