Analog & Digital Communication (IT-404) - Unit 3 Short Notes

UNIT 3: ANALOG & DIGITAL COMMUNICATION – EXAM-DRIVEN SHORT NOTES

I. SIGNALS AND SYSTEMS FUNDAMENTALS

A. Signal Classification

[!TIP] Exam Focus: Rectangular pulse x(t) = A for |t| ≤ τ/2, 0 otherwise is an energy signal (finite duration). A periodic square wave is a power signal.

B. System Classification

C. Standard Signals & Operations

• Unit Impulse (Dirac Delta) 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) = ∫_{-∞}^{t} δ(τ) dτ.

• Rectangular Pulse (Gate): rect(t/τ) = 1 for |t| ≤ τ/2, 0 otherwise.

• Triangular Pulse: Λ(t/τ) = 1 - |t|/τ for |t| ≤ τ, 0 otherwise.

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

• Signal Operations:

  • Time-shifting: x(t-t₀) → delay by t₀.

  • Time-scaling: x(at) → a>1 (compression), 0<a<1 (expansion).

  • Amplitude-scaling: A*x(t).

[!TIP] Common Pitfall: x(2t+3) is shift then scale: First shift left by 3 (x(t+3)), then compress by 2 (x(2(t+3)) = x(2t+6)). Alternatively, factor: x(2(t+1.5)) → scale by 2, then shift left by 1.5.

II. FOURIER ANALYSIS

A. Continuous-Time Fourier Transform (CTFT)

Definition: X(ω) = ∫_{-∞}^{∞} x(t) e^{-jωt} dt (Frequency domain representation). Inverse: x(t) = (1/(2π)) ∫_{-∞}^{∞} X(ω) e^{jωt} dω.

Key Fourier Transform Pairs:

1. δ(t) ↔ 1

2. u(t) ↔ πδ(ω) + 1/(jω) (Using u(t) = (1/2)(1+sgn(t)))

3. rect(t/τ) ↔ τ sinc(ωτ/2π) where sinc(x) = sin(πx)/(πx).

4. e^{-at}u(t) (a>0) ↔ 1/(a+jω)

B. Fourier Transform Properties

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

C. Application to LTI Systems

• Frequency Response: H(ω) = ∫_{-∞}^{∞} h(t) e^{-jωt} dt (Fourier Transform of impulse response h(t)).

• System Output: Y(ω) = H(ω) X(ω).

• Example: For h(t) = e^{-2t}u(t), H(ω) = ∫_{0}^{∞} e^{-2t} e^{-jωt} dt = 1/(2+jω).

  • Magnitude: |H(ω)| = 1/√(4+ω²), Phase: ∠H(ω) = -tan⁻¹(ω/2).

  • Plot: Low-pass filter (magnitude decreases as |ω| increases).

III. ANALOG MODULATION TECHNIQUES

A. Amplitude Modulation (AM)

1. Standard AM (DSB-FC):

   • Expression: s_AM(t) = A_c[1 + m_a cos(ω_m t)] cos(ω_c t), where m_a is modulation index (0 ≤ m_a ≤ 1 for linear modulation).

   • Spectrum: Carrier at ω_c, Upper Sideband (USB) at ω_c+ω_m, Lower Sideband (LSB) at ω_c-ω_m.

   • Power Relations:

     • Carrier Power: P_c = A_c²/2

     • Total Power: P_T = P_c (1 + m_a²/2)

     • Sideband Power: P_sb = P_c (m_a²/2)

     • Transmission Efficiency: η = P_sb / P_T = (m_a²/2) / (1 + m_a²/2). \boxed{\eta = \frac{m_a^2}{2 + m_a^2}}

   • Over-modulation (m_a > 1): Envelope distortion, carrier phase reversal.

2. Generation:

   • Square-law Modulator: Uses nonlinear device (diode) with i = a*v + b*v². Produces v²(t) term containing cos(ω_c t) and cos(2ω_c t). Filter out 2ω_c term to get DSB-SC. Limitation: Low power, harmonic distortion.

