Ferkans — Interactive Telecom Tutor

The Two Currencies of the AWGN Channel

Every communication scheme over an AWGN channel spends two resources: bandwidth and power. Bandwidth limits how many independent complex dimensions per second we can use; power limits how much energy per dimension we can pour in. The point is that these two resources are not interchangeable in a continuous way — they trade off along the Shannon curve $C = \log_2(1 + \text{SNR})$ bits per dimension, and every uncoded scheme sits below this curve by an amount that we can measure in two natural ways.

Either we fix the rate $R$ (bits per channel use) and ask how much more power the uncoded scheme needs than Shannon — this gives a power gap in dB at fixed rate. Or we fix the SNR and ask how many fewer bits per dimension the uncoded scheme carries — this gives a rate gap at fixed SNR.

A good code recovers some of that gap. Historically, practitioners quote the recovered power at fixed rate, and call it the coding gain. Understanding what drives coding gain, and how it trades against bandwidth expansion, is the first step toward the design criteria we will pursue throughout this book.

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Definition:
Spectral Efficiency

Let a coded modulation scheme transmit $k$ information bits per $N$ -dimensional signal vector at rate $1 / T_s$ signal vectors per second over a real bandwidth $W$ Hz. Its spectral efficiency is

$\eta \;=\; \frac{k}{N/2} \quad \text{bits per two-dimensional symbol,}$

or equivalently, $\eta$ bits per second per hertz of (complex-baseband) bandwidth whenever $N T_s^{-1} / 2 = W$ .

By convention in this book, we always measure $\eta$ in bits per 2D signal-space dimension, so that an uncoded $M$ -QAM scheme has $\eta = \log_2 M$ , 8-PSK has $\eta = 3$ , BPSK has $\eta = 1$ bit/2D (since BPSK uses only one real dimension per two available).

The choice of 2D as the reference dimension is historical and pragmatic: most passband schemes naturally deliver two real dimensions per symbol (in-phase and quadrature), and spectral-efficiency tables in standards documents are usually written in these units.

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Definition:
Coding Gain

Fix a target symbol- or codeword-error probability $P_e$ and a spectral efficiency $\eta$ . Let $(E_b/N_0)_{\rm ref}$ be the SNR required by a reference uncoded scheme of the same $\eta$ to reach $P_e$ , and let $(E_b/N_0)_{\rm coded}$ be the SNR required by the coded scheme to reach the same $P_e$ at the same $\eta$ . The coding gain of the coded scheme over the reference is

$\gamma_c \;=\; 10 \log_{10} \frac{(E_b/N_0)_{\rm ref}}{(E_b/N_0)_{\rm coded}} \quad \text{[dB]}.$

Two variants are in common use:

Asymptotic coding gain. Taking $P_e \to 0$ , the ratio tends to $d_{\min}^2 / d_{\min,\rm ref}^2$ times a rate penalty, giving a limiting asymptotic gain independent of the operating $P_e$ .
Real (operational) coding gain. At any finite $P_e$ the coding gain is smaller than the asymptotic value because the coded scheme's error probability is controlled by the full distance spectrum and the number of nearest neighbors, not just $d_{\min}$ .

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$E_b/N_0$ vs. $E_s/N_0$ : Two Normalizations, Same Story

The ratio $E_s/N_0$ is the SNR per transmitted symbol; $E_b/N_0$ is the SNR per information bit. They are related by

$\frac{E_s}{N_0} \;=\; R \cdot \frac{E_b}{N_0},$

where $R$ is the number of information bits per channel use. When comparing schemes of different spectral efficiency, $E_b/N_0$ is the meaningful axis (as in the Shannon limit curve). When comparing schemes at the same $\eta$ , either axis works and the horizontal shift is identical.

Uncoded AWGN BER Curves for BPSK, QPSK, and M-QAM

Bit-error probability on the AWGN channel for the standard constellations. At the same bit-error rate, different constellations require very different $E_b/N_0$ , and the horizontal gap between any two curves is the rate penalty of the higher-order constellation. Every dB we later recover with a code is measured against these baseline curves.

