Ferkans — Interactive Telecom Tutor

From Signal Space to Practical Constellations

Section 8.1 established the geometric framework. Now we populate the signal space with constellations that are used in every modern digital communication standard: PAM, QAM, and PSK. The key design goals are maximising $d_{\min}$ for a given average energy $E_s$ (to minimise error probability) and mapping bits to symbols so that the most likely errors cause the fewest bit errors (Gray mapping).

Definition:
Pulse Amplitude Modulation (M-PAM)

$M$ -ary Pulse Amplitude Modulation ( $M$ -PAM) uses $M$ equally spaced signal points on the real line:

$s_m = (2m - 1 - M)\, d, \qquad m = 1, 2, \ldots, M$

where $d = d_{\min}/2$ is half the minimum distance. The signal space is one-dimensional ( $N = 1$ ), with basis function

$\varphi_1(t) = \sqrt{\frac{2}{T_s}} \cos(2\pi f_c t), \qquad 0 \leq t \leq T_s$

The average energy per symbol is

$E_s = \frac{M^2 - 1}{3}\, d^2 = \frac{(M^2 - 1)\, d_{\min}^2}{12}$

Special cases: 2-PAM is BPSK; 4-PAM is used in PAM-4 signaling for high-speed interconnects.

Definition:
Quadrature Amplitude Modulation (M-QAM)

$M$ -ary Quadrature Amplitude Modulation ( $M$ -QAM) places $M = M_I \times M_Q$ signal points on a rectangular grid in the two-dimensional signal space. For square QAM ( $M_I = M_Q = \sqrt{M}$ ), the constellation points are

$\mathbf{s}_{m} = \begin{bmatrix} (2i - 1 - \sqrt{M})\, d \\ (2q - 1 - \sqrt{M})\, d \end{bmatrix}$

for $i, q = 1, \ldots, \sqrt{M}$ , with basis functions

$\varphi_1(t) = \sqrt{\frac{2}{T_s}} \cos(2\pi f_c t), \qquad \varphi_2(t) = -\sqrt{\frac{2}{T_s}} \sin(2\pi f_c t)$

The average energy is $E_s = \frac{2(M-1)}{3}\, d^2$ and the minimum distance is $d_{\min} = 2d$ .

Rectangular QAM decomposes into two independent PAM signals on the I and Q channels, simplifying both modulation and demodulation.

QPSK ( $M = 4$ ) is equivalent to two independent BPSK streams on the I and Q channels, doubling spectral efficiency at the same $E_b/N_0$ performance.

,

Definition:
M-ary Phase-Shift Keying (M-PSK)

$M$ -PSK places $M$ signal points equally spaced on a circle of radius $\sqrt{E_s}$ :

$\mathbf{s}_m = \sqrt{E_s} \begin{bmatrix} \cos(2\pi m/M) \\ \sin(2\pi m/M) \end{bmatrix}, \qquad m = 0, 1, \ldots, M-1$

The minimum distance is

$d_{\min} = 2\sqrt{E_s}\, \sin\!\left(\frac{\pi}{M}\right)$

All signals have the same energy $E_s$ (constant envelope), which is advantageous for power amplifier efficiency. However, for $M > 4$ , PSK has worse power efficiency than QAM because the circular arrangement wastes the interior of the constellation space.

,

Theorem: Gray Mapping Minimises BER

A Gray mapping assigns bit labels to constellation points such that adjacent points (nearest neighbours) differ in exactly one bit.

At high SNR, the dominant error event is a decision to a nearest neighbour. With Gray mapping, each such error causes exactly one bit error out of $\log_2 M$ bits, giving the approximation

$\text{BER} \approx \frac{1}{\log_2 M}\, \text{SER}$

where SER is the symbol error rate.

Without Gray mapping, a nearest-neighbour error can cause up to $\log_2 M$ bit errors, giving up to $\log_2 M$ times higher BER for the same SER.

Think of the bit labels as coordinates in a binary hypercube. Gray mapping arranges this hypercube so that geometric neighbours in the constellation are also neighbours in the hypercube (Hamming distance 1).

