SLYT839 july 2023 ADC32RF54

3 Calculating a system’s noise figure

You can use the Friis equation to calculate a receiver system’s noise figure. Assuming a simplified, ideal receiver with two amplifiers and one ADC, as shown in Figure 2, Equation 1 calculates the cascaded system noise factor as:

Equation 1.

F_{S y s t e m} = F_{1} + \frac{F_{2} - 1}{G_{1}} + \frac{F_{3} - 1}{G_{1} ∙ G_{2}} + \dots + \frac{F_{n} - 1}{G_{1} ∙ G_{2} \dots ∙ G_{n - 1}}

where F_x are the noise factors and G_x are the power gains.

The system noise figure in decibels is:

Equation 2.

{N F}_{S y s t e m} = 10 \log (F_{S y s t e m})

Figure 2 Typical receive signal chain.

There are two important things to highlight: the system noise figure is primarily dominated by the noise figure F₁ of the first element, as long as gain G₁ and G₂ are large enough to where the ADC noise figure F₃ is negligible.

Comparing two different ADCs with 20-dB vs. 25-dB noise figures in a system with two cascaded LNAs shows a drastic difference in system noise figures (see Table 1).

Table 1 System noise figure with two LNA stages.

	LNA1	LNA2	ADC1	ADC2
Noise figure	1 dB	3 dB	20 dB	25 dB
Gain	12 dB	15 dB	0 dB	0 dB
Resulting system noise figure			1.8 dB	2.9 dB

Getting the system listed in the ADC2 column (with a 5-dB worse noise figure) to a system noise figure below 2 dB would require an additional 10 dB of gain using a third LNA (noise figure = 3 dB), as shown in Table 2.

Table 2 highlights the impact of the ADC noise figure on the overall system noise figure. Adding a third LNA increases cost, board area (matching components, routing and power supply) and system power consumption, and further reduces the full-scale headroom.

Table 2 System noise figure using ADC2 with three LNA stages.

	LNA1	LNA2	LNA3	ADC2
Noise figure	1 dB	3 dB	3 dB	25 dB
Gain	12 dB	15 dB	10 dB	0 dB
Resulting system noise figure				1.4 dB

Assuming a target receiver sensitivity of –172 dBm, or very weak signals just 2 dB above the absolute noise floor (–174 dBm + 2 dB = –172 dBm), this receiver requires an noise figure better than 2 dB. Let’s use the above example with ADC1 (with a 20-dB noise figure, as listed in Table 1) and a cascaded system noise figure of 1.8 dB.

As shown in Figure 3 and Table 3, LNA1 with a gain of 12 dB raises both the input signal and noise by 12 dB while degrading the noise figure by 1 dB (noise figure_LNA1 = 1 dB). LNA2 raises both signal and noise by 15 dB. Even though LNA2 has a higher inherent noise Figure 3 dB, its impact is reduced to just 0.2 dB because of the 12-dB gain of LNA1.

Finally, the noise contribution of ADC1 (noise figure = 20 dB) reduces to just 0.6 dB, as it gets reduced by the 27-dB gain of both LNAs. Therefore, you end up with a system noise figure of 1.8 dB, which leaves approximately 0.2 dB of headroom to detect weak input signals.

Figure 3 Graphical illustration of the individual noise figure contributions in a receive signal chain.

Table 3 Calculations for individual noise figure contributions.

	LNA1	LNA2	ADC
Noise figure (dB)	1	3	20
Gain (dB)	12	15	0
Noise power (linear) 10^(noise figure/10)	1.26 10^1/10	2 10^3/10	100 10^100/10
Power gain (linear) 10^(gain/10)	15.85 10^12/10	31.62 10^15/10	1 10^0/10
Noise figure of LNA1 only (dB)	1	–	–
Noise figure of LNA1 + LNA2 only (dB)	1.2 10log[1.26+(2-1)/15.85]		–
Noise figure of LNA1 + LNA2 + ADC (dB)	1.8 10log[1.26 + (2-1)/15.85 + (100-1)/15.85/31.62]
Additional impact on system noise figure (dB)	1	0.2	0.6

High-speed data converters rarely list noise figure in the device-specific data sheet. The noise figure for an ADC can be calculated using Equation 3 using the common data-sheet parameters (see Table 4) for the ADC32RF54 RF-sampling ADC.

Table 4 Data sheet parameters of the ADC32RF54.

Parameter	Description	ADC32RF54 (1 times AVG)	ADC32RF54 (2 times AVG)
V	Input full-scale voltage peak to peak (V_pp)	1.1	1.35
R_IN	Input termination impedance (Ω)	100 Ω
FS	ADC sampling rate	2.6 GSPS
SNR	ADC SNR for small-input signals (dBFS), typically –20 dBFS	64.4	67.1

ADC Noise figure (dB) = P_SIG,dBm + 174 dBm – SNR (dBFS) – bandwidth (Hz)

Equation 3.

{N F}_{A D C} (d B) = 10 l o g (\frac{{(\frac{V}{2 \times \sqrt{2}})}^{2}}{R_{I N}} \times 1000) + 174 - S N R - 10 \log (\frac{F S}{2})

For the ADC32RF54, the noise figure calculates to:

Noise figure (1x AVG) = 20.3 dB

10log[(1.1/2/sqrt(2))²/100 x 1000] +174 – 64.4 – 10log[2.6e9/2]

Noise figure (2x AVG) = 19.3 dB

10log[(1.35/2/sqrt(2))²/100 x 1000] +174 – 67.1 – 10log[2.6e9/2]