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Lec 01 | Principles of Communication Systems-I | Basics | IIT KANPUR


Hello welcome to another module in this massive
open online course Let us start the discussion with the energy and power of a signal alright
Let us start 1st by considering energy of a signal and consider for this purpose a signal
x(t) okay let us start by considering a signal x(t) and naturally this signal we are considering
this is a function of time correct This is a function of time that is t denotes time
And the energy Ex of this signal x(t) can be defined as energy Ex of signal x(t) is
defined as we can define this as Ex equals integral minus inherit to infinity
magnitude x(t) square dt that is integral minus infinity to infinity magnitude x(t)
square dt alright For example
consider x(t) equals e to the power of minus t for t greater than or equal to 0 for t greater
than or equal to 0 and 0 for t less than 0 this is also termed as x(t) equals e to the
power of minus t into u(t) where u(t) is the unit step function that is u(t) is 1 if t
is greater than or equal to 0 and u(t) is 0 otherwise this is known as the unit step
function and these you should be familiar from a basic course on signal sins system This is the unit step function
since this signal is non zero only for t greater than equal to 0 Ex would be simply integral
since this is non0 only for x greater than equal to 0 so this is Ex integral 0 Ex is
integral 0 to infinity E to the power of minus t magnitude square dt that is integral 0 to
minus infinity e to the power of minus 2t dt which is equal to minus half e to the power
of minus 2t 0 to infinity that is equal to half So we obtain a total signal energy that is
Ex equal to half for x(t) equals e to the power of minus t into u(t) okay so that is
energy of this signal alright Now if Ex for a signal
for a signal x(t) if Ex is finite that is Ex is less than infinity that is strictly
less than infinity then x(t) is termed as a x(t) is termed as an energy signal
than x(t) is termed as energy signal that is if the energy is finite for instance similar
to what we have seen above that is energy of the signal is half which is a finite quantity
the energy is a finite quantity right The signal x(t) is termed as an energy signal
okay Now let us define the power of a signal x(t)
power the power of a signal x(t) is p of x equals
limit t tilde tends to infinity 1 over t tilde integral minus t tilde divided by 2 to t tilde
divided by 2 the magnitude x(t) square dt which is basically the energy if you look
at this quantity here this is the energy in a window of size t this integral minus t by
2 this is energy in window of size t tilde So this is basically your energy in a window
of size t tilde divided by t tilde as and the limit taken as t tilde tends to infinity
That is you take the energy in the window size t tilde normalize it divided by t tilde
and take the limit of this quantity as t tilde tends to infinity that is the definition of
the power of any signal that is the power of any signal x(t) okay Now if Px is finite similar to the energy
signal If Px now we have defined the power if Px is less than infinity that is power
of x(t) is is finite than x(t) is termed x(t) is termed as a power signal for instance alright
so if so this is termed as a power signal If the power of a signal is finite than x(t)
the power of the signal x(t) is finite then x(t) is termed as a power signal okay Now
observe our interesting property if the energy is finite if the signal is an energy signal
If x(t) is an energy signal then the power Px equals limit t tilde tending to infinity
1over t tilde integral minus t tilde divided by 2 to t tilde divided by 2 magnitude x(t)
square dt Now this integral minus t tilde divided by 2 to t tilde divided by 2 magnitude
x(t) square dt is less than or equal to the integral from minus infinity to infinity because
this is simply an energy in a window of size t tilde that is less than right that is less
than or equal to energy in the window of size that is the energy in a window from minus
infinity to infinity Therefore observe that this is less than or
equal to limit t tilde tending to infinity to 1 over t tilde minus infinity to infinity
magnitude x(t) square dt Now observe that this quantity is simply the energy of a signal
which is finite because this is an energy signal therefore this is equal to limit t
tilde tending to infinity Ex divided by t tilde So limit t tilde tend into infinity
of Ex divided by t tilde Ex is a constant divided by t tilde which is tending to infinity
therefore Ex by t tilde tends to 0 so this is equal to 0 And therefore what it means that the power
of an energy signal Px that is Px is less than or equal to 0 from the above argument
this we have obtained from the above inequality and we also know that Px is a positive quantity
so Px is greater than or equal to 0 since Px is a positive quantity or a nonnegative
quantity rather Px is nonnegative because it is integral minus t tilde divided by 2
to 2 tilde divided by 2 magnitude x(t) square dt divided by t tilde Everything is positive
so this is everything is nonnegative so this is a nonnegative quantity so the only possibility
is that Px is equal to 0 So Px is equal to 0 for an energy signal that is the power of
an energy signal is equal to 0 So this implies that basically this implies