Proof

Nearest-neighbour dominance

At high SNR, $P(\mathbf{r} \in \mathcal{R}_k \mid s_m\text{ sent})$ is dominated by the nearest neighbours of $\mathbf{s}_m$ . For a pair at distance $d$ , the pairwise error probability is $Q(d / \sqrt{2N_0})$ , and farther pairs contribute exponentially less.

Bit counting

Under Gray mapping, each nearest-neighbour pair differs in exactly one bit. Thus the expected number of bit errors per symbol error is 1 (at high SNR), giving $\text{BER} \approx \text{SER} / \log_2 M$ .

Optimality

For constellations where every point has the same number of nearest neighbours (e.g., square QAM, PSK), Gray mapping is optimal. For irregular constellations, it may not exist, and the problem of finding the minimum-BER labeling is NP-hard in general. $\blacksquare$

,

Constellation Diagram with Noisy Symbols

Visualise the constellation diagram for various modulation schemes. Ideal constellation points are shown as large markers; received symbols after AWGN corruption are shown as scattered dots. Toggle Gray bit labels to see the labeling strategy.

Parameters

Modulation scheme

SNR (dB)20

Number of symbols1000

Show Gray bit labels

Example: QPSK Bit Mapping and Error Probability

QPSK uses four constellation points at angles $\pi/4, 3\pi/4, 5\pi/4, 7\pi/4$ with energy $E_s$ per symbol.

(a) Write the constellation points in Cartesian form.

(b) Show that QPSK has the same BER as BPSK for the same $E_b/N_0$ .

(c) What is the spectral efficiency gain of QPSK over BPSK?

Solution

Constellation points

$\mathbf{s}_m = \sqrt{E_s}\begin{bmatrix} \cos(2\pi m/4 + \pi/4) \\ \sin(2\pi m/4 + \pi/4) \end{bmatrix}, \quad m = 0, 1, 2, 3$ $Explicitly:$ (\pm\sqrt{E_s/2},, \pm\sqrt{E_s/2})$.

BER equivalence with BPSK

QPSK carries $\log_2 4 = 2$ bits per symbol, so $E_s = 2E_b$ . Each I/Q component is independent BPSK with energy $E_s/2 = E_b$ .

The BER on each component is

$P_b = Q\!\left(\sqrt{\frac{2E_b}{N_0}}\right)$

which is identical to BPSK. QPSK achieves the same BER as BPSK while transmitting twice as many bits per symbol.

Spectral efficiency

BPSK: $\eta = 1$ bit/s/Hz. QPSK: $\eta = 2$ bits/s/Hz. QPSK doubles spectral efficiency with no penalty in $E_b/N_0$ .

This is why QPSK (or its rotated variant) is the baseline modulation in virtually every modern wireless standard. $\blacksquare$

Definition:
Spectral Efficiency of M-QAM

For an $M$ -ary modulation scheme transmitting $\log_2 M$ bits per symbol at symbol rate $R_s$ , the spectral efficiency is

$\eta = \frac{R_b}{W} = \frac{R_s \log_2 M}{W} \quad \text{bits/s/Hz}$

For Nyquist signaling with ideal rectangular spectral shaping ( $W = R_s$ ), the maximum spectral efficiency is $\eta = \log_2 M$ . With a raised-cosine roll-off $\alpha$ , the bandwidth is $W = (1+\alpha)R_s$ and

$\eta = \frac{\log_2 M}{1 + \alpha}$

Comparison of Standard Modulation Schemes

Scheme	Bits/symbol	$\eta$ (bits/s/Hz)	$E_b/N_0$ for BER $= 10^{-5}$ (dB)
BPSK	1	1	9.6
QPSK	2	2	9.6
8-PSK	3	3	13.0
16-QAM	4	4	13.5
64-QAM	6	6	17.8
256-QAM	8	8	21.5

⚠️Engineering Note

ADC Resolution and PA Linearity for High-Order QAM

Moving to higher-order QAM constellations (64-QAM, 256-QAM, 1024-QAM) imposes stringent hardware requirements:

ADC resolution. The effective number of bits (ENOB) of the ADC must resolve the amplitude levels of the constellation. For $M$ -QAM with $\sqrt{M}$ levels per dimension, we need roughly $\text{ENOB} \geq \lceil\log_2\sqrt{M}\rceil + 2$ to maintain acceptable quantisation noise. For 256-QAM this means ENOB $\geq 6$ ; for 1024-QAM, ENOB $\geq 7$ . At GHz sampling rates, achieving ENOB $> 6$ is a significant design challenge.

