Ferkans — Interactive Telecom Tutor

The Fundamental Design Trade-off

Every modulation scheme sits at a specific point in the bandwidth efficiency vs power efficiency plane. A system designer must choose a modulation format that meets both the spectral efficiency requirement (bits/s/Hz) and the power budget (required $E_b/N_0$ ). This section maps the standard modulation formats onto this plane and compares them against the ultimate limit: the Shannon capacity bound.

Definition:
Bandwidth Efficiency

The bandwidth efficiency (or spectral efficiency) of a modulation scheme is

$\eta = \frac{R_b}{W} = \frac{\log_2 M}{1 + \alpha} \quad \text{bits/s/Hz}$

where $R_b$ is the bit rate, $W = (1+\alpha)R_s/2$ is the occupied bandwidth, and $\alpha$ is the roll-off factor.

Higher-order modulations (larger $M$ ) increase $\eta$ but require higher $E_b/N_0$ for the same BER — the fundamental trade-off in modulation design.

Definition:
Power Efficiency

The power efficiency of a modulation scheme is characterised by the required $E_b/N_0$ (in dB) to achieve a target bit error rate, typically $\text{BER} = 10^{-5}$ or $10^{-6}$ .

For $M$ -QAM in AWGN, the approximate BER is

$P_b \approx \frac{4}{\log_2 M}\left(1 - \frac{1}{\sqrt{M}}\right) Q\!\left(\sqrt{\frac{3\log_2 M}{M-1} \cdot \frac{2E_b}{N_0}}\right)$

The required $E_b/N_0$ grows logarithmically with $M$ : doubling $\eta$ costs approximately 3 dB of additional $E_b/N_0$ .

The Shannon limit on power efficiency for reliable communication at spectral efficiency $\eta$ is

$\frac{E_b}{N_0}\bigg|_{\text{Shannon}} = \frac{2^\eta - 1}{\eta}$

At the ultimate limit $\eta \to 0$ : $E_b/N_0 \to \ln 2 = -1.59$ dB.

,

Theorem: Shannon Limit on Bandwidth Efficiency

For an AWGN channel with bandwidth $W$ and received SNR $\text{SNR} = E_s / (N_0 W)$ , the channel capacity is

$C = W \log_2\!\left(1 + \text{SNR}\right) \quad \text{bits/s}$

Expressing in terms of spectral efficiency $\eta = C/W$ and $E_b/N_0 = \text{SNR}/\eta$ :

$\eta = \log_2\!\left(1 + \eta \cdot \frac{E_b}{N_0}\right)$

This implicit equation defines the Shannon boundary in the $(\eta, E_b/N_0)$ plane. No modulation scheme can operate to the left of this boundary (higher $\eta$ for a given $E_b/N_0$ ) with arbitrarily low error probability.

Any practical modulation scheme operates at a Shannon gap $\Gamma$ (in dB) to the right of the boundary. The gap depends on the constellation and coding scheme.

Shannon's theorem says you can trade bandwidth for power (and vice versa) along the capacity curve, but you cannot escape the curve. The bandwidth efficiency plane is a convenient way to visualise where each modulation sits relative to this limit.

Proof

AWGN capacity

The AWGN channel capacity $C = W\log_2(1 + P/(N_0 W))$ is derived in Chapter 11. Here $P = E_s R_s = E_b R_b$ .

Spectral efficiency form

Dividing by $W$ : $\eta = \log_2(1 + P/(N_0 W))$ .

With $P = \eta W (E_b/N_0) N_0 W / W = \eta (E_b/N_0) N_0 W$ ... more directly:

$\text{SNR} = E_s/(N_0 W) = E_b \log_2 M \cdot R_s / (N_0 W) = \eta \cdot E_b/N_0$ when $W = R_s$ (Nyquist bandwidth).

Therefore $\eta = \log_2(1 + \eta \cdot E_b/N_0)$ .

Limiting cases

As $\eta \to 0$ : $E_b/N_0 \to \ln 2 = -1.59$ dB (power-limited regime).

