Formula Sheet: Modulation Techniques

Amplitude Modulation (AM)

Standard AM (Double-Sideband Full Carrier)

Modulated Signal: \[s_{AM}(t) = A_c[1 + m(t)]\cos(2\pi f_c t)\]

• A_c = carrier amplitude (V)
• m(t) = message signal (normalized)
• f_c = carrier frequency (Hz)
• t = time (s)

Single-Tone Modulation: \[s_{AM}(t) = A_c[1 + m_a\cos(2\pi f_m t)]\cos(2\pi f_c t)\]

• m_a = modulation index (dimensionless)
• f_m = modulating frequency (Hz)
• Condition: 0 ≤ m_a ≤ 1 for no distortion

Modulation Index: \[m_a = \frac{A_m}{A_c}\]

• A_m = message signal amplitude (V)
• A_c = carrier amplitude (V)

Alternative Expression (Peak Envelope): \[m_a = \frac{V_{max} - V_{min}}{V_{max} + V_{min}}\]

• V_max = maximum envelope voltage (V)
• V_min = minimum envelope voltage (V)

Percent Modulation: \[\text{Percent Modulation} = m_a × 100\%\]

AM Power Relations

Total Power (Single-Tone): \[P_t = P_c\left(1 + \frac{m_a^2}{2}\right)\]

• P_t = total transmitted power (W)
• P_c = carrier power (W)

Carrier Power: \[P_c = \frac{A_c^2}{2R}\]

• R = load resistance (Ω)

Sideband Power (Each): \[P_{SB} = \frac{P_c m_a^2}{4}\] Total Sideband Power: \[P_{SB,total} = \frac{P_c m_a^2}{2}\] Power Efficiency: \[\eta = \frac{P_{SB,total}}{P_t} = \frac{m_a^2}{2 + m_a^2}\]

• For m_a = 1: η_max = 33.33%

AM Bandwidth

Bandwidth (Double-Sideband): \[B_{AM} = 2f_m\]

• For complex message signal with maximum frequency f_m(max):

\[B_{AM} = 2f_{m(max)}\]

Double-Sideband Suppressed Carrier (DSB-SC)

Modulated Signal: \[s_{DSB-SC}(t) = A_c m(t)\cos(2\pi f_c t)\] Single-Tone DSB-SC: \[s_{DSB-SC}(t) = A_c A_m\cos(2\pi f_m t)\cos(2\pi f_c t)\] Expanded Form: \[s_{DSB-SC}(t) = \frac{A_c A_m}{2}[\cos(2\pi(f_c + f_m)t) + \cos(2\pi(f_c - f_m)t)]\] Total Power: \[P_{DSB-SC} = \frac{(A_c A_m)^2}{2R} = \frac{P_c m_a^2}{2}\] Power Efficiency: \[\eta_{DSB-SC} = 100\%\] Bandwidth: \[B_{DSB-SC} = 2f_m\]

Single-Sideband (SSB)

SSB Signal (Upper Sideband): \[s_{SSB}(t) = \frac{A_c m(t)}{2}\cos(2\pi f_c t) - \frac{A_c \hat{m}(t)}{2}\sin(2\pi f_c t)\]

• m̂(t) = Hilbert transform of m(t)

SSB Signal (Lower Sideband): \[s_{SSB}(t) = \frac{A_c m(t)}{2}\cos(2\pi f_c t) + \frac{A_c \hat{m}(t)}{2}\sin(2\pi f_c t)\] Single-Tone SSB: \[s_{SSB}(t) = \frac{A_c A_m}{2}\cos(2\pi(f_c ± f_m)t)\]

• + for USB (upper sideband)
• - for LSB (lower sideband)

SSB Power: \[P_{SSB} = \frac{(A_c A_m)^2}{4R} = \frac{P_c m_a^2}{4}\] Power Efficiency: \[\eta_{SSB} = 100\%\] Bandwidth: \[B_{SSB} = f_m\]

Vestigial Sideband (VSB)

Bandwidth: \[B_{VSB} = f_m + f_{vestige}\]

• f_vestige = vestigial sideband width (Hz)
• f_m <>_VSB <>_m

Angle Modulation

General Angle-Modulated Signal

General Form: \[s(t) = A_c\cos[\theta(t)] = A_c\cos[2\pi f_c t + \phi(t)]\]

• θ(t) = instantaneous phase angle (rad)
• φ(t) = phase deviation from carrier (rad)

Instantaneous Frequency: \[f_i(t) = \frac{1}{2\pi}\frac{d\theta(t)}{dt}\]

Frequency Modulation (FM)

FM Signal (Single-Tone): \[s_{FM}(t) = A_c\cos\left[2\pi f_c t + \beta\sin(2\pi f_m t)\right]\]

• β = modulation index (rad)

