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