SME
Communication Systems
Noise
Use this for “Find the noise floor of a 1 MHz receiver with a 6 dB noise figure, and the SNR of a −90 dBm signal.”
Chart
Noise floor: −114.0 + 6.0 = −108.0 dBm
Signal power: −90.0 dBm
Signal-to-noise ratio: 18.0 dB
This is an example. Type your own values to start a new problem.
N (dBm)
=
10
log₁₀
k
×
T (K)
×
B (MHz)
1
mW
Pₙ (dBm)
=
N (dBm)
+
NF (dB)
SNR (dB)
=
Pₛ (dBm)
−
(
Pₙ (dBm)
)
Formulas
N
=
10
log₁₀(1.381
×
10
-23
×
T
×
B
10
-3
)
−113.9741
=
10
log₁₀(1.381
×
10
-23
×
290
×
1,000,000
10
-3
)
Pₙ
=
N
+
NF
−107.9741 = −113.9741 + 6
SNR = Pₛ − Pₙ
+17.9741 = −90 − (−107.9741)
Formulas are worked in Hz. The step-by-step shows the conversions.
Assumptions
•
Thermal noise kTB at the input, with k = 1.381 × 10⁻²³ J/K; at T₀ = 290 K it is −174 dBm in each hertz.
•
NF is the receiver’s noise figure: the noise it adds, counted at its input, in dB.
•
Powers are in dBm, decibels above 1 mW; the SNR in dB is how far the signal stands above the noise floor.
Step-by-step
WE KNOW
T = 290 K, B = 1 MHz, NF = 6 dB, Pₛ = −90 dBm
FIND
thermal noise power (N), noise floor (Pₙ), signal-to-noise ratio (SNR)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to Hz).
B = 1 MHz = 1,000,000 Hz (1 MHz = 10⁶ Hz)
Step 2 · Find thermal noise power (N)
Use N = 10 log₁₀(1.381 × 10⁻²³ × T × B ÷ 10⁻³)
kTB is the noise power in watts; over 1 mW and 10 times its common log gives dBm.
N
=
10
log₁₀(1.381
×
10
-23
×
290
×
1,000,000
10
-3
)
N
=
−113.9741
dBm
Step 3 · Find noise floor (Pₙ)
Use Pₙ = N + NF
The receiver adds its own noise: in decibels its noise figure adds to the thermal noise.
Pₙ = −113.9741 + 6
Pₙ = −107.9741 dBm
Step 4 · Find signal-to-noise ratio (SNR)
Use SNR = Pₛ − Pₙ
A ratio of powers in decibels is the difference of their levels.
SNR = −90 − (−107.9741)
SNR = +17.9741 dB
Check
Put the numbers back in. Both sides should match.
−113.9741
=
10
log₁₀(1.381
×
10
-23
×
290
×
1,000,000
10
-3
)
✓
−107.9741 = −113.9741 + 6 ✓
+17.9741 = −90 − (−107.9741) ✓
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