NPTEL

NPTEL ONLINE CERTIFICATION COURSE

Course
on
Analog Communication
by
Prof. Goutam Das
G S Sanyal School of Telecommunications
Indian Institute of Technology Kharagpur

Lecture 11: Fourier Transform (Contd.-3)

Okay so we have discussed about this frequency shifting property right.

(Refer Slide Time: 00:30)

 ∞

∫−∞
G′( f ) = g(t)e −j2π( f−f0 )t dt

G′( f ) = G ( f − f0)

So that was the end result that we have got in the last class so basically what we have seen that if
I have a particular signal GT.

(Refer Slide Time: 00:39)

 g(t) ⇔ G ( f )
g(t)e j2π f0 t ⇔ G ( f − f0)
∞

∫−∞
G′( f ) = g(t)e j2π f0 t e −j2π ft dt
∞

∫−∞
G′( f ) = g(t)e −j2π( f−f0 )t dt

G′( f ) = G ( f − f0)

And I know the Fourier pair of that if I multiply that signal with e to the power J2piF0 T I get a
frequency shifting or frequency translation by an amount F0T whichever exponential I am
multiplying by okay so basically what happens the shape remains the same it just gets translated
so now let us try to see what is the implication of this thing okay the biggest implication is
suppose I have a signal G (t).

(Refer Slide Time: 01:13)

 g(t) ⇔ G (t)
F T [g(t)cos (2π f0t)]

[ ]
1
= F T g(t) (e j2π ft t + e −j2π f0 t)
2
1
2{
= F T [g(t)e j2π f0 t + g(t)e −j2π f0 t ]}
1
= [G ( f − f0) + G ( f + f0)]
2

Now let us say I will be multiplying this signal with cos this is the very typical actually this is
called modulation okay so let us say I will be multiplying by a cos, co sinusoidal signal 2 pi F0 t
okay,I'm doing this and this G(t) is my actual signal probably and this has a Fourier pair already
known which is gf so suppose I have a signal GT in time domain and I have a corresponding
Fourier pair which looks like this is the Gf, you see that because it is a real signal, I have
constructed the spectrum in such a way that it is even symmetric. So this is the amplitude
spectrum I have drawn correspondingly there will be a phase spectrum probably which is odd
symmetric something like that.
Okay, so I have got this G(f) now I know that this g(T) goes to GF okay, in frequency domain
now if I multiply by G(T) with cos w CT what will happen to this g(f)how that will look like how
the spectrum of this composite signal will look like so I have to just take a Fourier transform of
this okay.

Now what I will do, this cos I can write as ½ e to the power J2piF0T+ e to the power-J2piF0T
right, I can write this now because Fourier transform means this will get multiplied GT into this,
GT into this and then addition so linear combination Fourier transform gets distributed so I can
always write half it should be a Fourier transform of GT into e to the power J2piF0T plus Fourier
transform of GTi nbe to the power -J2p9F0T okay now go back to our previous results, so half if
GT has a Fourier transform of G F then if I multiply with this it should be G F- F0 and similarly
if I multiply by minus it should be – (F0), So that should be plus, so G (f+ F0) this is what I get
so basically what is happening first of all this gf strength is becoming half and it is getting
translated same G pattern is just getting translated to plus f0 so this is centered at plus f0 this is
centered at -f0 so suppose my f0 is something like this so what will happen this whole thing will
be little bit reduced and it will be centered around+ f0 and -f0 remember the shape of this
spectrum which actually specifies the signal quality that remains the same.

The shape does not change, the spectrum shape does not change I can again bring it back to 0 and
I will get the same thing. so this is in particular a very interesting property of Fourier transform so
what is happening, any signal if I know the Fourier transform of that and as long as the signal is
somewhat I should say band limited that means now the concept of band limited things will be
coming so what I know about the signal is if I see the spectrum of the signal the spectrum or
spectral density I should call that gf beyond some value it either completely vanishes beyond
some value B suppose either it completely vanishes or diminishes or beyond a certain level which
is insignificant okay.

So then we say that this particular signal because in time domain we cannot see this particular
signal does not contain any higher frequency component beyond some value B, so beyond B it
does not have those component, spectral component okay and then we call this signal as band
limited signals that means it is band limited up to B. so whatever spectral component it has or
frequency component it has it is limited up to bandwidth B or value, spectral value B so beyond
B does not have any spectral component or I should say any corresponding sinusoidal
component. So if I represent the signal like Fourier series or even in Fourier transform, so I do
not get any significant spectral component beyond this particular sinusoidal or exponential
whichever way I am representing okay, so once the signal is band limited if I try to multiply that,
that signal with a real Co-sinusoidal signal of frequency F0 where I have deliberately put this f0
to be greater than this B or sometimes much greater than this B then what will happen this signal
will get translated to that f0 okay.

