Outage Capacity

When Ergodic Averaging Is Not Possible

The ergodic capacity from Sections 13.2-13.3 relies on the codeword spanning many independent fading realizations, so the time average of the instantaneous rate concentrates around its expectation. But what happens when the channel changes slowly — so slowly that the entire codeword experiences a single fading realization?

In this quasi-static regime, there is no ergodic averaging. If the channel realization is bad (deep fade), no coding scheme can save us — the instantaneous mutual information is simply too low to support the target rate. The best we can do is characterize the probability that this happens and design the system to tolerate occasional failures. This leads to the concepts of outage probability and ϵ\epsilon-outage capacity.

Definition:

Outage Probability

For the quasi-static fading channel Y=HX+ZY = HX + Z with HH constant over the codeword, the outage probability at rate RR is

Pout(R)=Pr⁡ ⁣[12log⁡ ⁣(1+∣H∣2 SNR)<R].P_{\text{out}}(R) = \Pr\!\left[\frac{1}{2}\log\!\left(1 + |H|^2 \,\text{SNR}\right) < R\right].

This is the probability that the channel realization cannot support rate RR. When outage occurs, the decoder cannot reliably decode the message, and the codeword is lost.

The outage probability is a fundamental limit: no coding scheme operating at rate RR over a quasi-static fading channel can achieve an error probability lower than Pout(R)P_{\text{out}}(R) (in the limit of infinite blocklength within one fading block).

Outage probability

The probability that the instantaneous mutual information of a quasi-static fading channel falls below the target rate. Pout(R)=Pr⁡[12log⁡(1+∣H∣2SNR)<R]P_{\text{out}}(R) = \Pr[\frac{1}{2}\log(1 + |H|^2 \text{SNR}) < R].

Related: ϵ\epsilon-outage capacity, Rayleigh fading

Definition:

ϵ\epsilon-Outage Capacity

The ϵ\epsilon-outage capacity is the maximum rate such that the outage probability does not exceed ϵ\epsilon:

Cϵ=sup⁡{R≥0:Pout(R)≤ϵ}.C_\epsilon = \sup\left\{R \geq 0 : P_{\text{out}}(R) \leq \epsilon\right\}.

Equivalently, CϵC_\epsilon is determined by finding the ϵ\epsilon-quantile of the instantaneous capacity random variable 12log⁡(1+∣H∣2SNR)\frac{1}{2}\log(1 + |H|^2 \text{SNR}):

Cϵ=12log⁡ ⁣(1+F∣H∣2−1(ϵ)⋅SNR),C_\epsilon = \frac{1}{2}\log\!\left(1 + F_{|H|^2}^{-1}(\epsilon) \cdot \text{SNR}\right),

where F∣H∣2−1(ϵ)F_{|H|^2}^{-1}(\epsilon) is the ϵ\epsilon-quantile of ∣H∣2|H|^2.

For ϵ=0\epsilon = 0, the outage capacity is zero for any continuous fading distribution (since any rate has positive outage probability). For ϵ=1\epsilon = 1, it is infinite. Practical systems operate with ϵ∈[10−5,10−1]\epsilon \in [10^{-5}, 10^{-1}] depending on the application: voice calls tolerate ϵ∼0.01\epsilon \sim 0.01, while URLLC requires ϵ≤10−5\epsilon \leq 10^{-5}.

ϵ\epsilon-outage capacity

The maximum rate at which the outage probability does not exceed ϵ\epsilon. Defined as Cϵ=sup⁡{R:Pout(R)≤ϵ}C_\epsilon = \sup\{R : P_{\text{out}}(R) \leq \epsilon\}.

Related: Outage probability

Example: Outage Probability for Rayleigh Fading

For a Rayleigh fading channel (∣H∣2∼Exp(1)|H|^2 \sim \text{Exp}(1)), derive a closed-form expression for the outage probability Pout(R)P_{\text{out}}(R). Compute the 1%-outage capacity at SNR=20\text{SNR} = 20 dB.

Solution
Derive the outage probability

The outage event is 12log⁡(1+∣H∣2SNR)<R\frac{1}{2}\log(1 + |H|^2 \text{SNR}) < R, which is equivalent to ∣H∣2<22R−1SNR|H|^2 < \frac{2^{2R} - 1}{\text{SNR}}. Let γth=22R−1SNR\gamma_{\text{th}} = \frac{2^{2R} - 1}{\text{SNR}}. Then

Pout(R)=Pr⁡(∣H∣2<γth)=F∣H∣2(γth)=1−e−γth.P_{\text{out}}(R) = \Pr(|H|^2 < \gamma_{\text{th}}) = F_{|H|^2}(\gamma_{\text{th}}) = 1 - e^{-\gamma_{\text{th}}}.

