How do you derive Friis' Formula?

Answer 1

Answer

I believe it can be expressed in terms of noise temperature instead of noise ratio. The noise temperature is directly proportional to noise power and inverseley proportional to the product of the bandwidth in question and Boltzmann's constant (which is a different derivation altogether).

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I made an alternative derivation to the 'classical' one using noise temperature, where I avoid to use the noise temperature concept and only use noise power and noise factor:

Take a system which consists of two sub-blocks with power gain G1 resp. G2 (in case of a voltage amplifier for example G will be the square of the voltage gain A). In addition to the power amplification, each block introduces its own noise PN1 resp. PN2, in each case referred to the input of the block.

To the input of the first block is assigned a signal with noise power PNi1. The noise power at the output of the first block is PNo1 and at the output of the second block PNo2.

We can calculate the noise power at the output of the first block as:

PNo1 = G1 (PNi1 + PN1)

(i.e. the block amplifies the input noise and adds some noise of its own)

Similarly for the second block:

PNo2 = G2 (PNo1 + PN2) = G2 G1 Pni1 + G1 G2 PN1 + G2 PN2

We can also consider the two blocks together as a 'black box' with power gain G1*G2. We can then calculate the output noise power as follows:

PNo2 = G1 G2 (PNi1 + PN12)

Where PN12 is the effective noise power (input-referred) added by this 'black box' consisting of the combination of the two blocks. Combining these 3 equations gives us immediately:

PN12 = PN1 + PN2/G1

This already shows clearly the principle of the Friis equation, namely that the noise of subsequent stages is reduced by the gain of the preceding stages.

Now we will express this formula in function of the noise factor F.

Noise factor F is defined as SNRi/SNRo = (PSi/PNi) / (PSo/PNo). This combined with the fact that the signal power PSo = G1 * PSi leads us to :

F = 1 + PN / PNi <==> PN = (F-1) PNi

This also clearly shows us that an ideal noiseless stage would have a noise factor F of 1 (this is of course not possible in reality, so F will always be greater than 1).

If we write this equation for the first block:

PN1 = (F1-1) PNi1

And for the second block keeping in mind that the noise factor F2 is defined for this second block as if we were applying the input noise directly to it:

PN2 = (F2-1) PNi1

And last we write the equation for the 'black box' combination of both blocks:

PN12 = (F12-1) PNi1

We now set this last equation equal to the previous equation we derived for PN12 and fill in the other two:

(F12-1) PNi1 = PN1 + PN2/G1

F12-1 = (F1-1) + (F2-1)/G1

F12 = F1 + (F2-1)/G1

That gives us the Friis equation for a cascade of two blocks. This can easily be iteratively expanded to any number of stages to give:

Ftotal = F1 + (F2-1)/G1 + (F3-1)/G1G2 + (F4-1)/G1G2G3 + ...