So, in this matlab tutorial we're going
to extend what we learned last time
about the Fourier transform.
So, last time we took the Fourier
transform of a sine wave, and if you've
ever studied the Fourier transform in
school
you know that when you take the
Fourier transform of a sine wave you
should see a delta function or a spike
in the frequency domain, and what we were
seeing was something a little strange.
So, suppose I set my x to linspace(0,
5*pi,1000); and
then I plot(real(fft(sin(x))) Ok, so,
what we see here is actually two spikes
not one spike, and this is due to
artifacts, due to sampling, we don't have
an infinite sine wave which is assumed
when we calculate the fft of the sine
wave, and then we get a delta function in
the frequency domain. So, what we could do
if you want to have closer to an
internet sine wave you can do 0 to 10 pi,
so that's five full sine waves, and then do
500 samples.
Plot sine of X again, and so now you can
see something much nicer. So, now I want
to extend the idea of the Fourier
transform a little bit. So let's say
we're doing the Fourier transform of a
song or somebody playing the piano, the
frequencies if you've ever studied music
you know are at certain notes, are
multiples of each other, and so the
frequency is going to change as the song
progresses.
Now how do we show that using the
Fourier transform? The Fourier transform
takes the entire signal from time equals
minus infinity to infinity, and so what
we're really doing is we're taking all
the different frequencies over all of
time and then plotting that on the same
Fourier transform. And so the answer to
that question is we use something called
the short-time fourier transform. And so
the way that works is it's a two
dimensional heat map, or you can think of
it as a contour plot or something
similar, that shows time on one axis and
frequency on the other axis. It shows you
different frequency components at
different times, and the way it works is
it takes the Fourier transform of small
windows of time and then moves it along
the signal as time goes on.
So, it's only showing you the frequency
spectrum of a small window at each time.
Another name for the short-time fourier
transform is the spectrogram. So, if we look
here
there are different function signatures
that you could use. Notice that we
require the signal processing tool box
if we want to use this method.
A spectrogram look something like this, you
have time and frequency and it shows you
how the frequency, the main frequency
component is changing as time goes on.
So, how do we create, let's say we want to
test this function out, how do we create
a function that changes frequency as
time goes on?
So let's say we set X as before,
actually have already signed it, so it
goes from 0 to 5pi or 0 to 10pi, and it has
5,000 points. So, suppose we set the
frequency. So, a wave looks like this,
frequency*time. So, suppose we set
the frequency Omega, and I'm just typing
this out this isn't real matlab,
frequency as a function of time is just
equal to the time. So, we're going to
increase the frequency linearly as time
goes on.
So, if we combine those two we get
something like this,
y2 = sin(x.*x) because
this is element by element
multiplication.
Ok, so, now I do spectrogram of y2, and
you see that the frequency is increasing
linearly, so it takes an equal size step
at each window,
and notice that there are some artifacts
due to the fact that the frequency is
not equal within the window that we're
calculating the fft. So, you notice that
it looked kind of blockage, what you can
do is you can pass in the window size
along with the data. So, let's try window
size of 100, so now you can see it
looks much more like a line. So, let's try
that again with the window size of 500.
So, now you see the little red line is a
little bit skinnier but the time
windows are a little bit bigger, and so
this is what we call the uncertainty
principle and you may have heard this if
you take in quantum mechanics. The
uncertainty principle when you measure
the momentum and it's very accurate, you
can't measure the position as accurately.
So, it's the same when you're taking the
Fourier transform
because if we want to know the frequency
very accurately we need to have a very
long window of time, but that gives us
different frequencies at a lot of
different times. If we want to know the
frequency at a very specific time we
have to shorten the window, but then we
have a frequency that's not as
accurate.
