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• Pages 1-20 Basic mri:Physics" by Evert J Blink
• Watch videos #1, #4, #5, and #8 on on this page.
  • This whole series of 10 videos is really great. I have watched them all several times over the years.
• Pages 4-17 of Introduction to MRI Techniques by Lars G. Hanson
• Read the MRI Physics Primer
  • You can skip the following sections:
    • 4.1, 4.2, 4.3, 4.4
    • 5.2, 5.3, 5.4, 5.5
    • 6 (skip all)
    • 7 (skip all)

• Get a big-picture understanding of the spectral characteristics of time-series and how the Fourier analysis reveals those characteristics.

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To understand acquisition and analysis of EEG and fMRI we need to have, at minimum, a basic understanding of sine waves. This is not a trigonometry course so we will not concern ourselves with the math, per se, but it is absolutely essential that you understand waves and wave analysis at least at the conceptual level.

There are plenty of Intro to Trig tutorials online, but I've grabbed a couple from Khan Academy that you might find useful (each are about ~10 minutes long). Watching these is not mandatory nor part of your lab assignment, but I strongly suggest watching them on your own time if the sine wave stuff in this section makes you scratch your head in confusion. Again, performing the math is not necessary, but having a sense of what a sine wave represents is necessary.

A sine wave is a curve with repetitive oscillation of constant amplitude. The sine wave has three properties:

• frequency (): how fast the wave goes up and down. Measured in cycles per second, or “hertz” ().
• amplitude (): how big (i.e., tall) is the wave. Power is amplitude squared, and so these terms are sometimes used interchangeably.
• phase (): the timing of the sine wave. Measured in radians or degrees.

To better understand each of these properties, see the figures and play with the code in the following section

These three properties (frequency, amplitude, phase) describe the sine wave in the following formula.

1. Run the following code to create your first sine wave.

lab_2_1.m

% MATLAB code
% Create and plot a sine wave
 
% Define variables
srate = 1000; % sampling rate of 1 kHz
time  = 0:1/srate:2; 
freq  = 2; % in Hz
amp   = 3; % amplitude, or height of the sine wave
angle = 0; % phase angle of wave
theta = angle * (pi / 180); % phase of the sine wave (radians) converted from the angle parameter
 
% Create sine wave
sine_wave = amp.*sin(2*pi*freq.*time+theta); % Note that you need the .* for point-wise vector multiplication.
 
% The following code will plot your sine wave
figure
plot(time,sine_wave)
% set(gca,'ylim',[-4 4]) % this adjusts the y-axis limits for visibility
title('My first sine wave!')

2. Now try running it a few more times. Each time change the freq and/or amp variables to see how it affects your sine wave. It might be most informative to only change parameter one at a time.

3. To demonstrate how phase angle () affects the sine wave, run the following code which will create two waves offset by 90°. The blue wave has and the red wave has , but the amplitudes and frequencies are the same.

The code requires that the phase be reported in radians (the theta value in the code), but it is often easier to refer to it in “degrees”, as I did in the lecture. Conversion between the two is trivial.

So I have written this into the code so that you can enter a phase angle (0-360 degrees) for the angle value, and the code will convert that to radians for creation of the sine wave. In the code below, if we wanted a phase angle of 90° we would set angle to 90 and the program would convert that to theta=1.5708.

lab_2_2.m

% MATLAB code
% Plot two sine waves offset by 90°
 
% Define variables in common for both waves
srate = 1000; % sampling rate of 1 kHz
time  = 0:1/srate:2;
 
% Define variables for
% Sine wave 1 
freq  = 2; % in Hz
amp   = 3; % amplitude, or height of the sine wave
angle = 0; % phase angle of the wave
theta = angle * (pi / 180); % phase of the sine wave (radians) converted from the angle parameter
 
sine_wave_1 = amp.*sin(2*pi*freq.*time+theta); % Note that you need the .* for point-wise vector multiplication.
 
% Define variables for
% Sine wave 2 
freq  = 2; % in Hz
amp   = 3; % amplitude, or height of the sine wave
angle = 90; % phase angle of the wave
theta = angle * (pi / 180); % phase of the sine wave (radians) converted from the angle parameter
 
sine_wave_2 = amp.*sin(2*pi*freq.*time+theta); % Note that you need the .* for point-wise vector multiplication.
 
% The following code will plot your sine wave
figure
plot(time,sine_wave_1); hold on
plot(time,sine_wave_2,'r')
% set(gca,'ylim') % this adjusts the y-axis limits for visibility
title('More sine waves!')

3. Plot and save the output of the above code.

4. Before moving on to the next section play around with the amplitude (amp), frequency (freq), and phase angle (angle) values and see what effect they have on your waves. Use different parameters for the two different waves. Save one of these figures.

