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So now we are considering some external noise
which is incident from left hand side and altogether we want to express the system by
some equivalent noise temperature then how to calculate it So we have a 2 port network which is providing
some internal noise and it is also connected to some external source of noise which can
be given by T external So in that case then the total noise power at the output N s this
is equal to k T external by L this is for the contribution from left hand side plus
T physical into 1 by 1 minus L whatever we have seen we can represent it by some effective
noise temperature T s k Ts so N s this is equal to k T s Now consider one antenna so here you can see
here so N s it is equal to k into T s Now consider 1 antenna which is connected to a
transmitter now antenna it will have some efficiency Let us say antenna efficiency is
given as 90 percent then and if any power we feed to antenna it will be attenuated by
antenna and uhh this value it can be given by that 90 percent if we convert it to fraction
0 point 9 Or we can say L loss of the (anten) due to antenna this is 1 by Rho or 1 by 0
point 9 Then T physical is the antenna temperature let us say so this is the physical temperature
of the antenna and T external is the noise temperature of the transmitter just before
antenna So in that case we can calculate then what is noise power available from this antenna
and source which can be given by k T s Let us consider a receiver now and we are
going to calculate the overall receiver noise temperature So what we will be doing let us
first consider one amplifier a 2 port network again So the amplifier amplification factor is given
by G a it is sometimes called the available gain and T e is its effective noise temperature
Now it is connected to some external resistance which is given by R L at right hand side and
left hand side it is connected to a source of internal resistance R s And if the source
power is 0 in that case uhh the power which is incident from left hand side it is only
the noise power Also the input and output resistances they are also given for the amplifier
Now the noise voltage available uhh because of the noise power generated by R s that can
be given by e s equal to square root of 4 k T R so maximum noise power delivered to
a mesh load we are considering one mesh scenario that is P i this is equal to e s by twice
R s whole square into R s So you see here what we are considering so
R s this is the noise source I want to extract noise power from the source or we can see
in other way We are connecting a resistor here so some noise power will be delivered
to the resistor then we are considering what is the maximum noise power that is being delivered
to this resistor and that happens according to maximum power transfer theorem when this
R this is equal to R s or we can write down this is equal to R s So in that case if I
want to calculate P i this is simply I square into R then the current component went through
it this is e s by twice R s whole square multiplied by R s or we can write down this is equal
to e s square divided by 1 R s is cancelled out so 4 R s and this is equal to k T which
already we have seen Now gain of the amplifier G a we can express
in terms of the input (noid) noise power and output noise power So output noise power is
e 0 square divided by 4 into R out and the input noise power this is e s square divided
by 4 into R s So one interesting thing then Gain you see for C amplifier also if we want
what to calculate the overall gain of any C amplifier uhh it depends on R R l but simply
if I say what is the gain of a C amplifier we do not consider R l we take just the open
circuit at output voltage and then calculate what is the gain of the amplifier Now if you
change R l obviously the available power to R l it will change but when we call the G
a or gain of the amplifier then we do not consider R l but this gain it depends on the
source register R s So here also then this G a it depends on R
s but not on R l So output noise power if I want to calculate at output terminal let
us say we are denoting it by N 0 so that is equal to G a k T s so k T s this is the incident
power from left hand side which is being multiplied by gain of the amplifier G a In addition to
this the amplifier itself it will produce some noise which is given by N internal N
internal this factor we can also express in terms of T effective noise temperature which
is given T e so we can also write down that this is equal to G a k T s plus N internal
this is equal to G a k into T e or this is equal to G a k T s plus T e Let us represent it by effective temperature
G a k T 0 so T 0 this is called the operating temperature operating noise temperature So
you see here so this is the amplifier and left hand side we have a noise source connected
to this amplifier and right hand side we have a load resistor connected here then we calculate
it we expressed already a G a in terms of the input and output noise power and from
that we calculate that we calculated the output noise power from the device so output noise
power what we see that it contains uhh the contribution due to the external noise source
which is G a k T s In addition to that this amplifier itself produce some noise which
is given G a k T e so all together this effect we can represent by some operating noise temperature
