%% Problem Set #1
% Date: 09/22/08

%% Please return your homework in a .m file which can be run

%% 1. Perform the following computations

% a. 5+4
% b. 2/4
% c. 3%2
% d. 9*2
% e. (8+12)/8
% f. create a vector of 100 1s and add 2 to each element
% g. create a vector of 100 1s and have each subsequent element be 2 times
% the previous one

%% 2. Larmor Frequency

% What is the Larmor precession frequency of...
% a. 15N at 3T
% b. 29Si at 1.5T
% c. 1H at 400 Gauss
% d. 1H at the earth's magnetic field (approximately)

%% 3. Exponential Decay

%%% a. Simulate and plot an exponential decay with a time constant T2 = 100
%%% ms.

%%% b. Simulate and plot a signal which decays with two time constants
%%% T2_fast = 10 ms and T2_slow = 100 ms.

%% 4. Frequencies

%%% Hint:: be sure to use complex signals

%%% a. Simulate and plot 10 periods of a signal with a frequency 100 Hz.

%%% b. Simulate and plot 10 periods of a signal with two frequencies 10 Hz
%%% and -750 Hz.

%%% c. Simulate and plot (4b) again, but with a signal to noise ratio = 20.
%%% Hint::: signal-to-noise ratio = SNR = max signal divided by noise
%%% magnitude. You will need to simulate and add noise to the plot.

%%% d. Take the FFT of the signal you created in (4c) and display the
%%% result with a proper frequency axis.

%% 5. Resonance

%%% Resonances are the fundamentals of nuclear magnetic resonance.
%%% To simulate a resonance, we will need to know the
%%% frequency at which a nuclei is precessing (Larmor frequency).

%%% In addition, the resonances usually have associated time constants, so
%%% that if we make a measurement, the signal will decay exponentially to 
%%% zero in a time T2 (usually on the order of ms or s).

%%% Using your knowledge of exponential decay and complex signals
%%% from problems 3 and 4, simulate an NMR spectra with the
%%% following resonance. Display the time domain signal adjacent to this.

%%% v1 = -400 Hz, a1 = 1; T2 = 100 ms
%%% v2 = 500 Hz, a2 = 2; T2 = 10 ms
%%% v3 = 1000 Hz, a3 = 1.5, T2 = 100 ms

%%% Be sure to add some noise, so that it is properly faked!

%% 6. Statistics

%%% Using the find() command...
%%% What is the approx. probability that a normally distributed random 
%%% variable with unit standard deviation falls between 0.2 and 0.4?
%%% (HINT - what did "whos" tell us?)