As $E_b/N_0 \to \infty$ : $\eta \to \infty$ (bandwidth-limited regime, impractical).

Practical systems operate at $E_b/N_0$ between 0 and 20 dB, giving $\eta$ between 0.5 and 10 bits/s/Hz. $\blacksquare$

,

Bandwidth Efficiency Plane

The bandwidth efficiency plane plots spectral efficiency $\eta$ (bits/s/Hz) against required $E_b/N_0$ (dB) for a target BER. Standard modulation schemes appear as discrete points, and the Shannon limit defines the ultimate boundary. Observe how each modulation format trades power for bandwidth efficiency.

Parameters

Target BER

Show Shannon limit

Example: Choosing Modulation for a Given Link Budget

A satellite link has available $E_b/N_0 = 12$ dB and requires a data rate of $R_b = 10$ Mbps. The allocated bandwidth is $W = 5$ MHz (with $\alpha = 0.25$ roll-off).

(a) What spectral efficiency is required?

(b) Which modulation scheme from the comparison table can meet both the spectral efficiency and $E_b/N_0$ requirements?

(c) What is the Shannon gap of the chosen scheme?

Solution

Required spectral efficiency

$\eta_{\text{required}} = R_b / W = 10/5 = 2$ bits/s/Hz.

With $\alpha = 0.25$ , the symbol rate is $R_s = 2W/(1+\alpha) = 10/1.25 = 8$ Msymbols/s.

Bits per symbol: $\log_2 M = R_b/R_s = 10/8 = 1.25$ .

Since $\log_2 M$ must be an integer for standard modulations, we need $\log_2 M \geq 2$ , i.e., $M \geq 4$ (QPSK).

Check QPSK feasibility

QPSK: $\eta = 2/(1+0.25) = 1.6$ bits/s/Hz. Achievable rate: $1.6 \times 5 = 8$ Mbps $< 10$ Mbps.

Not sufficient. Try 8-PSK: $\eta = 3/1.25 = 2.4$ bits/s/Hz. Achievable rate: $2.4 \times 5 = 12$ Mbps $> 10$ Mbps.

8-PSK requires about 13 dB at BER $= 10^{-5}$ , which exceeds our 12 dB budget.

With coding, QPSK at rate $R = 10/12.8 \approx 0.78$ code rate can achieve 10 Mbps with lower $E_b/N_0$ requirement. A coded QPSK scheme (e.g., LDPC at rate 3/4) needs approximately 4-5 dB of $E_b/N_0$ , well within the 12 dB budget.

Shannon gap

At $\eta = 2$ bits/s/Hz, the Shannon limit is $E_b/N_0|_{\text{Shannon}} = (2^2 - 1)/2 = 1.5 = 1.76$ dB.

Uncoded QPSK requires 9.6 dB, giving a gap of 7.8 dB. Coded QPSK (LDPC rate 3/4) at $\sim$ 5 dB gives a gap of 3.2 dB. Modern turbo/LDPC codes typically achieve gaps of 1-2 dB. $\blacksquare$

Modulation Formats — Efficiency Summary

Format	$\eta$ (bits/s/Hz)	$E_b/N_0$ (dB)	Constant envelope?
BPSK	1	9.6	Yes
QPSK	2	9.6	Yes
8-PSK	3	13.0	Yes
16-QAM	4	13.5	No
64-QAM	6	17.8	No
256-QAM	8	21.5	No
MSK	1	9.6	Yes
GMSK ( $BT=0.3$ )	1.35	10.0	Yes
16-APSK	4	13.2	Quasi

Why This Matters: Modulation in MIMO Systems

In a MIMO system with $n_T$ transmit antennas, the modulation problem extends from choosing a point in a one- or two-dimensional signal space to choosing a vector in a $2n_T$ -dimensional space (I and Q per antenna). The signal-space geometry of this chapter generalises directly:

Spatial multiplexing: transmit independent QAM symbols on each antenna, using the MIMO channel's spatial dimensions to multiply throughput
Space-time codes: design constellations in the $n_T \times T$ matrix space to achieve both diversity and multiplexing gain

The bandwidth efficiency plane extends to MIMO: with $n_T$ antennas and rank- $r$ channel, the spectral efficiency scales as $\eta \approx r \log_2 M$ bits/s/Hz. The MIMO book develops these extensions in full detail.

