Filters and Composition - Electronics and Communication Engineering Video Lecture - Electronics and Communication Engineering (ECE)

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so we're going to continue talking about
LTI systems and how to work with them
we're thinking of an LTI model for a
channel something with input xn output
yn and we've seen that the output can be
obtained from the input through this
convolution operation right so for
instance this was one way to write it
and the shorthand notation here was a
star X evaluated a time in the am here
in general goes from minus infinity to
plus infinity okay in general you put a
unit sample function at the input the
response can extend from minus infinity
to infinity if you've got a non-causal
system if you've got a causal system and
you put a unit sample and then the
response starts from zero and goes on to
the future but it's often useful to be
able to represent and analyze non-causal
systems I'm if you have all the data
stored in your computer then you can
look forward and back from the time that
you're at and therefore you can do
things that can be looked at as
non-causal or analyzed as non-causal now
there's one issue that we we've kind of
swept under the rug which is you can
write nice-looking formulas but do they
mean anything so we went through a
plausible derivation but if you end up
with a summation from minus infinity to
infinity then you've got to ask yourself
when does this make sense okay because
you know that adding an infinite number
of things can cause problems so you need
conditions for all of this to make sense
for instance if H was one for all time
and X was one for all time there's no
way this is going to make sense because
it's going to blow up at every value of
n right so you clearly need conditions
so
we're looking for is conditions for
convolution to be well behaved and I'm
going to give you one important
condition well actually let me give you
two so here's one suppose we have a
causal system and an input that starts
from 0 at time 0 when I say starts out I
mean that for all prior times the input
is 0 so we've got an input that 0 for
all negative x and then
at n equal zero we start to get some
actual okay so let's see what happens to
this infinite sum in this particular
case if I have a causal system that
means that H of n is 0 for n less than 0
right that's what you should think of
right away when you're told the system
is causal the unit sample response can
only extend from 0 onwards so if the
unit sample response is 0 for a negative
time and the input starts at time 0 what
simplifications can you make to that
convolution representation somebody yeah
zero to infinity does it okay so the the
answer was that instead of starting at
minus infinity he only need to start the
summation at m equals zero because the H
of em is going to be 0 for all negative
values of M right so you can start that
summation at zero and then where does
the summation work in the summation end
m n yeah because this X here is 0 for
negative values of the argument negative
values of the argument our values of M
greater than n right so this only needs
to extend over a finite region well if
you have a finite sum then you're happy
nothing's going to blow up so this is
certainly one case where everything
works fine there's another case which
I'll describe which is where your input
is bounded and what we mean by that is
that your ex n has an absolute value
that's less than or equal to some
maximum value some finite maximum value
for all time so no matter what any pic
you're always within this interval okay
so your input is use your M here's your
plus M here's your minus M and your
input is constrained for all time to
just lie between these limits okay so
that's a bounded input and here's the
other part of the condition
absolute value of HM summed over all M
is finite okay we say that that H is
absolutely summable absolutely summable
ok
so what does that do for us well it
actually allows us to bound the output
so now it turns out y of n that's the
absolute value of the output well that's
less than or equal to the absolute value
of this convolution expression which is
in turn less than well oh it's equal to
that right isn't it it's equal to that
but it's less than or equal to this
right
absolute value of a sum is less than or
equal to the sum of the absolute values
and that's less than or equal to capital
m this is the max value that we allowed
up there times summation hm okay and so
this whole thing is finite so basically
you can bound the output you can
guarantee that the output is bounded
provided these two conditions are
satisfied so this is actually a very
important pair of conditions it's one
that we encounter all the time and in
practice and because of this result we
actually refer to an LTI system that
satisfies this absolute solubility
condition
we say that the LTI system is bounded
input bounded output stable bounded
input bounded output stable
and I've run out of space for my stable
there okay so if somebody tells you they
have a bounded input bounded output
stable system we're talking if we're
talking about an LTI system what they
mean is that that condition is satisfied
so that's an important condition ok I
have all that on a slide but if I race
it past you on a slide then it's hard to
to track but this is something that we
want to do okay so for instance if i had
an LTI system whose unit sample response
was the following let's say it's
0 for all x up to and including x 0 and
