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Okay now let us consider temperature T s instead
of that standard value 290 Kelvin And in that case let us see what happens SNR how to calculate
So SNR at the output this is equal to P s divided by k into T s plus F minus 1 T 0 into
B n So the first term this is due to this source temperature T s and the second term
this is due to the effectiveness temperature T e so if I now assume that T s equal to T
0 in that case P s by k into T 0 F B n So sometimes we simply write down that SNR at
the output this is equal to P s by T 0 F B n but that happens only if T s equal to T
0 And if the source temperature is different than that 290 Kelvin or T 0 so we have to
use this correction factor and this overall expression We have to consider the effect
of T s as well T e separately So considering a scenario like that we can
calculate the we can define the parameter which we call the operating noise factor N
op So how we do it operating noise factor of a combined system including the source
and the receiver is the ratio of the actual available output noise power density which
is given by N oa to the available output noise power density if the receiver had no internal
noise source So that means in this case what we will be doing we are considering actual
temperature not T 0 so T s and then calculating what is the available noise power density
and output And in the second case again we are recalculating it but we are assuming the
receiver itself is noiseless then what is the effect of that left hand side noise and
how it is being transferred to right hand side then we take the ratio So it is given by F op this is equal to N
oa which is the available output noise power density so it includes both the contribution
from left hand side and due to the receiver itself and divided by G a k T s so G a k T
s you see here we did not consider any internal contribution of noise so this is only due
to the left hand side So F op it takes into account the actual source
temperature T s So why we are doing it because let us say you have designed the receiver
you have the noise figure or let us say noise factor of whatever specified by manufacturer
that is for standard value 290 K And now for your system which may be it it may operate
let us say for space application it can operate at minus50 minus60 Kelvin minus50 minus60
degree centigrade or it can be at very high temperatures scenario so for that case you
calculate what is the SNR for your receiver chain So then SNR at the output this is equal to
G a P s this is the output power divided by output noise power we are expressing it in
terms of F op so then this is equal to simply F op G a k T s into B n over a noise bandwidth
B n or G a cancel out so P s by F op k T s B n now P s divided by k T s B n so this k
T s B n this is actually the noise present at my left hand side of the receiver over
a bandwidth B n So simply then P s divided by k T s B n this factor we can call SNR at
the input side so we can simply write down F operating that is equal to SNR at the input
divided by SNR at the output not at standard temperature but at any given temperature so
that is how we can define another parameter F operating noise operating noise factor So now in presence of receiver noise N internal
in the previous one when we calculated k we put simply K T s B n here we did not consider
effect of internal noise generated inside the receiver now we are introducing that In
that case F operating this is equal to N internal plus so this you see noise factor this is
SNRi by SNRo so in terms of noise simply then we can write down noise at the output divided
by noise at the input So noise at the output this is G a k T s plus internal noise divided
by noise at the input G a k T s so this is 1 plus T e by T s when we are putting N internal
equal to G a k T e Now already we have seen that T e in previous expression T e equal
to F minus 1 into T 0 so we can also express F operating that is equal to 1 plus we are
simply just putting the value of T e in the previous expression so F minus 1 into T 0
by T s In this expression it simplifies to F or F operating it becomes F only when T
s equal to T 0 Also we have seen it previously you see it
simplifies to this SNR at the output in this expression it simplifies to k T 0 F B n only
when T sequal to T 0 But this case also F operating equal to F that specified only when
TS equal to T 0 otherwise not So now let us say T s is not equal to T 0 so then we have
to consider all the effects of T s T e and T 0 is already given or specified for the
components whatever we are using so so many noise temperature we have to deal with We
can simply eliminate T e by introducing one model parameter so define a system noise factor
N system to eliminate T e for simplification so F system how we are defining F system that
is equal to T s plus T e by T 0 So putting the value of T e here F system this is F minus
1 plus T s by T 0 Or we can write down then SNR at the output
that is equal to P s divided by F system k T 0 B n You compare this expression with the
previous one for a noiseless receiver So for noiseless receiver SNR at the output is it
