Noise Figure of Receiver

The noise figure of a receiver is determined by the noise in the chain of circuit blocks in the signal path. Typical blocks in the chain are as follows:

• RF filter
• Low noise amplifier (LNA)
• Mixer
• Intermediate Frequency Filter (IF)
• Analog-to-Digital Converter (quantization noise)
• Coding Gain (This reduces the noise.)

The noise figure of a circuit block is defined as follows:

\[NF = \frac {SNR_{in}} {SNR_{out}}\]

Here is the same formula in decibel units:

\[NF (dB) = SNR_{in} (dB) - SNR_{out} (dB)\]

The noise figure of a passive block such as a filter or RF switch is the reciprocal of the loss which the manufacturer might specify in decibels. A loss in decibels is equal to the negative gain of the same block; if Loss = 3 dB then Gain = + 3 dB. We convert the loss from decibels to an ordinary multiplier as follows:

\[A_{filter} = 10 ^ {\frac {-Loss}{10}}\]

\[NF_{filter} = 1 / A_{filter}\]

\[NF_{dB} = 10 * log_{10}(NF_{filter})\]

We calculate the noise figure of a filter with a loss of 3 dB:

## [1] 0.501

What is the cummulative noise figure (NF) for the chain from Port A to Port G?

To determine the noise figure for a chain of three or more blocks, the procedure for two blocks is repeated in an iterative manner beginning with the final two blocks in the chain and adding blocks one at a time in reverse order. We generally state the noise figure as an ordinary multiplier but we show the result in decibels at the end also.

Block 5 is a passive filter. Block 6 is an amplifier after the IF filters.

\[NF_{5-6} = NF_5 + \frac {NF_6 - 1} {Ap_5}\]

nf_56 <- get_combined_nf(nf5, nf6, ap5)
print(as.numeric(formatC(nf_56, format = "f", digits = 3)))

## [1] 31.623

This is the noise figure expressed as a multiplier factor. This could be expressed alternatively in decibels as we do later at the final combination.

This is a reproducible slide show made with R-tools. Some calculations are hidden to keep the presentation understandable, but everything is available in an on-line Github repository.

nf_456 <- get_combined_nf(nf4, nf_56, ap4)
print(as.numeric(formatC(nf_456, format = "f", digits = 3)))

## [1] 25.533

ap3_db <- -6 ## dB
ap3 <- gain_absolute_from_decibels(ap3_db)
nf3 <- 1/ap3 ## filter formula

nf_3456 <- get_combined_nf(nf3, nf_456, ap3)
print(as.numeric(formatC(nf_3456, format = "f", digits = 3)))

## [1] 101.648

ap2 <- av2
nf2_db <- 2 ## This is given (the manufacturer or the equivalent)
nf2 <- nf_absolute_from_decibels(nf2_db)
nf_23456 <- get_combined_nf(nf2, nf_3456, ap2)
print(as.numeric(formatC(nf_23456, format = "f", digits = 3)))

## [1] 4.768

## Convert Noise Figure to decibels
nf23456_db <- 10 * log10(nf_23456)
print(as.numeric(formatC(nf23456_db, format = "f", digits = 3)))

## [1] 6.783

We repeat the procedure when we added the filter in Block-3 but the effect of the input filter is the same as just adding its loss in decibels to the Noise Figure. The combined result up to the analog-to-digital converter is:

## Noise factor multiplier
print(as.numeric(formatC(nf_123456, format = "f", digits = 3)))

## [1] 7.556

## Noise factor in decibels
print(as.numeric(formatC(nf_db, format = "f", digits = 3)))

## [1] 8.783

The effect of the RF filter in the first block in the chain degrades the noise factor by the same factor as the filter loss.

The quantization process of assigning available discrete levels to each signal sample adds noise for two reasons. The first is the distance between the analog signal level and the closest discrete level. The second reason corresponds to linearity errors which occur when measurements are not assigned to the closed level. Most linearity errors are caused by voltage reference inaccuracy. For example, if the voltage reference had a tolerance of 1% and there were 256 discrete levels, the reference is clearly not accurate enough to reliably assign to the closest level. The easiest way to calculate the noise factor of a converter block is if the manufacturer specifies the signal-to-noise ratio (SNR) at the converter output.

\[NF (dB) = Power_{in} (dB) - SNR_{ADC} - Noise_{in} (dB)\]

Contributions to the overall noise figure decreases as the signal moves along the chain except for the digital-to-analog converter at the end.

The example problem is similar to the example on page 202 of the Razavi textbook except for rounding effects. All results are expressed after rounding to a few decimal places.

Derivations of the formulas listed in this slide show are available. A future slide show will examine how the modify the chain in order to effectively incorporate the analog-to-digital converter.

August 2, 2017