then it takes the value 1 over adam from
then on okay so let's see one way to
write this is 1 over nu of n minus 1
right if I tell you that the H of n is
this that automatically takes care of
zeroing out everything from zero
backwards and then putting in 1 over N
from then on okay so is that a b-bow
stable system bounded input bounded
output stable yes how many think yes
you'd like to think it's stable because
the unit sample response is decaying but
actually it doesn't satisfy the absolute
solubility condition the sum of 1 over N
from 1 to infinity Zach it actually
blows up right doesn't converge if it
falls off any faster than that then
you're in good shape but this is this is
actually bad if you had something that
was one over N squared okay so this is
not be both stable
but if you had the one over N squared u
of n minus 1 it is bebo stable if you
had let's say one third to the n u of n
minus well who will say you event that's
bebo stable ok so this falls off as one
over N squared that's fast enough for
the sum to converge this falls off
exponentially fast or it's as a
geometric series right it's a
discrete-time exponential so that's fast
enough so that's also be both stable all
right so this is time domain we know how
to analyze any LTI system with this you
tell me what the unit sample response is
and I can tell you what the output is
for any given input but this would be a
nightmare if we had to do design with
this because convolution is not it's
simple enough to implement for a
particular case but it's not a simple
operation to think in terms of the
reason is that the output at any one
time is obtained by scrambling all the
inputs for all time combining them in
this weighted linear fashion and then if
you move to the next time step your
again scrambling all the inputs but with
the weight shifted a little bit so
you've got to start from scratch again
so it's very hard to know what you can
say in general using the time domain
it's not clear at all how are you how
you'll do design for instance of filters
to filter out noise and so on so this is
actually it's important it's a full
characterization of LTI systems but if
we have to stop there it's a fair bet
that
we wouldn't be anywhere near where we
are for engineered system certainly not
in digital communication so the key the
key thing is actually to start thinking
in terms of frequencies okay so we're
going to look we're going to spend the
next several lectures looking at the
frequency domain and so what is the
frequency domain well we're going to be
focusing essentially on sinusoidal
inputs and inputs that are related to
them but let me actually just start back
with something simpler
here's my question and I hope the answer
isn't already on the slides maybe it is
okay is it true that if my input was
periodic so for instance if i had an
input that lets say did this it ramped
up over maybe for time steps and then
start it again ramping up over for time
steps and continue that indefinitely
okay so here's my X of n it's an X event
that has some basic period that then
repeats periodically so it satisfies
that condition xn equals x and plus
capital P for some number capital P
that's the period what is capital P in
this case for
okay so every for this repeats is it
true that if I had a periodic input to
an LTI system that I'll get a periodic
output any reason I should expect that
yeah
can you speak up a little
it depends on the system telling its LTI
but nothing more so it depends on which
particular LTI system that would be my
intuition too yeah
well but that's an average that's one
number what I'm asking you is could this
output as a function of time also be
periodic can you guarantee it yeah
I'm not exactly understanding your
prescription so you are you telling me
how to prove that Y N equals y n plus
something for all n
we according to this well I raised the
convolution expression but the
convolution expression requires us to
take axes from minus infinity to plus
infinity right in general the output at
time n depends on all the inputs so it's
not clear how you can block things off
you had another idea
ok
okay good this is a time in barren
system right so if I shifted the input
by capital P the output should also get
shifted by capital P but shifting the
input by capital P gives me the same
input back again so the response that I
get for that shifted input must be the
old response that I had which tells me
that the output also has the property
that if I shifted by capital P i get the
same thing again so so this is
guaranteed to be period capital P now
actually there's a little twist to that
because it's we usually think of the
period as being the smallest interval
for which you can repeat it's
conceivable that this output has a
smaller interval and therefore the
period is some integer fraction of this
capital T but this is the general idea
so you see that just knowing that the
system is LTI you can already tell a lot
about what the response will look like
one of our favorite periodic inputs well
if I ask you to tell me what your
favorite periodic input is what might
would be any sorry constant is good
that's periodic a little less trivial
and constant
sinusoid right so here's the nice thing
about sinusoids it turns out for an LTI
system if i put in a sinusoid not only
is the output periodic with the same
period it's also a sinusoid so that's a
an even greater restriction here you see
in this particular case I have an input
that's periodic I'm guaranteed the
output is periodic with the same period
