Ferkans — Interactive Telecom Tutor

From Channel Estimates to Achievable Rates

In Chapter 3 we developed MMSE channel estimation and showed that the base station obtains an estimate $\hat{\mathbf{H}}_k$ of each user's channel. The natural next question is: how much data rate can the system deliver to each user, given imperfect channel knowledge?

Computing the exact ergodic capacity with imperfect CSI is, in general, an open problem. The channel estimate and the true channel are correlated, and the receiver must account for this correlation in its decoding. What we need is a tractable lower bound — tight enough to be useful for system design, yet simple enough to yield closed-form expressions.

The use-and-then-forget (UatF) bound provides exactly this. The idea is beautifully simple: treat the channel estimate as if it were perfect, and absorb all estimation error into an effective noise term. This "forgets" the statistical relationship between the estimate and the error, which can only reduce the achievable rate — hence the result is a lower bound. The bound becomes tight in the massive MIMO regime precisely because channel hardening makes the effective channel nearly deterministic.

Uplink System Model Recap

Consider a single-cell massive MIMO system with $N_t$ base station antennas and $K$ single-antenna users. In the uplink data transmission phase, the received signal at the base station is

$\mathbf{y} = \sum_{k=1}^{K} \sqrt{{P_t}_{k}} \, \mathbf{H}_{k} \, x_k + \mathbf{w},$

where $x_k$ is the unit-power data symbol from user $k$ , ${P_t}_{k}$ is the transmit power, and $\mathbf{w} \sim \mathcal{CN}(\mathbf{0}, \sigma^2 \mathbf{I}_{N_t})$ is the additive noise.

From Chapter 3, the MMSE channel estimate satisfies $\mathbf{H}_{k} = \hat{\mathbf{H}}_k + \tilde{\mathbf{H}}_k$ , where $\hat{\mathbf{H}}_k$ and $\tilde{\mathbf{H}}_k$ are independent (a property of MMSE estimation for Gaussian vectors). We define $\gamma_k \triangleq \mathbb{E}[\|\hat{\mathbf{H}}_k\|^2] / N_t$ as the normalized estimation quality.

Definition:
UatF Effective SINR

Let $\mathbf{v}_{k} \in \mathbb{C}^{N_t}$ be the combining vector for user $k$ , chosen as a function of the channel estimates $\{\hat{\mathbf{H}}_1, \ldots, \hat{\mathbf{H}}_{K}\}$ . After combining, the signal for user $k$ is

$\hat{x}_k = \mathbf{v}_{k}^{H} \mathbf{y} = \underbrace{\sqrt{{P_t}_{k}} \, \mathbf{v}_{k}^{H} \hat{\mathbf{H}}_k}_{\text{desired signal (known part)}} \, x_k + \underbrace{\sqrt{{P_t}_{k}} \, \mathbf{v}_{k}^{H} \tilde{\mathbf{H}}_k \, x_k + \sum_{j \neq k} \sqrt{{P_t}_{j}} \, \mathbf{v}_{k}^{H} \mathbf{H}_{j} \, x_j + \mathbf{v}_{k}^{H} \mathbf{w}}_{\text{effective noise (uncorrelated with desired)}}.$

The UatF effective SINR is defined as

$\text{SINR}_k^{\text{UatF}} = \frac{{P_t}_{k} \left| \mathbb{E}\!\left[\mathbf{v}_{k}^{H} \mathbf{H}_{k}\right] \right|^2}{\sum_{j=1}^{K} {P_t}_{j} \, \mathbb{E}\!\left[\left|\mathbf{v}_{k}^{H} \mathbf{H}_{j}\right|^2\right] - {P_t}_{k} \left|\mathbb{E}\!\left[\mathbf{v}_{k}^{H} \mathbf{H}_{k}\right]\right|^2 + \sigma^2 \, \mathbb{E}\!\left[\|\mathbf{v}_{k}\|^2\right]}.$

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Theorem: Use-and-Then-Forget Bound

Under the system model above, an achievable ergodic rate for user $k$ is

$R_k^{\text{UatF}} = \log_2\!\left(1 + \text{SINR}_k^{\text{UatF}}\right) \quad \text{[bits/s/Hz]},$

where $\text{SINR}_k^{\text{UatF}}$ is the UatF effective SINR from Definition DUatF Effective SINR.

