Noise, SNR and Decibel Calculations for Communication Engineers

Radio and communication engineers almost never multiply gains or divide powers directly. They work in decibels, where multiplication becomes addition and a range from picowatts to kilowatts fits into numbers between about -130 and +60. If you can move comfortably between watts, dB and dBm, and you know where the noise floor of a receiver sits, you can sanity-check almost any link on paper.

This tutorial covers the decibel rules, the thermal noise formula, signal-to-noise ratio, noise figure and the Friis formula for cascaded stages, each with a worked example.

Decibels: Power Ratios and Voltage Ratios

Power ratio:    G(dB) = 10 x log10(P2 / P1)
Voltage ratio:  G(dB) = 20 x log10(V2 / V1)   (same impedance)

Absolute power:
  dBm = 10 x log10(P / 1 mW)
  dBW = 10 x log10(P / 1 W)       dBm = dBW + 30

Back to linear:
  P(mW) = 10^(dBm / 10)

The factor 20 for voltage comes from power being proportional to voltage squared. Doubling voltage is +6 dB, doubling power is +3 dB. A dB value describes a ratio; dBm and dBW describe an absolute power. You may add dB to dBm (gain applied to a power), but adding two dBm values is meaningless.

Power ratio	dB	Absolute power	dBm
2	+3.01	1 mW	0
10	+10	10 mW	+10
100	+20	100 mW	+20
0.5	-3.01	1 W	+30
0.1	-10	1 microwatt	-30
0.001	-30	1 picowatt	-90

Worked Example: An Amplifier Chain

A 0.5 mW signal passes through a cable with 3 dB loss, then an amplifier with 20 dB gain. What is the output?

Input:  10 x log10(0.5) = -3.01 dBm
Output: -3.01 - 3 + 20 = +13.99 dBm
Linear: 10^(13.99/10) = 25.1 mW

Check in linear terms: 0.5 mW x 0.5 x 100 = 25 mW

Thermal Noise: The Floor You Cannot Remove

Any resistor above absolute zero generates random noise because of the thermal motion of electrons. The available noise power depends only on temperature and bandwidth, not on the resistor value.

N = k x T x B

k = 1.38 x 10^-23 J/K   (Boltzmann constant)
T = temperature in kelvin (290 K is the standard reference)
B = noise bandwidth in Hz

At 290 K:  kT = 4.0 x 10^-21 W/Hz = -174 dBm/Hz
So:        N(dBm) = -174 + 10 x log10(B)

Bandwidth	10 log10(B)	Noise floor at 290 K
1 Hz	0 dB	-174 dBm
125 kHz (LoRa channel)	51.0 dB	-123.0 dBm
1 MHz	60 dB	-114 dBm
20 MHz (Wi-Fi channel)	73.0 dB	-101.0 dBm

This is why narrowband systems such as LoRa can receive much weaker signals than Wi-Fi: a narrower channel lets in less noise.

Signal-to-Noise Ratio

SNR = P_signal / P_noise
SNR(dB) = 10 x log10(SNR) = P_signal(dBm) - P_noise(dBm)

In dB form, SNR is simply a subtraction. If a receiver sees -90 dBm of signal in a 1 MHz bandwidth and the noise is -114 dBm, the SNR is 24 dB, a power ratio of about 251. For voltages measured across the same impedance, use 20 log10 of the RMS voltage ratio.

Noise Factor, Noise Figure and Cascades

A real amplifier adds its own noise, so the SNR at its output is worse than at its input. The noise factor F measures how much worse. Noise figure NF is the same quantity in dB.

F  = SNR_in / SNR_out          (linear, F >= 1)
NF = 10 x log10(F)             (dB)

Receiver noise floor:
N_floor(dBm) = -174 + 10 x log10(B) + NF

Friis formula for cascaded stages (all linear):
F_total = F1 + (F2 - 1)/G1 + (F3 - 1)/(G1 x G2) + ...

Friis's formula shows that the noise of later stages is divided by the gain before them. A high-gain, low-noise first stage hides the noise of everything after it.

Worked Example: LNA Followed by a Mixer

Stage 1 is an LNA with NF 1 dB and gain 20 dB. Stage 2 is a mixer with NF 10 dB. Find the total noise figure.

F1 = 10^(1/10)  = 1.259
G1 = 10^(20/10) = 100
F2 = 10^(10/10) = 10

F_total = 1.259 + (10 - 1)/100 = 1.259 + 0.09 = 1.349
NF_total = 10 x log10(1.349) = 1.30 dB

Swap the order (mixer first, NF 10 dB, assume loss so G1 < 1):
the total can never be better than 10 dB.

Worked Example: Minimum Detectable Signal

A receiver has 200 kHz bandwidth, NF 6 dB, and needs 10 dB SNR to demodulate correctly. What is the weakest signal it can use?

Noise floor = -174 + 10 x log10(200000) + 6
            = -174 + 53.0 + 6
            = -115.0 dBm
Sensitivity = -115.0 + 10 = -105.0 dBm

Common Mistakes

Using 20 log for power ratios or 10 log for voltage ratios.
Adding dBm to dBm. Convert to milliwatts, add, then convert back. Two equal powers add to +3 dB, not double the dBm value.
Putting noise figure in dB into the Friis formula. Convert every NF and gain to linear first.
Using the receiver's channel spacing as noise bandwidth when its filter is actually narrower or wider.
Forgetting that -174 dBm/Hz assumes 290 K. A hot environment raises it slightly.

Summary

Decibels turn link calculations into additions: 10 log for power, 20 log for voltage, and dBm for absolute power referred to 1 mW. Thermal noise sets a floor of -174 dBm/Hz at room temperature, so the noise in any receiver is -174 + 10 log10(B) + NF. SNR in dB is just signal minus noise. The Friis formula tells you to put a good low-noise amplifier first. With these few relations you are ready to compute modulation power, antenna mismatch and full link budgets in the later articles.