Understanding Noise – Part 5: Noise Performance of Cascaded Systems

# Understanding Noise – Part 5: Noise Performance of Cascaded Systems

Thus far in this series on Understanding Noise we have looked at:

Modern radio receivers and electronic communications circuits consist of filters, amplifiers and mixers connected in series.  Each component contributes to the overall noise performance of the system as noise is added by each stage. This post will discuss how to calculate the contributions of each component to the overall noise performance of a cascaded system.  I will also show how system design, and the order that components are placed in can dramatically alter the performance characteristics of a receiver!

First let’s start by considering a generalized cascaded system, shown below with a total of k stages, each denoted by i.  Each stage has its own noise performance denoted by its own Noise Factor (Fni) as well as its own Gain (Gi).

The Noise Factor of the system is:

$F_{n} = \frac{P_{no}}{P_{ni}.G_{t}}$

$G_{t} = G_1.G_2.G_3... ...G_k$

For systems design tasks, using components with known characteristics, it would help if we were able to calculate the total noise performance of the system in terms of the noise performance of each of the components.  This leads us to the derivation of the Friis Noise Formulas.

## Understanding Pna in terms of Noise Factor

Recall from the previous post that Noise Factor of a system is defined by:

$F_n = \frac{P_{no}}{P_{ni}.G}$

But we also know that:

$P_{no} = (P_{ni} + P_{na}).G$

So we can replace Pno with:

$F_n = \frac{(P_{ni} + P_{na}).G}{P_{ni}.G}$

Simplifying the expression yields:

$F_n = 1 + \frac{P_{na}}{P_{ni}}$

Making Pna the subject of the equation:

$P_{na} = (F_n - 1).P_{ni}$

Does this equation look familiar?  It should, go back and check the equation expressing the effective noise temperature of a system in the previous post!

## Noise Factor of a Cascaded System

Consider the cascaded system below, constructed using a series of components, each described by the standard model.

If we look at output Noise Power Pno of the entire system we can see that:

$P_{no} = (...(((P_{ni}+P_{na1}).G_1 + P_{na2}).G_2 + P_{na3}).G_3 +... P_{nak}).G_k$

You can multiply this expression out to get:

$P_{no} = (P_{ni} +P_{na1}).(G_1.G_2.G_3... .G_k) + P_{na2}(G_2.G_3... .G_k) + ... P_{nak}.G_k$

Looking at the Noise Factor of the entire system:

$F_{nT} = \frac{(P_{ni} +P_{na1}).(G_1.G_2.G_3... .G_k) + P_{na2}(G_2.G_3... .G_k) + (P_{na3}.(G_3...G_k) + ...P_{nak}.G_k}{P_{ni}.(G_1.G_2.G_3...G_k)}$

Let’s break this out:

$F_{nT} = \frac{(P_{ni} +P_{na1}).(G_1.G_2.G_3... .G_k)}{P_{ni}.(G_1.G_2.G_3...G_k)} + \frac{P_{na2}(G_2.G_3... .G_k)}{P_{ni}.(G_1.G_2.G_3...G_k)} + \frac{(P_{na3}.(G_3...G_k)}{P_{ni}.(G_1.G_2.G_3...G_k)} + ... + \frac{P_{nak}.G_k}{P_{ni}.(G_1.G_2.G_3...G_k)}$

If we use the expression for Pna in terms of Noise Factor of a stage we can see that:

$F_{nT} = F_{n1} + \frac{F_{n2}-1}{G_1} +\frac{F_{n3}-1}{G_1.G_2} + ... + \frac{F_{nk}-1}{G_1.G_2.G_3...G_{(k-1)}}$

## Noise Temperature of a Cascaded System

We can also express a system’s noise performance in terms of equivalent noise temperature.  It makes sense to be able to calculate the equivalent noise temperature of a cascaded system.

Let’s start by looking at the model of a cascaded system with noise performance expressed in terms of equivalent noise temperature:

We know that for the first stage of the system the Noise Temperature of the output Tout is given by:

$T_{Out1} = T_o + T_{e1}$

Recall that this definition allows us to calculate the Noise Density at the output of the first stage as below:

$N_o = k.T_{Out1}.G_{1}$

where k is Boltzmann’s constant.

Looking at the expression for Tout of the entire system we can also see that:

$T_{out} = \frac{(T_{o} +T_{e1}).(G_1.G_2.G_3... .G_k) + T_{e2}(G_2.G_3... .G_k) + (T_{e3}.(G_3...G_k) + ...T_{ek}.G_k}{G_1.G_2.G_3...G_k}$

$T_{out} = T_{o} +T_{e1} + \frac{T_{e2}}{G_1} + \frac{T_{e3}}{G_1.G_2} + ... \frac{T_{ek}}{G_1.G_2.G_3...G_{k-1}}$

The total Equivalent noise temperature for the cascaded system is therefore:

$T_{e} = T_{e1} + \frac{T_{e2}}{G_1} + \frac{T_{e3}}{G_1.G_2} + ... \frac{T_{ek}}{G_1.G_2.G_3...G_{k-1}}$

Thats all for now!