SME
Communication Systems · Problem type
Shannon capacity
Use this for “A 3 kHz phone line has a 30 dB signal-to-noise ratio. Find its Shannon capacity.”
This is an example. Type your own values to start a new problem.
SNRdB (dB)
=
10
log₁₀
SNR
C (kb/s)
=
B (kHz)
×
log₂(1
+
SNR
)
Formulas
SNRdB = 10 log₁₀ SNR
30 = 10 log₁₀ 1,000
C
=
B
×
log₂(1
+
SNR)
29.9017 = 3 × log₂(1 + 1,000)
Assumptions
•
The noise is white and Gaussian across the band; SNR is signal power over noise power.
•
C is the most any code can send without errors; real links send less.
•
SNR in decibels is 10 log₁₀ SNR, so 30 dB is a power ratio of 1000.
Step-by-step
WE KNOW
B = 3 kHz, SNRdB = 30 dB
FIND
signal-to-noise ratio (SNR), channel capacity (C)
Step 1 · Find signal-to-noise ratio (SNR)
Use SNRdB = 10 log₁₀ SNR
Undo the decibels: divide by 10 and raise 10 to that power.
SNR
=
10
SNRdB ÷ 10
SNR
=
10
30 ÷ 10
SNR = 1,000
Step 2 · Find channel capacity (C)
Use C = B × log₂(1 + SNR)
Each hertz of band carries log₂(1 + SNR) bits per second at best.
C
=
3
×
log₂(1
+
1,000)
C
=
3
×
log₂(1,001)
C
=
3
×
9.967
C
=
29.9017
kb/s
Check
Put the numbers back in. Both sides should match.
30 = 10 log₁₀ 1,000 ✓
29.9017 = 3 × log₂(1 + 1,000) ✓
Topic check
See what you know: 5 questions on this topic
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