So if F0 is not greater than B then what might happen so suppose f0 is somewhere over here I
mean the signal is or somewhere over here so this signal will be something like this and it will be
centered again at -f0 it will look like this so these two things will overlap so that is something I
do not want, okay so why I do not want that will be clear later on but right now we are saying that
this is what if happens this condition is true, there is a signal that particular spectrum part is still
as it is,it might be a little bit reduced but whenever they are reduce their relative part is
equivalently reduced.

So that means the spectral feature or characteristics which define the signal that remains the
same, so that relative strength at every point are relatively equivalent compared to this one okay
so the signal pattern remains almost the same so what is happening I am just actually translating
them to a higher frequency it remains the relative shape remains the same I am just translating
them to a higher frequency and this particular part is called modulation why now we can define
what is modulation and what is the advantage of doing modulation.

Suppose, I have a channel we have talked about channel in the first few process and it is a shared
channel and everybody wish to use that same channel so suppose I want to transmit something on
the channel so I put some voice signal to my transducer it converts it into a electrical domain
signal it becomes a voice signal and then it is being through antenna it is being radiated into the
air and by corresponding recipient wants to receive this as long as I am the only one talking to
one particular guy this is all good.

(Refer Slide Time: 08:52)

But suppose two fellows wish to communicate now after suppose this is by speech signals it has a
random variation and the corresponding spectrum looks like this okay it will, it will be a band
emitted spectrum in speech we say that beyond three point four kilo Hertz there is nothing all are
insignificant so it is band limited up to 3.4 kilo Hz so whatever spectrum component we will
have, it will be up to 3.4 kilo Hertz.okay, so another guys so this is my g1 (T) which is my signal,
another guy wish to also communicate he also has generated a separate speech signal.

A different kind of speed signal so this we call as g2T which also looks like similar spectrum
because it is speed so it will still have same kind of frequency component but it might look little
bit the spectrum might look little different but it will be almost up to 3.4 KHZ. now if I just super
impose this G 1 and G2 and put through antenna at the same time what will happen these two
spectrum will coexist and then neither in time domain because time domain the signal will be just
added. If it is additive channel and in frequency domain also the frequency component will be
just added then it is very hard to separate them out because neither they are separated in time
domain nor they are separated infrequency domain so I will have no device or no mechanism to
separate them out so if my receiver wish to now listen he will get a jumbled signal or added
signal g1 with g2 and that will completely deliver him a completely different speech.

It will be, it will be just a distorted speech so he will not, neither he will be able to listen neither
the other guy wish to receive g2T will be able to receive, so that is where if I wish to multiplex
multiple users data into the same common channel I need something else the device that was
designed by the understanding of frequency shifting property is something like this let us
multiply this G1T with some cos 2pi F0T or F1 T let us say.

Then what will happen this particular frequency component will go around f1and -f1 and it will
sit nicely over there and when we are trying transmit G to T we do a separate multiplication, with
again a cos term but this time the cos frequency will be different > let us do it at F2 T well F2 is
predominantly different from F1 and this particular signal will nicely be sitting over here and at
-F2 the good part is now in the frequency domain they have separate location and at the receiver
side what I can do is I can just filter them out. So there is a device called filter which most of you
are familiar that just specifically takes some of the frequency or it just passes some of the
frequency component and it suppresses other frequency components so if this is the composite
signal coming to the filter and my filter is tuned at f1 so what will happen is a band pass filter so
it will just take this amount and it will reject this whereas the other guy can tune his filter at f2
and he can reject this so both of them will get their original signal as if it is transmitted on the air
single.

As long as we choose this f1, f2 carefully and as many we wish we can actually multiplex many
we can have corresponding f3 corresponding F4and so on we can actually multiplex multiple
users simultaneously transmitting over the channels and their separate existence is still remaining
because of this frequency translation property what is happening, whenever we multiply with cos
2 pi F1 T or F0 T or F2 T or F3 T we know that the spectral component or the relative spectrum
which is the characteristics of the signal remains the same it just gets translated to a different
frequency band and all We have to do at the receiver we have to carefully choose the frequency
band put a filter and take extract our own signal and reject all of the signal this is facilitated by
this communication technique or that is why this particular part is called modulation so what you
are trying to do whenever you are trying to use the media which is being used by all others you
translate or you multiply by co-sinusoidal signal to translate the frequency spectrum into a higher
frequency and then actually transmit it in the common media.
Everybody will be doing that as long as they are using separate bands to transmit they will have
separate identity, in time domain it might not have separate identity but because we have done
this, frequency domain they will always have their own separate identity that is fantastic! so this
particular thing is called frequency division multiplexing so what we are trying to do is because
now you can see all these things are so important.