The key insight is that $\mathbb{E}[\mathbf{v}_{k}^{H} \mathbf{H}_{k}]$ is a deterministic scalar — it does not depend on the instantaneous channel realization. The UatF bound treats this deterministic quantity as the "channel gain" and lumps everything else (estimation error, interference, noise) into an uncorrelated effective noise. Since the effective noise is uncorrelated with the desired signal, the worst-case distribution is Gaussian (by the maximum entropy property), which yields the log formula.

In the massive MIMO regime ( $N_t \to \infty$ ), channel hardening means $\mathbf{v}_{k}^{H} \mathbf{H}_{k} / \mathbb{E}[\mathbf{v}_{k}^{H} \mathbf{H}_{k}] \to 1$ , so the gap between the bound and the true capacity vanishes.

Proof

Step 1: Decompose the combined signal

Write $\hat{x}_k = \mathbf{v}_{k}^{H} \mathbf{y}$ and split the desired user's contribution using $\mathbf{H}_{k} = \hat{\mathbf{H}}_k + \tilde{\mathbf{H}}_k$ :

$\hat{x}_k = \sqrt{{P_t}_{k}} \, \mathbf{v}_{k}^{H} \hat{\mathbf{H}}_k \, x_k + \sqrt{{P_t}_{k}} \, \mathbf{v}_{k}^{H} \tilde{\mathbf{H}}_k \, x_k + \sum_{j \neq k} \sqrt{{P_t}_{j}} \, \mathbf{v}_{k}^{H} \mathbf{H}_{j} \, x_j + \mathbf{v}_{k}^{H} \mathbf{w}.$

Step 2: Replace the channel gain by its mean

Further split the first term:

$\sqrt{{P_t}_{k}} \, \mathbf{v}_{k}^{H} \hat{\mathbf{H}}_k \, x_k = \sqrt{{P_t}_{k}} \, \mathbb{E}[\mathbf{v}_{k}^{H} \mathbf{H}_{k}] \, x_k + \sqrt{{P_t}_{k}}\left(\mathbf{v}_{k}^{H} \hat{\mathbf{H}}_k - \mathbb{E}[\mathbf{v}_{k}^{H} \mathbf{H}_{k}]\right) x_k.$

The first part is a deterministic scalar times $x_k$ — this is the "useful" signal. The second part captures the randomness of the effective channel gain and is absorbed into the effective noise.

Step 3: Verify uncorrelatedness

The effective noise $n_k^{\text{eff}} = \hat{x}_k - \sqrt{{P_t}_{k}} \, \mathbb{E}[\mathbf{v}_{k}^{H} \mathbf{H}_{k}] \, x_k$ is uncorrelated with the desired signal $x_k$ because $\mathbb{E}[n_k^{\text{eff}} \, x_k^*] = 0$ (using independence of data symbols, independence of estimation error, and the fact that we subtracted the mean).

Step 4: Apply the worst-case noise argument

For a channel $\hat{x}_k = a_k \, x_k + n_k^{\text{eff}}$ with deterministic $a_k$ and uncorrelated noise, the mutual information $I(x_k; \hat{x}_k)$ is minimized (over all noise distributions with the given second moments) when $n_k^{\text{eff}}$ is Gaussian. The resulting rate is

$R_k = \log_2\!\left(1 + \frac{|a_k|^2}{\text{Var}(n_k^{\text{eff}})}\right) = \log_2\!\left(1 + \text{SINR}_k^{\text{UatF}}\right).$

Since the actual noise may not be Gaussian, this is a lower bound on the true achievable rate. $\blacksquare$

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When Is the UatF Bound Tight?

The UatF bound is tight when:

Channel hardening holds: $\mathbf{v}_{k}^{H} \mathbf{H}_{k} \approx \mathbb{E}[\mathbf{v}_{k}^{H} \mathbf{H}_{k}]$ with high probability, which occurs as $N_t \to \infty$ for i.i.d. Rayleigh channels.
The effective noise is approximately Gaussian: by the central limit theorem, the sum of many interference terms converges to Gaussian as $K$ grows.

For finite $N_t$ , the bound is loose. The gap can be reduced by using the instantaneous effective SINR $|\mathbf{v}_{k}^{H} \mathbf{H}_{k}|^2 / (\cdots)$ and taking $R_k = \mathbb{E}[\log_2(1 + \text{SINR}_k^{\text{inst}})]$ , but this no longer yields a closed-form expression.