but the actual shape of the wave form
can be all messed up relative to this it
may have no obvious visual relationship
to this but if you have a sinusoidal
input then it turns out that
that more is true okay so it turns out
if you put a sinusoid in what you get
out is a sinusoid of the same frequency
what might change is the amplitude of
the sinusoid and the phase angle on the
sinusoid but it'll be the same frequency
sinusoid that comes out so that's a
fairly dramatic restriction and that's
actually key to frequency domain methods
what it means is we can focus on what an
LTI system does one frequency at a time
I'll look to see what how it behaves
when I excited with a particular value
of this big big Omega here that's the
frequency look at the response i know
the response will be exactly at that
frequency so all i have to capture is
how much did this input get scaled by in
other words how much of the amplitude
change by and how much did the phase get
changed by i just need to know the
magnitude and phase transformation of a
cosine at each frequency if i know that
i know everything about the LTI system
and it decouples my design it allows me
to think frequency by frequency when I
design with an LTI system so this is
actually a great simplification one
other remark I've made here by the way
it's certainly the case in
continuous-time if i gave you x of t
equals cosine let's say omega naught t
plus theta this is always periodic right
this is periodic with period
2 pi over little Omega a little omega
naught because anytime i increase t by
an integer multiple of this i'm going to
get an integer multiple of 2 pi added
into the argument of the cosine so i'll
be back where i started right so this is
always periodic but with a discrete-time
sequence you actually have to be a
little more careful we can think of some
particular discrete time sequence we
refer to the omega in the
continuous-time case as the angular
frequency in radians per second here
we're thinking of this as angular
frequency in radians per sample for
instance because the thing it multiplies
is n but basically the units of big
Omega our angle well it turns out that
this may not be strictly periodic in the
sense that shifting it by an integer
will get you exactly the same waveform
and that's all related to whether this
frequency whether two pi over omega the
thing that you would like to compute as
a period is rational or not okay turns
out of 2 pi over big Omega is rational
then the period is the numerator of that
rational but otherwise it's not rational
but you can think of it as being samples
taken from some periodic quantity so
there's an underlying time varying
sorry there's an underlying
continuous-time periodic waveform and
you take samples of it and depending on
the frequency of the underlying sinusoid
the sequence of samples may be exactly
periodic or maybe close to it in the
sense of their samples taken from a
periodic signal in either case actually
we tend to not fuss about that we talk
about a cosine like this as a sinusoid
of frequency omega radians per sample
and we'll talk about the period as being
2 pi over omega even when it's not
strictly chaotic ok so 2 pi over omega
is our notional period
okay so a couple of examples here you
can leave you to look through those on
the slides but you can easily construct
examples where 2 pi / omega is rational
and draw a picture and convince yourself
that is actually periodic so in this
case for instance if big Omega is big
Omega is whatever multiplies n so it's
going to be 3 PI over 4 3 PI over 4 is
big Omega so now I need to look at 2 pi
over omega and so that's equal to 8 over
3 the numerator of that is what the
period is so if you were to actually
sketch this out you would find that
every eight samples it repeats whereas
if you look at this example the thing
that multiplies n is 3 over 4 so now the
period 2 pi over omega is 2 pi over 3
over 4 so now we're talking about 8 pi
over 3 that's not rational you're not
going to get a periodic sequence but we
still refer to 8 pi over 3 years the
period of this discrete time signal okay
just because it comes from sampling an
underlying continuous-time periodic
signal alright that's a detail I just
don't want you to trip up on on that
later on
so here's the the basic statement what I
said earlier if you have an LTI system
by the way I like to represent this with
an h dot saw that slip past there I
think I've talked about this before let
me not spend time on it notation
notation so if the input is a sinusoid
of some frequency and amplitude and
phase the output is guaranteed to be a
sinusoid of the same frequency
potentially different amplitude and
different phase ok so what I want to do
is establish that for you starting with
our time domain characterization which
is LTI convolution
it turns out actually sinusoids are not
the only things that have this property
in fact it might be good for me to show
you another example two so let me give
you another example of a waveform that
you can put at the input or a signal
that you can put in the input to an LTI
system and it comes out the same shape
despite all the convolution so here's
the example suppose xn is what I think
of as a discrete-time exponential so
this is our to the end for some real
number are so maybe maybe r is a half so
this is
okay so this is X n so discrete time
exponential okay you're used to thinking
of that as a geometric series but when
you talking about signals and systems
you'd like to think of that as a
discrete time exponential it does have