Compute the $\epsilon$-outage capacity

Setting Pout=ϵP_{\text{out}} = \epsilon and solving for RR:

1−e−γth=ϵ  ⟹  γth=−ln⁡(1−ϵ).1 - e^{-\gamma_{\text{th}}} = \epsilon \implies \gamma_{\text{th}} = -\ln(1 - \epsilon).

Therefore

Cϵ=12log⁡ ⁣(1−SNRln⁡(1−ϵ)).C_\epsilon = \frac{1}{2}\log\!\left(1 - \text{SNR} \ln(1-\epsilon)\right).

Numerical evaluation

At SNR=20\text{SNR} = 20 dB (SNR=100\text{SNR} = 100) and ϵ=0.01\epsilon = 0.01:

γth=−ln⁡(0.99)≈0.01005,\gamma_{\text{th}} = -\ln(0.99) \approx 0.01005, C0.01=12log⁡(1+100×0.01005)≈12log⁡(2.005)≈0.502 bits/use.C_{0.01} = \frac{1}{2}\log(1 + 100 \times 0.01005) \approx \frac{1}{2}\log(2.005) \approx 0.502 \text{ bits/use}.

Compare with the AWGN capacity at the same SNR: CAWGN=12log⁡(101)≈3.33C_{\text{AWGN}} = \frac{1}{2}\log(101) \approx 3.33 bits/use. The 1%-outage capacity is only about 15% of the AWGN capacity — a dramatic loss compared to the modest ergodic penalty. This illustrates the severity of quasi-static fading.

Definition:

Diversity Order

The diversity order (or diversity gain) of a communication scheme is defined as

d=−lim⁡SNR→∞log⁡Pout(R)log⁡SNR,d = -\lim_{\text{SNR} \to \infty} \frac{\log P_{\text{out}}(R)}{\log \text{SNR}},

where RR is held fixed. A diversity order of dd means Pout∼SNR−dP_{\text{out}} \sim \text{SNR}^{-d} at high SNR. Higher diversity order corresponds to a steeper decay of the outage probability with SNR, meaning the system is more robust against deep fades.

For a single Rayleigh link, Pout≈γth=(22R−1)/SNRP_{\text{out}} \approx \gamma_{\text{th}} = (2^{2R} - 1)/\text{SNR} at high SNR, giving diversity order d=1d = 1. Systems with LL independent diversity branches (e.g., LL receive antennas with MRC) achieve d=Ld = L, where Pout∼SNR−LP_{\text{out}} \sim \text{SNR}^{-L}.

Diversity order

The negative slope of the outage probability vs. SNR curve on a log-log scale. A system with diversity order dd has Pout∼SNR−dP_{\text{out}} \sim \text{SNR}^{-d} at high SNR.

Related: Outage probability

Theorem: Outage Probability with LL-fold Diversity

Consider a quasi-static fading channel with LL independent diversity branches (e.g., LL receive antennas with maximum ratio combining). If each branch has i.i.d. Rayleigh fading, then

Pout(R)=1−e−γth∑ℓ=0L−1γthℓℓ!,P_{\text{out}}(R) = 1 - e^{-\gamma_{\text{th}}} \sum_{\ell=0}^{L-1} \frac{\gamma_{\text{th}}^\ell}{\ell!},

where γth=(22R−1)/SNR\gamma_{\text{th}} = (2^{2R} - 1)/\text{SNR}. At high SNR, this simplifies to

Pout(R)≈γthLL!=(22R−1)LL!⋅SNRL,P_{\text{out}}(R) \approx \frac{\gamma_{\text{th}}^L}{L!} = \frac{(2^{2R} - 1)^L}{L! \cdot \text{SNR}^{L}},

yielding diversity order d=Ld = L.

With MRC over LL i.i.d. Rayleigh branches, the combined SNR is γtotal=∑ℓ=1L∣Hℓ∣2⋅SNR\gamma_{\text{total}} = \sum_{\ell=1}^L |H_\ell|^2 \cdot \text{SNR}, and ∑∣Hℓ∣2∼Gamma(L,1)\sum |H_\ell|^2 \sim \text{Gamma}(L, 1). The CDF of a Gamma distribution decays as γL\gamma^L near zero, so the probability of a deep fade (total SNR below threshold) decays as SNR−L\text{SNR}^{-L}. Each additional antenna provides one more order of magnitude of protection against deep fades per 10 dB of SNR.

Proof
Combined SNR distribution

With LL i.i.d. Rayleigh branches, the combined fading gain with MRC is γ=∑ℓ=1L∣Hℓ∣2\gamma = \sum_{\ell=1}^L |H_\ell|^2, which follows a Gamma(L,1)\text{Gamma}(L, 1) distribution (since each ∣Hℓ∣2∼Exp(1)|H_\ell|^2 \sim \text{Exp}(1)) with PDF

fγ(t)=tL−1e−t(L−1)!,t≥0.f_\gamma(t) = \frac{t^{L-1} e^{-t}}{(L-1)!}, \quad t \geq 0.

