UNIT 5: ANALOG & DIGITAL COMMUNICATION - EXAM-FOCUS NOTES

I. SIGNAL ANALYSIS & SYSTEM PROPERTIES (FOUNDATION)

Signal Classification & Operations

• Energy vs. Power Signals:

  • Energy Signal: Finite energy, zero average power. E.g., pulses, finite-duration signals.

$$E = \int_{-\infty}^{\infty} |x(t)|^2 dt < \infty$$

*   **Power Signal:** Finite average power, infinite energy. E.g., periodic signals, random signals.

$$P = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{T/2} |x(t)|^2 dt < \infty$$

> **Proof for Rectangular Pulse:** A rectangular pulse of amplitude `A` and duration `T₀` has finite energy `(A²T₀)` and zero average power → **Energy Signal**.

• Standard Signals:

  • Unit Impulse δ(t): δ(t)=0 for t≠0, ∫δ(t)dt=1. Sifting property: ∫x(t)δ(t-t₀)dt = x(t₀).

  • Unit Step u(t): u(t)=1 for t≥0, 0 for t<0. u(t) = ∫δ(τ)dτ from -∞ to t.

  • Rectangular Pulse (Gate): rect(t/T₀) or Π(t/T₀). Width T₀, amplitude 1.

  • Triangular Pulse: Λ(t/T₀) or tri(t/T₀).

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

• Signal Transformations (Example: y(t)=x(2t+3)):

  1. Time-shift: x(t+3) → shift left by 3.

  2. Time-scaling: x(2t) → compress by factor 2.

  • Order matters: For x(at+b), first shift by b/a, then scale by 1/a.

Fourier Transform (FT) & Properties

Definition:

$$X(ω) = \mathcal{F}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-jωt} dt$$

Inverse FT:

$$x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(ω) e^{jωt} dω$$

Physical Significance: Converts time-domain signal to frequency-domain representation (spectrum).

Core Properties (Must Know):

Time-scaling Proof: Let y(t)=x(at). Then Y(ω)=∫x(at)e^{-jωt}dt. Substitute τ=at, dt=dτ/a. Y(ω)=(1/a)∫x(τ)e^{-j(ω/a)τ}dτ = (1/a)X(ω/a) for a>0. For a<0, absolute value 1/|a| appears.

FT of Standard Signals:

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

2. Gate/Rect Function: rect(t/T₀) ↔ T₀ sinc(ωT₀/2π) where sinc(x)=sin(πx)/(πx).

3. Exponential e^{-at}u(t) (a>0): 1/(a+jω).

Linear Time-Invariant (LTI) System Analysis

• System Classification:

  • Linear: Superposition & homogeneity hold. E.g., y(t)=ax(t).

  • Non-linear: E.g., y(t)=x²(t).

  • Time-invariant: Delay doesn't change system behavior. x(t-t₀) → y(t-t₀).

  • Time-variant: E.g., y(t)=t x(t).

  • Causal: Output depends only on present & past inputs. h(t)=0 for t<0.

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

  • Stable (BIBO): Bounded input → bounded output. ∫|h(t)|dt < ∞.

  • Unstable: E.g., h(t)=e^{at}u(t) for a>0.

• Impulse Response h(t) & Frequency Response H(ω):

  H(ω) = \mathcal{F}\{h(t)\} = \int_{-\infty}^{\infty} h(t) e^{-jωt} dt.

  For LTI system: y(t) = x(t) * h(t).

• Convolution Integral: y(t) = ∫_{-∞}^{∞} x(τ) h(t-τ) dτ. Significance: Characterizes LTI system completely.

II. ANALOG MODULATION TECHNIQUES

Amplitude Modulation (AM)

• AM Wave Expression (Single-tone):

  s_AM(t) = A_c [1 + m cos(ω_m t)] cos(ω_c t)

  Where m = modulation index (0≤m≤1 for no distortion).

  = A_c cos(ω_c t) + (m A_c /2)[cos((ω_c+ω_m)t) + cos((ω_c-ω_m)t)].

• Generation:

  1. Square-law Modulator: Uses non-linear device (diode) with i = a v + b v². Output contains v_c² term → generates sidebands. Limitation: Only works for m<1.

     DiagramSquare-law AM modulator circuit

  2. Switching Modulator: Uses diode as switch driven by carrier. Output is ±m(t) at carrier harmonics. Filtering extracts AM wave. Works for m>1 (over-modulation).

     DiagramSwitching AM modulator circuit

• Power Relations:

  • Carrier Power: P_c = A_c² / (2R)

  • Total Power: P_T = P_c (1 + m²/2)

  • Sideband Power: P_sb = P_c (m²/2)

  • Transmission Efficiency (η): η = (P_sb / P_T) × 100% = [m²/(2+m²)] × 100%.

