Modulation

#1Why Modulation Is Necessary

A raw voice or data signal (baseband) cannot travel far on its own:

  Baseband Signal (Voice/Data)          Modulated RF Signal
  ───────────────────────────           ─────────────────────

  Amplitude
    │╲  ╱╲  ╱╲                          │    ╭─╮    ╭─╮
    │ ╲╱  ╲╱  ╲                         │   ╱   ╲  ╱   ╲     High-frequency
    │                                   │  ╱     ╲╱     ╲    carrier envelope
    └──────────────► t                  └───────────────────► t
    ~ 0–4 kHz                           ~ 700 MHz / 3.5 GHz / 28 GHz

#2The Carrier Wave

A sinusoidal carrier has three independently controllable properties:

  s(t) = A · cos(2πft + φ)

  A  = Amplitude  → controls signal strength / height of wave
  f  = Frequency  → controls how fast the wave oscillates
  φ  = Phase      → controls where in the cycle the wave starts

Modulation changes one or more of these three parameters to carry information.

#3Analog Modulation

When the message signal is analog (continuous), we speak of AM, FM, or PM.

3.1 Amplitude Modulation (AM)

The amplitude of the carrier follows the message signal.

  s_AM(t) = [A_c + m(t)] · cos(2πf_c t)

3.2 Frequency Modulation (FM)

The frequency of the carrier varies with the message signal; amplitude stays constant.

  s_FM(t) = A_c · cos(2πf_c t + 2πk_f ∫m(τ)dτ)

3.3 Phase Modulation (PM)

The phase of the carrier varies with the message signal.

  s_PM(t) = A_c · cos(2πf_c t + k_p · m(t))

Relationship: PM and FM are closely related — frequency deviation is the derivative of phase, and phase deviation is the integral of frequency deviation.

Analog Modulation Comparison

#4Digital Modulation

When the message is digital (bits), modulation maps bit patterns to discrete changes in amplitude, phase, or frequency.

4.1 Amplitude Shift Keying (ASK)

Different amplitudes represent bit 0 and bit 1.

  s_ASK(t) = A · b(t) · cos(2πf_c t),   where b(t) ∈ {0, 1}

4.2 Frequency Shift Keying (FSK)

Different frequencies represent bit 0 and bit 1.

4.3 Phase Shift Keying (PSK)

Different phases represent bit 0 and bit 1.

  s_PSK(t) = A · cos(2πf_c t + φ),   where φ depends on the bits

Digital Modulation Comparison

#5Spectral Efficiency & the Shannon Limit

Spectral efficiency measures how many bits per second we can push through each Hertz of spectrum:

                     Data Rate (bits/s)
  Spectral Efficiency = ─────────────────
                         Bandwidth (Hz)

  Units: bits/s/Hz

The Shannon-Hartley theorem defines the theoretical maximum data rate for a noisy channel:

  C = B · log₂(1 + SNR)

  C = Channel capacity (bits/s)
  B = Bandwidth (Hz)
  SNR = Signal-to-Noise Ratio (linear)

#6Quadrature Modulation & I/Q Representation

Modern digital modulation combines amplitude and phase simultaneously using In-phase (I) and Quadrature (Q) components.

  Any modulated signal can be written as:

  s(t) = I(t)·cos(2πf_c t) − Q(t)·sin(2πf_c t)

  I(t) = in-phase component
  Q(t) = quadrature component (90° shifted)

#7QAM — Quadrature Amplitude Modulation

QAM encodes bits into both amplitude and phase of the carrier. It is the dominant modulation family in modern wireless: Wi-Fi, 4G LTE, 5G NR, cable modems, optical networks.

7.1 Constellation Diagram

A constellation diagram plots each possible transmitted symbol as a point in the I/Q plane.

7.2 How Many Bits per Symbol?

  Bits per symbol = log₂(M), where M = number of constellation points

7.3 Why Square Grids?

Constellation points are arranged in a square grid because it maximizes the minimum Euclidean distance between neighboring points for a given average transmit power. Larger minimum distance → lower probability of the receiver confusing one symbol for another.

7.4 Gray Coding

Adjacent constellation points differ by only one bit, so if noise pushes a symbol across a decision boundary, only one bit is likely to be wrong.

#8M-ary Modulation Schemes

M-ary modulation uses M = 2^k symbols, where each symbol carries k bits.

#9Noise, Error Probability & BER

In practice, the received signal is corrupted by path loss, interference, and thermal noise (AWGN).

