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PE Exam Exam  >  PE Exam Notes  >  Electrical & Computer Engineering for PE  >  Cheatsheet: Modulation Techniques

Cheatsheet: Modulation Techniques

1. Amplitude Modulation (AM)

1.1 Basic AM Principles

Parameter Definition
AM Signal s(t) = Ac[1 + m(t)]cos(2πfct) where m(t) is message signal
Modulation Index μ = Am/Ac where Am is message amplitude, Ac is carrier amplitude
Percent Modulation %M = μ × 100%; must be ≤100% to avoid overmodulation
Bandwidth BW = 2fm where fm is maximum message frequency

1.2 AM Power Relations

Parameter Formula
Carrier Power Pc = Ac2/(2R)
Sideband Power PSB = (μ2/4)Pc (each sideband = μ2Pc/4)
Total Power Pt = Pc(1 + μ2/2)
Efficiency η = μ2/(2 + μ2) × 100%; maximum 33.33% at μ = 1

1.3 AM Variants

Type Characteristics
DSB-FC (Double Sideband Full Carrier) Standard AM; both sidebands + carrier; inefficient but simple demodulation
DSB-SC (Double Sideband Suppressed Carrier) Carrier suppressed; 100% efficient; requires coherent detection
SSB (Single Sideband) One sideband only; BW = fm; power efficient; complex generation
VSB (Vestigial Sideband) One full sideband + partial second sideband; used in analog TV

2. Frequency Modulation (FM)

2.1 FM Fundamentals

Parameter Definition
FM Signal s(t) = Accos[2πfct + 2πkf∫m(t)dt]
Instantaneous Frequency fi(t) = fc + kfm(t)
Frequency Deviation Δf = kfAm where Am is message amplitude
Modulation Index β = Δf/fm where fm is message frequency
Deviation Ratio D = Δfmax/fm(max)

2.2 FM Bandwidth

Method Formula
Carson's Rule BW ≈ 2(Δf + fm) = 2fm(β + 1)
Universal Curve (98% power) BW = 2Δf(1 + 1/β) for β >> 1; BW = 2fm for β <>
Narrowband FM (NBFM) β < 0.5;="" bw="" ≈="">m (similar to AM)
Wideband FM (WBFM) β > 1; BW ≈ 2Δf

2.3 FM Spectrum and Bessel Functions

  • FM spectrum contains carrier and infinite sidebands at fc ± nfm (n = 1, 2, 3...)
  • Sideband amplitudes given by Bessel functions Jn(β)
  • Carrier amplitude: AcJ0(β); nth sideband pair amplitude: AcJn(β)
  • Total power constant regardless of modulation: Pt = Ac2/(2R)
  • Number of significant sidebands ≈ β + 1

2.4 FM vs PM Comparison

Aspect FM (Frequency Modulation)
Varying Parameter Instantaneous frequency proportional to m(t)
Phase Deviation φ(t) = 2πkf∫m(t)dt
Noise Performance Superior noise immunity; SNR improvement with increased Δf
Applications FM radio (88-108 MHz), TV audio, two-way radio

2.5 FM Demodulation

  • Frequency discriminator: converts frequency variations to amplitude variations
  • Phase-locked loop (PLL): tracks carrier frequency using VCO
  • Slope detector: uses tuned circuit on linear portion of response
  • Ratio detector: amplitude limiting with balanced output

3. Phase Modulation (PM)

3.1 PM Fundamentals

Parameter Definition
PM Signal s(t) = Accos[2πfct + kpm(t)]
Phase Deviation Δφ = kpAm where kp is phase sensitivity (rad/V)
Modulation Index β = Δφ = kpAm
Instantaneous Phase φi(t) = 2πfct + kpm(t)

3.2 PM Characteristics

  • Bandwidth: BW ≈ 2(Δφmax + 1)fm using Carson's rule
  • PM index independent of message frequency (unlike FM)
  • FM from PM: integrate message signal before phase modulation
  • PM from FM: differentiate message signal before frequency modulation
  • Phase deviation proportional to message amplitude, not frequency

4. Digital Modulation Techniques

4.1 Amplitude Shift Keying (ASK)

Parameter Description
Signal s(t) = Aicos(2πfct) where Ai ∈ {A1, A2, ..., AM}
Binary ASK (OOK) On-Off Keying: A1 = A for '1', A0 = 0 for '0'
Bandwidth BW = 2Rb where Rb is bit rate (bps)
Error Probability Pe = (1/2)erfc(√(Eb/(2N0))) for coherent detection