The waves we've seen so far are “simple” waves because they are perfectly sinusoidal (aka “pure”). That is, the wave is completely described by a single value for each of the three parameters; amplitude, frequency, and phase. A complex wave, on the other hand, is made up of more than one pure sine wave. For example a pure sine wave is created when you press a single key on a piano. The sound waves created are oscillating at only one frequency. However, if you were to play a chord (many keys simultaneously) then the sound wave would be a complex combination of the sine wave generated by each of the keys. The sound waves would be a combination of many frequencies.

The math to calculate this new complex wave is extremely complicated. It requires that you <checks notes> add numbers together. Hmm, okay I guess it's not terribly complicated.

5. Let's re-create the waves we created in the last section, but this time we'll also create a third wave that represents the linear combination of the two (i.e., the sum of adding them together). This complex wave will be plotted in green.

lab_2_3.m

% MATLAB code
% Combine two sine waves
 
% Define general variables
srate = 1000; % sampling rate of 1 kHz
time  = 0:1/srate:2;
 
% Define variables for
% Sine wave 1 
freq  = 2; % in Hz
amp   = 3; % amplitude, or height of the sine wave
angle = 0; % phase angle of wave
theta = angle * (pi / 180); % phase of the sine wave (radians) converted from the angle parameter
 
sine_wave_1 = amp.*sin(2*pi*freq.*time+theta); % Note that you need the .* for point-wise vector multiplication.
 
% Create name for figure legend
name1 = sprintf('f=%0.1f, a=%0.1f, pa=%0.1f',freq,amp,angle);
 
% Define variables for
% Sine wave 2 
freq  = 2; % in Hz
amp   = 3; % amplitude, or height of the sine wave
angle = 90; % phase angle of wave
theta = angle * (pi / 180); % phase of the sine wave (radians) converted from the angle parameter
sine_wave_2 = amp.*sin(2*pi*freq.*time+theta); % Note that you need the .* for point-wise vector multiplication.
 
% Create name for figure legend
name2= sprintf('f=%0.1f, a=%0.1f, pa=%0.1f',freq,amp,angle);
 
% Create Sine wave 3 by simply adding sine_wave_1 and sine_wave_2
sine_wave_3 = sine_wave_1 + sine_wave_2;
 
% Create name for figure legend
name3 = 'combined wave';
 
% The following code will plot your sine wave
figure
plot(time,sine_wave_1); hold on
plot(time,sine_wave_2,'r')
plot(time,sine_wave_3,'g')
legend(name1,name2,name3);
% set(gca,'ylim') % this adjusts the y-axis limits for visibility
title('Complex waves!');

This creates the following figure (without the annotations):

6. Given what we've learned, what do you think would happen if we combined two identical waves (same frequency, amplitude, and phase)? To find out, modify and your code from lab1_3.m so that wave 1 and wave 2 are identical.

The figure you create should have a new (green) wave with the same frequency and phase as the others, but double the amplitude. This is called constructive interference; the superposition (fancy way of saying that the two waves are occupying the same space; this is really just the combination of the two waves, which we now know is simply the point by point sum) is larger than either of the contributing waves.

7.Let's look at another extreme. This time let's see what happens when we combine two waves with equal amplitude and frequency, but a phase offset of 180° (). Modify your code from lab1_3.m so that wave1 and wave 2 are 180° out of phase.

In this case we can see that our two waves cause destructive interference and the superposition is smaller than either of the individual waves. Actually, in this case because the waves are perfectly out of phase with one another (“anti-phase”) the resultant wave is a flat line.

The following video further demonstrates the effects of constructive/destructive interference.

So far we've just been combining two different sine waves, but everything we've learned applies just the same regardless of how many waves we're talking about. For instance, I could combine the following five sine waves (frequencies and amplitudes are summarized in the table)

and I'd get this complex wave.

What if we wanted to take any given signal and know what simple waves needed to be summed together in order to create it? We would use the "Fourier transform", of course! Joseph Fourier (1768-1830) claimed that any complex signal can be described as the sum of sine waves of varying amplitude and frequency. That is, signals could be decomposed into a set of “basis functions” that, if linearly added, would create the complex wave. This decomposition is called the Fourier transform.

We will not concern ourselves with the math of the Fourier transform, but let's see what kind of results it gives us. To do so, we really just need to look at the signals shown at the end of the last section. Let's say we started with this complex signal and wanted to know what basis functions it was made from.

Using the Fourier transform to decompose this signal we learn that it is made up of the following simple sine waves, each with their own frequency, amplitude, and phase. We would say that this description is in the time-domain because the complex wave is represented by a series of sine waves that each vary over time.

We can also represent this exact same information in the frequency-domain. That is, we can describe the signal in terms of how much each frequency contributes to the total, which is what's done in the plot below. The x-axis now represents frequency rather than time and the y-axis still represents amplitude. Compare the bar graph to the table at the end of the last section. You'll see that the Fourier analysis has accurately identified which frequencies are present and at what amplitudes.

In summary, these two plots represent the exact same signal. The one on the left represents the signal in the time domain, whereas the one on the right represents the signal in the frequency-domain. But importantly, neither is any more informative than the other. They contain the same information and (theoretically) we can go back and forth between them.