  Max η = 33.33% at m=1. Improved via SSB (only one sideband transmitted).

• AM Detector (Envelope Detector):

  • Circuit: Diode + RC low-pass filter.

  • Operation: Diode conducts when v_in > v_cap, charging capacitor. Discharges through R when v_in < v_cap. RC must satisfy: 1/ω_c << RC << 1/ω_m for no distortion.

  DiagramEnvelope detector circuit & waveforms

Suppressed Carrier AM (SC-AM)

Coherent Detection: Requires carrier synchronization at receiver. Demodulation via multiplication with local oscillator & low-pass filtering.

Angle Modulation (FM & PM)

• Frequency Modulation (FM):

  s_FM(t) = A_c cos[ω_c t + k_f ∫ m(τ) dτ]

  Modulation Index (β): β = Δf / f_m = (k_f A_m) / f_m (for single-tone m(t)=A_m cos(2πf_m t)).

  Deviation Ratio: β_max = Δf_max / f_m,max (max frequency deviation / max modulating freq).

  • Spectrum (Single-tone): Infinite sidebands (Jₙ(β) coefficients). Significant power in sidebands where |n| ≤ β+1.

  • Bandwidth (Carson's Rule): BW ≈ 2(Δf + f_m) = 2f_m (β + 1).

  Example: Δf=75 kHz, f_m=15 kHz → BW≈180 kHz.

• Generation Methods:

  1. Direct (Varactor Diode VCO): Varying capacitance of varactor diode with m(t) changes VCO frequency directly.

     DiagramDirect FM generator with varactor

  2. Indirect (Armstrong): Generate PM first (s_PM = A_c cos(ω_c t + k_p m(t))), then use frequency multipliers to increase Δf.

• Demodulation (Detection):

  1. Foster-Seeley Discriminator: Uses two tuned circuits (primary & secondary) 90° out of phase. Output proportional to frequency deviation.

     DiagramFoster-Seeley discriminator with phase diagrams

  2. Ratio Detector: Similar but uses diode in series with capacitor. Advantage: No carrier recovery needed, better for noisy signals.

     DiagramRatio detector circuit

  3. Balanced Slope Detector: Two slope detectors in push-pull. Simple but poor linearity.

• Phase Modulation (PM): s_PM(t) = A_c cos[ω_c t + k_p m(t)]. Instantaneous phase varies with m(t). Relation to FM: If m(t) is integrated before PM, it becomes FM.

• Pre-emphasis & De-emphasis:

  • Need: FM has good noise immunity at low freq, poor at high freq. Pre-emphasis boosts high-freq signals before transmission; de-emphasis attenuates them at receiver.

  • Transfer Functions (RC Network):

    • Pre-emphasis (High-pass): H_pre(f) = 1 + j2πfτ (τ=RC). |H|↑ with f.

    • De-emphasis (Low-pass): H_de(f) = 1 / (1 + j2πfτ). |H|↓ with f.

  • Noise Improvement: SNR improvement ≈ (1 + (2πf_m τ)²).

III. RECEIVERS & RADIO FREQUENCY (RF) SYSTEMS

Receiver Types & Performance

• TRF (Tuned Radio Frequency) Receiver:

  • Block Diagram: RF Amp → Mixer (with LO) → IF Amp → Detector → Audio Amp.

  • Limitations:

    1. Selectivity: Poor, as filtering at RF is difficult.

    2. Stability: Oscillator frequency drift causes tuning issues.

    3. Image Frequency: f_image = f_signal + 2f_IF. Hard to filter out at RF.

  • Superheterodyne Receiver:

    • Block Diagram & Function:

      1. RF Amplifier: Amplifies & initial filtering.

      2. Mixer + LO: Heterodynes to fixed IF (e.g., 455 kHz, 10.7 MHz).

      3. IF Amplifier: High gain, excellent selectivity (fixed-tuned).

      4. Detector: Demodulates (AM: envelope; FM: discriminator).

      5. Audio Amplifier: Drives speaker.

    DiagramSuperheterodyne receiver block diagram
    • Image Frequency Rejection: f_image = f_signal + 2f_IF. Rejected by RF front-end filter. Image Rejection Ratio (IMRR): IMRR = (f_image - f_signal) / (f_IF).

Receiver Characteristics (Key Metrics)

Frequency Translation & Multiplexing

• Heterodyning: Mixing two frequencies f₁ and f₂ → outputs |f₁±f₂|. Used for frequency conversion.

• Frequency Division Multiplexing (FDM): Each signal modulates different carrier frequency (non-overlapping bands). Used in analog radio/TV.

  DiagramFDM spectrum

• Time Division Multiplexing (TDM): Signals share channel in time slots. Requires synchronization. Used in digital telephony (PCM).