an exponential fall off what if that
goes in to the system well the output is
given by this expression always for an
LTI system so let's just plug in what X
is that summation over all m hm r to the
n minus M right I'm just substituting in
this expression for X of n the end piece
of this doesn't depend on the summation
index so I can actually simplify this
further and here's what I have
so look what's happened I sent in a
discrete time exponential and out comes
the same discrete-time exponential but
scaled by some number this is just a
number right it's an infinite sum it's
just a number works out to be a number
you'll have to look for conditions under
which it's guaranteed to exist and
certainly if the system is bebo stable
and well if the system is causal and the
exponential is decaying then this is
guaranteed to exist okay so here's an
example of another kind of signal
related to the sinusoid as will see that
has the property that you put it through
the LTI system despite all this
convolution stuff that's going on I
don't even have to know what that H is I
can tell you right away that what comes
out is the same exponential but scaled
okay so now we're going to try and
establish the property for sinusoids and
the way to do that the efficient way to
do that is actually to work with
Exponential's again except they're not
Exponential's of the type that I have
there they're complex Exponential's
now when you learn complex numbers in
high school maybe you thought you
wouldn't have to deal with them again
and then you came to calculus and you've
had complex numbers and you thought
maybe after that you don't have to deal
with them again so I'm here to tell you
that you're always going to have to deal
with complex numbers so you have to get
comfortable with them so we're talking
about essentially points on the plane
point with the real part and an
imaginary part so here's the complex
number see here's the real part here's
the imaginary part I never thought I'd
need to define J but it actually turns
out if not everyone in the room is an
electrical engineer then maybe you're
used to thinking of I as being the
square root of minus one electrical
engineer is like Jay because they
reserved I for currents right so Jay is
the square root of minus one and all
electrical engineering
the key identity that you need is
Euler's identity okay let me actually
write it on the board and leave it there
because we'll be coming back to it
multiple times
well actually I can have it on this
sport because it's really the same it's
really this picture e to the J theta is
some complex number its real part is
cosine theta and it's a measuring part
is sine theta that's all that owner's
identity is saying right here's a
complex number its real part is cosine
theta its imaginary part is sine theta
so what's the magnitude of e to the J
theta every complex number has a
magnitude e to the J theta has magnitude
what one because it's cosine squared
plus sine squared right and the angle of
e to the J theta
the angle of a complex number is just
the angle from the real axis up to this
vector so the angle is
um
I'm hearing more complicated answers
than I expected theta right you're right
it's arctan sine theta over cosine a'
which is theta right
alright so this is Euler's identity and
this is really critical and if you can
remember this then you can remember all
sorts of other identities that might
trip you up from time to time so for
instance let's see
if I have e to the J theta 1 times e to
the J theta 2 does that simplify to this
that's okay right that's just combining
the exponents well expand this out using
Euler's identity expand this out using
Euler's identity expand this out using
Euler's identity and you discover for
instance that cosine theta 1 plus theta
2 equals cosine theta 1 cosine theta2
minus sign one sine theta 1 sine theta 2
okay you know all these identities but
if you're ever pressed to derive them
the place to start as owner's identity
if you're pressed to derive Euler's
identity then maybe go back to Taylor
series but if you want to carry one
thing in your head carry Euler's
identity all the rest follows
alright
so here's the well actually it may make
that point a little later these are easy
e to the j 0 is just this unit length
vector line along the real axis is just
a number one e to the j pi it's the unit
vector lying along the negative real
axis so that's the number minus one
right
so here's how we use complex
exponentials to prove the result that i
claimed earlier and they me that
sinusoids n gives you sinusoids out
sinusoid of a given frequency and gives
you a sinusoid of the same frequency out
what we're going to do is actually
combine sines and cosines into one
calculation this may look a little funny
because now we're suddenly putting a
complex input into this LTI system but
if you think about the mat that we did
for LTI it didn't really care whether we
were feeding in real numbers or complex
numbers so we could have a signal that
has a real part in the complex part at
each time convolution would work exactly
the same way we arrived at convolution
through linearity and time invariance
arguments and all those work for complex
signals so we could actually be putting
in a signal of this type and seeing what
comes out all right well this is very
close to the calculation we did earlier
with a real exponential the only
difference now is it's a complex
exponential but here's the computation
alright we start as always with the time
domain representation that's convolution
