okay enough of learning about how for
your transforms
work and their properties it's time to
apply fourier methods
to physical things before we begin
though
i'll just point out that i've pasted
this table of
fourier transform pairs and fourier
transform properties
in various places through the course
notes oftentimes if you're looking for
the solution to some problem it'll be
buried in this table somewhere okay
i want to discuss the concept of a
linear time invariant system
because this is a great place to use
fourier transforms
so some sort of system it's got some
input signal of
assumed here as a signal as a function
of time so it could be some electrical
signal
uh but it could also be a pattern so an
image for example that we're going to
process so it could be a function of
space
anyhow this input goes into some
system and gives us an output so you can
imagine here for example an electrical
signal going in
being filtered and then we have some
output and this is an lti system a
linear time invariant system
the two important properties are its
linearity which means if we add another
signal here
then the output is the sum of the
processed inputs
so y in is processed to y out on its own
zn is processed to z out on its own
if we add them together here we add them
together over here so it's linearity
the second one is the time and variance
so we have some input and some output
if we time shift the input we get the
time shifted output
that is to say the properties of the
system are constant so if it were
a filter we're not changing capacitors
or if it's an oscillator
we're not adding mass to it or anything
like that
so linear time and variance systems have
these properties
okay how would we describe a linear time
invariant system
mathematically well we have some input
here
the output there and it's processed by
some function g
of t and the way it's processed is
through a convolution
so the output y out is the convolution
of g
with y in which should be written as an
integral like this
what is g though how do we understand
what g is well
let's hypothetically imagine that our
input here is a delta spike
so it's a short sharp spike of input
in this case y in here would be a delta
function
so y out would be the convolution of gt
with a delta function
which would be just g of t so we call
g of t the impulse response function
it's the response of our system if we
hit with a sharp
sharp blow and impulse if we put an
impulse in here
it's processed by the system the output
would be g of t
g of t is known as the impulse response
function
now let's look at this whole thing in
the fourier domain we take fourier
transforms of
y in g of t and y out now it's going to
look like this we have some
fourier domain function going in it's
processed by the g
and we have y out in the frequency
domain
well this convolution we take the
fourier transform of that
that now looks like a product so that y
out is just the product
of g capital g of omega with capital y n
of omega
in other words g is basically a filter
the function g
changes the amplitude and phase of every
frequency
in y in so g of omega is a filter and
sometimes called the transfer function
so we can refer to g
volume as a transfer function so in
summary
lti an lti system is a filter in
frequency space for the transfer
function g of omega
in the time domain this corresponds to a
convolution and the convolution member
is some sort of a weighted moving
average of whatever function is going in
here
and the function little g of t is called
the impulse response function