is uhh equal to P s by this factor but here this is T s by F system not F operating if
we have a noisy receiver which is the practical scenario so F system where it is related to
F op by F op into T s by T 0 So if we know the F value given for T 0 then we can easily
calculate F system from this expression F op into T s by T 0 Now noise figure for a cascaded system we
already calculated equivalent noise temperature for a cascaded system now we are going to
calculate noise figure or noise factor for cascaded system So basically overall SNR calculation
we can use any of these 2 approaches noise figure approach or noise temperature approach
Noise temperature approach already we know now we are going to see the noise figure one
So we are considering a simple scenario we have let us say just 2 amplifiers in Cascade
it can be any other components also simply we want we have to calculate its gain The
first one let us say gain is G a1 and it is providing some internal temperature N internal
1 so if I express in terms of noise temperature you remember we expressed it by T e 1 then
N internal 1 that is equal to k T e 1 The second component it is G a2 gain and internal
noise generated N internal 2 So N 1 whatever falling from left inside to the second component
so this is the output noise power from the first component So if I now first consider
a noiseless component that means N internal 1 or N internal 2 is 0 in that case N 1 this
is equal to then N i multiplied by gain of this amplifier so that is the noise available
at the output we consider a noiseless system So that output is falling on the second amplifier
so N 1 equal to G 1 G a1 N I similarly N 2 equal to G a2 N 1 that is equal to G a2 G
a1 N 1 Now let us consider the effect of N internal 1 and N internal 2 so considering
noise introduced by the elements N 2 that is equal to N internal 2 plus G a2 multiplied
by whatever noise coming from left hand side N internal 1 plus G a1 N i or we can write
down this is equal to then N internal 2 plus G a2 into N internal 1 plus G a1 k T 0 so
N i we are representing by k T 0 Then the standard cascaded noise factor this
is also the Friis formula second form for noise factor F equal to N 2 so output noise
power divided by G a2 G a1 k T 0 So you can we can now put the values here N 2 already
we calculated in the previous one N internal 2 plus G a2 N internal 1 G a2 G a1 k T 0 divided
by G a2 G a1 k T 0 So if I consider the first 2 terms N internal 1 and G a1 k T 0 so G a2
this is cancel out and this is for the we are considering this sorry second and third
term Now if I consider the first term N internal 2 with that we are simply adding and subtracting
G a2 k T 0 T 0 so original term is N internal 2 then plus minus G a2 k T 0 divided by this
so it simplifies to F 1 N internal 1 plus G a1 k T 0 by G a1k T 0 this is F 1 plus F
2 minus 1 by G a1 So this is for a 2 component system if we
have n number of components we can simply add other terms so then the third term it
will be F 3 minus 1 divided by G a1 G a2 similarly you can define it for nth number of terms
So again what we see that uhh the contribution of first component it becomes very important
so for all other components it is being divided by the gain of other components preceding
components So that is why whenever we use an amplifier it should be a low noise amplifier
just immediately after the antenna to decrease the overall noise figure of the receiver
So now we learn how to calculate the overall noise figure of a receiver noise factor of
a and if we simply convert it to decibel scale then we will call overall noise figure of
the receiver So it will determine the noise floor below which the receiver it cannot detect
any noise power any signal power So minimum SNR that should be higher than
this also we have we learned what is link margin this value is not sufficient for any
wireless system so above that we have to maintain link margin of 3 dB to 10 dB Now let us take
one example numerical with numerical values let us say one antenna it is directly pointing
towards Earth it can uhh it can be a radiometer itself this whole receiver chain together
with antenna So in radiometer application it is actually one type of passive imaging
what is done simply the power whatever radiated by the other surface it is it is connected
by antenna and then it is converted to some effective noise temperature And now if we
point towards different directions so depending on the dielectric constituents of the Earth
surface and the roughness of other surfaces the effective noise temperature will change
since the input power will change so that is how we can map any given Earth surface
and it is very popular in uhh remote sensing applications We can monitor sea level we can monitor ice
level and many other things so this is one such application So antenna is likely pointing
towards Earth and the Earths temperature physical temperature is given as 300 Kelvin so it is
not also it is not a perfect black body because its (emis) it is given by characterised by
some emissivity Epsilon so emissivity for black body equal to 1 Now emissivity it can
be given by 1 minus Rho where Rho is the reflectivity so if Epsilon is 0 point 9 that means Rho