substitute in for X of n minus M X of n
is this signal here I should have
labeled it as X of n so just stick X of
n minus M here and then you discover
that there's a part of this that doesn't
depend on the summation index so you
pull that out so you're left with this
summation inside and this piece out so
look what's happened you've put in a
complex exponential and out comes the
same complex exponential scale by
something what is this complex
exponential well it's the trill part is
a cosine signal and the imaginary part
is a sign signal alright so this object
is something that will encounter and use
a lot it's referred to as the frequency
response of the system you've heard the
term used undoubtedly numba settings but
here's the definition
it's a function only of big Omega
because once you've summed over M little
m everything else has gone away so only
depends on big Omega and here's what it
is it's the summation over all values of
M H of m e to the minus J Omega M all
right that's the frequency response
so if you give me the unit sample
response of a system I can find for you
what the frequency responses
so let's actually do an example maybe
let's see let's take one of your let's
take an averaging filter that you've
looked at
to make it easy annual sketch it as a
function of M doesn't matter what that's
called so here's one third one third one
third okay we've come to recognize this
as a unit sample response of a
three-point averaging filter it's a
causal three-point averaging
okay so what's the frequency response of
this system well its age of big Omega
and then I just follow this prescription
so the only values of H that are nonzero
are for M equals 0 1 & 2 so it's going
to be one-third 1 plus e to the minus J
Omega plus e to the minus j 2 omega and
i'm done right
now to actually get a feel for this what
you want to do is write it in different
forms for instance a very useful way to
write this is in terms of the magnitude
and the angle so here's here's a way to
represent any complex number this is a
complex number right I can write it as
magnitude times e to the J angle so that
will actually turn out to be a much more
efficient way to think about frequency
responses this is the magnitude of the
frequency response and this is the phase
alright
so how does that bring us back to sines
and cosines
let me actually go a little bit out of
order here and
here's the basic statement
from the result that we have up there
and I'll show you on the next slide how
to derive it what you can show is that
if you put a cosine into the system what
comes out is the same cosine except its
amplitude is scaled by the magnitude of
the frequency response and the phase
angle is increased by the angle of the
frequency response okay so this is
really if you know this as a function of
frequency if you know the magnitude and
phase angle of the frequency response as
a function of frequency you can describe
the response of the system to any cosine
input what if the cosine was a little
more complicated than the one there
suppose I had
suppose I had cosine Omega naught n plus
let's say PI over 4 going in what comes
out what do you think comes out
anyone
I showed you a particular case here how
much new work would you have to do if it
was actually a slightly shifted cosine
take a guess yeah
yeah you just you just change this by
adding in an extra PI over 4 so what
comes out that's what you said right Zoe
said okay sometimes I'm guessing because
I don't hear well but
so here is plus the angle and then plus
the additional PI over 4 and this
actually will follow just from time
invariance of the system so you can
actually so this is actually pretty
general if you're going to remember one
thing about frequency response in terms
of what the operational significance is
for instance in the laboratory
experiment this is the result to
remember this is why frequency response
is important okay and the proof very
easy once you have the basic result with
exponential inputs the proof is easy
because a cosine can be written as the
sum of these two Exponential's how do i
get that just from Euler's identity
right use Euler's identity for each of
these for e to the j e to the minus j
when you add oilers for this and orders
for this the sign terms cancel out is
you get to co sign in the numerator and
you divide by 2 so if you're stuck for a
derivation of a result like this go back
to oilers so cosine be written this way
so when I feed a cosine big Omega naught
n at the input what I'm actually feeding
in is a linear combination of two
Exponential's but I know how to write
the response to an exponential if this
exponential this sum of exponential goes
in what comes out is the corresponding
some of responses so it will be this
exponential times the half they're
scaled by the frequency response okay
that's what frequency response does to
an exponential
and this exponential comes out the same
but scaled by the frequency response
evaluated at the frequency of that
exponential ok so maybe you've having
trouble visualizing the result that I'm
invoking but it's the one that we just
proved if you have e to the a e to the J
Omega and going in let's say at some
particular frequency Omega naught to a
system with frequency response H of
Omega what comes out is the same
exponential that went in scaled by the
frequency response evaluated at the
frequency that we're talking about Omega
naught okay so when I label the system
with the frequency response this is a
general Omega but the value of it that
