Ferkans — Interactive Telecom Tutor

Why Bounds Matter

Exact symbol-error probability expressions exist only for a handful of highly symmetric constellations (orthogonal, simplex, M-PSK, square QAM). For the vast majority of practical signal sets — and certainly for any lattice- or code-based constellation — closed-form $P_e$ is out of reach. Bounds that are provably tight for high SNR (union bound, nearest-neighbor bound) and simple enough to evaluate by hand fill this gap. They are the daily working tools of every communication engineer.

Definition:
Pairwise Error Probability

Given constellation points $\mathbf{x}_m, \mathbf{x}_j$ in AWGN, the pairwise error probability (PEP) is the probability that a minimum-distance decoder — restricted to just these two signals — decides $j$ given that $m$ was transmitted: $P(m\to j) \;=\; \Pr\bigl(\|\mathbf{Y} - \mathbf{x}_j\|^2 < \|\mathbf{Y}-\mathbf{x}_m\|^2 \,\bigm|\, \mathcal{H}_m\bigr).$ For $\mathbf{Z} \sim \mathcal{N}(\mathbf{0}, \tfrac{N_0}{2}\mathbf{I}_N)$ this simplifies to $P(m\to j) \;=\; Q\!\left(\frac{\|\mathbf{x}_m - \mathbf{x}_j\|}{\sqrt{2 N_0}}\right) \;=\; Q\!\left(\sqrt{\frac{d_{m,j}^2}{2 N_0}}\right).$

The PEP depends only on the distance $d_{m,j} = \|\mathbf{x}_m - \mathbf{x}_j\|$ — a consequence of the isotropy of the AWGN noise vector.

Theorem: Union Bound on Symbol Error Probability

For a constellation with equiprobable symbols in AWGN, $P_e \;\leq\; \frac{1}{M}\sum_{m=0}^{M-1}\sum_{j\neq m} P(m\to j) \;=\; \frac{1}{M}\sum_{m=0}^{M-1}\sum_{j\neq m} Q\!\Bigl(\tfrac{d_{m,j}}{\sqrt{2 N_0}}\Bigr).$ A looser — but useful — universal bound is obtained by replacing every pairwise distance with the minimum distance: $P_e \;\leq\; (M-1)\, Q\!\Bigl(\tfrac{d_{\min}}{\sqrt{2 N_0}}\Bigr), \qquad d_{\min} = \min_{m\neq j} d_{m,j}.$

The event "the ML detector makes an error" is the union $\bigcup_{j\neq m} \{\|\mathbf{y}-\mathbf{x}_j\|<\|\mathbf{y}-\mathbf{x}_m\|\}$ . Replacing the union probability by the sum of individual PEPs (the union bound) yields a bound that is tight when the component events are approximately disjoint — which is the case at high SNR.

Proof

Error event as a union

Given $\mathcal{H}_m$ , the ML decoder makes an error iff $\|\mathbf{Y}-\mathbf{x}_j\| < \|\mathbf{Y}-\mathbf{x}_m\|$ for some $j \neq m$ . Hence $\{\text{error}\mid \mathcal{H}_m\} = \bigcup_{j\neq m} A_{m,j}$ where $A_{m,j} = \{\|\mathbf{Y}-\mathbf{x}_j\|^2 < \|\mathbf{Y}-\mathbf{x}_m\|^2\}$ .

Apply Boole's inequality

$\Pr\bigl(\bigcup_{j\neq m} A_{m,j}\bigr) \leq \sum_{j\neq m} \Pr(A_{m,j}) = \sum_{j\neq m} P(m\to j)$ .

Average over $m$ and loosen to $d_{\min}$

Averaging over $m$ with uniform priors and using $Q$ monotonicity — so $Q(d_{m,j}/\sqrt{2N_0}) \leq Q(d_{\min}/\sqrt{2N_0})$ — gives the loose bound with $(M-1)$ terms. $\blacksquare$

Theorem: Nearest-Neighbor (High-SNR) Bound

Let $N_{\text{nn}}(m)$ be the number of constellation points at minimum distance from $\mathbf{x}_m$ , and $\bar N_{\text{nn}} = \tfrac{1}{M}\sum_m N_{\text{nn}}(m)$ . Then at high SNR $P_e \;\approx\; \bar N_{\text{nn}} \cdot Q\!\left(\frac{d_{\min}}{\sqrt{2 N_0}}\right).$ More precisely, the ratio of $P_e$ to this expression tends to 1 as $\text{SNR} \to \infty$ .

