Part V · Chapter 15

Modulation & Demodulation

AM, FM, PM, the superheterodyne receiver, and digital modulation schemes (ASK, FSK, PSK, QAM) — the technology behind every wireless link.

1. Why Modulate?

Baseband signals (voice: 300–3400 Hz, audio: 20–20 kHz) have wavelengths too long for practical antennas. Modulation shifts information onto a high-frequency carrier \(c(t) = A_c\cos(2\pi f_c t + \phi_c)\), enabling efficient transmission. The three parameters — amplitude \(A_c\), frequency \(f_c\), and phase \(\phi_c\)— give rise to AM, FM, and PM respectively.

A quarter-wave antenna at 1 MHz would be 75 m long; at 2.4 GHz (Wi-Fi) it is just 3 cm. Modulation onto a carrier also allows frequency-division multiplexing — many channels sharing a single medium by occupying different frequency bands.

2. Amplitude Modulation (AM)

DSB-SC

\[ s(t) = A_c m(t)\cos(2\pi f_c t) \]

Double sideband, suppressed carrier. Power-efficient but requires coherent detection.

DSB-LC (Standard AM)

\[ s(t) = A_c[1+m\,x(t)]\cos(2\pi f_c t) \]

Modulation index \(m \leq 1\) prevents over-modulation. Envelope detection possible — simple receivers.

SSB

\[ s_{SSB}(t) = \tfrac{1}{2}m(t)\cos(2\pi f_c t) \]

Single sideband — half bandwidth of DSB. Most bandwidth-efficient. Used in HF voice radio.

AM Waveform & Envelope

3. Frequency Modulation (FM)

In FM, the instantaneous frequency tracks the message:

\[ f_i(t) = f_c + k_f\,x(t) \qquad s(t) = A_c\cos\!\left(2\pi f_c t + 2\pi k_f \int_0^t x(\tau)\,d\tau\right) \]

The modulation index \(\beta = \Delta f / f_m = k_f A_m / f_m\) determines bandwidth. Carson's rule estimates the occupied bandwidth: \(B_{FM} \approx 2(\Delta f + f_m) = 2f_m(1+\beta)\).

FM provides noise immunity through the capture effect and FM improvement: above the threshold SNR, FM output SNR improves as \(3\beta^2(\beta+1)\) times the baseband SNR — far better than AM for high-fidelity audio. Commercial FM uses \(\Delta f = 75\) kHz, \(f_m = 15\) kHz → \(\beta = 5\).

4. Digital Modulation

ASK

Amplitude Shift Keying

1/sym bit(s)

Simple, noise-prone

FSK

Frequency Shift Keying

1/sym bit(s)

Robust, wideband

PSK/BPSK

Phase Shift Keying

1/sym bit(s)

Good BER vs SNR

QAM-16

Quadrature Amplitude

4/sym bit(s)

High spectral eff.

QAM encodes bits in both amplitude and phase by placing symbols on a 2D constellation. QAM-16 uses 4×4 = 16 points → 4 bits/symbol. QAM-64 (6 bits/symbol) is used in LTE; QAM-256 (8 bits/symbol) in cable modems and 5G. Higher-order QAM demands better SNR:

\[ P_e \approx \frac{4(M-1)}{M\log_2 M} Q\!\left(\sqrt{\frac{3 E_s}{(M-1) N_0}}\right) \qquad \text{for M-QAM} \]

BER vs SNR — the fundamental performance metric of any digital link

5. Superheterodyne Receiver

The superheterodyne architecture, invented by Edwin Armstrong in 1918, dominates radio receiver design. Instead of amplifying at the received frequency (which changes per station), the signal is mixed with a local oscillator (LO) to produce a fixed intermediate frequency (IF)where all filtering and amplification occurs.

Antenna→RF Amp→Mixer→IF Filter→IF Amp→Detector→Audio

↑ Local Oscillator (LO) mixes with RF to produce IF = |f_RF − f_LO|

The image frequency (at \(f_{LO} + f_{IF}\) or \(f_{LO} - f_{IF}\)) must be rejected by a pre-mixer bandpass filter. The IF is typically 455 kHz (AM) or 10.7 MHz (FM). Modern software-defined radios (SDR) digitize near the antenna and perform all mixing in software.

Python Simulation

AM-DSB-LC waveform with envelope, FM signal, AM/FM spectra, QAM-16 ideal constellation, and QAM-16 with AWGN noise at 18 dB SNR.