   • Switching Modulator (Ring Modulator): Uses diodes as switches driven by carrier. Output is product m(t)*c(t) (DSB-SC). Efficient, no carrier leakage.

3. Suppressed Carrier Variants:

   • DSB-SC: s(t) = m(t) cos(ω_c t). Bandwidth = 2f_m. Requires coherent detection (phase sync).

   • SSB-SC: Transmits only one sideband (USB or LSB). Bandwidth = f_m, Power savings (no carrier, one sideband).

     • Generation:

       • Filter Method: Generate DSB-SC, then use sharp bandpass filter.

       • Phasing Method (Hilbert Transform): s(t) = m(t)cos(ω_c t) ∓ ĥm(t)sin(ω_c t), where ĥm(t) is Hilbert transform (90° phase-shifted) of m(t).

   • VSB-SC: Transmits one full sideband + vestige of other. Used in TV broadcasting (to preserve low-frequency video components). Filter has a gradual transition band.

B. Angle Modulation (FM & PM)

1. Fundamentals:

   • General Expression: s(t) = A_c cos[ω_c t + φ(t)].

   • Frequency Modulation (FM): φ(t) = k_f ∫ m(τ) dτ. Instantaneous frequency: ω_i(t) = ω_c + k_f m(t).

     • Modulation Index (β): β = Δf / f_m = (k_f A_m)/ω_m. Δf is peak frequency deviation.

     • Single-tone FM: s(t) = A_c cos(ω_c t + β sin ω_m t).

   • Phase Modulation (PM): φ(t) = k_p m(t). β = k_p A_m.

   • Relationship: FM is the integral of PM. For single-tone, β_FM = β_PM / (ω_m).

   • Deviation Ratio (for wideband FM): D = Δf / f_{max} (where f_{max} is max modulating frequency).

2. Generation:

   • Direct Method: VCO (Voltage Controlled Oscillator) where ω_out ∝ v_in. Armstrong Method: Phase modulator + differentiator → FM.

   • Indirect Method: Direct PM + frequency multiplication (to increase Δf).

3. Demodulation:

   • Ratio Detector: Uses a transformer and diodes. Output across R1 (no carrier recovery needed). Advantage: Less sensitive to amplitude variations.

   • Foster-Seeley Discriminator: Uses a tuned transformer (primary & secondary 90° out of phase). Output across R is proportional to dφ/dt (frequency). Requires carrier for operation.

   • Balanced Slope Detector: Two slope detectors in balance. Simple but poor linearity.

4. Pre-emphasis & De-emphasis:

   • Need: FM has better SNR for high-frequency components? Actually, noise power increases with frequency. Pre-emphasis boosts high-frequency modulating signal before transmission to improve post-demodulation SNR.

   • Transfer Functions:

     • Pre-emphasis (High-pass): H_{pre}(f) = 1 + j(f/f_x) (simple RC differentiator).

     • De-emphasis (Low-pass): H_{de}(f) = 1 / (1 + j(f/f_x)) (RC integrator).

   • f_x is the emphasis frequency (typically 2.12 kHz or 75 µs time constant).

[!TIP] Key Difference: Ratio Detector does not require a separate carrier recovery circuit (uses the transmitted carrier), while Foster-Seeley does.

IV. RADIO RECEIVERS

A. Superheterodyne Receiver

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

• RF Amplifier: Selects desired station, provides initial gain.

• Mixer & Local Oscillator: Heterodynes RF to fixed Intermediate Frequency (IF). f_IF = |f_RF - f_LO|.

• IF Amplifier: Provides most gain and selectivity (using fixed-tuned filters).

• Detector: Demodulates IF signal (AM envelope detector, FM discriminator).

• Audio Amplifier: Boosts audio signal for speaker.

