ELEC5508 Chap.5 Channel Capacity and Water-Filling
Channel Capacity and Water-Filling
This chapter reads Shannon capacity through bandwidth-limited, power-limited and fading regimes. It distinguishes ergodic averaging from slow-fading outage and makes channel knowledge part of every adaptation claim. Water-filling then allocates a fixed power budget across OFDM or MIMO modes by solving an active set and disabling modes whose effective-noise floor exceeds the common level.
The chapter treats capacity as a boundary under explicit assumptions rather than a promised application rate. Bandwidth, noise spectral density, power constraint and CSI location must stay attached to the formula. In fading, a long code can average many states while a short slow-fading transmission can enter outage. Those are different operational questions.
Water-filling adds another boundary: instantaneous adaptation is impossible when the transmitter does not know current mode quality.
What this chapter covers
- 01
AWGN capacity and logarithmic power gain
- 02
Bandwidth-limited and power-limited regimes
- 03
Ergodic capacity and slow-fading outage
- 04
Receiver and transmitter channel knowledge
- 05
Water-filling across parallel modes
- 06
Channel inversion and fixed-rate trade-offs
Worked example · free
Solve a water-filling active set
- +1Assuming all active gives water level 16/3 = 5.333.
- +1The third tentative power is negative, so remove that mode.
- +1Recompute the two-mode level as (6+1+2)/2 = 4.5.
- +1Allocate [3.5, 2.5, 0].
- +1All powers are non-negative and sum to 6.
Key terms
- Channel capacity
- The maximum reliable information rate under a stated channel model and resource constraint.
- Outage
- The event that instantaneous channel capacity falls below the selected transmission rate.
- Ergodic capacity
- A long-term capacity averaged across channel realisations when coding spans their variation.
- Water level
- The common threshold that determines positive power above each effective-noise floor.
- Channel inversion
- Power adaptation that compensates weak channels to target a more constant received quality.
Channel Capacity and Water-Filling FAQ
Why does doubling power not double capacity?
The AWGN capacity expression is logarithmic in SNR when bandwidth is fixed. Additional power therefore has diminishing rate return at high SNR. At low SNR or in the power-limited wideband regime, the relationship is closer to linear.
When is water-filling useful?
It needs parallel modes and transmitter knowledge of their current qualities. The gain is strongest when power is scarce or mode strengths differ substantially. At high SNR, active allocations approach equality and the advantage over uniform power shrinks.
What is the difference between ergodic and outage capacity?
Ergodic capacity assumes coding can average across many channel states. In slow fading, one transmission may see one nearly constant random state, so a fixed rate has a non-zero probability of exceeding instantaneous capacity. That probability is outage.
How can I verify a water-filling solution?
Draw the effective-noise floors and begin with an assumed active set. Solve the common water level from the total-power constraint, calculate each tentative allocation, and remove any channel whose result is negative. Recompute after every removal rather than clipping a negative value and keeping the old level.
A valid final allocation has no negative powers, spends exactly the available total power and gives the same water level across all active channels. Substitute the powers into the stated capacity expression and compare with equal power under the same bandwidth, noise and channel-state assumptions.
Then state the transmitter-information requirement, because instantaneous allocation is not justified by receiver channel knowledge alone.
Exam move
Annotate every capacity formula with bandwidth, noise model and CSI assumption. Practise active-set water-filling until negative tentative powers trigger an automatic removal and recomputation. Begin with the AWGN expression and explain its two limiting regimes in words. For fixed bandwidth, increasing power gives a logarithmic return; in a wideband low-SNR limit, capacity becomes power constrained.
Work several decibel conversions so a supplied SNR never enters a linear formula in the wrong form. Next, separate fast or averaged fading from slow-fading outage. For an outage question, write the target rate, instantaneous capacity event and accepted probability. Add diversity as a lower-tail mechanism rather than calling it an extra spatial stream.
For water-filling, draw effective-noise floors and use a strict active-set routine. Assume a set, solve the common level, remove any negative allocation and recompute. Check non-negativity and total power at the end. Compare the result with equal power at low and high SNR and explain why the gain shrinks when every mode lies well below the level.
Then state the CSI boundary: receiver knowledge supports detection, while instantaneous transmitter knowledge is needed for current-state allocation. Finish by contrasting water-filling with inversion. One favours good states to maximise average rate; the other spends power on weak states to support a fixed target and may fail in deep fades. A complete answer names the relevant objective before selecting either rule.
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