Ferkans — Interactive Telecom Tutor

From AWGN to Fading

The AWGN capacity formula assumes a deterministic channel gain. In wireless, the channel gain $|h|^2$ fluctuates randomly due to fading. The instantaneous SNR $\gamma = |h|^2 \bar{\gamma}$ becomes a random variable, and we must redefine "capacity" carefully. Two fundamentally different notions arise depending on the delay constraint: ergodic capacity (long codewords spanning many fading realisations) and outage capacity (short codewords experiencing a single fading state).

Definition:
Ergodic Capacity

The ergodic capacity of a fading channel is the maximum long-term average rate achievable when codewords span many independent fading realisations (ergodic regime):

$C_{\text{erg}} = \max_{P(\gamma)} E\!\left[\log_2\!\left(1 + \gamma \frac{P(\gamma)}{\bar{P}}\right)\right]$

where $\gamma$ is the instantaneous SNR, $P(\gamma)$ is the power allocation policy, and the expectation is over the fading distribution. Ergodic capacity requires infinite delay tolerance — the codeword must be long enough to experience the full fading statistics.

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Theorem: Ergodic Capacity with Receiver CSI Only

When only the receiver knows the channel state (CSIR), the transmitter uses constant power $\bar{P}$ , and the ergodic capacity is

$C_{\text{CSIR}} = E\!\left[\log_2(1 + \gamma)\right] = \int_0^\infty \log_2(1 + \gamma)\, f_\gamma(\gamma)\, d\gamma$

where $f_\gamma(\gamma)$ is the PDF of the instantaneous SNR.

For Rayleigh fading ( $\gamma \sim \text{Exp}(\bar{\gamma})$ ):

$C_{\text{CSIR}} = \int_0^\infty \log_2(1+\gamma) \frac{1}{\bar{\gamma}} e^{-\gamma/\bar{\gamma}} d\gamma = \frac{e^{1/\bar{\gamma}}}{\ln 2} E_1\!\left(\frac{1}{\bar{\gamma}}\right)$

where $E_1(x) = \int_x^\infty e^{-t}/t\, dt$ is the exponential integral.

With only CSIR, the transmitter cannot adapt its power or rate to the channel state, so it transmits at constant power. The capacity is the average of the instantaneous AWGN capacities weighted by the fading distribution. By Jensen's inequality, $C_{\text{CSIR}} = E[\log_2(1+\gamma)] < \log_2(1+\bar{\gamma})$ : fading always reduces ergodic capacity compared to an unfaded AWGN channel with the same average SNR.

Proof

Achievability

Use a codebook designed for AWGN at rate $R$ . The receiver, knowing $h$ , can compute the effective SNR $\gamma$ for each codeword symbol. Over many fading realisations, the average mutual information is $E[\log_2(1+\gamma)]$ . By the coding theorem for ergodic channels, rates up to this value are achievable.

Converse

Without CSIT, the transmitter cannot adapt, so the input distribution is independent of $\gamma$ . Gaussian input maximises the per-realisation mutual information, and the ergodic average $E[\log_2(1+\gamma)]$ is an upper bound. $\blacksquare$

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Theorem: Ergodic Capacity with Full CSI (Water-Filling in Time)

When both transmitter and receiver know the channel state, the transmitter adapts its power $P(\gamma)$ to the instantaneous SNR. The optimal policy is water-filling in time:

$\frac{P(\gamma)}{\bar{P}} = \left(\mu - \frac{1}{\gamma}\right)^+$

where $(x)^+ = \max(0, x)$ and $\mu$ is chosen so that

$E\!\left[\left(\mu - \frac{1}{\gamma}\right)^+\right] = 1$

(average power constraint). The resulting capacity is

$C_{\text{CSIT}} = E\!\left[\log_2\!\left(\mu\gamma\right)^+\right] = \int_{\gamma \geq 1/\mu} \log_2(\mu\gamma)\, f_\gamma(\gamma)\, d\gamma$

The transmitter allocates more power to stronger channels and no power to deeply faded channels (below threshold $1/\mu$ ).

Water-filling is like pouring water into a vessel with an uneven bottom: the water level $\mu$ is constant, more water fills the shallow parts (good channels) and deep parts (poor channels) get no water. Paradoxically, the optimal strategy sends more power when the channel is already good, not when it is bad.

Proof

Lagrangian optimisation

Maximise $E[\log_2(1 + \gamma P(\gamma)/\bar{P})]$ subject to $E[P(\gamma)/\bar{P}] \leq 1$ and $P(\gamma) \geq 0$ .