Historical Note: Origin of the UatF Bounding Technique

2000-2016

The use-and-then-forget terminology was coined by Marzetta, Larsson, Yang, and Ngo in their 2016 textbook, but the underlying technique is older. The idea of treating the channel estimate as the true channel and absorbing estimation error into effective noise dates back to Medard (2000), who studied the capacity of channels with imperfect CSI. Hassibi and Hochwald (2003) applied similar ideas to MIMO training design. The key contribution of the massive MIMO literature was recognizing that channel hardening makes this bound asymptotically tight, turning an approximation tool into a principled design framework.

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Use-and-Then-Forget (UatF) Bound

A lower bound on the achievable ergodic rate obtained by treating the channel estimate as if it were the true channel and absorbing the estimation error into an effective noise term whose distribution is then pessimized to Gaussian. Tight under channel hardening.

Channel Hardening

The phenomenon whereby $\mathbf{v}_{k}^{H} \mathbf{H}_{k} / \mathbb{E}[\mathbf{v}_{k}^{H} \mathbf{H}_{k}] \to 1$ almost surely as $N_t \to \infty$ . The effective scalar channel after combining becomes nearly deterministic, eliminating small-scale fading from the user's perspective.

Effective SINR

The signal-to-interference-plus-noise ratio computed after linear combining, using the UatF decomposition. For user $k$ with combining vector $\mathbf{v}_{k}$ , the effective SINR determines the achievable rate via $R_k = \log_2(1 + \text{SINR}_k)$ .

Example: UatF Bound for a Single User with MRC

Consider a single-user system ( $K = 1$ ) with i.i.d. Rayleigh fading: $\mathbf{H}_{1} \sim \mathcal{CN}(\mathbf{0}, \beta_{1} \mathbf{I}_{N_t})$ . After MMSE estimation with $\tau_p$ pilot symbols at power ${P_t}_{p}$ , the estimate satisfies $\hat{\mathbf{H}}_1 \sim \mathcal{CN}(\mathbf{0}, \gamma_1 N_t \mathbf{I})$ where $\gamma_1 = \beta_{1}^{2} \tau_p {P_t}_{p} / (\beta_{1} \tau_p {P_t}_{p} + \sigma^2)$ . With MRC combining $\mathbf{v}_{1} = \hat{\mathbf{H}}_1$ , compute $\text{SINR}_1^{\text{UatF}}$ .

Solution

Compute the numerator

$\mathbb{E}[\mathbf{v}_{1}^{H} \mathbf{H}_{1}] = \mathbb{E}[\hat{\mathbf{H}}_1^H \mathbf{H}_{1}] = \mathbb{E}[\|\hat{\mathbf{H}}_1\|^2] = N_t \gamma_1,$ $using$ \mathbb{E}[\hat{\mathbf{H}}1^H \tilde{\mathbf{H}}1] = 0 $(MMSE orthogonality). The numerator is$ {P_t}{1} |N_t \gamma_1|^2 = {P_t}{1} N_t^{2} \gamma_1^2$.

Compute the denominator terms

Interference power: $K = 1$ , so there is no multi-user interference.

Estimation error contribution: $\mathbb{E}[|\hat{\mathbf{H}}_1^H \mathbf{H}_{1}|^2] = N_t^{2} \gamma_1^2 + N_t \gamma_1 \beta_{1}$ (using the fourth-moment formula for complex Gaussian vectors). After subtracting the numerator term $N_t^{2} \gamma_1^2$ , the remaining variance from the desired signal is ${P_t}_{1} N_t \gamma_1 \beta_{1}$ .

Noise contribution: $\sigma^2 \, \mathbb{E}[\|\hat{\mathbf{H}}_1\|^2] = \sigma^2 N_t \gamma_1$ .

Assemble the SINR

$\text{SINR}_1^{\text{UatF}} = \frac{{P_t}_{1} N_t^{2} \gamma_1^2}{{P_t}_{1} N_t \gamma_1 \beta_{1} + \sigma^2 N_t \gamma_1} = \frac{N_t \gamma_1}{\beta_{1} + \sigma^2/{P_t}_{1}}.$ $Notice that the SINR grows linearly with$ N_t$ — this is the massive MIMO array gain. Doubling the antennas doubles the SINR (3 dB gain).