Frequency Deviation: \[\Delta f = k_f A_m\]

• Δf = peak frequency deviation (Hz)
• k_f = frequency deviation constant (Hz/V)
• A_m = message signal amplitude (V)

Instantaneous Frequency: \[f_i(t) = f_c + k_f m(t)\] FM Modulation Index: \[\beta = \frac{\Delta f}{f_m}\]

• Also called deviation ratio for single-tone modulation

General Message Signal: \[\beta = \frac{\Delta f_{max}}{f_{m(max)}}\] Phase Deviation: \[\phi(t) = \beta\sin(2\pi f_m t)\]

Phase Modulation (PM)

PM Signal (Single-Tone): \[s_{PM}(t) = A_c\cos[2\pi f_c t + k_p A_m\cos(2\pi f_m t)]\]

• k_p = phase deviation constant (rad/V)

PM Modulation Index: \[\beta_p = k_p A_m\] Phase Deviation: \[\phi(t) = k_p m(t)\] Relationship Between FM and PM:

• FM: φ(t) proportional to integral of m(t)
• PM: φ(t) proportional to m(t)

FM/PM Bandwidth

Carson's Rule (Practical Bandwidth): \[B_{FM} = 2(\Delta f + f_m) = 2f_m(\beta + 1)\]

• Contains approximately 98% of signal power
• Most commonly used bandwidth estimate

Narrowband FM (NBFM):

• Condition: β < 0.3="" (or="" β="" ≪="">

\[B_{NBFM} ≈ 2f_m\] Wideband FM (WBFM):

• Condition: β > 1

\[B_{WBFM} ≈ 2\Delta f = 2\beta f_m\] Universal Bandwidth Rule: \[B = 2(\beta + 2)f_m\]

• More conservative estimate, contains >99% power

Bessel Function Representation

FM Signal Expansion: \[s_{FM}(t) = A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[2\pi(f_c + nf_m)t]\]

• J_n(β) = Bessel function of first kind, order n
• n = harmonic number (integer)

Sideband Amplitude: \[A_n = A_c J_n(\beta)\] Number of Significant Sidebands: \[N ≈ \beta + 1\]

• Rule of thumb: sidebands with J_n(β) > 0.01 are significant

Bessel Function Properties:

• J_-n(β) = (-1)ⁿJ_n(β)
• J₀(0) = 1
• For small β: J₀(β) ≈ 1, J₁(β) ≈ β/2

Power Conservation: \[\sum_{n=-\infty}^{\infty}J_n^2(\beta) = 1\]

FM Power

Total Power: \[P_{FM} = \frac{A_c^2}{2R}\]

• Independent of modulation index β
• Constant envelope modulation

Power in n-th Sideband Pair: \[P_n = \frac{A_c^2 J_n^2(\beta)}{R}\]

Digital Modulation Techniques

Amplitude Shift Keying (ASK)

Binary ASK (OOK - On-Off Keying): \[s(t) = \begin{cases} A_c\cos(2\pi f_c t) & \text{for bit 1} \\ 0 & \text{for bit 0} \end{cases}\] General ASK Signal: \[s_i(t) = A_i\cos(2\pi f_c t), \quad 0 ≤ t ≤ T_b\]

• A_i = amplitude level i
• T_b = bit duration (s)

Bandwidth (ASK): \[B_{ASK} = 2R_b = \frac{2}{T_b}\]

• R_b = bit rate (bits/s)

Frequency Shift Keying (FSK)

Binary FSK Signal: \[s(t) = \begin{cases} A_c\cos(2\pi f_1 t) & \text{for bit 1} \\ A_c\cos(2\pi f_2 t) & \text{for bit 0} \end{cases}\] Frequency Separation: \[\Delta f = |f_2 - f_1|\] Minimum Frequency Spacing (Orthogonal FSK): \[\Delta f_{min} = \frac{1}{2T_b} = \frac{R_b}{2}\] Bandwidth (FSK): \[B_{FSK} = |f_2 - f_1| + 2R_b = \Delta f + 2R_b\] Alternative Expression: \[B_{FSK} = 2(\Delta f + f_m)\]

• f_m = modulating frequency = R_b/2

Phase Shift Keying (PSK)

Binary PSK (BPSK) Signal: \[s(t) = A_c\cos(2\pi f_c t + \phi_i), \quad 0 ≤ t ≤ T_b\]

• φ_i ∈ {0, π} for BPSK

\[s(t) = \begin{cases} A_c\cos(2\pi f_c t) & \text{for bit 1} \\ -A_c\cos(2\pi f_c t) & \text{for bit 0} \end{cases}\] M-ary PSK Signal: \[s_i(t) = A_c\cos\left(2\pi f_c t + \frac{2\pi(i-1)}{M}\right), \quad i = 1, 2, ..., M\]