4.2 Frequency Shift Keying (FSK)

Parameter Description
Binary FSK Signal s(t) = Acos(2πfit) where fi = fc ± Δf
Frequency Separation Minimum orthogonal spacing: Δf = n/(2Tb) where n is integer, Tb is bit duration
Bandwidth BW = 2(Δf + Rb) for binary FSK
Error Probability Pe = (1/2)erfc(√(Eb/(2N0))) for coherent; Pe = (1/2)e-Eb/(2N0) for non-coherent
M-ary FSK M frequency tones; Rb = (log2M)/Ts

4.3 Phase Shift Keying (PSK)

4.3.1 Binary PSK (BPSK)

Parameter Description
Signal s(t) = Acos(2πfct + φi) where φi ∈ {0, π}
Bandwidth BW = 2Rb
Error Probability Pe = (1/2)erfc(√(Eb/N0))
Symbol States 2 phase states separated by 180°

4.3.2 Quadrature PSK (QPSK)

Parameter Description
Phase States φi ∈ {π/4, 3π/4, 5π/4, 7π/4} or {0, π/2, π, 3π/2}
Bits per Symbol 2 bits/symbol; symbol rate = Rb/2
Bandwidth BW = Rb (half of BPSK for same bit rate)
Error Probability Pe ≈ erfc(√(Eb/N0)) for high SNR

4.3.3 M-ary PSK

  • Phase states: φi = 2π(i-1)/M for i = 1, 2, ..., M
  • Bits per symbol: k = log2M
  • Symbol rate: Rs = Rb/k
  • Bandwidth: BW = 2Rb/log2M
  • Symbol error: Ps ≈ 2erfc(√(Es/N0)sin(π/M)) for high SNR

4.4 Quadrature Amplitude Modulation (QAM)

Parameter Description
Signal s(t) = AIcos(2πfct) - AQsin(2πfct)
Constellation M amplitude/phase combinations on I-Q plane
Common Forms 16-QAM, 64-QAM, 256-QAM (M = 2k)
Bits per Symbol k = log2M
Bandwidth BW = 2Rb/log2M
Applications Cable modems, digital TV, wireless systems (LTE, WiFi)

4.5 Differential PSK (DPSK)

  • Information encoded in phase transitions, not absolute phase
  • Binary DPSK: phase shift of 0° for '0', 180° for '1' relative to previous symbol
  • No carrier phase recovery required
  • Error probability: Pe = (1/2)e-Eb/N0
  • π/4-DQPSK: phase changes limited to ±π/4, ±3π/4; reduces envelope variations

4.6 Minimum Shift Keying (MSK)

Parameter Description
Frequency Separation Δf = 1/(4Tb); minimum spacing for orthogonality
Phase Continuity Continuous phase FSK; no phase discontinuities
Modulation Index h = 0.5
Bandwidth BW = 1.5Rb (99% power); more compact than FSK
Error Probability Pe = (1/2)erfc(√(Eb/N0))

4.7 Gaussian MSK (GMSK)

  • MSK with Gaussian pulse shaping filter before modulation
  • Filter bandwidth: BT product (B = 3-dB bandwidth, T = bit duration)
  • BT = 0.3 for GSM; reduces spectral sidelobes
  • Further bandwidth reduction compared to MSK
  • Slight performance degradation due to ISI from filtering

5. Performance Metrics

5.1 Key Performance Parameters

Metric Definition
Bit Error Rate (BER) Pe = (number of bit errors)/(total bits transmitted)
Symbol Error Rate (SER) Ps = (number of symbol errors)/(total symbols transmitted)
Eb/N0 Energy per bit to noise power spectral density ratio
Es/N0 Energy per symbol to noise ratio; Es = Eb × log2M
SNR Signal-to-Noise Ratio = Ps/Pn

5.2 Bandwidth Efficiency

Parameter Formula
Spectral Efficiency η = Rb/BW (bits/s/Hz)
M-ary Improvement η = log2M/2 for PSK/QAM with Nyquist filtering

5.3 Power Efficiency Comparison

Modulation Approximate Eb/N0 for BER = 10-5
BPSK 9.6 dB
QPSK 9.6 dB
8-PSK 14 dB
16-QAM 14.5 dB
64-QAM 20 dB
FSK (coherent) 13.5 dB
FSK (non-coherent) 14.5 dB

6. Pulse Shaping and ISI

6.1 Intersymbol Interference (ISI)

  • ISI occurs when symbols overlap due to channel dispersion or filtering
  • Nyquist criterion for zero ISI: H(f) = T for |f| ≤ 1/(2T), with specific symmetry
  • Raised cosine filter satisfies Nyquist criterion

6.2 Raised Cosine Filter

Parameter Description
Roll-off Factor α (0 ≤ α ≤ 1); controls bandwidth vs. time-domain decay trade-off
Bandwidth BW = (1 + α)/(2T) = Rs(1 + α)/2
α = 0 Ideal brick-wall (sinc pulse); minimum BW = 1/(2T); impractical
α = 1 Maximum excess BW = 100%; faster time decay; easier implementation
Root Raised Cosine Split between transmitter and receiver; matched filtering