T 0 so N 0 then that is equal to G a k T 0 So T 0 T e G a all functions of R s and but
they do not depend on R l And in this case also we assume that the output
R l is mesh to this and so it is the worst case scenario this N 0 this is the highest
available noise power from the amplifier on the noisy source Now let us consider several
2 port components they are connected in Cascade so that means what we are considering let
us say we have 3 amplifiers the first one gain is given by G a 1 and the effect noise
temperature effective noise temperature is given by T e1 It is followed by another amplifier for which
gain is G a 2 effective temperature is T e 2 Again we have one more G e 3 effective noise
temperature T e 3 and we want to calculate what is the overall noise contribution of
this system so if we know for 3 we can do it for N components in Cascade Then N internal
we can write down this is equal to so you see this first amplifier whatever noise it
will produce let us say that is given by N internal 1 so this N internal 1 it will be
amplified by G a 2 further it will be amplified by G a 3 but whatever noise produced by the
third component this component is not being amplified by left hand that is G a 1 or G
a 2 Then the total noise contribution N internal for the first one this is G a 3 G a 2 into
G a 1 k T e 1 So k T e 1 this is the noise produced by the
first one and it is being amplified by all these 3 amplifiers Then for the second one
G a 3 G a 2 into k into k T e 2 and for the third one this is G a 3 k T e 3 So if we represent
it this overall combination if we contribution if we represent it by some effective temperature
let us say T e in that case we can write down this is equal to k T e is the overall equivalent
noise temperature then k T e into G a 3 G a 2 G a 1 so therefore T e this is equal to
N internal divided by k G a 3 G a 2 G a 1 So what we are assuming as if one resistor
of equivalent noise temperature T e is connected to left of this system and system itself is
noise less so that is why we are multiplying by 3 gain factors G a 3 G a 2 and G a 1 So in terms of the individual noise temperature
we can also express it so you just put that value of N internal whatever we obtained in
this first equation So if we put this value here so it becomes T e 1 plus T e 2 by so
remaining term is G a 1 plus T e 3 remaining terms G a 1 G a 2 For N components if we have
N number of components simply we can extend it so it will become for Nth term T e n and
below we have (G 1) G a 1 G a 2 to till G a (N 1) so this is called the Friis law for
cascaded system So what we see that if we have many Cascade
connections the contribution of noise from the first component it is the it is dominant
over all others That is why when we design any receiver the first component is chosen
to have which will have minimum contribution of noise so that is why we use low noise amplifier
typically which will provide very low noise and after that whatever components we are
having their contribution is smaller compared to the first component So now let us consider
a receiver it is modelled by a constant gain G 0 over a band B n we are introducing now
some bandwidth given by B n Then the receiver input power is given by P s and it receives
a noise power P n from the antenna then considering matched scenario so always by default we will
be considering matched scenario so whatever power is coming from left hand side so it
is being observed right hand side half of that observed in the load So then SNR SNR at the output of the receiver
that is equal to you see this is G 0 SNR is signal to noise ratio so signal at left hand
side is P s it is being multiplied by G 0 so out at output signal available signal power
is G 0 into P s And what is the output noise power so left hand side we have receive P
n noise it is being amplified by G 0 times by the receiver chain plus we have some internal
noise contribution because of the receiver cover bandwidth B n B n multiplied by n internal
and it can be given by equal to SNR at the input divided by 1 plus B n N internal divided
by G 0 P n How we uhh come here simply you divide both denominator numerator and denominator
by G 0 P n so then the top one it becomes P s by P n which is SNR at the input So this equation is called Radar equation
for receiver it is one important quantity parameter Now considering available power
gain G a and this above definition we can also express SNR at the output so in terms
of operating temperature this is equal to G a into P s divided by G a k T 0 B n so this
is the operating temperature and simply then this is equal to P s by k T 0 B n so you remember
recall this T 0 is the operating temperature Now noise bandwidth so what if uhh we are
many times calling noise bandwidth B n and we are using it to calculate overall noise
power then what value should we take should we take for B n for any given channel Let
us say uhh we are using a channel uhh bandwidth 1 gigahertz so should we take simply B n equal
to 1 gigahertz So whatever definition the significance of
B n till now we have used it is like one simple window function which has some value over
the given bandwidth and out of this band it has 0 value but it is not the practical scenario
So let us consider any practical case so in this expression when we use some value of
B n so that means let us say we are considering the ideal (scenario) scenario this is the