I'm interested in is the value of at the
frequency of the input so if the input
is e to the J Omega naught n what comes
out is that same e to the J Omega naught
n but scaled by the frequency response
at that frequency so that's what I'm
invoking here okay invoking it twice
well this is just the real part of this
quantity because it's a complex number
and its complex conjugate and so on so
you put it all together and you actually
very directly have this result so we're
using complex inputs is just a trick for
getting the results that we'd really
like to get for real inputs if you
didn't want to do that you could
actually just put in the cosine Omega
naught n and crank it all the way
through and you would get it so you
could say YN
equals summation over all M H of em then
we have X of n minus M here right so
it's going to be cosine Omega naught n
minus M going in and now use appropriate
algebraic identities and you get the
same result so we didn't have to use
complex exponential inputs to get the
result it's just a convenient way of
getting it okay again if you're going to
carry one result in your head in the
complex case this is the one to carry
all right it's very simple it says
complex exponential n you get the same
exponential out okay so let's play a
little more with this particular filter
since we see that magnitude and angle
are so important
we're talking we're back to this
three-point averaging right here's the
frequency response and I'd like to get
the magnitude out and you could
certainly use Euler's identity on each
of these pieces group all the real parts
all the imaginary parts and so on it
ends up actually being a bit of a mess
to try and write down cleanly so let me
show you a trick that works for this
kind of thing and it'll give you some
practice in thinking about these complex
Exponential's do you agree that I've
just rewritten the same thing as I had a
above right okay does this simplify
can you write it as something real we've
got a complex quantity and it's complex
conjugate there so we should be able to
collapse them into real right the way
you recognize the complex conjugate is
that the J has gone to a minus J so what
is this whole thing somebody
where did that come from can I have a
hand just oh yeah
yeah so say one plus two cosine Omega
right so that's simplified nicely okay
so are we in a position to say what the
magnitude of H is
I've got an H that is represented as the
product of this oh by the way sorry I
this should just be a 3 here right not a
one-third everyone shook their head in
agreement when I wrote that down but
what's the magnitude of the product of
two complex numbers is it the product of
the individual ones the magnitude of H
is going to be let's see 1 3rd magnitude
e minus J Omega times magnitude 1 plus 2
cosine Omega right
the magnitude of a product of complex
numbers is just the product of the
individual magnitudes what's the
magnitude of this one so we're actually
done we actually have a very simple
expression for the magnitude of the
frequency response of this moving
average filter that's all it is oh well
sorry I need the absolute value okay so
let's see I think I have that sketch
done in one of these
I actually have three moving average
filters drawn out here the one that I've
just worked out is this case this is the
3-point moving average filter a height
of one third for each of these at 0 1
and 2 and everything else is 0 here is
the frequency response magnitude the
notation that's used on the figure is
slightly different so some people
including me and the next course I teach
will write the frequency response as
this instead of H of Omega but that's
really unnecessarily fussy it's
important when you're talking about Z
transforms at the same time that you're
talking about Fourier transforms but
first it's not important so you'll see
slightly different notation probably in
the notes as well but just think of that
as just H of Omega okay so here's the
filter we were talking about here
supposedly is the frequency response
magnitude what we should be seeing is
the magnitude of this quantity and let
me see if you believe it so what we have
is
um
so what I have is at Omega equals zero
I've got something that starts at one
and then when i get out to minus pi or
pie this has come down to the value
minus the third and so this is what i
have for
this is the quantity within the absolute
sign right before I take an absolute
value and when I take the absolute value
this flips over and that's really what
you're looking at you're looking at
frequency response magnitude which is
this okay and you've got to figure out
the phase accordingly and that i'll
leave you to do in recitation but i want
to ask you one last thing why do i not
bother to plot the frequency response
beyond minus pi to pi so the reason is
that the frequency response tells us
what the response is to inputs of this
type right the frequency response says
if this goes in how much you get scaled
by when it comes out well if i increase
omega naught by an integer multiple of 2
pi i get the same exponential back again
right so there's no new information out
outside of minus PI to PI another way to
think of it is we're really talking
about complex numbers and their angles
once you've made a full circle from
minus PI to PI there's no new space to
cover okay so you'll see frequency
response is only plotted from minus PI
to PI all the interesting action is
there all right we'll develop more
intuition for this in recitation and in
the next couple of lectures