that Px equals 0 which basically implies that for energy signal that is that is power of
an energy signal So we get our 1st principle that is the power of the energy signal that
is where signal x(t) is an energy signal that it is energy Ex is finite then its power is
0 Now let us look at the energy of a power signal is x(t) is a power signal on the other
hand if x(t) is a power signal
if x(t) is a power signal okay alright then the energy in a window of size t tilde we
know this is approximately equal to the power Px into t tilde because look at this we have
the power that is Px equals limit t telding to t tilde tends to inifinity energy in window
of size t tilde divided by t tilde which means the energy in a window of size t tilde is
approximately t tilde times the power Px equals them times the power Px which implies total
energy equals limit t tilde at tending to infinity energy in
window of size tilde of size t tilde that is equal to limit t tilde tends to infinity
Px times t tilde Px is a constant and t tilde tends to infinity which means this is equal
to infinity Therefore how we have proved it is basically
we have considered the energy in a window of size t tilde and we have said that is Px
times t tilde therefore as the window tends to infinity t tilde tends to infinity right
Uhh the window tends to infinity the size of the window tends to infinity naturally
Px into t tilde that tends to infinity because the power is a constant correct Therefore
the power in unit time is constant as time tends to infinity the total energy tends to
infinity naturally therefore energy of a power signal is infinity This implies that energy
energy of a power signal the energy of a power signal
the energy of a power signal is infinity so we have that the energy of a power signal
is infinity Now what kind of a signal is power signal
we have special we have power signals a special kind of a power signal other is a periodic
signal alright So a periodic signal that is a good example of a power signal is a periodic
signal Let us therefore discuss periodic signals are very important in the context of communication
alright and signal processing and several other applications in electrical engineering
so let us start this discussion about periodic signals okay So let us consider a periodic signal okay Periodic signal we know that x(t) is
a signal with period to x(t) is periodic with period T if x(t) equals x(t) plus kT x(t)
plus kT for all T that is for all T and for all this is the symbol for all for all integers
that is for all that is we call x(t) to be a periodic signal
with period capital T if x(t) is equal to x(t) plus some integer k times capital T where
capital T is the period for a for all times small t and for all integers K alright So
basically it means something very simple that is if you take x(t) and if you shift it if
you consider x(t) at any integer that is any integer multiple of capital T later that t
plus k times capital T than the signal x(t) remains unchanged that is x(t) is a periodic
signal alright For instance some of the classic 1 of the
very popular examples of a periodic signal is a sinusoidal signal that is if you have
sin 2 pi Ft this is a periodic this is a periodic signal alright and what
is the period we know that the period F is the frequency F is the frequency of the sinusoidal
signal the period T equal 1 over F So this is basically your 1 over 2 F this point is
1 over F this point is 3 over 2 F this point is your 2 over F and so on okay So this is
basically the Sinusoidal signal alright okay and this is periodic with capital T and this
is also termed as the fundamental period capital T equals 1 over F is also termed as the fundamental
as the fundamental period of the Sinusoidal signal And you can see that it is periodic with T
because sin of sin of 2 pi Ft equals sin of that is if you consider sin of 2 pi Ft plus
some multiple of the period some multiple kT which is equal to sin of 2 pi Ft plus 2
pi kF into T but we know T equal to 1 by F which implies F into capital T equals 1 so
this is basically sin of 2 pi Ft plus k times 2 pi sin at any multiple that is k times 2
pi that is sin of x plus k times 2 pi simply sin of x so this is sin of 2 pi Ft so this
is basically periodic So we have shown this as periodic with period
T which is equal to 1 over F So the Sinusoidal
signal is periodic with P equal to t capital T equals 1 over F Similarly when we say a
Sinusoidal signal does not necessarily mean only a sin signal it means it can also be
a cosine signal cosine 2 pi Ft also cosine which is a Sinusoidal signal and in general
1 can consider any sin 2 pi Ft plus 5 that is 5 is the phase and A is the amplitude So we can have a phase
and an amplitude okay So the Sinusoidal signal A sine 2 pi Ft plus 5 has phase 5 and amplitude
A okay So this is a general Sinusoidal signal which we have shown that is Sinusoidal signal
with frequency F as a pure Sinusoid with frequency F has a fundamental period that is a period
of Sinusoidal signal is capital T equals 1 over F that is the signal is periodic 1 over
F okay Now what is the power of a periodic signal How to find the power of a power of
a periodic signal Now the power of a periodic signal this can
be found as follows Let T be the period of the periodic signal then we have from the
definition of power we have Px equals limit t tilde tends to infinity sorry t tilde tends