Outage probability via Gamma CDF

The outage probability is

Pout=Pr⁡ ⁣[12log⁡(1+γ⋅SNR)<R]=Pr⁡ ⁣[γ<γth]=Fγ(γth).P_{\text{out}} = \Pr\!\left[\frac{1}{2}\log(1 + \gamma \cdot \text{SNR}) < R\right] = \Pr\!\left[\gamma < \gamma_{\text{th}}\right] = F_\gamma(\gamma_{\text{th}}).

The CDF of Gamma(L,1)\text{Gamma}(L, 1) is the regularized incomplete gamma function:

Fγ(t)=1−e−t∑ℓ=0L−1tℓℓ!.F_\gamma(t) = 1 - e^{-t} \sum_{\ell=0}^{L-1} \frac{t^\ell}{\ell!}.

High-SNR approximation

At high SNR, γth=(22R−1)/SNR→0\gamma_{\text{th}} = (2^{2R} - 1)/\text{SNR} \to 0. Using the Taylor expansion e−t≈1−t+⋯e^{-t} \approx 1 - t + \cdots for small tt:

Fγ(γth)≈γthLL!=(22R−1)LL!⋅SNRL.F_\gamma(\gamma_{\text{th}}) \approx \frac{\gamma_{\text{th}}^L}{L!} = \frac{(2^{2R} - 1)^L}{L! \cdot \text{SNR}^{L}}.

This confirms the diversity order is d=Ld = L. ■\blacksquare

Outage Probability vs. SNR

Plot the outage probability Pout(R)P_{\text{out}}(R) as a function of SNR\text{SNR} (in dB) for different numbers of diversity branches LL. On a log scale, the slope of each curve at high SNR equals −L-L (the diversity order). Adjust the target rate and number of branches to observe the diversity-outage tradeoff.

Parameters
1
4

Plot curves for $L = 1, 2, \ldots, L_{\max}$

30

Definition:

Diversity-Multiplexing Tradeoff (DMT)

For a quasi-static fading channel, the diversity-multiplexing tradeoff characterizes the fundamental relationship between the diversity order dd and the multiplexing gain rr, defined as

r=lim⁡SNR→∞R(SNR)12log⁡(SNR),r = \lim_{\text{SNR} \to \infty} \frac{R(\text{SNR})}{\frac{1}{2}\log(\text{SNR})},

where R(SNR)R(\text{SNR}) is the operating rate as a function of SNR. The multiplexing gain measures what fraction of the AWGN capacity growth rate the scheme captures.

For a system with ntn_t transmit and nrn_r receive antennas operating over i.i.d. Rayleigh fading, the optimal DMT curve is

d∗(r)=(nt−r)(nr−r),0≤r≤min⁡(nt,nr).d^*(r) = (n_t - r)(n_r - r), \quad 0 \leq r \leq \min(n_t, n_r).

This piecewise-linear curve connects the extreme points:

  • Maximum diversity: d∗(0)=ntnrd^*(0) = n_t n_r (reliability-focused),
  • Maximum multiplexing: d∗(min⁡(nt,nr))=0d^*(\min(n_t, n_r)) = 0 (rate-focused).

The DMT is a high-SNR characterization that captures how fast the error probability decays (diversity) versus how fast the rate grows (multiplexing) as SNR increases. It was introduced by Zheng and Tse (2003) and provides a unified framework for comparing space-time codes.

Connection to Book CM: Coded Modulation

The diversity-multiplexing tradeoff framework is studied in depth in the Coded Modulation book (Book CM), where space-time codes are designed to achieve specific points on the DMT curve. The Alamouti code achieves full diversity d=2nrd = 2n_r at multiplexing gain r=1r = 1 for a 2×nr2 \times n_r system. Lattice-based space-time codes (e.g., the Golden code) can achieve the entire DMT frontier for 2×22 \times 2 MIMO. Understanding the outage framework from this section is essential for that material.

Example: How Many Antennas for 99.999% Reliability?

A quasi-static Rayleigh fading channel operates at rate R=1R = 1 bit/channel use with SNR=20\text{SNR} = 20 dB. Determine the number of receive antennas LL (with MRC) needed to achieve Pout≤10−5P_{\text{out}} \leq 10^{-5}.

Solution
Compute the SNR threshold

At R=1R = 1 bit/use: γth=(22⋅1−1)/SNR=3/100=0.03\gamma_{\text{th}} = (2^{2 \cdot 1} - 1)/\text{SNR} = 3/100 = 0.03.

