Ferkans — Interactive Telecom Tutor

Exploiting Channel Knowledge at the Transmitter

In the previous section, we showed that CSIR alone achieves the ergodic capacity with constant-power transmission. But what if the transmitter also knows the channel state? Intuitively, the transmitter should invest more power when the channel is good (high $|H|^2$ ) and save power when the channel is bad (low $|H|^2$ ). This is precisely the water-filling strategy we met for parallel Gaussian channels in Chapter 12 — except now we are "filling" over random fading states rather than deterministic sub-channels.

The gain from CSIT over CSIR-only turns out to be small for ergodic capacity (especially at high SNR), but the conceptual framework is important because the same water-filling structure appears in MIMO channels (Section 13.5) and multiuser systems (Book MIMO).

Definition:
Power Adaptation Policy

A power adaptation policy is a function $P(H)$ that assigns a transmit power level to each fading state $H$ , subject to the average power constraint

$\mathbb{E}_H[P(H)] \leq P.$

With CSIT, the transmitter uses power $P(H_i)$ at time $i$ when the fading state is $H_i$ . The input signal is $X_i \sim \mathcal{N}(0, P(H_i))$ , so the instantaneous rate for state $H = h$ is

$r(h) = \frac{1}{2}\log\!\left(1 + \frac{|h|^2 P(h)}{N}\right).$

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

Consider the scalar fading channel $Y = HX + Z$ with CSIR and CSIT, average power constraint $\mathbb{E}[P(H)] \leq P$ , and ergodic fading. The capacity is

$C_{\text{CSIT}} = \mathbb{E}_H\!\left[\frac{1}{2}\log\!\left(1 + \frac{|H|^2 P^*(H)}{N}\right)\right],$

where the optimal power allocation is the water-filling solution

$P^*(h) = \left(\nu - \frac{N}{|h|^2}\right)^{\!+},$

and the water level $\nu$ is chosen so that $\mathbb{E}_H[P^*(H)] = P$ .

The fading channel with full CSI is equivalent to a continuum of parallel AWGN channels indexed by the fading state $h$ , each with noise power $N$ and gain $|h|^2$ . Water-filling allocates more power to states with higher $|h|^2/N$ (better channels). States with $|h|^2 < N/\nu$ receive zero power — the transmitter stays silent during deep fades.

Show Hint

Write the capacity as $\max_{P(h) \geq 0:\, \mathbb{E}[P(H)] \leq P} \mathbb{E}[\frac{1}{2}\log(1 + |H|^2 P(H)/N)]$ .

Form the Lagrangian with multiplier $\lambda$ for the power constraint and optimize pointwise over $P(h)$ .

The KKT conditions yield the water-filling structure.

Proof

Formulate the optimization

The capacity is the solution to

$C_{\text{CSIT}} = \max_{\substack{P(h) \geq 0 \\ \mathbb{E}[P(H)] \leq P}} \mathbb{E}_H\!\left[\frac{1}{2}\log\!\left(1 + \frac{|H|^2 P(H)}{N}\right)\right].$

This is a functional optimization over the power allocation $P(\cdot)$ . The objective is concave in $P(h)$ for each $h$ (composition of concave $\log$ with affine function of $P$ ), and the constraint is linear, so the problem is convex.

Lagrangian and KKT conditions

Form the Lagrangian with multiplier $\lambda \geq 0$ :

$\mathcal{L} = \mathbb{E}\!\left[\frac{1}{2}\log\!\left(1 + \frac{|H|^2 P(H)}{N}\right)\right] - \lambda\left(\mathbb{E}[P(H)] - P\right).$

For each realization $h$ , the optimality condition is

$\frac{\partial}{\partial P(h)}\left[\frac{1}{2}\log\!\left(1 + \frac{|h|^2 P(h)}{N}\right) - \lambda P(h)\right] = 0,$

which gives

$\frac{|h|^2/(2N\ln 2)}{1 + |h|^2 P(h)/N} = \lambda.$

Solve for $P^*(h)$

Rearranging:

$P^*(h) = \frac{1}{2\lambda \ln 2} - \frac{N}{|h|^2}.$

Setting $\nu = \frac{1}{2\lambda \ln 2}$ and enforcing the non-negativity constraint $P(h) \geq 0$ , we obtain the water-filling solution:

$P^*(h) = \left(\nu - \frac{N}{|h|^2}\right)^{\!+}.$

The water level $\nu$ is determined by the complementary slackness condition $\mathbb{E}[P^*(H)] = P$ :

$\mathbb{E}\!\left[\left(\nu - \frac{N}{|H|^2}\right)^{\!+}\right] = P. \qquad \blacksquare$

Example: Water-Filling over Rayleigh Fading States

For a Rayleigh fading channel ( $|H|^2 \sim \text{Exp}(1)$ ) with $\text{SNR} = P/N = 10$ dB, compute the water level $\nu$ numerically and determine the fraction of time the transmitter is silent.