6.3 Eye Diagram Interpretation

  • Eye opening height: noise margin
  • Eye opening width: timing margin and jitter tolerance
  • Eye closure: indicates ISI
  • Optimal sampling time: center of widest eye opening
  • Slope at zero crossing: sensitivity to timing errors

7. Advanced Modulation Concepts

7.1 Orthogonal Frequency Division Multiplexing (OFDM)

Parameter Description
Principle Multiple orthogonal subcarriers with overlapping spectra
Subcarrier Spacing Δf = 1/Ts where Ts is symbol duration
Implementation IFFT at transmitter, FFT at receiver
Cyclic Prefix Guard interval to combat ISI and maintain orthogonality; length ≥ channel delay spread
Applications WiFi (802.11a/g/n/ac/ax), LTE, 5G NR, DVB-T, DSL

7.2 Spread Spectrum Techniques

7.2.1 Direct Sequence Spread Spectrum (DSSS)

  • Data multiplied by high-rate pseudo-noise (PN) sequence
  • Processing gain: Gp = BWspread/BWdata = Rc/Rb
  • Chip rate Rc >> bit rate Rb
  • Interference rejection and low probability of intercept
  • Used in GPS, CDMA cellular systems

7.2.2 Frequency Hopping Spread Spectrum (FHSS)

  • Carrier frequency hops according to PN sequence
  • Fast hopping: multiple hops per symbol
  • Slow hopping: multiple symbols per hop
  • Resistance to narrowband interference and jamming
  • Used in Bluetooth, legacy 802.11

7.3 Continuous Phase Modulation (CPM)

  • Constant envelope modulation with continuous phase trajectory
  • Modulation index h = Δf × T (frequency deviation × symbol period)
  • Full response CPM: phase influenced by all previous symbols
  • Partial response CPM: finite memory length
  • MSK is special case of CPM with h = 0.5

7.4 Constellation Diagrams

Aspect Interpretation
Symbol Points Represent ideal transmitted symbols in I-Q plane
Distance Between Points Related to noise immunity; larger distance = better BER
Gray Coding Adjacent symbols differ by 1 bit; minimizes bit errors
Scatter Plot Received symbols show noise and distortion effects

8. Carrier and Symbol Synchronization

8.1 Carrier Recovery

Method Application
Squaring Loop BPSK: square signal to create 2fc component, divide by 2
Costas Loop BPSK/QPSK: phase-locked loop with I-Q structure
Fourth Power Loop QPSK: raise signal to 4th power to remove modulation
Decision-Directed Use symbol decisions to estimate phase error

8.2 Symbol Timing Recovery

  • Early-Late Gate: compare energy in early vs. late sampling windows
  • Zero-Crossing Detection: detect symbol transitions
  • Maximum Likelihood: Gardner detector for timing error estimation
  • Mueller-Muller: decision-directed timing recovery

9. Modulation Selection Criteria

9.1 Design Trade-offs

Criterion Considerations
Power Efficiency BPSK/QPSK best; higher-order QAM requires more power for same BER
Bandwidth Efficiency Higher-order modulation (QAM, M-PSK) more efficient; trade-off with power
Complexity Coherent detection more complex than non-coherent; higher M increases complexity
Constant Envelope FM, FSK, MSK, CPM allow efficient non-linear amplifiers
Noise Immunity FM superior in analog; BPSK best in digital for same power

9.2 Application Examples

Application Modulation Used
AM Radio DSB-FC AM (535-1605 kHz)
FM Radio WBFM (88-108 MHz, Δf = 75 kHz)
GSM GMSK (BT = 0.3)
LTE QPSK, 16-QAM, 64-QAM (OFDM)
5G NR Up to 256-QAM (OFDM)
WiFi 6 (802.11ax) BPSK to 1024-QAM (OFDM)
Satellite QPSK, 8-PSK (power limited)
Cable Modem 64-QAM, 256-QAM (DOCSIS)

10. Important Formulas Summary

10.1 Analog Modulation

Parameter Formula
AM Total Power Pt = Pc(1 + μ2/2)
AM Efficiency η = μ2/(2 + μ2)
FM Modulation Index β = Δf/fm
Carson's BW Rule BW = 2(Δf + fm)

10.2 Digital Modulation

Parameter Formula
Bit Rate Rb = (log2M)/Ts bits/s
Symbol Rate Rs = 1/Ts = Rb/log2M symbols/s
Eb/N0 to SNR Eb/N0 = SNR × (BW/Rb)
Es to Eb Es = Eb × log2M
BPSK BER Pe = (1/2)erfc(√(Eb/N0))
Spectral Efficiency η = Rb/BW bits/s/Hz
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Apr 20, 2026 Last updated
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