The Lagrangian is

$\mathcal{L} = E\!\left[\log_2\!\left(1 + \gamma\frac{P(\gamma)}{\bar{P}}\right) - \lambda \frac{P(\gamma)}{\bar{P}}\right]$

Taking the derivative with respect to $P(\gamma)/\bar{P}$ and setting to zero:

$\frac{\gamma}{(1 + \gamma P(\gamma)/\bar{P})\ln 2} = \lambda$

With the substitution $\mu = 1/(\lambda \ln 2)$ and the non-negativity constraint:

$\frac{P(\gamma)}{\bar{P}} = \left(\mu - \frac{1}{\gamma}\right)^+$

$\blacksquare$

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Definition:
Outage Probability and $\varepsilon$ -Outage Capacity

For slow fading (quasi-static) channels where the codeword experiences a single fading realisation, Shannon capacity in the strict sense is zero (any fixed rate $R > 0$ will be unsupported by some fading states). Instead, we define:

Outage probability at rate $R$ :

$P_{\text{out}}(R) = P\!\left(\log_2(1 + \gamma) < R\right) = P\!\left(\gamma < 2^R - 1\right)$

$\varepsilon$ -outage capacity: the maximum rate $C_\varepsilon$ such that $P_{\text{out}}(C_\varepsilon) \leq \varepsilon$ :

$C_\varepsilon = \log_2(1 + F_\gamma^{-1}(\varepsilon))$

where $F_\gamma^{-1}$ is the inverse CDF of $\gamma$ .

For Rayleigh fading: $F_\gamma(\gamma) = 1 - e^{-\gamma/\bar{\gamma}}$ , so $C_\varepsilon = \log_2(1 - \bar{\gamma}\ln(1-\varepsilon))$ .

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Ergodic vs Outage Capacity

Compare ergodic capacity (achievable with long codewords) and $\varepsilon$ -outage capacity (relevant for delay-limited systems) as functions of average SNR. Select the fading distribution and observe how the gap between ergodic and outage capacity widens with more severe fading.

Parameters

Fading distribution

Average SNR (dB)15

Ricean K-factor (dB)6

Water-Filling Power Allocation in Time

Visualise the water-filling power allocation across time slots with varying channel gains. The water level $\mu$ is shown as a horizontal line. Channels below the cutoff $1/\mu$ receive no power (the transmitter stays silent during deep fades).

Parameters

Average SNR (dB)15

Number of time slots20

Outage Probability vs Transmission Rate

Animate how the outage probability curve shifts as the average SNR changes. For a fixed target outage probability $\varepsilon$ , the $\varepsilon$ -outage capacity increases with SNR. Observe the steep waterfall behaviour characteristic of fading channels.

Parameters

Fading distribution

Average SNR (dB)15

Example: Outage Capacity of a Rayleigh Channel

A Rayleigh fading channel has average SNR $\bar{\gamma} = 20$ dB. Compute the 1%-outage capacity and compare with the ergodic capacity and the AWGN capacity at the same average SNR.

Solution

1%-outage capacity

For Rayleigh fading, $F_\gamma(\gamma) = 1 - e^{-\gamma/\bar{\gamma}}$ .

$\gamma_{0.01} = F_\gamma^{-1}(0.01) = -\bar{\gamma}\ln(1-0.01) = -100 \times \ln(0.99)$

$= 100 \times 0.01005 = 1.005$ (linear).

$C_{0.01} = \log_2(1 + 1.005) = \log_2(2.005) = 1.00$ bits/s/Hz.

Ergodic capacity

$C_{\text{erg}} = \frac{e^{1/100}}{\ln 2} E_1(1/100) \approx 6.0$ bits/s/Hz.

(Numerical evaluation of the exponential integral.)

AWGN capacity

$C_{\text{AWGN}} = \log_2(1+100) = 6.66$ bits/s/Hz.

Summary: $C_{0.01} = 1.00 \ll C_{\text{erg}} = 6.0 < C_{\text{AWGN}} = 6.66$ .

The 1%-outage capacity is dramatically lower than the ergodic capacity because the channel must support the rate even during deep fades (bottom 1% of the SNR distribution). $\blacksquare$

Quick Check

Which capacity notion is appropriate for a voice call with a strict 20 ms latency requirement over a channel with coherence time 100 ms?

Ergodic capacity

Outage capacity

AWGN capacity

Zero, since Shannon capacity of a slow fading channel is zero

Correction:

Outage capacity

Correct. The codeword (20 ms) is shorter than the coherence time (100 ms), so it experiences essentially one fading realisation. Outage capacity with an appropriate $\varepsilon$ (e.g., 1%) is the right metric.