Python

script.py134 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec

plt.style.use('dark_background')
fig = plt.figure(figsize=(14, 13), facecolor='#0a0a0a')
gs = gridspec.GridSpec(3, 2, figure=fig, hspace=0.52, wspace=0.4)

fs = 10000
t = np.linspace(0, 0.1, int(fs * 0.1), endpoint=False)
fc = 500.0   # carrier Hz
fm = 40.0    # message Hz
m_idx = 0.8  # AM modulation index

msg = np.cos(2 * np.pi * fm * t)
carrier = np.cos(2 * np.pi * fc * t)

# ── Plot 1: AM (DSB-LC) signal and envelope ───────────────────────────
ax1 = fig.add_subplot(gs[0, 0])
am_signal = (1 + m_idx * msg) * carrier
env_pos = 1 + m_idx * msg
env_neg = -(1 + m_idx * msg)
ax1.plot(t * 1e3, am_signal, color='#6ee7b7', lw=0.8, alpha=0.7, label='AM signal')
ax1.plot(t * 1e3, env_pos, color='#f87171', lw=1.5, ls='--', label='Envelope')
ax1.plot(t * 1e3, env_neg, color='#f87171', lw=1.5, ls='--')
ax1.set_title(f'AM-DSB-LC  (m={m_idx}, fc={fc} Hz)', color='#34d399', fontsize=11)
ax1.set_xlabel('Time (ms)', color='#9ca3af', fontsize=9)
ax1.set_ylabel('Amplitude', color='#9ca3af', fontsize=9)
ax1.tick_params(colors='#6b7280', labelsize=8)
ax1.legend(fontsize=8, facecolor='#111', edgecolor='#374151', labelcolor='#d1d5db')
ax1.set_xlim(0, 60)
ax1.set_facecolor('#0f0f0f')
for sp in ax1.spines.values(): sp.set_edgecolor('#374151')

# ── Plot 2: FM signal ─────────────────────────────────────────────────
ax2 = fig.add_subplot(gs[0, 1])
kf = 150.0   # Hz/V frequency deviation
phase_fm = 2 * np.pi * fc * t + 2 * np.pi * kf * np.cumsum(msg) / fs
fm_signal = np.cos(phase_fm)
ax2.plot(t * 1e3, fm_signal, color='#a78bfa', lw=0.9, label='FM signal')
ax2.plot(t * 1e3, msg * 0.8, color='#fbbf24', lw=1.5, ls='--', alpha=0.7, label='Message (scaled)')
ax2.set_title(f'FM Signal  (kf={kf} Hz/V, fc={fc} Hz)', color='#34d399', fontsize=11)
ax2.set_xlabel('Time (ms)', color='#9ca3af', fontsize=9)
ax2.set_ylabel('Amplitude', color='#9ca3af', fontsize=9)
ax2.tick_params(colors='#6b7280', labelsize=8)
ax2.legend(fontsize=8, facecolor='#111', edgecolor='#374151', labelcolor='#d1d5db')
ax2.set_xlim(0, 60)
ax2.set_facecolor('#0f0f0f')
for sp in ax2.spines.values(): sp.set_edgecolor('#374151')

# ── Plot 3: AM spectrum ───────────────────────────────────────────────
ax3 = fig.add_subplot(gs[1, 0])
for sig, col, lbl in [(am_signal, '#34d399', 'AM-DSB-LC'),
                       (msg * carrier, '#38bdf8', 'DSB-SC')]:
    N_fft = len(sig)
    freq = np.fft.rfftfreq(N_fft, 1/fs)
    S = np.abs(np.fft.rfft(sig)) * 2 / N_fft
    ax3.plot(freq, 20 * np.log10(S + 1e-6), color=col, lw=1.4, label=lbl)
ax3.set_xlim(0, 800)
ax3.set_ylim(-50, 10)
ax3.set_title('AM Spectrum (DSB-LC vs DSB-SC)', color='#34d399', fontsize=11)
ax3.set_xlabel('Frequency (Hz)', color='#9ca3af', fontsize=9)
ax3.set_ylabel('Magnitude (dB)', color='#9ca3af', fontsize=9)
ax3.tick_params(colors='#6b7280', labelsize=8)
ax3.legend(fontsize=8, facecolor='#111', edgecolor='#374151', labelcolor='#d1d5db')
ax3.set_facecolor('#0f0f0f')
for sp in ax3.spines.values(): sp.set_edgecolor('#374151')

