UNIT 4: Analog & Digital Communication - Exam-Focused Short Notes

I. SIGNALS & SYSTEMS FUNDAMENTALS

A. Signal Classification & Operations

1. Signal Classification:

• Continuous-Time vs. Discrete-Time: Defined by the nature of the independent variable (time t continuous or discrete n).

• Deterministic vs. Random: Predictable exactly vs. described by statistical properties.

• Periodic vs. Aperiodic: x(t) = x(t+T) for some T>0 vs. no such T.

• Energy vs. Power Signals:

  • Energy Signal: Total energy E = ∫_{-∞}^{∞} |x(t)|² dt is finite and average power P=0. E.g., rectangular pulse.

  • Power Signal: Average power P = lim_{T→∞} (1/2T) ∫_{-T}^{T} |x(t)|² dt is finite and energy E=∞. E.g., sinusoid, periodic signals.

  [!TIP] Proof for Rectangular Pulse: For x(t) = A for |t| ≤ T₀/2, E = A²T₀ (finite), P = 0. Hence, energy signal.

2. Basic Signal Operations on x(t):

• Time-shifting: y(t) = x(t - t₀) → shift right by t₀.

• Time-scaling: y(t) = x(at) → compression if |a|>1, expansion if |a|<1.

• Time-reversal: y(t) = x(-t) → mirror about y-axis.

• Example: For y(t)=x(2t+3), first shift left by 3 (x(t+3)), then compress by 2 (x(2(t+3))).

3. Standard Signals:

• Unit Impulse δ(t): ∫_{-∞}^{∞} δ(t)φ(t) dt = φ(0) (sifting property). δ(at) = (1/|a|)δ(t).

• Unit Step u(t): u(t) = 1 for t≥0, 0 for t<0. du(t)/dt = δ(t).

• Rectangular Pulse (Gate Function): rect(t/T) = 1 for |t|≤T/2, else 0.

• Triangular Pulse: Λ(t/T).

• Signum sgn(t): +1 for t>0, -1 for t<0, 0 at t=0.

B. System Classification

• Linear: Superposition holds (Additivity + Homogeneity). E.g., y(t) = ax(t) + bx₁(t).

• Non-linear: E.g., y(t) = x²(t).

• Time-invariant (TI): A shift in input causes identical shift in output. x(t-t₀) → y(t-t₀). Test: x(t) → y(t), check if y(t-t₀) equals response to x(t-t₀).

• Time-variant: E.g., y(t) = t·x(t).

• Causal: Output depends only on present & past inputs. y(t₀) depends on x(τ) for τ ≤ t₀. For LTI, h(t)=0 for t<0.

• Non-causal: E.g., y(t) = x(t+1) (future input).

• BIBO Stable: Every bounded input yields bounded output. For LTI, necessary & sufficient: ∫_{-∞}^{∞} |h(t)| dt < ∞.

• LTI System: Characterized by Impulse Response h(t). Output y(t) = x(t) * h(t). Frequency Response H(ω) = F{h(t)}.

C. Fourier Transform (FT) & Properties

Definition: X(ω) = ∫_{-∞}^{∞} x(t) e^{-jωt} dt. Inverse: x(t) = (1/2π) ∫_{-∞}^{∞} X(ω) e^{jωt} dω. Physical Significance: Reveals frequency content (amplitude & phase spectrum) of a signal.

FT of Standard Signals:

1. Unit Step u(t): U(ω) = πδ(ω) + (1/jω).

2. Gate Function (Rectangular Pulse): x(t) = A·rect(t/τ) → X(ω) = Aτ sinc(ωτ/2).

3. Exponential: x(t)=e^{-at}u(t) (Re(a)>0) → X(ω)=1/(a+jω).

4. Sinusoid: cos(ω₀t) ↔ π[δ(ω-ω₀) + δ(ω+ω₀)].

Key Properties (State & Prove where asked):

[!TIP] Time-scaling Proof: F{x(at)} = ∫ x(at) e^{-jωt} dt. Substitute τ=at, dt=dτ/a. Get (1/|a|) ∫ x(τ) e^{-j(ω/a)τ} dτ = (1/|a|)X(ω/a).