At high SNR, errors to points beyond the nearest neighbors are exponentially less likely than errors to the nearest neighbors (the $Q$ decays as $e^{-d^2/(4 N_0)}$ ). The dominant contribution to $P_e$ comes from the $\bar N_{\text{nn}}$ minimum-distance neighbors, so the union bound is asymptotically tight with the correct constant.

Proof

Isolate the minimum-distance contribution

Split the union bound into terms at $d_{m,j} = d_{\min}$ and terms at $d_{m,j} > d_{\min}$ . The first group contributes $\bar N_{\text{nn}} \cdot Q(d_{\min}/\sqrt{2 N_0})$ .

Asymptotic dominance

Using the tail approximation $Q(x) \sim \tfrac{1}{x\sqrt{2\pi}} e^{-x^2/2}$ , the ratio of any term at $d > d_{\min}$ to the minimum-distance term is at most $e^{-(d^2 - d_{\min}^2)/(4 N_0)} \to 0$ as $N_0 \to 0$ . Hence the remaining terms vanish relative to the leading term.

Lower bound via a single pair

For a matching lower bound, $P_e \geq \tfrac{1}{M}\sum_m \max_{j\neq m} P(m\to j) \geq \tfrac{1}{M}\sum_m P(m \to \text{nn}(m))$ . Combined with the upper bound, this gives asymptotic equality. $\blacksquare$

Example: Exact $P_e$ for BPSK and QPSK

Derive the exact symbol-error probability for BPSK ( $\mathbf{x}_{0,1} = \pm\sqrt{E_s}$ ) and QPSK in AWGN.

Solution

BPSK is a pure binary problem

With $d_{0,1}^2 = 4E_s$ and $N_0$ noise variance, the exact error probability is $P_e = Q(\sqrt{2E_s/N_0})$ . This expression is exact because $M=2$ — no union bound is needed.

QPSK = two parallel BPSKs

QPSK separates into two independent BPSK channels on I and Q, each with per-symbol energy $E_s/2$ . The I-channel bit error probability is $Q(\sqrt{E_s/N_0})$ , same for Q. A QPSK symbol is correctly detected iff both bits are correct: $P_e^{\text{QPSK}} = 1 - \bigl(1 - Q(\sqrt{E_s/N_0})\bigr)^2 \approx 2\,Q(\sqrt{E_s/N_0})$ at high SNR. The approximation drops the $Q^{2}$ cross-term.

Theorem: Exact SER for M-PSK

For equi-energy $M$ -PSK with $\mathbf{x}_m = \sqrt{E_s}(\cos(2\pi m/M),\sin(2\pi m/M))$ in AWGN, the exact symbol-error probability is $P_e^{\text{MPSK}} \;=\; 1 - \int_{-\pi/M}^{\pi/M} p_\Theta(\theta)\,d\theta,$ where $p_\Theta$ is the density of the angle of $\mathbf{Y}$ given $\mathbf{x}_0$ was sent. For large $\text{SNR}$ this integrates to $P_e \approx 2\,Q\!\bigl(\sqrt{2\text{SNR}}\sin(\pi/M)\bigr)$ .

By rotational symmetry, assume $\mathbf{x}_0$ was sent; the received point is correctly decoded iff its phase lies in the wedge $|\theta| < \pi/M$ . The exact angular density has a closed form in terms of erfc and is given in the proof.

Proof

Angular density of the received point

Conditional on $\mathbf{x}_0 = (\sqrt{E_s},0)$ , write $\mathbf{Y}$ in polar form $(\rho,\theta)$ . The joint density of $(\rho,\theta)$ factors under the Gaussian distribution. Marginalizing over $\rho \geq 0$ gives $p_\Theta(\theta) = \frac{e^{-\text{SNR}}}{2\pi} + \frac{\sqrt{\text{SNR}}\cos\theta}{\sqrt{\pi}} e^{-\text{SNR}\sin^2\theta} \Phi(\sqrt{2\text{SNR}}\cos\theta)$ where $\Phi$ is the standard Gaussian CDF.