# ── Plot 4: FM spectrum ───────────────────────────────────────────────
ax4 = fig.add_subplot(gs[1, 1])
N_fft = len(fm_signal)
freq = np.fft.rfftfreq(N_fft, 1/fs)
S_fm = np.abs(np.fft.rfft(fm_signal)) * 2 / N_fft
ax4.plot(freq, 20 * np.log10(S_fm + 1e-6), color='#a78bfa', lw=1.4, label='FM Spectrum')
ax4.set_xlim(0, 1200)
ax4.set_ylim(-60, 10)
ax4.set_title('FM Spectrum (Bessel sidebands)', color='#34d399', fontsize=11)
ax4.set_xlabel('Frequency (Hz)', color='#9ca3af', fontsize=9)
ax4.set_ylabel('Magnitude (dB)', color='#9ca3af', fontsize=9)
ax4.tick_params(colors='#6b7280', labelsize=8)
ax4.legend(fontsize=8, facecolor='#111', edgecolor='#374151', labelcolor='#d1d5db')
ax4.set_facecolor('#0f0f0f')
for sp in ax4.spines.values(): sp.set_edgecolor('#374151')

# ── Plot 5: QAM-16 constellation (ideal) ─────────────────────────────
ax5 = fig.add_subplot(gs[2, 0])
levels = [-3, -1, 1, 3]
ideal_pts = np.array([[i + 1j*j for i in levels] for j in levels]).flatten()
ax5.scatter(ideal_pts.real, ideal_pts.imag,
            color='#34d399', s=60, zorder=5, label='Ideal QAM-16')
for pt in ideal_pts:
    ax5.annotate('', xy=(pt.real, pt.imag), fontsize=6)
ax5.axhline(0, color='#374151', lw=0.8)
ax5.axvline(0, color='#374151', lw=0.8)
ax5.set_title('QAM-16 Constellation', color='#34d399', fontsize=11)
ax5.set_xlabel('In-Phase (I)', color='#9ca3af', fontsize=9)
ax5.set_ylabel('Quadrature (Q)', color='#9ca3af', fontsize=9)
ax5.tick_params(colors='#6b7280', labelsize=8)
ax5.set_xlim(-4.5, 4.5); ax5.set_ylim(-4.5, 4.5)
ax5.set_aspect('equal')
ax5.set_facecolor('#0f0f0f')
for sp in ax5.spines.values(): sp.set_edgecolor('#374151')

# ── Plot 6: QAM-16 constellation with AWGN noise ─────────────────────
ax6 = fig.add_subplot(gs[2, 1])
rng = np.random.default_rng(42)
n_sym = 400
# Pick random QAM-16 symbols
sym_idx = rng.integers(0, 16, n_sym)
tx = np.array([levels[s % 4] + 1j * levels[s // 4] for s in sym_idx])
snr_db = 18
snr_lin = 10 ** (snr_db / 10)
noise_std = np.sqrt(np.mean(np.abs(tx)**2) / (2 * snr_lin))
rx = tx + noise_std * (rng.standard_normal(n_sym) + 1j * rng.standard_normal(n_sym))
ax6.scatter(rx.real, rx.imag, color='#38bdf8', s=12, alpha=0.6, label=f'Rx (SNR={snr_db} dB)')
ax6.scatter(ideal_pts.real, ideal_pts.imag, color='#f87171', s=50, zorder=5, marker='x', lw=2, label='Ideal')
ax6.axhline(0, color='#374151', lw=0.8)
ax6.axvline(0, color='#374151', lw=0.8)
ax6.set_title(f'QAM-16 with AWGN (SNR={snr_db} dB)', color='#34d399', fontsize=11)
ax6.set_xlabel('In-Phase (I)', color='#9ca3af', fontsize=9)
ax6.set_ylabel('Quadrature (Q)', color='#9ca3af', fontsize=9)
ax6.tick_params(colors='#6b7280', labelsize=8)
ax6.legend(fontsize=8, facecolor='#111', edgecolor='#374151', labelcolor='#d1d5db')
ax6.set_xlim(-5, 5); ax6.set_ylim(-5, 5)
ax6.set_aspect('equal')
ax6.set_facecolor('#0f0f0f')
for sp in ax6.spines.values(): sp.set_edgecolor('#374151')

fig.suptitle('Modulation & Demodulation', color='#34d399', fontsize=14, fontweight='bold', y=0.98)
plt.savefig('output.png', dpi=130, bbox_inches='tight', facecolor='#0a0a0a')
print('Saved output.png')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

← Ch 14: Filters Next: Rectifiers →