Gibbs Phenomenon Video Lecture - Electrical Engineering (EE)

hey guys and welcome to my short
tutorial on Gibbs phenomenon which is
the unusual behavior of Fourier series
around discontinuities or the jumps to
explain Gibbs phenomenon let's consider
this phreatic function the period here
is for an Omega nought is 2 pi divided
by the period which is pi divided by 2
to find C ka or Fourier series
coefficients we need to find this
integral I already solve this example in
the Fourier series lecture here's the
final answer as was mentioned before a
pre otic signal can be represented as a
linear combination of complex
exponential functions where C KS are the
weights for different frequencies so if
you replace this CK here and find the
Sigma from minus infinity to plus
infinity we get these shape
now let's implement this summation in
the computer to see the behavior
specifically I'm going to use MATLAB
software to find this Sigma and plot the
function but there is one big problem
here the problem is computers cannot
understand infinity what is infinity how
big the Infinity is we don't know in
fact if you want to implement this Sigma
in computers you need to have infinite
memory which is not feasible so to deal
with this problem let's consider Sigma
from minus n to n n is a finite number
that we can define for MATLAB for
example ten hundred or thousand now
because n is finite we can implement
this Sigma in MATLAB but think about it
here we go from minus n to n but in the
above equation Sigma goes from minus
infinity to plus infinity basically we
are ignoring lots of terms or we can say
we truncate the infinite Sigma to have a
finite Sigma therefore this Sigma is an
approximation of XT let's call it X hat
T as you can guess by choosing a larger
n X hat T is getting closer to XT or we
can say X hat T becomes a better
approximation of X T now let's look at X
hat T for different n for the first case
I set n to 1 basically the Sigma goes
from minus 1 to 1 and here's what we get
from MATLAB the purple signal is XT or
the original signal in the time domain
the white signal is X hat T or the
approximation now let's increase n to 5
in this case Sigma goes from minus 5 to
5 and here is what we get from MATLAB I
you can see the approximation ie the
white signal is getting closer to XT or
the purple signal
now let's set n 200 here's the result as
you can see the approximation is very
close to XT it's interesting to see
there is a large ripple or oscillation
near discontinuities or simply when
there is a jump in XT we can see ripples
let me zoom in to show you the ripples
better in fact we have overshoot and
undershoot around discontinuities now
let's increase n to 1000 here's what we
get from MATLAB as you can see the
ripples still exist but they are
compressed toward discontinuities let me
zoom in to see the ripples better so
here is the conclusion as we increase n
the ripples get compressed toward dicks
discontinuities but for any finite n
again I'm going to emphasize on finite
for any finite n the peak amplitude of
repo remains constant this behavior is
known as Gibbs phenomenon okay that's
all I want to say for this lecture
thanks a lot for giving me a time and
watching this tutorial please don't
forget to subscribe to my channel and
see you guys in the next video
you
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