Derive the expression for overall noise figure of a cascaded system.
The noise facto can be expressed in term of decibels. The representation of noise factor in decibel is known as noise figure.
The noise figure \(F_{dB}=10logF\)
The noise figure \(F=10log\frac{S/Nattheinput}{S/Nattheoutput}=10log\left(\frac{S}{N}\right)_i-10log\left(\frac{S}{N}\right)_0\) … (1)
Hence noise figure F_{dB}=log\left(\frac{S}{N}\right)_idB-log\left(\frac{S}{N}\right)_0dB … (2)
The ideal value of noise figure is zero.
Fig. 19 In cascaded configuration of two amplifiers.
In Fig. 19, two amplifiers are connected in cascading form. The source voltage \(V_s\) is connected with the amplifier 1 by the source resistance R.
Let us consider the gain and noise factor of the amplifiers are \(G_1\), \(G_2\), \(F_1\land F_2\) respectively.
The input noise power of the source is
\(P_{on}=kT_0B\) … (3)
Where k is the Boltzmann’s constant and Bis the bandwidth of the system.
The input noise power of amplifier 1 at room temperature is \(P_1=F_1kT_0B\).
The output noise power of amplifier 1 at room temperature is
\(P_{n01}=F_1G_1kT_0B+\left(F_1-1\right)kT_0B\) . … (4)
The first term and second term of equation (4) are representing the amplified noise power of amplifier 1 and noise contributed by the 2nd amplifier.
The noise power at the output of the 2nd amplifier is the product of the gain \(G_2\) with the input noise power of amplifier 2.
Therefore the final output noise power is given by
\(P_{n02}=F_1G_1G_2kT_0B+G_2\left(F_2-1\right)kT_0B\) . … (5)
The overall gain of the system is
\(G=G_1G_2\) … (6)
The overall noise factor is defined as
\(F=\frac{P_{n02}}{G_1G_2P_{on}}\) … (7)
Using equation (3-7), we have the overall noise factor
\(F=\frac{F_1G_1G_2kT_0B+G_2\left(F_2-1\right)kT_0B}{G_1G_2kT_0B}=F_1+\frac{\left(F_2-1\right)}{G_1}\) … (8)
If the number of amplifier is cascaded more than two we have,
\(F=F_1+\frac{\left(F_2-1\right)}{G_1}+\frac{\left(F_3-1\right)}{G_1G_2}+\frac{\left(F_4-1\right)}{G_1G_2G_3}+\ldots\ldots\) … (9)
This formula is known as the Friiss formula to find the noise factor in a cascaded system.
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