Because we have understanding of Fourier series and transform and because of that we could get
another representation of signal and from the Fourier transform only we could derive that there is
a property called frequency shifting property by which we can actually keep the signal intact but
translate it into a higher frequency and because we have some device called filters which can
separate out some portion of frequency.

So we can actually give separate entity for separate signal in the frequency domain so the entire
manipulation is done in the frequency domain if we would not have understood Fourier transform
would not have understood the frequency representation of a signal we could not have produce
these things okay if we were not able to understand the mathematics behind that frequency
shifting property we could not have produced this thing.

So it is very important means now you can appreciate it is very important for communication any
simple technique that has been applied in communication it is deeply rooted in Fourier series or
Fourier transform, it is absolutely necessary that you understand those concepts very clearly okay
so now we have understood some portion of it we have talked about this modulation why this
modulation is required we have told that one particular thing is this multiplexing.

So this is now probably clear you have stated earlier, but now it is all mathematically clear that
what we are trying to do and why that is important there is another aspect of this frequency
shifting property or this modulation that is we have already talked about that that whenever you
put multiply with cos this one you can see the frequency component that it is having whenever
we are transmitting it is around f0 some +/- so earlier it was having, if you just see this one it was
having frequency from 0 to some 3.4 kilo Hertz.
Now suppose I put them at it is a very high frequency let us have some 400 megahertz so what
will happen it will just go center around 400 megahertz and it will just be around that 400
megahertz 400+ 3.4 and 400 -3.4 okay so it will be around that 400 megahertz only so the
frequency component it predominantly had that has been transmitted in the air will be around that
400 megahertz whenever it is 400 megahertz the frequency component.

The corresponding wavelength will be very short okay 3.4 kilo Hertz one by that you will see the
corresponding wavelength is quite big whereas 400 megahertz you should do one by that you will
see because of that 10 to the power 6 it will be very small and we have also discussed that
whenever you are putting a transmitting antenna or receiving antenna to actually transfer the
energy properly we need to have the antenna size or to capture the energy properly through the
antenna we need to have an antenna size which is comparable to the wavelength of the
correspond frequency that may translate or transferring okay.

So it is very much essential that the frequency is higher to facilitate the antenna size to be
smaller, so the with modulation we are also achieving that particular part because what is
happening here if you see there are very small frequency component which will have huge
wavelength because it is inversely proportional and the antenna size will become very huge
whereas by this, by just this simple technique by multiplying the signal with a cos it serves two
purposes one is it translate it to a very high frequency immediately corresponding wavelength
will be very smaller.

So I can devise very small antenna for my transmitter and receiver and on top of that we are now
creating in the frequency domain multiple such places where we can start multiplexing multiple
signals simultaneously and transmit them through the air media simultaneously and just use a
filter to take my own signal and reject other signal so that is actually the story behind modulation
you will see that later on okay.

But right now we are happy with this frequency shifting property which will help us to do
modulation okay the next property that we are trying to provide is.
(Refer Slide Time: 19:37)

∫−∞
g(t) * w(t) = g(τ)ω(t − τ)dτ

F [g1(t) * g2(t)] ⇔ G1( f )G2( f )

Something called convolution I am not going to prove that but I will just state what do we mean
by convolution so this convolution is given by this star symbol so that means the signal is
convoluted with another signal GT is convoluted with another signal WT what does that means
convolution actually means this, so what is happening so you take this signal GT Tau is just do
not worry about tau, that the dummy variable of, in this integration okay.

So take the signal GT and for w, you can see it is already -tau so basically you reverse this so you
take W- tau so like u -T, I was taking earlier so take w- tau and then time shifted by t amount
okay so do this and then keep varying this tau multiply this and integrate so that is called the
convolution so basically any signal you take and the other signal you invert it first in time domain
and then time shift and keep varying this time shift and you integrate it multiplication and
integration .

So basically one particular signal you will take that will be suppose this is my GT let us say this
is like this and suppose the other one w T is something like this, now what you do first you flip
this w T so from there it would be W- T or- tau so this is w tau this is G tau okay so time I am
defining respect to tau and what you do for a particular so then you time shift it at tau = T you put
it okay so this will be just at t okay. So whatever that value of T wherever that T will be it will
look like that so at, so this will be wt-tau so what different value of 7 this will this signal will be
I mean going to 0 at different location and then you multiply these two things and integrate it, so
basically you will be keep on shifting this for different value of T and you will be getting a
different value of those integrations so this is called convolution and there is a well known result
in Fourier transform so I will give this as homework you have to prove that if I have two signal G
1 T & G 2 T if I convolute them and try to take a Fourier transform of them if I individually know
the Fourier transform of G 1 T which is suppose G1F and G 2 T which is G 2 F we can always
prove that this should be G 1 F into G Z F.

So in time if we convolute in frequency it becomes multiplication and if in frequency if we

convolute in time it will be multiplication that comes from another property of Fourier transform
which is called a duality so the duality property if you wish to know that is also very simple you
can just very quickly give them.