Solution

Power constraint equation

Let $\gamma = |H|^2$ with PDF $f(\gamma) = e^{-\gamma}$ . The power constraint is

$\int_{\gamma_0}^{\infty} \left(\nu - \frac{N}{\gamma}\right) e^{-\gamma} \, d\gamma = P,$

where $\gamma_0 = N/\nu$ is the cutoff below which the transmitter is silent.

Evaluate the integral

Splitting: $\nu \cdot e^{-\gamma_0} - N \int_{\gamma_0}^{\infty} \frac{e^{-\gamma}}{\gamma}\, d\gamma = P.$

The second integral is the exponential integral $E_1(\gamma_0)$ . With $\text{SNR} = 10$ (10 dB), numerical solution of the transcendental equation gives $\nu/N \approx 11.08$ and $\gamma_0 = 1/(\nu/N) \approx 0.0903$ .

Silent fraction

The transmitter is silent when $|H|^2 < \gamma_0$ . For $|H|^2 \sim \text{Exp}(1)$ :

$\Pr(|H|^2 < \gamma_0) = 1 - e^{-\gamma_0} \approx 1 - e^{-0.0903} \approx 8.6\%.$

At $\text{SNR} = 10$ dB, the transmitter stays silent about 8.6% of the time. At lower SNR, this fraction increases because deep fades are too costly to overcome.

Water-Filling over Fading States

Visualize the water-filling power allocation $P^*(h) = (\nu - N/|h|^2)^+$ over the fading gain $|H|^2$ . The water level (horizontal dashed line) determines which fading states receive power. Adjust the average SNR to see how the cutoff and power allocation change.

Parameters

\text{SNR}

(dB)10

Average signal-to-noise ratio in decibels

Fading distribution

Theorem: CSIT Gain Vanishes at High SNR

For the ergodic fading channel,

$C_{\text{CSIT}} - C_{\text{CSIR}} \to 0 \quad \text{as } \text{SNR} \to \infty.$

More precisely, the difference is $O(1/\text{SNR})$ at high SNR. The capacity gain from CSIT is most significant at low SNR.

At high SNR, the water level $\nu$ becomes very large relative to $N/|h|^2$ for almost all fading states. This means the water-filling allocation $P^*(h) = \nu - N/|h|^2 \approx \nu$ is nearly constant — the transmitter allocates roughly the same power to all states. But constant power is exactly the CSIR-only strategy. So at high SNR, CSIT provides negligible additional benefit for ergodic capacity.

Proof

High-SNR water-filling

As $P \to \infty$ , the water level $\nu \to \infty$ and the cutoff $\gamma_0 = N/\nu \to 0$ . For any $\gamma > 0$ :

$P^*(\gamma) = \nu - N/\gamma = \nu(1 - N/(\gamma \nu)) \approx \nu.$

The power allocation becomes approximately $P^*(h) \approx P$ for all $h$ , matching the constant-power CSIR strategy.

Capacity difference

Writing the capacity difference:

$C_{\text{CSIT}} - C_{\text{CSIR}} = \mathbb{E}\!\left[\frac{1}{2}\log\!\left(\frac{1 + |H|^2 P^*(H)/N}{1 + |H|^2 P/N}\right)\right].$

Since $P^*(H)/P \to 1$ for almost all $H$ as $P \to \infty$ , the ratio inside the log approaches 1, and the difference vanishes. $\blacksquare$

CSIR Only vs. Full CSI (CSIR + CSIT)

