noise floor of attenuator

Question

NF of attenuator is same as the attenuation value as the noise floor remains same but signal level gets attenuated? But NF of amplifier depends on the device as it increases both signal and noise?

Answer 1

The Noise Figure of a passive attenuator is the attenuation value of the attenuator. This is because noise figure is by definition the difference in the SNR of the output and the SNR of the input, with the input terminated with the characteristic impedance of the system (typically 50 ohms). Using this definition, the noise figure of an amplifier similarly is the additional increase in noise beyond the gain of the amplifier when the input of the amplifier is terminated.

The ever-present thermal noise floor is a white noise source with a power spectral density given by kT with k as Boltzmann’s Constant and T as temperature. At room temperature in power units of dBm (dB relative to 1 mW) this ends up being a power spectral density of -174 dBm/Hz. In the case of an attenuator, thermal noise is the only noise source given it is a passive device. If the attenuator is matched to 50 ohms, and with the input tied to a 50 ohm source resistor only and no other gain at the input, the thermal noise at the input will be -174 dBm/Hz (kTB) at room temperature.

The thermal noise at the output of the 50 ohm attenuator will also still be -174 dBm/Hz! The attenuator cannot somehow attenuate the thermal noise that is present in all parts of the system-- it certainly attenuates any noise that is applied to the input, but in terms of power density, the same level of thermal noise is present at the output as a local independent noise source.

Thus in terms of (SNR out) - (SNR in): Consider a noise-free signal at the input with it's noise floor then limited by thermal noise alone; this sets the input SNR. The signal will certainly be attenuated by the attenuation value, but as we explained above the noise can't get pushed down lower than its theoretical limit (the thermal noise floor). Since the noise is not affected as it is the same value at the output and input in this case, the SNR is therefore changed by the attenuator value, which by definition is the Noise Figure.

Cascaded Noise Figure is then considering what occurs when you place an amplifier in front of an attenuator. Now in this case amplified noise is applied to the input, not the thermal noise floor specifically. This amplified noise like any other waveform can of course be attenuated, down to the limit of the thermal noise floor. Consider these two cases and how the overall Noise Figure compares:

The formula for cascaded Noise Figure is as follows below for two components, expandable for any number of components. I'll give the equation first but then demonstrate with two cases where the explanation above becomes very clear.

$$NF_{cascade} = NF_{N-1} + \frac{NF_N-1}{G_{N-1}}$$

This states that the overall noise figure of N cascaded devices is the noise figure of the first component plus the noise figure of the subsequent components reduced by the cumulative gain of to each components input. The gain (G) and noise figure (NF) are given in power ratio (non-dB) quantities.

Consider this with these two components:

Amplifier with 20 dB gain, and 3 dB Noise Figure 10 dB Attenuator

Case 1: Amplifier followed by Attenuator

According to the formula:

$NF_1= 10^{3/10} = 2$

$NF_2 = 10^{10/10} = 10$

$G_1 = 10^{20/10} = 100$

$NF_{cascade} = 2 + \frac{10-1}{100} = 2.09$

The Noise Figure in dB is $10 \log_{10}(2.09) = 3.2 \text{ dB} $

Notice the attenuator had very little effect due to the amplifier in front of it! This is the motivation for Low Noise Amplifiers (LNA) at the front-end of the receiver with very low loss components in front of it (typically a necessary filter).

Case 2: Attenuator followed by Amplifier

$NF_1= 10^{10/10} = 10$

$NF_2 = 10^{3/10} = 2$

$G_1 = 10^{-10/10} = 0.1$

$NF_{cascade} = 10 + \frac{2-1}{0.1} = 20$

The Noise Figure in dB is $10 \log_{10}(20) = 13 \text{ dB} $

Intuition:

Consider a received signal that has a 30 dB SNR, with the noise limited by thermal noise alone. In case 1, we amplify both the signal and the noise by the amplifier with 20 dB gain. This amplifier has a 3 dB noise figure which means the signal goes up by 20 dB, but the noise goes up by 23 dB! (Due to the added electronics noise that is captured in the Noise Figure metric). So now the SNR at the output of the amplifier is 17 dB, but the thermal noise is amplified and now much higher (23 dB higher!) than the actual thermal noise floor present at the amplifiers output (we have smothered it with an amplified version of the noise). The attenuator can now push both amplified waveforms (signal and noise components down by it's attenuation or 10 dB. Thus the signal that we increased by 20 dB is now reduced by 10 dB, and the noise that we increased by 23 dB is reduced by 10 dB. However the underlying thermal noise is now closer to the amplified noise level, 13 dB away, contributing a small amount to total noise (explaining the additional 0.2 dB in the 3.2 dB Noise Figure result).

Notice the power summation of the amplified thermal noise with the thermal noise 13 dB lower is a total noise increase of 0.2 dB (it all adds up, demystifying cascaded noise figure!):

$$10 \log_{10}(1+10^{-13/10}) = 0.2 \text{ dB}$$