II. ANALOG MODULATION: AMPLITUDE MODULATION (AM)

A. AM Theory & Generation

Expression (Single-tone): s_AM(t) = A_c[1 + m_a cos(ω_m t)] cos(ω_c t)

• A_c: Carrier amplitude.

• m_a: Modulation Index (0 ≤ m_a ≤ 1 for linear modulation).

• ω_m, ω_c: Modulating & carrier angular frequencies.

Modulation Index (m_a):

m_a = (V_max - V_min) / (V_max + V_min) from envelope.

• m_a = 0 → No modulation (pure carrier).

• 0 < m_a < 1 → Linear modulation.

• m_a > 1 → Over-modulation (envelope distortion, carrier reversal).

Generation Methods:

1. Square-law Modulator: Uses non-linear device (diode) with i = a₁v + a₂v². The v² term produces cos²(ω_c t) → (1+cos(2ω_c t))/2, generating sidebands. Limitation: Only works for small m_a (<0.5) to avoid higher-order terms.

2. Switching Modulator: Uses diode as switch (ON/OFF). Input [A_c + m(t)] multiplied by sgn(cos(ω_c t)). Output is ±[A_c + m(t)]. Fourier series of switching function yields carrier & all odd harmonics. Filtering extracts carrier & sidebands → DSB-SC.

3. Balanced Modulator: Two switching modulators in push-pull. Cancels carrier, outputs DSB-SC signal: s(t) = m(t) cos(ω_c t).

B. AM Spectrum & Power Relations

Spectrum: Carrier at f_c, Upper Sideband (USB) at f_c+f_m, Lower Sideband (LSB) at f_c-f_m. Bandwidth: BW = 2f_m (twice the highest modulating frequency).

Power Calculation:

s_AM(t) = A_c cos(ω_c t) + (m_a A_c/2) cos((ω_c-ω_m)t) + (m_a A_c/2) cos((ω_c+ω_m)t)

• Carrier Power: P_c = (A_c²)/2R (assuming load R).

• Sideband Power (each): P_sb = (m_a² A_c²)/(8R).

• Total Power: P_T = P_c + 2P_sb = (A_c²/2R)(1 + m_a²/2).

• Transmission Efficiency: η = (2P_sb / P_T) × 100% = [m_a² / (2 + m_a²)] × 100%.

[!TIP] Efficiency Limitation: Max η = 50% at m_a=1. Improvement: Use DSB-SC (no carrier, 100% efficiency) or SSB (only one sideband, 50% efficiency of DSB-SC).

C. AM Demodulation & Receivers

1. AM Envelope Detector:

• Circuit: Diode + RC low-pass filter.

• Operation: Diode rectifies. RC time constant τ = RC must satisfy:

  1. 1/ω_c ≪ τ ≪ 1/ω_m (discharge slow enough to follow envelope, fast enough to discharge between carrier peaks).

  2. For m_a ≤ 1: τ ≥ 1/ω_c and τ ≤ (1+ m_a)/(ω_c m_a).

• Output: v_o(t) ≈ A_c[1 + m_a cos(ω_m t)] (envelope).

[!TIP] Distortion occurs if: m_a > 1 (over-modulation) or τ too large (diode doesn't conduct every cycle) or too small (ripple not filtered).

2. Receiver Architectures:

3. Receiver Performance Parameters:

• Sensitivity: Minimum input signal power required for acceptable output SNR. Sensitivity ∝ (SNR_required) × (k T B F).

• Selectivity: Ability to separate adjacent channels. Measured by bandwidth of IF filter at -60 dB or -3 dB.

• Fidelity: Accuracy of audio reproduction. Limited by IF & audio bandwidth and linearity.

III. ANGLE MODULATION: FREQUENCY & PHASE MODULATION

A. Frequency Modulation (FM) - Theory

Expression (Single-tone): s_FM(t) = A_c cos[ω_c t + β sin(ω_m t)]

• β = k_f A_m / ω_m = Δf / f_m is Modulation Index.

• Δf = k_f A_m is Peak Frequency Deviation.

• Deviation Ratio: D = Δf_max / f_m_max (for multi-tone). Determines bandwidth category (Narrowband if D<<1, Wideband if D>>1).

Spectrum & Bandwidth (Carson's Rule):

BW ≈ 2(Δf + f_m) = 2f_m (β + 1). More precisely, infinite sidebands with amplitudes given by Bessel functions J_n(β).