Image Frequency: f_image = f_LO + f_IF (for high-side injection) or f_LO - f_IF (low-side). An undesired signal at f_image also produces f_IF output. Rejection achieved by RF front-end filtering.

B. TRF (Tuned Radio Frequency) Receiver

Block Diagram: RF Amp (tuned) → Detector → Audio Amp. All stages tuned to f_c.

• Operation: Each RF stage is tuned to the carrier frequency.

• Limitations vs. Superhet:

  1. Poor Selectivity: Requires multiple high-Q RF stages, difficult to achieve constant bandwidth over tuning range.

  2. Poor Sensitivity: Gains at RF are low; high gain at RF causes instability (oscillations).

  3. Instability: High gain at high frequencies prone to feedback oscillations.

  4. No Image Rejection: No frequency conversion to a fixed IF.

C. Receiver Performance Parameters

• Sensitivity: Minimum input signal power required at antenna to produce a usable output (specified SNR at output). Sensitivity ∝ (SNR_required * k T B) / (G * modulation efficiency).

• Selectivity: Ability to separate adjacent channels. Measured by bandwidth of IF filters (e.g., 6 dB bandwidth).

• Fidelity: Faithfulness of audio reproduction. Limited by bandwidth of entire receiver (IF, audio stages).

V. PULSE MODULATION TECHNIQUES

A. Basic Pulse Modulation

• PAM Types:

  • Natural Sampling: Top of pulses follow m(t).

  • Flat-top Sampling: Pulse held constant between samples (easier to handle, requires reconstruction filter).

B. Delta Modulation (DM)

• Principle: 1-bit quantizer. Compares input with integrated output (previous quantized level + step Δ).

  • If m(t) > pred. output: Output +Δ (bit 1).

  • If m(t) < pred. output: Output -Δ (bit 0).

• Encoder/Decoder: Simple, low bit rate (1/Ts).

• Waveforms: Staircase approximation of m(t).

• Limitations:

  1. Slope Overload Distortion: When dm/dt > Δ/Ts (step too small to follow steep slope).

  2. Granular (Idle) Noise: When dm/dt ≈ 0, output oscillates between ±Δ due to quantization noise.

C. Adaptive Delta Modulation (ADM)

• Principle: Variable step size Δ.

  • Increase Δ for steep slopes (reduce slope overload).

  • Decrease Δ for gentle slopes (reduce granular noise).

• Merits over DM: Reduces both distortions.

• Demerits: More complex logic to adapt step size (e.g., based on recent bit pattern).

D. Time Division Multiplexing (TDM)

• Synchronous TDM: Each signal gets a fixed time slot in a repeating frame.

• Block Diagram:

  • Multiplexer: Parallel-to-serial switch, samples each input in round-robin.

  • Demultiplexer: Serial-to-parallel switch, distributes bits to appropriate output lines using sync.

• Applications: PCM telephony (E1/T1 lines), digital audio (AES/EBU).

VI. SAMPLING THEORY AND QUANTIZATION

A. Nyquist-Shannon Sampling Theorem (Low-pass)

• Statement: A bandlimited signal with max frequency f_max can be perfectly reconstructed from its samples if sampled at f_s ≥ 2 f_max.

• Nyquist Rate: f_N = 2 f_max.

• Proof Concept:

  1. Sampling x(t) by δ_T(t) = Σ δ(t-nT) (with T=1/f_s) in time domain is multiplication.

  2. In frequency, X_s(ω) = (1/T) Σ X(ω - kω_s) (periodic repetition of X(ω) with period ω_s = 2π f_s).

  3. To avoid aliasing (overlap of spectral replicas), need ω_s ≥ 2ω_max → f_s ≥ 2 f_max.

• Consequences of Undersampling (f_s < 2f_max): Spectral replicas overlap → aliasing (irreversible distortion). Anti-aliasing filter (low-pass) required before sampling.

B. Sampling of Bandpass Signals

• Bandpass Sampling Theorem: For signal bandlimited to [f_L, f_H], minimum f_s can be as low as 2B (where B = f_H - f_L) if f_L is sufficiently high.