Common Mistake: Ergodic Capacity Assumes Infinite Delay Tolerance

Mistake:

Quoting ergodic capacity as the achievable rate for a delay-sensitive application (e.g., VoIP, video conferencing) over a slowly fading channel.

Correction:

Ergodic capacity is achievable only when the codeword spans many independent fading realisations — requiring the coding delay to be much larger than the coherence time. For delay- sensitive applications with codewords shorter than the coherence time, outage capacity is the appropriate metric. The ergodic capacity can dramatically overestimate what is achievable in practice under strict delay constraints.

Why This Matters: Adaptive Modulation as Finite-Rate Water-Filling

Optimal water-filling requires continuously adapting the power and rate to the instantaneous channel state. In practice, systems like LTE and 5G NR implement a discrete approximation through adaptive modulation and coding (AMC):

The receiver estimates the channel quality and feeds back a Channel Quality Indicator (CQI)
The transmitter selects a Modulation and Coding Scheme (MCS) from a finite table (e.g., 29 entries in 5G NR)
Higher MCS indices correspond to higher spectral efficiency (more power-demanding modulations and higher code rates)

This discrete adaptation captures most of the water-filling gain. At high SNR, the loss from using a finite MCS table instead of continuous adaptation is typically less than 0.5 dB.

See full treatment in Adaptive Modulation and Coding in OFDM

⚠️Engineering Note

CSI Feedback Overhead in FDD Systems

Water-filling requires the transmitter to know the instantaneous channel state. In TDD systems, uplink-downlink channel reciprocity provides CSIT directly. In FDD systems, the receiver must quantise and feed back the CSI, creating overhead:

LTE: 4-bit wideband CQI + 2-bit sub-band differential CQI, reported every 5-80 ms. Total overhead: ~1-5% of uplink capacity.
5G NR: Type I/II codebook-based PMI feedback. Type II uses up to 22 bits per sub-band for 32-antenna systems.
Massive MIMO (FDD): Full channel feedback scales as $O(N_t \cdot K)$ bits per coherence interval, which becomes prohibitive for $N_t > 64$ . This is a key driver for TDD-based massive MIMO deployments.

When feedback is delayed by $\Delta t$ , the effective CSIT quality degrades as $\rho(\Delta t) = J_0(2\pi f_D \Delta t)^2$ where $f_D$ is the maximum Doppler frequency. At vehicular speeds ( $f_D = 500$ Hz) with 5 ms delay, $\rho \approx 0.45$ — severely degraded. This limits the water-filling gain and motivates robust power allocation strategies.

Practical Constraints

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FDD feedback overhead: 1-5% of uplink capacity (LTE/5G NR)
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Massive MIMO FDD: feedback $O(N_t)$ , impractical for $N_t > 64$
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Feedback delay > 2 ms at 60 km/h causes ~50% CSIT quality loss

📋 Ref: 3GPP TS 38.214 §5.2.2 (CSI reporting)

Ergodic Capacity

The maximum average rate achievable over a fading channel when codewords span many independent fading realisations: $C_{\text{erg}} = E[\log_2(1+\gamma)]$ with CSIR only.

Outage Capacity

The maximum rate $C_\varepsilon$ such that the probability of the channel not supporting this rate is at most $\varepsilon$ : $P(\log_2(1+\gamma) < C_\varepsilon) \leq \varepsilon$ . Appropriate for delay-limited systems over slow fading.

Water-Filling

The optimal power allocation strategy that maximises capacity under an average power constraint. More power is allocated to channel states with higher gain, and no power to states below a cutoff threshold. The power plus inverse channel gain equals a constant "water level" $\mu$ .

Capacity of Flat-Fading Channels

From AWGN to Fading

Definition: Ergodic Capacity

Theorem: Ergodic Capacity with Receiver CSI Only

Achievability

Converse

Theorem: Ergodic Capacity with Full CSI (Water-Filling in Time)

Lagrangian optimisation

Definition: Outage Probability and ε\varepsilonε-Outage Capacity

Ergodic vs Outage Capacity

Parameters

Water-Filling Power Allocation in Time

Parameters

Outage Probability vs Transmission Rate

Parameters

Example: Outage Capacity of a Rayleigh Channel

1%-outage capacity

Ergodic capacity

AWGN capacity

Quick Check

Common Mistake: Ergodic Capacity Assumes Infinite Delay Tolerance

Why This Matters: Adaptive Modulation as Finite-Rate Water-Filling

CSI Feedback Overhead in FDD Systems

Ergodic Capacity

Outage Capacity

Water-Filling

Definition:
Ergodic Capacity

Definition:
Outage Probability and $\varepsilon$ -Outage Capacity