[!TIP] Carson's Rule gives ~98% power. For β=5, BW ≈ 12f_m (vs. 2f_m for AM).

B. FM Generation (Modulators)

1. Direct Method (VCO/Reactance Modulator): f_out = f_c + k_f m(t). Use voltage-controlled oscillator (varactor diode in LC tank). Advantage: Simple, wideband possible. Disadvantage: Poor frequency stability.

2. Indirect Method (Armstrong): First generate PM (s_PM = A_c cos(ω_c t + k_p m(t))), then use frequency multipliers (non-linear devices) to increase Δf and f_c. β_FM = n·β_PM after n-th multiplier. Advantage: High stability (uses stable crystal oscillator for carrier). Disadvantage: Complex, bandwidth expansion.

3. Using PLL: VCO output fed back via phase detector. Modulating signal controls VCO frequency indirectly via loop. Provides stable FM with good suppression of VCO noise.

C. FM Demodulation

Principle: Convert instantaneous frequency variations to amplitude variations.

1. Discriminators:

• Foster-Seeley Discriminator:

  • Circuit: Tuned transformer (primary: RF input; secondary: two tuned circuits at f_c & f_c + Δf, 180° out of phase) → diodes → load.

  • Operation: At f_c, secondary voltages cancel (zero output). For f > f_c or f < f_c, unbalance occurs → DC output proportional to Δf.

  • Phase Diagram: Shows vector sum of two secondary voltages.

• Ratio Detector:

  • Circuit: Similar to Foster-Seeley but secondary winding center-tapped, diodes in series with capacitor C₁, C₂ in parallel with load.

  • Operation: Output taken across C₁ & C₂ in series. No limiter needed (inherently limits amplitude variations). Output v_o ∝ (f - f_c)/(f + f_c) (linear for small Δf).

  • Advantage over Foster-Seeley: Insensitive to amplitude noise, no separate limiter.

• Balanced Slope Detector: Simple version: two tuned circuits (one at f_c - Δf, one at f_c + Δf) with opposite slopes. Their outputs subtracted.

2. Pre-emphasis & De-emphasis:

• Need: FM has threshold effect; noise power increases with frequency (1/f² noise from demodulator). High-freq components of modulating signal are more degraded.

• Solution: Pre-emphasize (boost) high-freq components at transmitter, de-emphasize (attenuate) at receiver.

• Transfer Functions (RC networks):

  • Pre-emphasis (High-pass): H_pre(f) = 1 + j2πfτ (τ = 75 μs for FM radio). Gain increases at 6 dB/octave above f_c = 1/(2πτ).

  • De-emphasis (Low-pass): H_de(f) = 1 / (1 + j2πfτ). Inverse of pre-emphasis.

  • Overall: Flat response, but SNR improvement at high frequencies by factor (1 + (2πfτ)²).

IV. DIGITAL COMMUNICATION

A. Sampling & Quantization (PCM)

1. Sampling Theorem for Low-pass Signals:

• Statement: A bandlimited signal with f_max Hz can be reconstructed perfectly from its samples if sampled at f_s ≥ 2f_max (Nyquist rate).

• Proof Sketch: x(t) bandlimited to [-W, W] → X(f)=0 for |f|>W. Sampling at f_s=2W → X_s(f) is periodic with period 2W. No overlap if f_s ≥ 2W. Ideal reconstruction via low-pass filter: x(t) = Σ x(nT_s) sinc((t-nT_s)/T_s).

[!TIP] Aliasing: If f_s < 2f_max, spectra overlap → irreversible distortion. Anti-aliasing filter (LPF with f_c ≈ f_max) needed before sampling.

2. Pulse Code Modulation (PCM):

• Block Diagram: Sampler → Quantizer → Encoder → Channel.

• Quantization: Map continuous amplitude to finite levels.

  • Uniform Quantization: Step size Δ constant. Range [-V_max, V_max] → L = 2V_max/Δ levels.

  • Quantization Error/Noise: e_q = x_q - x, uniformly distributed in [-Δ/2, Δ/2] for large L.

  • Power: P_q = Δ²/12.