equal to 1 minus 0 point 9 which is 0 point 1 so a good emitter is a bad reflector So
this antenna is connected to a circulator by using a section of transmission line may
be a cable or waveguide section Starting from antenna to circulator they are at a physical
temperature given by 180 Kelvin it is quite low but it may happen in space The loss of this connecting section is given
as 1 dB we also have some loss from this circulator which is given by 1 dB and look at the antenna
antenna efficiency is 95 percent or Rho e it is given as 0 point 95 Immediately after
the circulator we have a low noise amplifier so this low gain of this low noise amplifier
is given as 20 dB and the corresponding noise figure not factored in dB Noise figure is
4 dB again we have some loss due to connecting sections that is 1 dB and next we have one
mixer it is down converting the millimetre wave signal coming from left hand side So
for the Earth this LNA and the connecting cables they are at physical temperature to
50 Kelvin but right hand side maybe they are sealed inside one package for them they are
at thermal equilibrium of physical temperature 400 Kelvin it is quite high So just after mixer we have a connecting section
then the IF amplifier which is obviously low frequency amplifier so for this low frequency
amplifier noise figure is given as 3 dB gain 60 dB Now when we convert this noise figure
to noise factor obviously log base is 10 we have to take 10 to the power but we are dealing
with power not voltages So that means we have to divide all these values by 10 not 20 for
example if I want to convert this 4 dB noise filter to corresponding noise factor what
we have to do 4 by 10 point 4 so it will be 10 to the power 0 point 4 Now what we have
to do we have to calculate what is the overall SNR that also we can calculate if we can calculate
whatever the overall effective noise temperature T e of this system So we will start from uhh from the source
basically before antenna we have source the Earth surface so the earth surface temperature
is given 300 Kelvin Now what happen you can see if I look at the power whatever received
by antenna some part is due to emission due to the Earth itself and another part which
is coming from the dwelling radiation from atmosphere that is being reflected by Earth
surface coming to antenna But since it is Rho is 0 point 1 it is a poor reflector and
looking at the roughness of Earth surface we are simply neglecting that reflected power
it is in practice also it is negligibly small We will be considering then only the effect
due to this emission from Earth surface itself which is which we are considering as a grey
body So this effective radiation will be represented
uhh we will be representing by resistor so this is the figure so for this part what we
are doing we are considering the first part starting from Earth surface to circulator
which are at a physical temperature 180 Kelvin Now how to calculate T external T external
is simply Epsilon into T physical so temperature of Earth surface is given as 300 Kelvin into
Epsilon 0 point 9 T external it becomes to 70 Kelvin we are using temperature approach
Now antenna temperature T antenna so antenna loss is given by 1 by Rho e because gain a
0 point 95 then loss it is 1 by 0 point 95 then followed by antenna we have a section
of waveguide waveguide for which again loss is 1 dB so we calculate then what is the corresponding
gain so it is 1 by 10 that is 0 point 1 so 10 to the power 0 point 1 So overall gain G 1 into G 2 for this 2 cascaded
system so G 1 into G 2 so G 1 you see 0 point 95 G 2 0 point 794 G 12 then it become 0 point
754 Now loss L 12 it is simply 1 by overall gain which is 1 point 33 so we calculate then
T antenna it is equal to we already have seen the formula T e equal to T physical into L
minus 1 so put it here T antenna it becomes 59 Kelvin Then the source temperature T s
for these 3 components altogether this is equal to G 12 multiplied by T external because
you see noise generated by this external source or the resister it is being propagated through
antenna and that waveguide infection so T s equal to G 12 into T external plus T antenna
or put the values it becomes 248 Kelvin so T s is 248 Kelvin Here we did not consider any direct contribution
of sunlight actually sky is very cold if you point your antenna towards earth you will
receive more power compared to if you just point your antenna to the cold sky So cold
sky temperature is typically very low equivalent noise temperature is 50 Kelvin so this source
part calculation is over next part we have to calculate T e and hence the operating So
we have to consider the contribution due to LNA then connecting section Mixer connecting
section and then the IF amplifier oh before that we have circulator also So among these
mixer specification it is given its temperature ratio you see here L equal to 5 dB that means
conversion loss is 5 dB and temperature ratio is 1 point 5 so actual its operating temperature
is 290 multiplied by 1 point 5 Okay let us take 5 minutes break then we will continue
the calculation

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