Information-Theoretic Foundations

The Shannon limit on bandwidth efficiency invoked in this section is a consequence of the AWGN channel coding theorem, which establishes that rates up to $C = W\log_2(1 + \text{SNR})$ are achievable with vanishing error probability using sufficiently long codes. The converse — that rates above $C$ are not achievable — is equally important. Both are developed with full proofs in the Information Theory and Applications (ITA) book and summarised in Chapter 11 of this book.

Key Takeaway

The Shannon gap is the additional $E_b/N_0$ (in dB) required by a practical modulation/coding scheme compared to the Shannon limit at the same spectral efficiency. Uncoded modulation typically has a gap of 8-10 dB. Modern LDPC and turbo codes reduce this gap to 1-2 dB. Closing the Shannon gap is the central goal of coded modulation design (Chapter 12).

Why This Matters: Adaptive Modulation and Coding in 5G

Modern wireless systems do not use a fixed modulation scheme. Instead, they adapt the modulation order and code rate to the instantaneous channel conditions:

When the channel is strong (high SNR): use 256-QAM with high code rate for maximum spectral efficiency
When the channel is weak (low SNR): fall back to QPSK with low code rate for reliability

In 5G NR, this is implemented through the MCS table (Modulation and Coding Scheme), which maps a Channel Quality Indicator (CQI) to a specific combination of modulation order and code rate. The MCS index ranges from 0 (QPSK, rate 0.12) to 27 (256-QAM, rate 0.93), covering spectral efficiencies from 0.23 to 7.41 bits/s/Hz.

This adaptive approach moves along the bandwidth efficiency plane in real time, tracking the Shannon boundary as closely as the finite MCS table allows.

See full treatment in Adaptive Modulation and Coding in OFDM

Quick Check

The Shannon limit at $\eta = 0$ is $E_b/N_0 = -1.59$ dB. What does this fundamental limit represent?

The minimum SNR for any reliable communication

The minimum energy per bit for reliable communication at vanishing rate

The noise figure of an ideal receiver

The coding gain of turbo codes

Correction:

The minimum energy per bit for reliable communication at vanishing rate

As spectral efficiency approaches zero (infinite bandwidth available), reliable communication requires at least $E_b/N_0 = \ln 2 = -1.59$ dB. This is the ultimate power efficiency limit set by Shannon theory.

Spectral Efficiency

The data rate achievable per unit bandwidth, measured in bits/s/Hz. For $M$ -ary modulation with roll-off $\alpha$ : $\eta = \log_2 M / (1 + \alpha)$ .

Shannon Gap

The excess $E_b/N_0$ (in dB) required by a practical modulation and coding scheme relative to the Shannon capacity limit at the same spectral efficiency. Modern codes achieve gaps of 1-2 dB.

Spectral Efficiency and Power Efficiency Trade-offs

The Fundamental Design Trade-off

Definition: Bandwidth Efficiency

Definition: Power Efficiency

Theorem: Shannon Limit on Bandwidth Efficiency

AWGN capacity

Spectral efficiency form

Limiting cases

Bandwidth Efficiency Plane

Parameters

Example: Choosing Modulation for a Given Link Budget

Required spectral efficiency

Check QPSK feasibility

Shannon gap

Modulation Formats — Efficiency Summary

Why This Matters: Modulation in MIMO Systems

Information-Theoretic Foundations

Key Takeaway

Why This Matters: Adaptive Modulation and Coding in 5G

Quick Check

Spectral Efficiency

Shannon Gap

Definition:
Bandwidth Efficiency

Definition:
Power Efficiency