• Signal-to-Quantization-Noise Ratio (SQNR):

  • For sine wave x(t)=A sin(ωt), P_signal = A²/2.

  • L ≈ 2A/Δ → Δ ≈ 2A/L.

  • SQNR = P_signal / P_q = (A²/2) / (Δ²/12) = 3A²/Δ² = 3L²/4.

  • In dB: SQNR_dB = 10 log₁₀(3L²/4) ≈ 10 log₁₀(L²) = 20 log₁₀(L).

  • Since L = 2^n (n bits), SQNR_dB ≈ 6.02n + 1.76 dB.

[!TIP] SQNR increases by ~6 dB per bit. Non-uniform quantizers (μ-law, A-law) improve SQNR for low-amplitude signals (used in telephony).

B. Delta Modulation (DM) & Adaptive DM

1. Delta Modulation (DM):

• Principle: 1-bit quantizer. Compares current sample with previous quantized value + step size Δ.

• Block Diagram: Integrator (accumulator) → 1-bit quantizer (comparator) → Encoder. Output 1 if x(t) > x_q(t-Δt), else 0.

• Waveforms: Staircase approximation of input.

• Errors:

  • Slope Overload: Input slope > Δ/Δt → tracking error (large).

  • Granular Noise: Input slope small → idle channel noise (chatter).

[!TIP] DM vs PCM: DM uses 1-bit/sample (low rate) but requires high f_s (≥ 2f_max for no slope overload). PCM uses n bits/sample.

2. Adaptive Delta Modulation (ADM):

• Principle: Variable step size Δ adapted based on recent output bits.

  • If previous bits 11 → increase Δ (reduce slope overload).

  • If previous bits 00 or 01 → decrease Δ (reduce granular noise).

• Advantages over DM: Better performance for both high & low slope signals, reduced average power, improved SQNR for same bit rate.

C. Digital Passband Modulation

1. Binary Modulation:

2. Differential Modulation:

• DPSK (Differential Phase Shift Keying):

  • Principle: Encode bit b_k as phase change relative to previous bit: θ_k = θ_{k-1} + π(1 - b_k).

  • Transmitter: s(t) = A_c cos(ω_c t + θ_k) during kT_b ≤ t < (k+1)T_b.

  • Receiver: Differential Detector (delay line T_b + multiplier + integrator). Compares phase of two successive symbols.

  • Advantage over BPSK: No absolute carrier phase synchronization needed. Only phase difference matters. Disadvantage: ~3 dB worse SNR for same BER.

  [!TIP] DBPSK Reliability Example: If carrier phase slips by 180° during transmission, BPSK receiver flips all bits (catastrophic). DPSK receiver unaffected (only differences matter).

3. M-ary Modulation:

• M-ary PSK (MPSK):

  • M phases equally spaced on circle: θ_k = (2πk)/M, k=0,...,M-1.

  • Bandwidth Efficiency: η = (log₂M) / (2) bits/s/Hz (for coherent MPSK).

  • Comparison with BPSK: Higher data rate for same BW, but BER increases with M for given E_b/N₀. Requires better SNR.

• Quadrature Amplitude Modulation (QAM):

  • Principle: Two orthogonal carriers (cos(ω_c t), sin(ω_c t)), each ASK-modulated by I & Q signals. s(t) = I(t)cos(ω_c t) - Q(t)sin(ω_c t).

  • Generation & Detection (Coherent):

    • Tx: Split data → two streams → ASK modulators (I & Q) → summer.

    • Rx: Coherent mixers with cos(ω_c t) & sin(ω_c t) → LPF → decision devices.

  • Constellation Diagram: Points in I-Q plane. E.g., 16-QAM (4×4 grid), 64-QAM (8×8 grid).

  • Comparison with QPSK: QPSK is 4-PSK (points on circle, constant amplitude). QAM has both amplitude & phase states → higher BW efficiency (same M as QPSK? No: 4-QAM = QPSK). For M>4, QAM more BW-efficient but more susceptible to amplitude noise & non-linearities.

• Offset QPSK (OQPSK):

  • Concept: In QPSK, I & Q bits change simultaneously → 180° phase jumps possible. In OQPSK, Q channel delayed by T_b relative to I. So only one bit changes at a time → max phase shift = 90°.

