Noise Factor of a Lossy Network:
When a signal source is matched through a lossy network, such as a connecting cable, the available signal power at the output of the network is reduced by the insertion loss of the network. The output noise remains unchanged at kT0Bn, since available noise power is independent of source resistance. In effect, the network attenuates the source noise, but at the same time adds noise of its own. The S/N ratio is therefore reduced by the amount that the output power is attenuated.
Denoting the power insertion loss ratio as L, the output S/N ratio will be 1/L times the input S/N ratio and from the definition of noise factor given by
Noise Temperature:
The concept of noise temperature is based on the available noise power which is repeated here for convenience:
Hence, the subscript a has been include to indicate that the noise temperature is associated only with the available noise power. In general, Ta will not be the same as the physical temperature of the noise source. As an example an antenna pointed at deep space will pick up a small amount of cosmic noise. The equivalent noise temperature of the antenna that represents this noise power may be a few tens of kelvins, well below the physical ambient temperature of the antenna. If the antenna is pointed directly at the sun, the received noise power increases enormously and the corresponding equivalent noise temperature is well above the ambient temperature
When the concept is applied to an amplifier, it relates to the equivalent noise of the amplifier referred to the input. If the amplifier noise referred to the input is denoted by Pna, the equivalent noise temperature of the amplifier referred to the input is
We know noise factor,
.Substituting this value in equation (1) Friis’s formula can be expressed in terms of equivalent noise temperetures. Denoting by Te the overall noise of the cascaded system referred to the input and by 
and so on, the noise temperatures of the individual stages,then Friis’s formula is easily rearrange to give Narrowband Band-pass noise:
Band-pass filtering of signals arises in many situations, the basic arrangement being shown in fig-1. The filter has an equivalent noise band-width BN and a center frequency fc. A narrowband system is one in which the Centre frequency is much greater than bandwidth, which is the situation to be considered here.
The signal source is shown as a voltage generator of internal resistance Rs. System noise is referred to the input as a thermal noise source at a noise temperature Ts. The available power spectral density is
For the ideal band-pass system shown, the spectral density is not altered by transmission through the filter, but the filter bandwidth determines the available noise power as 
BN. So far, this is a result that has already been encountered in general. The output waveform has the form of a modulated wave and can be expressed mathematically as
This represents the noise in term of a randomly varying voltage envelope 
and a random phase angle 
.
These components are readily identified as part of the waveform, as shown in fig-2 but an equivalent although not so apparent can be obtained by trigonometric expansion of the waveform as
Here,
is a random noise voltage termed the in-phase component because it multiplies a cosine term used as a reference phasor and 
is a similar random voltage termed the quadrature component because it multiplies a sine term, which is therefore 900 out of phase, or in quadrature with, the reference phasor. The reason for using this form of equation is that, when dealing with modulated signals, the output noise voltage is determined bu these two components. The two noise voltage 
and appear to modulate a carrier at frequency 
and are known as the low-pass equivalent noise voltage. The carrier may be chosen anywhere within the pass band, but the analysis is simplified by placing it at the center as shown. This is illustrated in fig-3.
This importance result, which is illustrated in fig-3, will be encountered again in relation to modulated signals.