9.1 Decision Regions

The receiver divides the I/Q plane into regions. Wherever a received point lands, it is decoded as the nearest ideal constellation point. If noise pushes a point across a boundary, a symbol error occurs.

9.2 Bit Error Rate (BER)

BER is the probability that a transmitted bit is received incorrectly.

  BER ≈ Q(√(2·E_b/N_0))     for BPSK in AWGN

  E_b = Energy per bit
  N_0 = Noise power spectral density
  Q() = Q-function (tail probability of Gaussian)

Higher-order modulation needs higher SNR to achieve the same BER.

#10Modulation Across Cellular Generations

#11Adaptive Modulation & Coding (AMC)

Modern cellular systems do not use one fixed modulation. They adapt dynamically based on channel quality:

  Excellent Channel (near gNB, LOS)     Poor Channel (cell edge, fading)
  ─────────────────────────────────     ─────────────────────────────────
  High SNR → 256-QAM                    Low SNR → QPSK or 16-QAM
  8 bits/symbol                         2–4 bits/symbol
  Peak throughput                       Robust, reliable link

How AMC Works

1. UE measures channel quality → reports CQI
2. gNB looks up the best MCS for that CQI
3. gNB transmits using that MCS
4. If BLER is too high, MCS is reduced; if too low, MCS is increased

  CQI 1–6      → QPSK + heavy coding
  CQI 7–10     → 16-QAM + moderate coding
  CQI 11–14    → 64-QAM + light coding
  CQI 15       → 256-QAM + minimal coding

#12Modulation in OFDM

OFDM does not change the modulation scheme — it changes how modulation is applied.

  Serial bit stream → Split into N parallel streams → Modulate each subcarrier

  Subcarrier 0:  ━━●━━  (BPSK)
  Subcarrier 1:  ━━●━━  (QPSK)
  Subcarrier 2:  ━━●━━  (16-QAM)
  Subcarrier 3:  ━━●━━  (64-QAM)
  ...
  Subcarrier N:  ━━●━━  (QPSK)

  Combined OFDM symbol = sum of all modulated subcarriers

#135G NR Modulation Specifics

13.1 Supported Modulation Orders

13.2 π/2-BPSK

A variant of BPSK used in 5G NR uplink for coverage-limited scenarios. It reduces peak-to-average power ratio (PAPR), extending UE battery life and coverage.

13.3 256-QAM and Beyond

• LTE introduced 64-QAM.
• 5G NR introduced 256-QAM for both DL and UL.
• 5G-Advanced explores 1024-QAM for extreme throughput in ideal conditions.

#146G Outlook — Beyond Classical QAM

14.1 Probabilistic Constellation Shaping (PCS)

Instead of transmitting all constellation points equally often, PCS uses inner (low-power) points more frequently. This increases achievable rate at a given SNR, moving closer to the Shannon limit.

14.2 Geometric Shaping

Arrange constellation points in non-square geometries (circular, triangular) optimized for specific channels.

14.3 AI-Optimized Modulation

Neural networks learn optimal constellation shapes for specific channel conditions and adapt modulation in real time based on environment sensing.

14.4 THz & Optical Wireless

THz frequencies enable ultra-wide bandwidth → very high symbol rates. 4096-QAM and beyond may be practical in extremely clean LOS channels.

#15Practical Field Considerations

15.1 Common Pitfalls

15.2 Modulation vs Coding

Modulation determines how many bits per symbol; coding determines how much redundancy is added:

  Spectral Efficiency ≈ bits_per_symbol × coding_rate

  Example: 64-QAM (6 bits/symbol) × 0.85 coding rate = 5.1 bits/s/Hz

#16Further Reading

3GPP Specifications

• TS 38.211 — NR Physical Channels and Modulation
• TS 38.212 — NR Multiplexing and Channel Coding
• TS 38.213 — NR Physical Layer Procedures
• TS 36.211 — LTE Physical Channels and Modulation

Research Papers & Resources

• "Digital Communications" — John G. Proakis & Masoud Salehi
• "Wireless Communications: Principles and Practice" — Theodore S. Rappaport
• "Fundamentals of Wireless Communication" — David Tse & Pramod Viswanath
• "Probabilistic Shaping for 5G and Beyond" — IEEE Communications Surveys & Tutorials

Podcast

Beamforming Deep Dive — Telecom Leaders Podcast

Companion Resources

• Beamforming.md — Antenna arrays, beamforming, and spatial processing
• SpatialMultiplexing.md — MIMO layers, precoding, and spatial multiplexing
• UE-RACH-Procedure.md — Random access, SSB association, and power ramping