  • Advantage over QPSK: Reduced amplitude fluctuations during phase transitions → lower out-of-band spectrum (smaller sidelobes), better for non-linear amplifiers. Slightly lower error probability in some channels.

V. PULSE MODULATION & MULTIPLEXING

A. Pulse Modulation Techniques

[!TIP] Comparison: PAM is amplitude-sensitive. PWM & PPM are constant-amplitude → better for optical/optical links (noise affects intensity).

B. Multiplexing

1. Frequency Division Multiplexing (FDM):

• Concept: Each signal modulated to different carrier frequency band. Bands separated by guard bands.

• Application: Analog radio/TV broadcasting, cable TV.

• Block Diagram: Modulators → Frequency Division Multiplexer (adder) → Channel → Demultiplexer (filters) → Demodulators.

2. Time Division Multiplexing (TDM):

• Concept: Signals share same channel in time slots. Synchronous (fixed slots) or asynchronous (statistical).

• Block Diagram (Synchronous TDM):

  • Mux: n sampling gates sequentially connect inputs to output line.

  • Demux: Synchronous switch at receiver separates slots.

• Applications: Digital telephony (PCM T1/E1), digital radio (TDMA), computer networks.

• Key Parameter: Frame rate = f_s (sampling rate per channel). Frame duration = 1/f_s.

VI. SPECIAL TOPICS & SHORT NOTES

A. VSB-SC (Vestigial Sideband Suppressed Carrier)

• Concept: One sideband transmitted almost completely, other sideband transmitted partially (vestige). Carrier suppressed (like DSB-SC).

• Need: For TV broadcast (video signals have significant low-frequency content, including DC). SSB requires perfect filter with sharp cutoff → impractical. VSB uses compromise filter with gradual transition band.

• Filter Characteristic: Passband for USB, gradual roll-off for LSB (transmits vestige of LSB). Carrier reinserted at receiver (coherent detection).

• Comparison with SSB: VSB has wider bandwidth than SSB but narrower than DSB. Easier filter design. Used in analog TV (VSB filter ~0.75 MHz for 4.2 MHz video).

B. Fourier Transform Properties (Detailed Review)

See Section I.C. Ensure proofs for: Time-shifting, Frequency-shifting, Time-scaling, Duality, Convolution ↔ Multiplication.

C. System Classification (Detailed Review)

See Section I.B. Be ready with examples:

• Linear: y(t) = 3x(t) + 2x(t-1).

• Non-linear: y(t) = x²(t) + t x(t).

• Time-invariant: y(t) = ∫ x(τ) dτ (from -∞ to t).

• Time-variant: y(t) = t·x(t).

• Causal: y(t) = x(t-1).

• Non-causal: y(t) = x(t+1).

• BIBO Stable LTI: h(t) = e^{-t}u(t) (∫|h|dt=1). Unstable: h(t)=u(t) (∫|h|dt=∞).

D. Sensitivity, Selectivity, Fidelity (Receiver Parameters)

• Sensitivity: P_min (dBm) = kTBF + (SNR_required)_{dB} + NF. Lower P_min → better sensitivity.

• Selectivity: Determined by IF filter bandwidth & shape factor (BW₆₀dB / BW₃dB). Lower shape factor → better selectivity.

• Fidelity: Determined by audio bandwidth (e.g., 5 kHz vs 15 kHz for high-fidelity) and linearity (distortion). Wider audio BW → higher fidelity.

Final Exam Strategy:

1. Definitions First: Always start with crisp definitions (e.g., Modulation Index, Carson's Rule, Nyquist Rate).

2. Diagrams are Key: Draw block diagrams (Superhet, PCM, QAM) and waveforms (AM envelope, FM spectrum, DM error) clearly.

3. Derivations: Practice proofs for FT properties, sampling theorem, SQNR, AM power relations.

4. Comparisons: Know AM vs DSB-SC vs SSB; BPSK vs DPSK; QPSK vs OQPSK; QAM vs MPSK; TDM vs FDM.

5. Numericals: Be fluent in m_a calculation, AM power/efficiency, Carson's BW, SQNR, deviation ratio, data rate for M-ary schemes (R = (log₂M)/T_b).

\boxed{\text{Revise past papers chronologically: Jun 2025 → Jun 2023 → Nov 2023 → Jun 2022.}}