Common Mistake: UatF Does Not Account for Correlation Structure

Mistake:

Applying the UatF bound with the same formula to spatially correlated channels without modifying the expectations. With correlated channels ( $\mathbf{H}_{k} \sim \mathcal{CN}(\mathbf{0}, \mathbf{R}_k)$ ), the expectations $\mathbb{E}[\mathbf{v}_{k}^{H} \mathbf{H}_{k}]$ and $\mathbb{E}[|\mathbf{v}_{k}^{H} \mathbf{H}_{j}|^2]$ have different structure than in the i.i.d. case.

Correction:

The UatF bounding technique is valid for any channel distribution. However, the closed-form expressions derived under i.i.d. Rayleigh must be rederived for correlated channels. With spatial correlation $\mathbf{R}_k$ , the MMSE estimate becomes $\hat{\mathbf{H}}_k \sim \mathcal{CN}(\mathbf{0}, \gamma_k \mathbf{R}_k)$ where $\gamma_k$ now depends on the correlation structure and pilot contamination pattern. Always check which channel model underlies a given rate expression before applying it.

Quick Check

Why is the UatF bound a lower bound on the true achievable rate?

Because it overestimates the noise power

Because it pessimizes the effective noise distribution to Gaussian

Because it ignores the estimation error entirely

Because it uses Jensen's inequality on the log function

Correction:

Because it pessimizes the effective noise distribution to Gaussian

The actual effective noise is a sum of non-Gaussian terms. By assuming the worst-case (Gaussian) distribution with the same variance, the mutual information can only decrease. Hence the resulting rate is a lower bound.

Quick Check

As $N_t \to \infty$ with i.i.d. Rayleigh fading, what happens to $\mathbf{v}_{k}^{H} \mathbf{H}_{k} / \mathbb{E}[\mathbf{v}_{k}^{H} \mathbf{H}_{k}]$ when $\mathbf{v}_{k} = \hat{\mathbf{H}}_k$ (MRC)?

It diverges to infinity

It converges to 1 almost surely

It oscillates randomly around 1

It converges to 0

Correction:

It converges to 1 almost surely

By the law of large numbers, $\hat{\mathbf{H}}_k^H \mathbf{H}_{k} / N_t \to \gamma_k$ almost surely. Since $\mathbb{E}[\hat{\mathbf{H}}_k^H \mathbf{H}_{k}] / N_t = \gamma_k$ , the ratio converges to 1. This is channel hardening.

Why This Matters: UatF Bound and 5G NR System Design

The UatF bound is not merely a theoretical tool — it is the standard method for evaluating massive MIMO performance in 3GPP studies. 5G NR system-level simulations compute user rates using the UatF SINR formula with MMSE channel estimation. The bound's closed-form nature allows rapid evaluation of scheduling, power control, and pilot assignment algorithms without resorting to Monte Carlo simulation of the full mutual information. This is why the massive MIMO literature almost universally reports UatF rates rather than true ergodic rates.

Key Takeaway

The UatF bound transforms the intractable problem of computing capacity with imperfect CSI into a simple SINR formula. It works by treating the estimated channel as deterministic and absorbing estimation error into effective noise. The bound is tight under channel hardening ( $N_t \gg 1$ ), making it the workhorse of massive MIMO rate analysis.

The Use-and-Then-Forget Bound

From Channel Estimates to Achievable Rates

Uplink System Model Recap

Definition: UatF Effective SINR

Theorem: Use-and-Then-Forget Bound

Step 1: Decompose the combined signal

Step 2: Replace the channel gain by its mean

Step 3: Verify uncorrelatedness

Step 4: Apply the worst-case noise argument

When Is the UatF Bound Tight?

Historical Note: Origin of the UatF Bounding Technique

Use-and-Then-Forget (UatF) Bound

Channel Hardening

Effective SINR

Example: UatF Bound for a Single User with MRC

Compute the numerator

Compute the denominator terms

Assemble the SINR

Common Mistake: UatF Does Not Account for Correlation Structure

Quick Check

Quick Check

Why This Matters: UatF Bound and 5G NR System Design

Key Takeaway

Definition:
UatF Effective SINR