Power amplifier linearity. Non-constant-envelope modulations (all QAM with $M > 4$ ) require linear power amplifiers. The peak-to-average power ratio (PAPR) of $M$ -QAM grows with $M$ : 16-QAM has PAPR $\approx 2.55$ dB; 64-QAM has PAPR $\approx 3.68$ dB. The PA must back off from its saturation point by at least the PAPR, reducing power efficiency. Digital pre-distortion (DPD) is used in modern base stations to linearise the PA and recover some of the lost efficiency.

EVM requirements. 3GPP TS 38.104 specifies Error Vector Magnitude limits: 17.5% for QPSK, 12.5% for 16-QAM, 8% for 64-QAM, and 3.5% for 256-QAM. These increasingly tight requirements drive the cost and complexity of RF front-ends.

Practical Constraints

•
ENOB >= 6 for 256-QAM at GHz sampling rates
•
PA backoff >= PAPR of constellation
•
EVM <= 3.5% for 256-QAM (3GPP TS 38.104)

📋 Ref: 3GPP TS 38.104

Common Mistake: QPSK vs 4-QAM — Subtleties

Mistake:

Assuming that QPSK and 4-QAM are always identical and interchangeable in all contexts.

Correction:

While QPSK and square 4-QAM have the same constellation geometry (four points at $(\pm 1, \pm 1)/\sqrt{2}$ ), they differ in standards implementations:

Bit mapping may differ: some standards define QPSK with a specific Gray labeling convention (e.g., 3GPP uses a particular mapping) that does not match a generic 4-QAM labeling.
Detection: QPSK is always detected as two independent BPSK channels (I and Q), while a general 4-point constellation could have a different geometry (e.g., diamond 4-QAM).
Terminology: in the literature, 4-QAM sometimes refers to a constellation rotated by $45°$ relative to QPSK.

Always check the standard's constellation diagram and bit mapping.

Quick Check

16-QAM transmits 4 bits per symbol. Compared to QPSK (2 bits per symbol), how much additional $E_b/N_0$ (in dB) does 16-QAM require for the same BER $= 10^{-5}$ ?

0 dB

About 4 dB

About 8 dB

About 1.5 dB

Correction:

About 4 dB

16-QAM requires about 13.5 dB vs 9.6 dB for QPSK, a difference of approximately 3.9 dB. This is the power penalty for doubling spectral efficiency from 2 to 4 bits/s/Hz.

Pulse Amplitude Modulation (PAM)

A one-dimensional modulation scheme that encodes information in the amplitude of a pulse. $M$ -PAM uses $M$ equally spaced amplitude levels, transmitting $\log_2 M$ bits per symbol.

Quadrature Amplitude Modulation (QAM)

A two-dimensional modulation scheme that places constellation points on a rectangular grid using independent amplitude modulation on the in-phase and quadrature carriers. $M$ -QAM transmits $\log_2 M$ bits per symbol.

Gray Mapping

A binary labeling of constellation points such that nearest neighbours differ in exactly one bit, minimising the number of bit errors per symbol error at high SNR.

Constellation Diagram

A plot of the signal-space coordinates of all $M$ constellation points. The horizontal axis is the in-phase (I) component and the vertical axis is the quadrature (Q) component.

Historical Note: Evolution of QAM

1960-present

QAM was developed in the 1960s for telephone-line modems. C. R. Cahn (1960) and J. C. Hancock (1960) independently proposed combined amplitude-phase modulation. The first practical implementations appeared in the V.29 modem standard (1976) using 16-QAM at 9600 bps. Today, 256-QAM is standard in LTE, 1024-QAM is used in Wi-Fi 6, and 4096-QAM is specified in Wi-Fi 7 — a testament to advances in signal processing and receiver design.

Pulse Amplitude Modulation (PAM) and QAM