• M = number of phase states

Phase Separation (M-PSK): \[\Delta\phi = \frac{2\pi}{M}\] Bandwidth (BPSK): \[B_{BPSK} = 2R_b\] Bandwidth (M-PSK): \[B_{MPSK} = \frac{2R_b}{\log_2 M}\]

Quadrature Phase Shift Keying (QPSK)

QPSK Signal: \[s_i(t) = A_c\cos\left(2\pi f_c t + \frac{(2i-1)\pi}{4}\right), \quad i = 1, 2, 3, 4\]

• Phase values: π/4, 3π/4, 5π/4, 7π/4 (or 45°, 135°, 225°, 315°)

Alternative Form: \[s(t) = A_I\cos(2\pi f_c t) - A_Q\sin(2\pi f_c t)\]

• A_I = in-phase component amplitude
• A_Q = quadrature component amplitude

Bandwidth (QPSK): \[B_{QPSK} = R_b = \frac{1}{T_b}\]

• Half the bandwidth of BPSK for same bit rate

Symbol Rate: \[R_s = \frac{R_b}{2}\]

• R_s = symbol rate (symbols/s)

Quadrature Amplitude Modulation (QAM)

QAM Signal: \[s(t) = A_I(t)\cos(2\pi f_c t) - A_Q(t)\sin(2\pi f_c t)\] M-ary QAM: \[s_i(t) = A_{I,i}\cos(2\pi f_c t) - A_{Q,i}\sin(2\pi f_c t)\]

• M = number of constellation points
• Common values: M = 4, 16, 64, 256

Bandwidth (M-QAM): \[B_{QAM} = \frac{2R_b}{\log_2 M}\] Symbol Rate (M-QAM): \[R_s = \frac{R_b}{\log_2 M}\]

Digital Modulation Performance

Bit Rate to Symbol Rate: \[R_b = R_s \log_2 M\] Bandwidth Efficiency: \[\eta_B = \frac{R_b}{B} = \frac{\log_2 M}{B/R_s} \text{ (bits/s/Hz)}\] Energy per Bit: \[E_b = P T_b = \frac{P}{R_b}\]

• E_b = energy per bit (J)
• P = average power (W)

Energy per Symbol: \[E_s = P T_s = \frac{P}{R_s} = E_b \log_2 M\]

• E_s = energy per symbol (J)
• T_s = symbol duration (s)

Signal-to-Noise Ratio

AM Signal-to-Noise Ratio

Output SNR (Standard AM): \[\left(\frac{S}{N}\right)_{out} = \frac{m_a^2}{2 + m_a^2}\left(\frac{S}{N}\right)_{in}\]

• For m_a = 1:

\[\left(\frac{S}{N}\right)_{out} = \frac{1}{3}\left(\frac{S}{N}\right)_{in}\] DSB-SC SNR: \[\left(\frac{S}{N}\right)_{out,DSB-SC} = \left(\frac{S}{N}\right)_{in}\] SSB SNR: \[\left(\frac{S}{N}\right)_{out,SSB} = \left(\frac{S}{N}\right)_{in}\]

FM Signal-to-Noise Ratio

FM Output SNR: \[\left(\frac{S}{N}\right)_{out,FM} = 3\beta^2(\beta + 1)\left(\frac{S}{N}\right)_{in}\] Simplified Form (β ≫ 1): \[\left(\frac{S}{N}\right)_{out,FM} ≈ 3\beta^2\left(\frac{S}{N}\right)_{in}\] FM Improvement Factor: \[I_{FM} = \frac{(S/N)_{out}}{(S/N)_{in}} = 3\beta^2(\beta + 1)\] SNR in dB: \[\text{SNR (dB)} = 10\log_{10}\left(\frac{S}{N}\right)\]

Digital Modulation BER

Bit Error Rate (BPSK): \[P_e = Q\left(\sqrt{2\frac{E_b}{N_0}}\right) = \frac{1}{2}\text{erfc}\left(\sqrt{\frac{E_b}{N_0}}\right)\]

• P_e = probability of bit error
• E_b/N₀ = energy per bit to noise power spectral density ratio
• Q(x) = complementary error function

Q-Function: \[Q(x) = \frac{1}{\sqrt{2\pi}}\int_x^{\infty}e^{-u^2/2}du\] Relationship to erfc: \[Q(x) = \frac{1}{2}\text{erfc}\left(\frac{x}{\sqrt{2}}\right)\] BER (QPSK): \[P_e ≈ Q\left(\sqrt{2\frac{E_b}{N_0}}\right)\]

• Same as BPSK for Gray coding

BER (FSK, Coherent): \[P_e = Q\left(\sqrt{\frac{E_b}{N_0}}\right)\] BER (FSK, Non-coherent): \[P_e = \frac{1}{2}e^{-E_b/(2N_0)}\]