(Refer Slide Time: 23:02)

 g(t) ⇔ G ( f )
∞

∫−∞
G( f ) = g(t)e −j2π ft dt
∞

∫−∞
G (t) = g( f )e −j2π ft d f, −f = x
∞

∫−∞
= g(−x)e +j2πxt d x

So we know that GT I have a Fourier transform which is called G F okay so in that case what we
can write is this G F is nothing but -infinity to plus infinity GT e to the power - j2 pi Ft dt right,
now so this is something we already know we also know because it is a Fourier transform so we
can also represent this G T as -infinity plus infinity gf e to the power +J2pi F T d T sorry d F right
we know this.
Now what we can do, instead of F if we just put -F so then from this equation we can get GT I
can write -infinity plus infinity so f is replaced by-f. So what will happen this is G -F and here we
will just get e power -J 2pi FT right, so this is something we will get okay, now all you have to do
is, okay so what we have done we have just replaced this okay so we can do one more thing
instead of doing it over here I can,I can also so do it over here so basically what we are trying to
do I have got this right now you just replace T by F so T = F if I just put that so what will happen
this GF will become GT and I get -infinity to plus infinity this becomes G F e to the power -J2piF
becomes replaced by T and T becomes replaced by F. So I get F T and T becomes G f right, so
this is something I get okay so I get Gt is equal to -infinity to + infinity G F e to the power -J2pi F
T d F okay so now you can see that, if I have this as
frequency domain okay so this becomes T right so this is almost becoming inverse Fourier
transform but I only have one problem this there is a minus over here inverse Fourier transform is
having plus so what I can do is instead of this-f I can put -F right.

So immediately what will happen if F is replaced by minus F or I can write minus F means,
minus F as some X so immediately what will happen so this will become X so this is becoming
G- X because itsF, F is - X, G- X e to the power minus this would become plus because - F
becomes X so J2pi T X and d this would be dx, - and + so that will just change the limit so this
will still remain the same okay so G T becomes this now you can replace X by F immediately.

(Refer Slide Time: 27:00)

 ∞

∫−∞
G (t) = g(−f )e j2π ft d f

G (t) ⇔ g(−f )
g(t) ⇔ G (t)

What you get our G T is becoming just -infinity to plus infinity G -F e to the power J2piFTd f
right so what is happening now I can see this G T and G -F are Fourier transform to each other so
G -F if I take inverse Fourier transform I get G T so that becomes a Fourier pair so basically we
have started with a Fourier pair GT which has a Fourier transform of G F if I just, this whatever
time domain function I have got that if I represent in frequency domain if I just take negative of
that, that will have whatever frequency domain function I have, if I just represent it in time
domain I will be getting that so that is called the duality of Fourier transform so that means
suppose I have already got a Fourier transform let us say I do a Fourier transform of some, this
kind of pulse okay which is defined from - T/ 2 to + T /2 take that as homework if you do a
Fourier transform easily you can just put the fundamental Fourier transform this things you will
see that it will be a sin function it is defined by this it is Fourier transform will be GF should be
tau sin C pi F tau, sin C means sine of this argument divided by this argument.
So sin pi F tau divided by pi F tau so this gf becomes this, which will be
defined a,s at t equal to 0 it looks like this at means tau T = 0, it is having a value of tau
at 1/tau, it goes to 0 again at 2 / tau it goes to 0 and so on, at- 1 /tau, it goes to 0 and so on, okay
so if I already have got a GT which is looking like this and if I already have got a GF which looks
like this, now if I give a sinc function in time domain that means this becomes GT and if I wish to
actually get a Fourier transform this would be just similar like this this duality formula tells me
that.

So if the frequency domain 1 okay frequency domain 1 is now becoming my time domain
function so now the sinc function which was the frequency domain representation that becomes
my time function this will be just same time domain function will be the Fourier transform now
this will be frequency domain okay so this will be now this is in time domain this is in frequency
domain if I take this in time domain it will just be frequency domain the time domain1 becomes
the frequency domain 1 only thing is that it must be having this -F but because this function is
symmetric over T or F, So it will remain the same.

so they become ups Fourier conjugate of each others that means a square box function if I take, if
I do a Fourier transform I get a sinc function, if I take a sinc function in time domain if I do a
Fourier transform I get a box function in frequency domain, so if I know one I will be always
knowing the conjugate of that so that is the duality property of Fourier transform.

so with all these basics what we will try to do is we will try to now go into the measurement part
which we have ignored means we have done that for Fourier series but for Fourier transform the
measurement part that means the energy or power we have still not devised that so now we will
try to see for a signal the very important property is the measurement of the signal which is
energy so we will try to see how to evaluate the energy of a signal so that will be our next target.

thank you.