  DiagramTDM frame structure

IV. PULSE MODULATION & SAMPLING

Sampling Theory

• Nyquist Sampling Theorem (Low-pass):

  A bandlimited signal with max frequency f_max can be reconstructed from samples if sampled at f_s ≥ 2 f_max (Nyquist Rate).

  • Proof Sketch: FT of sampled signal is periodic replication of original spectrum. To avoid overlap (aliasing), replication spacing f_s must be ≥ 2f_max.

  • Reconstruction: Using ideal low-pass filter (sinc interpolation).

• Aliasing: Occurs when f_s < 2f_max. High-frequency components fold back to lower frequencies. Consequence: Irreversible distortion.

  Anti-aliasing Filter: Low-pass filter before sampler with f_c ≤ f_s/2.

• Bandpass Sampling: For signal with bandwidth B centered at f_c, f_s ≥ 2B but f_s can be much less than 2f_c.

Pulse Modulation Techniques (Analog)

Pulse Code Modulation (PCM)

• Complete Block Diagram:

  Signal → Anti-aliasing Filter → Sampler → Quantizer → Encoder → Channel

  ← Reconstruction Filter ← Decoder ←

  DiagramPCM system block diagram

• Quantization:

  • Uniform Quantizer: Equal step size Δ. Mid-tread: Zero at origin. Mid-riser: Zero at decision boundary.

  • Quantization Error (e_q): e_q = x_q - x (quantized - original). Assumed uniformly distributed in [-Δ/2, Δ/2].

  • Quantization Noise Power: σ_q² = Δ² / 12.

  • Signal Power (for full-scale sine wave): P_x = (A_max²)/2.

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

    SQNR = P_x / σ_q² = (3 × 2^{2n}) (for uniform quantizer, n bits).

    SQNR_dB ≈ 6.02n + 1.76 dB.

• Encoding: Binary (most common). n bits → 2^n levels.

• Regeneration: Digital nature allows perfect regeneration at repeaters → excellent noise immunity.

Delta Modulation (DM) & Adaptive DM (ADM)

• Delta Modulation (DM):

  • Principle: 1-bit quantizer. Compares input with integrated (predicted) signal. Outputs +Δ or -Δ (1 bit/sample).

  • Block Diagram: Summer → 1-bit quantizer → encoder → accumulator (integrator) in feedback.

  DiagramDM encoder/decoder block diagram

  • Distortions:

    1. Slope Overload: Input slope > DM step size slope (Δ/T_s). Causes granular noise at low slopes.

    2. Granular Noise: Small input variations cause switching noise.

  • Advantage: Simple, low bit rate.

  • Disadvantage: Δ fixed → poor performance for varying signal slopes.

• Adaptive Delta Modulation (ADM):

  • Need: Overcome DM's fixed step size problem.

  • Adaptation Logic: Step size Δ changes based on recent output pattern (e.g., if consecutive +1s or -1s, increase Δ; if alternating, decrease Δ).

  • Merits over DM: Reduced slope overload & granular noise. Better SQNR for same bit rate.

  • Demerits: More complex circuitry.

V. DIGITAL MODULATION (DIGITAL PASSBAND)

Binary Modulation Schemes

• Binary Phase Shift Keying (BPSK):

  • Signal: s(t) = ±√(2E_b/T_b) cos(ω_c t) for bits 1 and 0.

  • Generation: Multiply binary data (±1) with carrier.

  • Coherent Detection: Multiply with synchronized carrier, integrate over T_b, sample & decide.

  • Bandwidth: BW ≈ 2/T_b (same as BASK).

  • Constellation: Two points on real axis at ±√E_b.

  • Probability of Error (Coherent): P_e = Q(√(2E_b/N_0)).

• Differential BPSK (DBPSK/DPSK):

  • Concept: Bit 1 → phase change of 180°, 0 → no change (relative to previous bit).

  • Transmitter: s(t) = cos(ω_c t + θ_n), where θ_n = θ_{n-1} + π(1 - b_n).

  • Receiver (Differential Detector): Compares phase of current & previous symbol intervals. No need for absolute carrier phase sync.

  • Comparison with BPSK:

    • Advantage: Eliminates need for coherent carrier recovery (simpler receiver).

    • Disadvantage: ~3 dB worse P_e than coherent BPSK for same E_b/N_0.

• Binary Frequency Shift Keying (BFSK):

  • Generation: Binary data selects between two oscillators at f₁ and f₂.

  • Coherent Detection: Two matched filters/correlators for each frequency.

  • Non-coherent Detection: Envelope detectors (simpler, worse performance).

  • Bandwidth: BW ≈ |f₁ - f₂| + 2/T_b. Minimum separation for orthogonality: |f₁ - f₂| = 1/T_b.

M-ary Modulation Schemes

• M-ary PSK (MPSK):

  • Concept: M phases for log₂M bits/symbol. E.g., QPSK (4-PSK): 4 phases (45°, 135°, 225°, 315°).