1 gigahertz channel bandwidth it is given B n and then model it by a constant gain G
a when we uhh talk about the gain of the amplifier or gain of the receiver G 0 whatever and out
of this band this G a is exactly 0 But for any practical system consider any uhh band
pass filter or anything so for that it is not constant but it becomes a function of
frequency and we have contribution due to you see other frequency components so it will
it will vary So gain G a it is a function of frequency
and bandwidth B n then what value we should take for this type of practical scenario so
what we do here there is a way of calculation For a real system obviously it will be defined
by its transfer function H which is a function of j omega and the corresponding temperature
which previously we consider constant over the frequency but practically it should be
a function of frequency Then actual output noise power we should consider all the frequency
bands starting from 0 to infinity and we integrate it over 0 to infinity H j Omega square k T
0 d f Now if T 0 is constant over the band and let us say mid band gain is given by G
G a equal to H j Omega 0 square in that case so we are assuming T 0 is constant otherwise
it is becoming complicated for simplification this assumption So B n this is equal to then total noise power
P n divided by k t 00 G a or 0 to infinity this integration value divided by H j Omega
0 whole square So from this expression we can calculate B n value but even then you
see here we consider T 0 is constant over this given bandwidth Next noise factor so
total noise contribution it can be calculated in terms of noise temperature also sometimes
another parameter is used to quantify the noise contribution that is called noise figure
we can calculate it for any given 2 port devices so usually if it is a lossy device with some
insertion loss let us say L in dB and then noise figure in dB it is simply becomes becomes
L If you buy any active component like an amplifier or some active mixer usually the
noise figure value will be written or given by the manufacturer it can be also measured
in uhh labs by using in lab experiment by using spectrum analyser and standard noise
source values or VNA Vector Network Analyser So in general then what is first let us define
noise factor of a uhh 2 port network Noise factor F this is the ratio of signal to noise
ratio at the input terminal divided by signal to noise ratio at the output terminal so SNR
at the input divided by SNR at the output Now signal to noise ratio obviously is better
at left hand side compared to right hand side at the output why Because at output side we
have additional noise generated by the component itself so that is why F this is always greater
than 1 If we express it in dB we call it noise figure and that is always a positive quantity
so for a given system then already we did it let me express mathematically So then the noise factor F this is equal to
SNR at the input terminal divided by SNR at the output terminal so at the input terminal
let us say the power already whatever we assumed in previous example P s is the incident power
and let us say N i is the incident uhh noise from left hand side we are considering per
unit hertz So this is the SNR at the input side now we can calculate SNR at the output
side so then the available power is the application factor G a multiplied by P s divided by total
noise total noise is some noise internally generated inside the 2 port device plus the
contribution from left hand side which is G a into N i So this is equal to then G a
k T 0 plus N internal divided by G a k T 0 Now it is a function of temperature you see
what value we should take for T 0 so if the noise from my left hand side it changes obviously
this value will change So then uhh if we buy let us say one amplifier
from market some noise figure or noise factor is given with that then what it denotes Because
it becomes a function of input noise So usually a standard temperature is considered to calculate
noise factor and hence figure that is 290 Kelvin So if I do not consider simply actual
scenario practical scenario by noise factor we mean the noise contributed by the component
it has been calculated at a given T 0 equal to 290 Kelvin So from this then you see if
I put N internal that is given by G a k T e T e is the equivalent noise temperature
so then it becomes simply 1 plus T e by T 0 So T e this is the effective noise temperature
of the 2 port device itself T 0 uhh this is for standard calculation we will consider
290 Kelvin Then we can express T e also in terms of F
so T e equal to F minus 1 into T 0 this is another important expression So if F is given
we can calculate what is the effective noise temperature T e if T e is given we can calculate
what is the noise factor F so they are related by this important relationship This noise
factor depends on source impedance and operating frequency obviously if we have a lossy network
as I discussed it will provide you some insertion loss or power will attenuated by that 2 port
network If the loss is given by L then simply its noise factor is equal to L And now we
are considering a different scenario let us say this source temperature is not T 0 it
is given by T s whatever we considered in the previous case Okay so let us take a break
then uhh we will consider a practical scenario with a given temperature T s thank you

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