Property	CSIR Only	Full CSI (CSIR + CSIT)
Optimal input power	Constant: $P(h) = P$ for all $h$	Water-filling: $P^*(h) = (\nu - N/\|h\|^2)^+$
Capacity formula	$\mathbb{E}[\frac{1}{2}\log(1 + \|H\|^2 \text{SNR})]$	$\mathbb{E}[\frac{1}{2}\log(\nu \|H\|^2/N)]$ (for active states)
Behavior in deep fades	Transmits at full power (wastes energy)	Stays silent (conserves power for better states)
Low-SNR gain from CSIT	N/A	Significant — water-filling concentrates power on best states
High-SNR gain from CSIT	N/A	Negligible — water-filling approaches constant power
Feedback required?	No	Yes — channel quality indicator from receiver to transmitter

Ergodic Capacity: CSIR Only vs. Full CSI

Compare the ergodic capacity with CSIR only (constant power) and with full CSI (water-filling) as a function of $\text{SNR}$ . Also shows the AWGN capacity as an upper bound. Notice that the CSIT gain is largest at low SNR and diminishes at high SNR.

Parameters

Max SNR (dB)30

Fading distribution

Historical Note: Goldsmith and Varaiya: Water-Filling over Fading

1997

The optimal power allocation for fading channels with CSIT was derived by Andrea Goldsmith and Pravin Varaiya in their 1997 paper "Capacity of Fading Channels with Channel Side Information." They showed that the water-filling solution, previously known for parallel Gaussian channels (a deterministic setting), applies identically when the "parallel channels" are random fading states.

Goldsmith and Varaiya also analyzed several suboptimal strategies: channel inversion (forcing a constant received SNR), truncated inversion, and on-off power control. Their framework became a cornerstone of wireless information theory and motivated the development of adaptive modulation and coding (AMC) schemes used in all modern wireless standards.

Common Mistake: Channel Inversion Wastes Power on Deep Fades

Mistake:

Using channel inversion $P(h) = N/(|h|^2)$ to maintain a constant received SNR of 1 for all fading states, thinking this simplifies the system design.

Correction:

Channel inversion allocates disproportionate power to deep fades. For Rayleigh fading, $\mathbb{E}[N/|H|^2] = \int_0^\infty (N/\gamma) e^{-\gamma}\, d\gamma = \infty$ . The average power is infinite — channel inversion is infeasible under an average power constraint for Rayleigh fading.

Even for fading distributions where channel inversion is feasible (e.g., when $|H|^2$ is bounded away from zero), the capacity with channel inversion is strictly lower than water-filling because it wastes power compensating for deep fades instead of exploiting good channel states.

Quick Check

In the water-filling power allocation over fading states, what happens to the fraction of time the transmitter is silent as $\text{SNR} \to \infty$ ?

It increases toward 1

It stays constant

It decreases toward 0

It oscillates depending on the fading distribution

Correction:

It decreases toward 0

As $\\text{SNR} \\to \\infty$ , the water level $\\nu \\to \\infty$ , so $\\gamma_0 = N/\\nu \\to 0$ . The probability $\\Pr(|H|^2 < \\gamma_0) \\to 0$ . At high SNR, the transmitter transmits in almost every fading state, and water-filling converges to constant-power allocation.

Key Takeaway

With full CSI (CSIR + CSIT), the ergodic capacity is achieved by water-filling over fading states: $P^*(h) = (\nu - N/|h|^2)^+$ . The structure is identical to water-filling over parallel channels (Chapter 12), with fading states playing the role of sub-channels. The gain from CSIT is most significant at low SNR and vanishes at high SNR, where constant power is nearly optimal.

Water-Filling over Fading States

Visualization of the water-filling power allocation across different fading states. As the water level rises, power is first allocated to strong channels. Weak channels below the noise floor receive no power — it is better to stay silent than to waste power on a deeply faded state.

Ergodic Capacity with Full CSI (Water-Filling)

Exploiting Channel Knowledge at the Transmitter

Definition: Power Adaptation Policy

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

Formulate the optimization

Lagrangian and KKT conditions

Solve for $P^*(h)$

Example: Water-Filling over Rayleigh Fading States

Power constraint equation

Evaluate the integral

Silent fraction

Water-Filling over Fading States

Parameters

Theorem: CSIT Gain Vanishes at High SNR

High-SNR water-filling

Capacity difference

CSIR Only vs. Full CSI (CSIR + CSIT)

Ergodic Capacity: CSIR Only vs. Full CSI

Parameters

Historical Note: Goldsmith and Varaiya: Water-Filling over Fading

Common Mistake: Channel Inversion Wastes Power on Deep Fades

Quick Check

Key Takeaway

Water-Filling over Fading States

Definition:
Power Adaptation Policy