Correct-decision event

The decision region $\mathcal{R}_0$ in polar coordinates is $\{|\theta| < \pi/M\}$ (all radii). Hence $P_c = \int_{-\pi/M}^{\pi/M} p_\Theta(\theta)\,d\theta$ and $P_e = 1 - P_c$ .

High-SNR asymptotic

At high SNR, errors are dominated by the two nearest neighbors at distance $d_{\min} = 2\sqrt{E_s}\sin(\pi/M)$ . Applying the nearest-neighbor bound with $\bar N_{\text{nn}} = 2$ gives $P_e \approx 2Q\!\bigl(d_{\min}/\sqrt{2N_0}\bigr) = 2Q\!\bigl(\sqrt{2\text{SNR}}\sin(\pi/M)\bigr)$ . $\blacksquare$

Example: Exact SER for Square M-QAM

Compute the exact symbol-error probability for square $M$ -QAM with $M = L^2$ (so $L = \sqrt M$ is an integer) and grid spacing $\Delta$ .

Solution

Reduce to two independent PAMs

Square $M$ -QAM is the Cartesian product of two $L$ -PAM constellations on I and Q with spacing $\Delta$ . The I and Q noise components are independent $\mathcal{N}(0, N_0/2)$ .

Error probability of L-PAM

For an $L$ -PAM with interior points and edge points, the conditional error probabilities are $2Q(\Delta/\sqrt{2N_0})$ (interior) and $Q(\Delta/\sqrt{2N_0})$ (edge). Averaging: $P_e^{L\text{-PAM}} = 2\bigl(1 - \tfrac{1}{L}\bigr) Q\!\bigl(\Delta/\sqrt{2N_0}\bigr)$ .

Combine I and Q

A QAM symbol is correct iff both I and Q are correct: $P_c = (1 - P_e^{L\text{-PAM}})^2$ , so $P_e^{M\text{-QAM}} = 1 - \Bigl[1 - 2\bigl(1-\tfrac{1}{\sqrt M}\bigr) Q\!\bigl(\sqrt{\tfrac{3E_s}{(M-1)N_0}}\bigr)\Bigr]^2$ using $\Delta^2 = 6E_s/(M-1)$ .

Union Bound vs. Exact Monte-Carlo SER

Compare three curves for a selected constellation: (i) the exact analytical $P_e$ where available, (ii) the loose union bound $(M-1)Q(d_{\min}/\sqrt{2N_0})$ , and (iii) the nearest-neighbor tight bound $\bar N_{\text{nn}}Q(d_{\min}/\sqrt{2N_0})$ . At high SNR, the nearest-neighbor curve closes the gap to exact.

Parameters

Constellation

SNR min (dB)0

SNR max (dB)25

SER Curves for M-PSK

Compute the symbol-error probability of $M$ -PSK for $M \in \{2,4,8,16\}$ over a range of symbol SNRs, using the exact angular-integral formula. The curves illustrate the 3-dB penalty incurred on each doubling of $M$ beyond QPSK.

Parameters

SNR min (dB)0

SNR max (dB)25

Show Gray-coded BER

SER Curves for Square M-QAM

Exact symbol-error curves for $M$ -QAM with $M \in \{4,16,64,256\}$ . The 6-dB penalty per quadrupling of $M$ (at fixed $E_s$ ) reflects the shrinking of $d_{\min}^2 = 6E_s/(M-1)$ .

Parameters

SNR min (dB)0

SNR max (dB)30

⚠️Engineering Note

BER Is Not SER: The Gray-Coding Gift

Link-budget engineers often quote "BER = $10^{-6}$ " but the curves in standards documents show SER. Under Gray coding — where nearest constellation neighbors differ in exactly one bit — a single symbol error causes on average one bit error out of $\log_2 M$ , so $\text{BER} \approx \text{SER}/\log_2 M$ at high SNR. For 64-QAM ( $\log_2 M = 6$ ), that is a factor of 6 reduction. Without Gray coding (e.g. natural binary labeling) a single symbol error can flip $\Theta(\log_2 M)$ bits, and the high-SNR SER $\approx$ BER slope is lost. Every modern standard (LTE, NR, DVB, Wi-Fi) uses Gray-labeled QAM.