Pulse Modulation

Pulse Amplitude Modulation (PAM)

PAM Signal: \[s_{PAM}(t) = \sum_{n=-\infty}^{\infty}m(nT_s)p(t - nT_s)\]

• m(nT_s) = sampled message value
• p(t) = pulse shape
• T_s = sampling period (s)

Sampling Frequency: \[f_s = \frac{1}{T_s}\] Nyquist Sampling Theorem: \[f_s ≥ 2f_{m(max)} = 2B\]

• B = message bandwidth (Hz)
• f_m(max) = maximum message frequency (Hz)

Pulse Width Modulation (PWM)

Pulse Width: \[\tau(t) = \tau_0 + k_{PWM} \cdot m(t)\]

• τ(t) = pulse width (s)
• τ₀ = unmodulated pulse width (s)
• k_PWM = PWM sensitivity constant

Pulse Position Modulation (PPM)

Pulse Position: \[t_p = nT_s + k_{PPM} \cdot m(nT_s)\]

• t_p = pulse position (s)
• k_PPM = PPM sensitivity constant

Pulse Code Modulation (PCM)

Number of Quantization Levels: \[L = 2^n\]

• L = number of levels
• n = number of bits per sample

Quantization Step Size: \[\Delta = \frac{V_{max} - V_{min}}{L} = \frac{V_{max} - V_{min}}{2^n}\]

• Δ = step size (V)

PCM Bit Rate: \[R_b = n \cdot f_s\]

• R_b = bit rate (bits/s)
• n = bits per sample
• f_s = sampling rate (samples/s)

Quantization Noise Power: \[P_q = \frac{\Delta^2}{12}\] Signal-to-Quantization Noise Ratio: \[\left(\frac{S}{N_q}\right) = \frac{P_s}{\Delta^2/12}\]

• P_s = signal power (W)

SNR for Uniform PCM (dB): \[\text{SNR}_{\text{dB}} = 6.02n + 1.76 \text{ dB}\]

• Approximately 6 dB improvement per bit

Delta Modulation (DM)

Delta Modulation Step: \[\Delta m = ±\delta\]

• δ = step size (V)
• ± determined by comparison with previous sample

Slope Overload Condition: \[\left|\frac{dm(t)}{dt}\right|_{max} ≤ \frac{\delta}{T_s}\]

• To avoid slope overload

Maximum Signal Frequency (to avoid slope overload): \[f_{m(max)} = \frac{\delta}{2\pi A T_s}\]

• A = signal amplitude (V)

Spread Spectrum Modulation

Direct Sequence Spread Spectrum (DSSS)

Processing Gain: \[G_p = \frac{B_{ss}}{B_i} = \frac{R_c}{R_b}\]

• G_p = processing gain (dimensionless or dB)
• B_ss = spread spectrum bandwidth (Hz)
• B_i = information bandwidth (Hz)
• R_c = chip rate (chips/s)
• R_b = bit rate (bits/s)

Processing Gain (dB): \[G_p(\text{dB}) = 10\log_{10}\left(\frac{R_c}{R_b}\right)\] Spread Spectrum Bandwidth: \[B_{ss} = 2R_c\] Jamming Margin: \[M_j = G_p - L_{\text{implementation}} - \text{Required SNR}\]

• M_j = jamming margin (dB)
• L_{implementation} = implementation losses (dB)

Frequency Hopping Spread Spectrum (FHSS)

Number of Frequency Channels: \[N = \frac{B_{ss}}{B_i}\]

• N = number of frequency channels

Hop Duration: \[T_h = \frac{k}{R_b}\]

• T_h = hop duration (s)
• k = number of bits per hop

Hopping Rate: \[R_h = \frac{1}{T_h}\]

• R_h = hopping rate (hops/s)

Processing Gain (FHSS): \[G_p = \frac{B_{ss}}{B_i}\]

Modulation Parameters Summary

Modulation Index Definitions

• AM: m_a = A_m/A_c (0 ≤ m_a ≤ 1)
• FM: β = Δf/f_m
• PM: β_p = k_pA_m

Bandwidth Summary

• AM: B = 2f_m
• DSB-SC: B = 2f_m
• SSB: B = f_m
• FM (Carson): B = 2(Δf + f_m)
• NBFM: B ≈ 2f_m
• WBFM: B ≈ 2Δf
• ASK: B = 2R_b
• FSK: B = Δf + 2R_b
• BPSK: B = 2R_b
• QPSK: B = R_b
• M-PSK: B = 2R_b/log₂M
• M-QAM: B = 2R_b/log₂M

Power Efficiency Summary

• AM (m_a = 1): η = 33.33%
• DSB-SC: η = 100%
• SSB: η = 100%