Linear Shift Invariant Systems Video Lecture | Signals and Systems - Electrical Engineering (EE)

welcome back when I study systems or
system theory we are typically
interested in a number of property such
systems might have for example we are
interested if the systems are stable and
important properties since you might say
that in general and stable systems are
of not much use whether their causal
invertible and so on all these
properties are independent of each other
one property does not require another
property or does not lead to another
property out of all possible to these
systems were primarily interested in
systems that have two important
properties they're both linear and
specially invariant thus forming lsi
systems will be introducing them in this
segment these two properties define an
important subset of all possible to the
systems linearity means that if a sum of
signals is that the input of a linear
system the system can process each
signal separately and add up the
processed signals
specially variance means that it is
irrelevant where the origin of the
coordinate system is located as well see
in the next two segments we can describe
and implement LSI systems efficiently in
the spatial domain through convolution
we can also describe them in the
frequency domain as well see in week 3
they're there for friendly systems in
that they reveal their properties to us
in a rather straightforward way but at
the same time extremely useful and
widely used in a number of applications
so let us see how LSI systems can be
defined mathematically and also let us
look at some simple examples
so far we talked about signals
2-dimensional multi-dimensional signals
in general we have examples of some
important signals such as the Delta and
the cosine that we will be using
throughout this class we are interested
in in processing or manipulating such
signals through systems so x + 1 + 2 is
the input image to such a system Y is
the output which is equal to a
transformed version of the input a
manipulated version of the input some
examples of such signals y n 1 + 2 is
equal to 255 - x + 1 + 2 assuming these
images are 8 bits per pixel therefore
the range from 0 to 255 what such a
system does is changing the polarity of
the input is turning black values into
white and white values into black so a
negative into a positive or a positive
into a negative I can have another
system but perform this transformation
to the input values according to this
function and what such a system does is
stretching the intensity values of the
input image we'll see actually quite a
few systems of this nature when we talk
about image enhancement similarly you
can have a system that provides an
output which is equal to the average of
the input values in a neighborhood
denoted here by n
so this denotes a neighborhood of input
values and yet as a final example I can
have a system that gives me as an output
the median of the input values again in
the neighborhood when we talk about
systems we are interested in a number of
their properties some of which are
mentioned here where in the city the
system is stable whether it has memory
or his memory less whether it's causal
whether it is linear and whether it's
spatially invariant all these five here
listed properties are independent of
each other a system can have all of them
none of them just one or two and so on
and out of all these five properties we
are going to focus next on systems that
have the last two properties and to see
such systems will be referring to them
as linear and spatially invariant
systems are quite useful are used very
widely and it's relatively
straightforward to describe such systems
both in the spatial domain as well as in
the frequency domain a two dimensional
system is linear if it satisfies the
homogeneity property shown here in other
words if a table of the system I put a
weighted sum of two signals the weight
is alpha one and alpha 2 and if at the
output I find the weighted sum of the
individual output so alpha one the
response of the system to X 1 plus alpha
2 the response of the system to X 2 then
the system is linear this is clearly a
very useful properties because in many
applications I have to process the some
weighted sum of individual images and it
might be easier to process each image
individually and
up the responses or litigated the
reverse way in some sense I can take any
signal and decompose it into simpler
signal simpler images which then I
process individually and again a add up
add up the individual responses from
this equation it's clear that if alpha 2
is equal to zero then the response of
the system to alpha 1 X 1 + 1 + 2 is
simply alpha 1 the response of the
system to X 1 so if an image is
multiplied by a scalar I don't need to
be concerned I can process the image and
then multiply the output by the scalar
similarly if alpha 1 equals minus alpha
2 and x1 equals x2 since this property
holds for any X 1 X 2 and any weight
then I see that the response to alpha 1
X 1 minus alpha 1 X 1 which is equal to
zero right so it's the response to the
zero signal is equal to alpha 1 the
response to X 1 minus alpha 1 the
response to X 1
which is equal to zero so in other words
we see that if the system is linear when
I put a 0 at the input I find a 0 at the
output now this is a property that can
be shared also by nonlinear systems
therefore is a therefore is a property
but I cannot use to prove that the
system is linear but they can use to
prove that the system is nonlinear and
there's a simple example we can consider
the system that we looked at in the
previous slide so using the notation
here y + 1 + 2 is the response of the
system to X as an input right and we
define the system as 255 - x + 1 and -
right so such a system takes an 8-bit
image and invert it finds the negative
of it right so clearly if I put a 0 at
the input of the system the output
equals 255 which is different than 0 and
therefore this system that finds the
negative of an image is nonlinear
generally speaking is rather
straightforward to utilize this
homogeneity property this equation that
you see here on top and prove or
disprove that the system is linear and
this property and everything that we
covered here in the slide applies to
two-dimensional systems and signals as
well as one dimensional three
dimensional higher dimensional signals
and systems in general let us consider
again a two dimensional system T X and
one and two is the input Y and one and
two the output for such a system if when
I shift the input by k1 k2 I find that
the output is shifted by the same amount
k1 k2 then the system is specially
invariant another way to express this is
that to say that the system does not
care about the location of the axis does
not care well the zero zero point is
located this property is important by
itself but even more important when
combined with linearity as we are going
to see right away
now this property is independent of
linearity so if we consider the system
we looked at earlier which for an input
X and one and two generates as output
255 minus X and one and two so this
system finds the negative of an 8-bit
image we saw that this system is
nonlinear
now if I shift the input to the system
by X and 1 minus k1 so I shifted by k1
k2 and I put this as input to the system
the output is equal to 255 minus X and 1
minus K 1 and 2 minus k2 which is
clearly equal to the shifted output
therefore this system is specially
invariant si so the system effects the
negative of animate is nonlinear but is
spatially invariant as another example
if I look at the system that multiplies
the input by a time varying gain so this
see is a game that changes according to
the location of the pixel and 1 & 2
it is rather straightforward for you to
verify that such a system is linear but
is not especially invariant
or it is specially bearing that's
another way to express it again this
particular property holds true
everything we talked about here for
constant for one dimensional three
dimensional multi-dimensional in general
systems and signals let us look now at
systems that are both linear and
specially invariant lsi systems such
systems are used widely and we have
developed very useful and convenient
tools to describe and analyze them now
such systems can be completely described
by a signal that we'll be referring to
as the impulse response of the system as
the name implies if I put a delta at the
input of such system and measure the
output which I'll denote by H and 1 and
2 this is again the response of the
system to an impulse and we will refer
to it as the impulse response of the lsi
system
this is not just a mathematical
construct but in many cases I can
utilize the system such as a camera and
point the camera to a printout that it
has black background and the white spot
in the middle or I can point the
telescope that is orbiting outside the
atmosphere such as the Hubble Space
Telescope to a distant star in the dark
sky measure the response and artist
impulse response of the system
knowing the impulse response now again I
can completely describe the system which
means for any input X + 1 + 2 I can find
the output of the system and this is
equal to the convolution of the input
with the impulse response of the system
so this double star here denotes the 2
dimensional discrete convolution
so linear systems or filters as we often
we fail to perform convolution discrete
convolution the convolution of X with H
as we will show is equal to this
superposition sum K 1 minus F into
infinity K 2 minus F into infinity X k1
k2 H M 1 minus K 1 & 2 minus K 2 I'll
show some examples of performing
convolution in the following slides it's
easy to verify by substituting variables
that the convolution has the commutative
property so in other words convolution
of X with H equals convolution of age
with X