• Condition: f_s ≥ 2B and f_s chosen so that spectral replicas do not overlap. f_s must satisfy: f_s = 2f_H / n for some integer n such that f_L ≥ (n-1)f_s / 2.

C. Quantization

• Need: Convert continuous amplitude samples to discrete levels for digital transmission (PCM).

• Uniform Quantization: L levels, step size Δ = (x_max - x_min)/L.

  • Quantization Error (Noise) e_q: e_q = x_q - x, where x_q is quantized value.

  • Assumed uniformly distributed over [-Δ/2, Δ/2] for large L and high probability of overload.

  • Variance (Power): σ_q² = Δ²/12.

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

  • For a full-scale sinusoidal signal x(t) = A sin(ωt), signal power P_x = A²/2.

  • Δ = 2A/L (for L levels over [-A, A]).

  • σ_q² = (2A/L)² / 12 = A²/(3L²).

  • SQNR = P_x / σ_q² = (A²/2) / (A²/(3L²)) = (3/2) L².

  • In dB: SQNR_dB = 10 log10(3L²/2) ≈ 10 log10(L²) + 1.76 = 20 log10(L) + 1.76.

  • Since L = 2^n (n bits), \boxed{SQNR_{dB} \approx 6n + 1.76}. Approx. 6 dB per bit.

VII. DIGITAL MODULATION TECHNIQUES

A. Binary Modulation

1. Amplitude Shift Keying (ASK):

   • Coherent ASK (On-Off Keying): s(t) = A_c cos(ω_c t) for bit 1, 0 for bit 0.

   • Non-coherent ASK: Envelope detection (simple, but poorer performance).

   • Bandwidth: Approximately 2/T_b (same as baseband).

2. Frequency Shift Keying (FSK):

   • Generation: Two oscillators at f1 and f2, switched by data. Or VCO controlled by data.

   • Spectrum: Two discrete tones. Bandwidth (approx.): B ≈ |f1 - f2| + 2f_m = 2Δf + 2f_m (using Carson's rule for continuous phase).

   • Continuous Phase FSK (CPFSK): Phase is continuous at bit transitions (e.g., MSK).

3. Binary Phase Shift Keying (BPSK):

   • Expression: s(t) = ±A_c cos(ω_c t) (0° for 1, 180° for 0).

   • Generation: Multiply binary data ±1 by carrier.

   • Coherent Detection: Correlator or matched filter with reference carrier. Optimal.

   • Bandwidth: Same as ASK (≈ 2/T_b).

B. Differential and Quadrature Modulation

1. Differential BPSK (DBPSK):

   • Principle: s(t) = A_c cos(ω_c t + θ_n), where θ_n = θ_{n-1} + π(1 - b_n) (phase change depends on current bit b_n).

   • Receiver (Differential Detection): Delay current symbol by T_b, multiply with previous symbol. Output polarity gives b_n.

   • Advantage over BPSK: No need for absolute carrier phase synchronization. Disadvantage: ~3 dB worse BER in AWGN.

2. Quadrature Phase Shift Keying (QPSK):

   • Generation: Two BPSK streams on quadrature carriers (cos(ω_c t) and sin(ω_c t)).

     • I(t) = A_c * d_I(t) (even bits), Q(t) = A_c * d_Q(t) (odd bits).

     • s(t) = I(t)cos(ω_c t) - Q(t)sin(ω_c t).

   • Constellation: 4 points (±1, ±1) on a circle (constant envelope). 2 bits/symbol.

   • Bandwidth: Half of BPSK for same bit rate (since symbol rate R_s = R_b / 2). B ≈ R_s = R_b / 2.

   • Offset QPSK (OQPSK): Stagger I and Q channels by T_b/2. Prevents 180° phase jumps (which cause spectral regrowth). Better for nonlinear amplifiers.