Use the high-SNR approximation

We need Pout≈γthL/L!≤10−5P_{\text{out}} \approx \gamma_{\text{th}}^L / L! \leq 10^{-5}:

  • L=1L = 1: Pout≈0.03=3×10−2P_{\text{out}} \approx 0.03 = 3 \times 10^{-2}. Too high.
  • L=2L = 2: Pout≈(0.03)2/2=4.5×10−4P_{\text{out}} \approx (0.03)^2/2 = 4.5 \times 10^{-4}. Still too high.
  • L=3L = 3: Pout≈(0.03)3/6=4.5×10−6P_{\text{out}} \approx (0.03)^3/6 = 4.5 \times 10^{-6}. Meets the target.
Conclusion

Three receive antennas with MRC are sufficient to achieve Pout≤10−5P_{\text{out}} \leq 10^{-5} at this operating point. Each additional antenna improves the outage probability by roughly a factor of γth≈0.03\gamma_{\text{th}} \approx 0.03 (15 dB on a log scale). This illustrates the enormous value of diversity in the outage regime.

Common Mistake: Confusing Ergodic and Outage Capacity

Mistake:

Using the ergodic capacity formula Cerg=E[12log⁡(1+∣H∣2SNR)]C_{\text{erg}} = \mathbb{E}[\frac{1}{2}\log(1 + |H|^2 \text{SNR})] for a slow-fading system where the codeword does not span multiple fading states, and concluding that "reliable communication at this rate is possible."

Correction:

The ergodic capacity requires coding over many fading realizations. In quasi-static fading (one realization per codeword), the ergodic capacity is not achievable with vanishing error probability. Instead, any rate R>0R > 0 incurs a nonzero outage probability. The correct metric is the ϵ\epsilon-outage capacity, which can be dramatically lower than the ergodic capacity. For Rayleigh fading at 20 dB with 1% outage, C0.01≈0.5C_{0.01} \approx 0.5 bits/use vs. Cerg≈3.1C_{\text{erg}} \approx 3.1 bits/use — a 6x difference.

⚠️Engineering Note

Outage Capacity and URLLC Design

Ultra-reliable low-latency communication (URLLC) in 5G NR targets 10−510^{-5} block error rate (BLER) with 1 ms latency. This is fundamentally an outage capacity problem: the short packet duration means the codeword cannot span many fading states (quasi-static regime), and the reliability requirement sets ϵ=10−5\epsilon = 10^{-5}.

The outage capacity framework reveals that achieving such extreme reliability at reasonable rates requires:

  1. Diversity — multiple antennas, frequency diversity via wideband transmission, or retransmissions (HARQ) across independent fading blocks.
  2. Conservative rate selection — operating far below the ergodic capacity.
  3. Short-packet corrections — the finite blocklength penalty (Chapter 16 of this book) further reduces the achievable rate below CϵC_\epsilon.
Practical Constraints
  • •

    URLLC mini-slot in 5G NR is 2-7 OFDM symbols — insufficient for ergodic averaging

  • •

    Target BLER of 10−510^{-5} requires diversity order d≥3d \geq 3 at typical operating SNR

  • •

    HARQ retransmissions provide time diversity but add latency (each retransmission costs ~1 ms)

📋 Ref: 3GPP TS 38.300, Section 12

ϵ\epsilon-Outage Capacity vs. SNR

Plot the ϵ\epsilon-outage capacity as a function of SNR\text{SNR} for different values of the outage probability ϵ\epsilon and different diversity orders LL. Compare with the ergodic capacity and the AWGN capacity to see how severely quasi-static fading limits the achievable rate.

Parameters
0.01
1
30

Quick Check

For a Rayleigh fading channel with SNR=20\text{SNR} = 20 dB and no diversity (L=1L = 1), what is the approximate 1%-outage capacity?

About 3.3 bits/use (close to AWGN capacity)

About 3.1 bits/use (close to ergodic capacity)

About 0.5 bits/use

About 1.5 bits/use

Correction:
About 0.5 bits/use

With gammatextth=−ln(0.99)approx0.01\\gamma_{\\text{th}} = -\\ln(0.99) \\approx 0.01 and textSNR=100\\text{SNR} = 100, C0.01=frac12log(1+100times0.01)approx0.5\C_{0.01} = \\frac{1}{2}\\log(1 + 100 \\times 0.01) \\approx 0.5 bits/use.

Key Takeaway

In quasi-static fading, the ergodic capacity is unachievable. The ϵ\epsilon-outage capacity CϵC_\epsilon is determined by the ϵ\epsilon-quantile of the instantaneous capacity and can be dramatically lower than the ergodic capacity. Diversity (multiple antennas, frequency, time) is the primary tool for improving outage performance, reducing PoutP_{\text{out}} from SNR−1\text{SNR}^{-1} to SNR−L\text{SNR}^{-L} with LL independent branches.

Ergodic Capacity with Full CSI (Water-Filling)MIMO Channel Capacity