%%
% One challenge of teaching Fourier transform concepts is that each concept can be
% (and is) interpreted and explained in many different ways. It isn't always
% obvious how the different explanations for the same concepts are connected.
%
% For example, in my <https://blogs.mathworks.com/steve/2010/02/22/aliasing-and-the-discrete-time-fourier-transform/ 
% last Fourier transform post> I talked about _aliasing_.  I
% said that if you sample a continuous-time cosine at a sampling frequency
% $F_s$, then you can't distinguish between a cosine with frequency $\alpha$ and
% a cosine with frequency $F_s k \pm \alpha$.
%
% <https://blogs.mathworks.com/steve/2010/02/22/aliasing-and-the-discrete-time-fourier-transform/#comment-22765 
% In response>, Dave S. wanted to know how this related to what he learned about
% aliasing: that aliasing is a "problem that occurs when the sampling rate of
% your discrete signal is not at least twice the [...] highest existing
% frequency [in the continuous-time signal]."
%
% I know I promised to introduce the discrete Fourier transform next, but I'd
% like to change my mind and try to answer Dave's question instead.  There are
% two key pieces of the question to address: What is the nature of the
% "problem," and what is the significance of "twice the highest frequency"?
%
% I thought about drawing some new frequency-domain diagrams showing overlapping
% triangles like you'd see in
% <http://www.amazon.com/Discrete-Time-Signal-Processing-Prentice-Hall/dp/0131988425/ref=sr_1_2?ie=UTF8&s=books&qid=1267639933&sr=8-2 Oppenheim and Schafer>, but then I thought it
% might be better to just continue the <https://blogs.mathworks.com/steve/2010/02/22/aliasing-and-the-discrete-time-fourier-transform/ 
% sampled cosine example from last time>.
%
% Let's start with a continuous-time cosine signal at 60 Hz.

f = 60;  % Hz
tmin = -0.05;
tmax = 0.05;
t = linspace(tmin, tmax, 400);
x_c = cos(2*pi*f * t);
plot(t,x_c)
xlabel('t (seconds)')

%%
% Let's sample $x_c$ with a sampling frequency of 800 Hz.
T = 1/800;
nmin = ceil(tmin / T);
nmax = floor(tmax / T);
n = nmin:nmax;
x1 = cos(2*pi*f * n*T);
hold on
plot(n*T,x1,'.')
hold off

%%
% The sampling frequency of 800 Hz is well above 120 Hz, which is twice the
% frequency of the cosine. And you can see that the samples are clearly
% capturing the oscillation of the continuous-time cosine.
%
% Let's try a lower sampling frequency.

T = 1/400;
nmin = ceil(tmin / T);
nmax = floor(tmax / T);
n = nmin:nmax;
x1 = cos(2*pi*f * n*T);
plot(t, x_c)
hold on
plot(n*T, x1, '.')
hold off

%%
% The samples above are still adequately capturing the shape of the cosine.  Now
% let's drop the sampling frequency down to exactly 120 Hz, twice the frequency
% of the 60 Hz cosine. (And I'll switch to using circle markers to make the
% samples easier to see.)

T = 1/120;
nmin = ceil(tmin / T);
nmax = floor(tmax / T);
n = nmin:nmax;
x1 = cos(2*pi*f * n*T);
plot(t, x_c)
hold on
plot(n*T, x1, 'o')
hold off

%%
% See how the samples jump back and forth between 1 and -1? And how they capture
% only the extremes of each period of the cosine oscillation?  This is the
% significance of "twice the highest frequency of the signal" value for sampling
% frequency.  If you'll allow a "hand-wavy" explanation here, I'll say that this
% sampling frequency of 120 Hz is just enough to capture the cosine oscillation.
%
% But aliasing is worse that "just" losing information.  When we drop the
% sampling frequency too low, the samples start to look increasingly like they
% came from a *different, lower-frequency signal*.
%
% Let's try 70 Hz.

T = 1/70;
nmin = ceil(tmin / T);
nmax = floor(tmax / T);
n = nmin:nmax;
x1 = cos(2*pi*f * n*T);
plot(t, x_c)
hold on
plot(n*T, x1, 'o')
hold off

%%
% The samples above look like they actually could have come from a *10 Hz cosine
% signal*, instead of a 60 Hz cosine signal.  Take a look:

T = 1/70;
x_c = cos(2*pi*10 * t);
nmin = ceil(tmin / T);
nmax = floor(tmax / T);
n = nmin:nmax;
x1 = cos(2*pi*f * n*T);
plot(t, x_c)
hold on
plot(n*T, x1, 'o')
hold off

%%
% That's the heart of the "problem" of aliasing. Because the sampling frequency
% was too low, a high-frequency cosine looked like a low-frequency cosine after
% we sampled it.
%
% Later on in <https://blogs.mathworks.com/steve/category/fourier-transforms/ 
% this series> I plan to come back again to the concept of aliasing
% and show some examples of how it looks in an image.
%
% OK, now I'll start working on the upcoming discrete Fourier transform (DFT) post.


%%
% _Copyright 2010 The MathWorks, Inc._