  • Constellation: Points equally spaced on circle.

  • Bandwidth Efficiency: η = log₂M / (bits/sec/Hz). QPSK: 2 bits/symbol → same BW as BPSK but double data rate.

  • Drawback: As M increases, points closer → more sensitive to noise.

• Quadrature Amplitude Modulation (QAM):

  • Principle: Two orthogonal carriers (cos(ω_c t) and sin(ω_c t)). Independent amplitude modulation on I (in-phase) and Q (quadrature) channels.

  • Generation: s(t) = I(t) cos(ω_c t) - Q(t) sin(ω_c t). I(t), Q(t) are PAM signals from separate bit streams.

  • Detection: Coherent detection with two matched filters (I & Q channels).

  • Constellation: Grid of points (e.g., 16-QAM: 4×4 grid; 64-QAM: 8×8 grid). M = 2^{2k} for square QAM (k bits/symbol per dimension).

  • Comparison with QPSK: QAM uses both amplitude & phase → more points in same area → higher data rate for same BW, but lower noise immunity (closer constellation points).

• Offset QPSK (OQPSK):

  • Concept: In QPSK, I & Q bits change simultaneously → 180° phase jumps possible. In OQPSK, I & Q channels offset by T_b (half symbol period).

  • Reason for Reduced Error: Prevents 180° jumps → lower sidelobes in spectrum, less envelope variation → better for non-linear amplifiers (e.g., satellite).

VI. ADVANCED TOPICS & SYNTHESIS

Performance & Trade-offs

Key Trade-off: Bandwidth efficiency vs. Power efficiency (noise immunity).

Frequency Synthesis (Indirect using PLL)

• PLL Components: Phase Detector (PD) → Loop Filter (LPF) → VCO → Feedback (divider).

• Working Principle (FM Generation):

  1. PD compares phase of reference f_ref and VCO output f_out (after /N division).

  2. Error voltage filters & controls VCO.

  3. In lock: f_out = N × f_ref.

  4. To generate FM: Modulate VCO control input. PLL locks to average frequency, output is clean FM at N×f_ref.

  DiagramPLL block diagram for FM synthesis

Integrated System Perspective

• Complete Digital Communication System:

  Source → Source Encoder → Channel Encoder → **Modulator** → Channel → Demodulator → Channel Decoder → Source Decoder → Destination

• Role of Modulation:

  • Bandwidth Matching: Convert baseband to passband for efficient transmission.

  • Power Efficiency: Choose modulation robust to channel noise (e.g., BPSK for power-limited AWGN).

  • Complexity Trade-off: Higher-order modulation (QAM) increases data rate but needs better SNR & linear amplifiers.

EXAM TIPS & COMMON PITFALLS

[!TIP] Fourier Transform: Always write the integral definition. For time-scaling proof, remember the absolute value 1/|a|.

[!TIP] AM Power: Total power P_T = P_c (1 + m²/2). Sideband power = m² P_c / 2. Efficiency η = m²/(2+m²).

[!TIP] FM Bandwidth: Use Carson's Rule BW ≈ 2(Δf + f_m). Do NOT use 2(β+1)f_m unless single-tone.

[!TIP] PCM SQNR: SQNR_dB ≈ 6.02n + 1.76 dB for uniform quantizer. Remember n = number of bits.

[!TIP] Sampling Theorem: f_s ≥ 2 f_max for baseband. For bandpass, f_s ≥ 2B but f_s can be much lower.

[!TIP] BPSK vs DPSK: DPSK avoids carrier sync but has ~3 dB worse P_e.

[!TIP] QAM vs QPSK: QPSK is 4-QAM with points on circle (constant envelope). Square QAM has amplitude variation.

[!TIP] Superheterodyne: Always mention fixed IF for high selectivity & stability. Image frequency = f_signal + 2f_IF.

[!TIP] DM Distortions: Clearly distinguish slope overload (large signal slope) from granular noise (small signal).

[!TIP] Pre-emphasis/De-emphasis: Transfer functions are reciprocal (HPF & LPF). Improves SNR at high frequencies by (1+(2πf_mτ)²).

BOXED KEY FORMULAS

• Energy/Power: E = ∫|x(t)|² dt, P = lim_{T→∞} (1/T) ∫|x(t)|² dt.

• FT Time-scaling: x(at) ↔ (1/|a|) X(ω/a).

• AM Efficiency: η = [m²/(2+m²)] × 100%.

• FM Bandwidth (Carson): BW ≈ 2(Δf + f_m).

• PCM SQNR: SQNR = 3 × 2^{2n} (linear), ≈ 6.02n + 1.76 dB.

• BPSK P_e (Coherent): P_e = Q(√(2E_b/N_0)).

• Nyquist Rate: f_s ≥ 2 f_max.