to infinity 1 over t tilde minus t tilde divided by 2 to t tilde divided by 2 magnitude x(t)
square dt now what we are going to do is choose t tilde equals m times capital T that is we
choose t tilde to be m times capital T so naturally as m tends to infinity t tilde tends
to infinity So instead of tending t tilde into infinity we can equivalently tend this
integer m so what we are doing is we are choosing t tilde to be an integer multiple m times
T where T is the period right So this is realize note that T is the period of the signal okay
So if t tilde so if t tilde m tends to infinity this implies t tends mT tends to infinity
this implies t tilde tends to infinity So naturally I can use this so instead of
limit t tilde tending to infinity I can here equivalent represent this as limit m tending
to infinity 1 over mT minus mT divided by 2 to mT divided by 2 magnitude x(t) square
dt okay Now at this this is integral look from minus mT by divided by 2 to mT divided
by 2 this contains m periods right The total duration is mtimes capital T so it contains
m periods of the signal that is this is the energy in m periods Now this signal is periodic
so energy in m periods is m times the energy in a single period because this is a periodic
signal okay so that is the property we are going to solve This is basically
energy in m periods equals m into because the signal
is periodic energy in m periods is m times the energy in a single period so this is limit
m tend into infinity 1 over mT minus T by 2 to T by 2 that is single period mtimes the
energy in a single period That is magnitude x(t)square dt now you can see the m is canceled So this is equal to and therefore there is
no m therefore limit m tend into infinity is simply 1 over T integral minus T divided
by 2 to T divided by 2 magnitude x(t) square dt this is the power Px of the periodic this
is the power of a periodic signal this is the the power of a periodic signal What is this
This is simply the power of a periodic signal x(t) is simply the energy in a single window
that is the energy in a single window of size T that is energy in a single period capital
T divided by the period T that is the size of the window capital T So the power of a periodic signal for the
special case of a periodic signal this is energy in
window of size T divided by T Realize that this T this T is the period of the signal
so you simply take the energy of the periodic signal in 1 period divided by the period that
is capital T and you get the power of the periodic signal that is Px okay So let us take again a simple example let
us again go back to our standard periodic signal that is the Sinusoid example again
x(t) equals A cosine 2 pi Ft let us consider this periodic this again we said the cosine
signal with amplitude A phase 5 equals to 0 we said cosine signal is also a Sinusoid
so this is Sinusoid and more importantly we know the period T this is the period capital
T which is equal to 1 over F therefore the power of this signal power is equal to 1 over
T minus T by 2 to T by 2 A square cosine square that is 1 over T 1 over the period capital
T where T equals to 1 over F minus T by 2 the capital T by 2 a square cosine square
2 pi Ft which is equal to now I can bring the A square outside A square is constant A square divided by capital
T integral minus T by 2 to T by 2 cosine square 2 pi Ft is 1 over cosine 4 pi Ft that is cos
square theta is 1 plus cos 2 theta divided by 2 so cos square 2 pi Ft is 1 plus cosine
4 pi Ft divided by 2 So this is now the integral of half between minus T by 2 to T by 2 that
is straightforward that is basically that is half times T plus half integral of cosine
2 pi Ft is basically sine integral cosine 4 pi Ft is sine 4 pi Ft divided by 4pi F evaluated
between minus T by 2 to T by 2 which is A square by T into half T So that is A square
by 2 plus A square by uhh plus A square by T times 1 by 8 pi F sine 4 pi Ft at capital
T by 2 F into T is 1 so this is sine 2 pi minus sine Uhh 4 pi Ft and t equal to minus
capital T divided by 2 equals sine minus 2 pi and you can see that sine 2 pi equals sine
minus 2 pi because there is a difference of integer But anyway integer multiple of 2 pi So this
is basically so this quantity here equals 0 So this is A square divided by power of
the Sinusoidal signal of Amplitude A power of your cosine power of your A cosine 2 pi
Ft that is Sinusoidal signal Uhh A cosine 2 pi Ft is fine In general you can show show
that the power of any Sinusoidal signal with amplitude A and phase five is equal to A square
divided by 2 It can be shown that power of A this is equal to A square by 2 where
A is the amplitude that is it does not depend on the phase the power is simply a square
divided by 2 where A is the amplitude of the Sinusoidal signal alright So this is simple module where we have started
with the definition of an energy defined the energy of a signal defined what is an energy
signal The power of a signal what is the power of a signal Also looked at the periodic signal
and defined the power of a periodic signal and illustrated how to compute the power of
a simple or a very common and frequently used periodic signal and signal processing and
communication that is the Sinusoidal signal whose power is A square divided by 2 where
A is the amplitude of the Sinusoidal so we will stop here and look at the other aspects
in the subsequent modules thank you

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