3. Quadrature Amplitude Modulation (QAM):

   • Principle: Independent amplitude modulation of I and Q channels. s(t) = I(t)cos(ω_c t) - Q(t)sin(ω_c t).

   • Constellation: Square grid (e.g., 16-QAM: 16 points, 4 bits/symbol).

   • Generation & Detection: I-Q Modulator/Demodulator block diagram.

     • Modulator: Data split → I & Q → amplitude scale → multiply with cos/sin → add.

     • Demodulator: Coherent detection: multiply with cos/sin, low-pass filter, sample.

   • Comparison with QPSK:

     • QPSK is 4-QAM (points at corners of square, constant radius).

     • Higher-order QAM (16, 64) has varying amplitude → more sensitive to noise and nonlinear distortion.

     • QAM has higher bandwidth efficiency (bits/s/Hz) but worse BER.

C. M-ary Modulation

• M-ary PSK: M equally spaced phases on a circle. Bandwidth efficiency = log2(M) bits/symbol. BER increases with M for same E_b/N_0.

• General Concept: Increase M → more bits per symbol → higher bandwidth efficiency, but requires higher E_b/N_0 for same BER.

VIII. PERFORMANCE ANALYSIS & MULTIPLEXING

A. Signal-to-Noise Ratio (SNR) Analysis (Coherent Detection)

• DSB-SC:

  • Demodulator output: y(t) = (2/A_c) [m(t) * cos(ω_c t)] * cos(ω_c t) = m(t) + (1/A_c)[m(t) cos(2ω_c t)].

  • After LPF: y(t) = m(t). Noise power at output: N_0 (baseband).

  • SNR_out = (P_m) / (N_0 B), where P_m is message power.

• SSB (USB):

  • s(t) = m(t)cos(ω_c t) ∓ ĥm(t)sin(ω_c t).

  • Coherent demodulation with cos(ω_c t) and sin(ω_c t) recovers m(t) and ±ĥm(t).

  • SNR_out = (P_m) / (N_0 B) (same as DSB-SC, but bandwidth B = f_m instead of 2f_m → better SNR for same noise power spectral density).

• Comparison: SSB has half the bandwidth of DSB-SC for same message, so for same total noise power N_0 B, SSB has 3 dB better SNR.

B. Multiplexing Techniques

• Frequency Division Multiplexing (FDM):

  • Concept: Each signal modulates a different carrier frequency (f_c1, f_c2, ...). Spectra are separated in frequency domain.

  • Application: Analog systems (AM radio bands, cable TV).

  • Process: Modulate → translate spectra → sum. At receiver: bandpass filter → demodulate.

• Time Division Multiplexing (TDM): (See Section V.D) → Digital domain.

C. Bandwidth and Spectral Efficiency

• Bandwidth Calculations (approx.):

  • AM (DSB-FC): B = 2 f_m.

  • DSB-SC: B = 2 f_m.

  • SSB: B = f_m.

  • FM (Carson's Rule): B ≈ 2(Δf + f_m).

  • BPSK/QPSK: B ≈ R_s (for practical pulses). R_s = R_b / log2(M).

  • FSK: B ≈ |f1-f2| + 2f_m = 2Δf + 2f_m.

• Bandwidth Efficiency (η): η = (bit rate) / (bandwidth) = R_b / B (bits/s/Hz).

  • BPSK: η ≈ 1 (if B ≈ R_b).

  • QPSK: η ≈ 2 (since R_b = 2 R_s, B ≈ R_s).

  • M-ary: η = log2(M) (theoretical max).

[!TIP] Exam Formula: For raised cosine filtering with roll-off α, B = R_s (1+α)/2 for QPSK. So η = 1/(1+α) (e.g., α=0.2 → η ≈ 0.83 bits/s/Hz for QPSK? Wait: R_b = 2 R_s, B = R_s(1+α), so η = 2R_s / (R_s(1+α)) = 2/(1+α). For α=0.2, η = 2/1.2 ≈ 1.67 bits/s/Hz).