Parameters

Include 8-PSK curve

Theorem: Uncoded BER for Gray-Labeled Square M-QAM

For a square $M$ -QAM constellation with $M = 2^m$ , $m$ even, Gray labeling, and maximum-likelihood detection over AWGN, the bit-error probability satisfies

$P_b(E_b/N_0) \;\approx\; \frac{4}{\log_2 M}\!\left(1 - \frac{1}{\sqrt{M}}\right) Q\!\left(\sqrt{\frac{3 \log_2 M}{M - 1} \cdot \frac{E_b}{N_0}}\right),$

where the approximation holds at high SNR and absorbs a double-count of corner points. In particular, the exponent in the $Q$ function is $\tfrac{3 \log_2 M}{M-1}$ , which decreases as $M$ grows — this is the quantitative statement of the bandwidth-for-power tradeoff.

Each Gray-labeled symbol error typically flips a single bit, which is why the factor $1/\log_2 M$ appears. The argument of the $Q$ function is $d_{\min} / (2\sigma)$ , and for square QAM $d_{\min}^2 = 6 E_s / (M-1)$ . Substituting $E_s = \log_2 M \cdot E_b$ yields the displayed expression.

Show Hint

Compute $d_{\min}$ for a square $M$ -QAM of energy $E_s$ — use that the constellation is a rectangular grid of spacing $d_{\min}$ .

Express the symbol-error probability as a union bound over the $\le 4$ nearest neighbors of a typical interior point.

Convert symbol error to bit error by dividing by $\log_2 M$ , which is valid only under Gray labeling at high SNR.

Proof

Minimum distance of square M-QAM

Place the $M$ -QAM constellation on the grid $\{-(\sqrt{M}-1), \ldots, -1, 1, \ldots, \sqrt{M}-1\}$ on each axis times $d_{\min}/2$ . Its average energy is

$E_s \;=\; \frac{d_{\min}^2}{4} \cdot \frac{2(M-1)}{3} \;=\; \frac{d_{\min}^2 (M-1)}{6}.$

Solving for the minimum distance, $d_{\min}^2 \;=\; \tfrac{6 E_s}{M-1}$ .

Symbol-error probability via nearest-neighbor union bound

An interior constellation point has four nearest neighbors at distance $d_{\min}$ . A boundary point has fewer. Under ML detection on AWGN with noise variance $\sigma^2 = N_0/2$ per real dimension, each pairwise decision is

$\mathrm{Pr}\{\mathbf{x} \to \hat{\mathbf{x}}\} = Q\!\!\left(\frac{d_{\min}}{2\sigma}\right) = Q\!\!\left(\sqrt{\frac{d_{\min}^2}{2 N_0}}\right).$

Averaging the number of nearest neighbors over the constellation gives the factor $4(1 - 1/\sqrt{M})$ , which accounts for the reduced neighborhood at boundary points.

Per-bit conversion under Gray labeling

With Gray labeling, at high SNR the dominant symbol error events flip exactly one bit out of $\log_2 M$ . Dividing by $\log_2 M$ and substituting $E_s = \log_2 M \cdot E_b$ yields

$P_b \;\approx\; \frac{4}{\log_2 M}\!\left(1 - \frac{1}{\sqrt{M}}\right) Q\!\left(\sqrt{\frac{3 \log_2 M}{M - 1} \cdot \frac{E_b}{N_0}}\right).$

$\blacksquare$

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Example: The Cost of Going from 16-QAM to 64-QAM

A system currently transmits at $\eta = 4$ bits/2D using 16-QAM at a target $P_b = 10^{-5}$ . The operator wants to go to $\eta = 6$ bits/2D by switching to 64-QAM while keeping the same $P_b$ . How much additional $E_b/N_0$ will this require, approximately?

Solution

Express the argument of $\ntn{qfn}$ in each case

From the theorem above, the argument of the $Q$ function is $\sqrt{\tfrac{3 \log_2 M}{M-1} \cdot E_b/N_0}$ . For the same $P_b \approx 10^{-5}$ we need the argument to be approximately the same value, call it $\beta$ . So

$\frac{3 \log_2 16}{15} \cdot \left(\frac{E_b}{N_0}\right)_{16} \;=\; \frac{3 \log_2 64}{63} \cdot \left(\frac{E_b}{N_0}\right)_{64}.$

Take the ratio

The factor $\tfrac{3 \log_2 M}{M-1}$ evaluates to $12/15 = 0.8$ for $M = 16$ and $18/63 = 2/7 \approx 0.286$ for $M = 64$ . Their ratio is $0.8/0.286 \approx 2.8$ , which in decibels is $10 \log_{10} 2.8 \approx 4.5$ dB.