Practical Constraints

•
Gray labeling exists only for constellations with 'neighbor-of-neighbor' graphs that are bipartite (PSK, PAM, square QAM all work)
•
Non-square QAM (e.g. 32-QAM cross) has no perfect Gray labeling — standards use near-Gray approximations

📋 Ref: 3GPP TS 38.211, Section 5.1.3; ITU-T G.9960

High-SNR SER Approximations for Common Constellations

Constellation	$d_{\min}^2/E_s$	High-SNR SER	Spectral Eff. (b/s/Hz)
BPSK	$4$	$Q(\sqrt{2\text{SNR}})$	1
QPSK	$2$	$2\,Q(\sqrt{\text{SNR}})$ (approx.)	2
8-PSK	$2(1-\cos\pi/4)$	$2\,Q(\sqrt{2\text{SNR}}\sin\pi/8)$	3
16-QAM	$0.4$	$3\,Q(\sqrt{\text{SNR}/5})$	4
64-QAM	$\approx 0.095$	$3.5\,Q(\sqrt{\text{SNR}/21})$	6
256-QAM	$\approx 0.0235$	$3.75\,Q(\sqrt{\text{SNR}/85})$	8

Common Mistake: The Loose Union Bound Exceeds 1 at Low SNR

Mistake:

Plotting $(M-1)Q(d_{\min}/\sqrt{2N_0})$ against SNR on log-log axes, a student reports "error probability = 5" at low SNR and concludes the simulation is broken.

Correction:

A probability bound can exceed 1 — it is simply uninformative in that regime. Always cap bounds at 1: $P_e \leq \min\bigl(1, (M-1)Q(\cdot)\bigr)$ . At very low SNR, $P_e$ approaches $1 - 1/M$ (random guessing) while the raw union bound grows linearly with $M$ , so the cap is essential.

Quick Check

For 16-QAM with symbol energy $E_s$ , what is $d_{\min}^2$ ?

$E_s/5$

$2E_s/5$

$E_s/4$

$6E_s$

Correction:

2E_s/5

Using $d_{\min}^2 = 6E_s/(M-1) = 6E_s/15 = 2E_s/5$ .

Pairwise Error Probability (PEP)

The probability $P(m\to j)$ that a minimum-distance decoder restricted to two candidate symbols $\{\mathbf{x}_m, \mathbf{x}_j\}$ decides $j$ given that $m$ was sent. In AWGN with isotropic noise of variance $N_0/2$ , $P(m\to j) = Q(d_{m,j}/\sqrt{2N_0})$ .

Error Probability: Union Bounds and Exact Formulas

Why Bounds Matter

Definition: Pairwise Error Probability

Theorem: Union Bound on Symbol Error Probability

Error event as a union

Apply Boole's inequality

Average over $m$ and loosen to $d_{\min}$

Theorem: Nearest-Neighbor (High-SNR) Bound

Isolate the minimum-distance contribution

Asymptotic dominance

Lower bound via a single pair

Example: Exact PeP_ePe​ for BPSK and QPSK

BPSK is a pure binary problem

QPSK = two parallel BPSKs

Theorem: Exact SER for M-PSK

Angular density of the received point

Correct-decision event

High-SNR asymptotic

Example: Exact SER for Square M-QAM

Reduce to two independent PAMs

Error probability of L-PAM

Combine I and Q

Union Bound vs. Exact Monte-Carlo SER

Parameters

SER Curves for M-PSK

Parameters

SER Curves for Square M-QAM

Parameters

BER Is Not SER: The Gray-Coding Gift

High-SNR SER Approximations for Common Constellations

Common Mistake: The Loose Union Bound Exceeds 1 at Low SNR

Quick Check

Pairwise Error Probability (PEP)

Definition:
Pairwise Error Probability

Example: Exact $P_e$ for BPSK and QPSK