Interpret

Going from 16-QAM to 64-QAM costs roughly 4.5 dB in $E_b/N_0$ for the same $P_b$ . This is the rate penalty we spend for every additional bit per dimension we demand — and it is the gap that a good code hopes to recover.

Definition:
Power-Limited vs. Bandwidth-Limited Regimes

We say an operating point is power-limited if $\eta \ll 1$ (many degrees of freedom, little power per dimension) and bandwidth-limited if $\eta \gg 1$ (few dimensions, lots of power per dimension). The boundary $\eta \approx 1$ bit/2D separates the two regimes.

In the power-limited regime, the Shannon limit approaches $E_b/N_0 \to \ln 2 \approx -1.59$ dB and additional bandwidth is essentially free, so codes with large bandwidth expansion (e.g., low-rate turbo, LDPC, and orthogonal signaling) are the right choice. In the bandwidth-limited regime, every doubling of $\eta$ costs roughly 3 dB in power and bandwidth is scarce; here the goal of coding is to recover power without consuming bandwidth — this is the territory of coded modulation.

The split between "binary coding over a binary input channel" and "coded modulation over a QAM channel" tracks this boundary almost exactly. Below 1 bit/2D, concatenation of a binary capacity-approaching code with BPSK is already near-optimal; above 1 bit/2D, doing the same with a QAM mapper loses a sizeable chunk of capacity unless the mapping is designed with care.

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Key Takeaway

Coding gain and bandwidth expansion are two separate knobs. A code of rate $R < 1$ expands bandwidth by $1/R$ and can therefore recover power in the power-limited regime. A coded modulation scheme keeps the bandwidth fixed and recovers power by exploiting the signal-space structure directly — this is the only option in the bandwidth-limited regime, where binary coding over a QAM mapper already costs too much.

Quick Check

Two uncoded systems operate at the same bit-error rate on an AWGN channel. System A is 4-QAM; system B is 64-QAM. Approximately how much more $E_b/N_0$ does System B require than System A?

about 2 dB

about 5 dB

about 8 dB

about 12 dB

Correction:

about 8 dB

The argument of the $Q$ function in the high-SNR BER expression scales as $\tfrac{3 \log_2 M}{M-1}$ : it equals $2$ for 4-QAM and $2/7$ for 64-QAM, a ratio of $7$ , i.e. about $8.5$ dB. In practice the gap at $P_b = 10^{-5}$ is approximately $8$ dB.

Common Mistake: Coding gain is measured against a matched-rate uncoded reference

Mistake:

Reporting that a code achieves "5 dB of coding gain" without specifying the reference constellation and the target error rate.

Correction:

Coding gain is always relative. A 5 dB gain over uncoded 16-QAM at $P_b = 10^{-5}$ is not the same thing as 5 dB of gain over uncoded QPSK. Always state (i) the baseline constellation, (ii) the target error probability, and (iii) the spectral efficiency $\eta$ at which the comparison is made.

Coding gain

The horizontal distance in dB between the $P_b$ -vs- $E_b/N_0$ curve of a coded scheme and that of a reference uncoded scheme of the same spectral efficiency, at a fixed target error probability.

Spectral efficiency

Number of information bits transmitted per two-dimensional signal-space symbol, or equivalently per hertz of complex baseband bandwidth. Denoted $\eta$ in this book.

Related: Coding Gain, Capacity Gap

Why This Matters: From the Spectral-Efficiency Plane to 5G NR MCS Tables

Modern cellular standards encode the coded-modulation tradeoff as an MCS (modulation and coding scheme) table: every MCS index picks a constellation (QPSK, 16-QAM, 64-QAM, 256-QAM) and an LDPC code rate, together implementing a specific $(\eta, E_b/N_0)$ operating point. The base station chooses the MCS by estimating the channel SNR and selecting the highest- $\eta$ entry whose required SNR is below the measured one. The structure of the table — a staircase of (constellation, code rate) pairs climbing through the spectral-efficiency plane — is a direct engineering reflection of the theory we develop in this chapter.

Coding Gain vs. Bandwidth Efficiency