DIGITAL COMMUNICATION (EC-502) - Unit 5 Short Notes

UNIT 5: DIGITAL COMMUNICATION - EXAM-FOCUSED SHORT NOTES

I. SAMPLING & PULSE MODULATION (Source Coding Fundamentals)

A. Sampling Theorem

• Nyquist Sampling Theorem (Lowpass): A bandlimited signal with highest frequency component $$\displaystyle f_m $$ Hz can be completely reconstructed from its samples if sampled at a rate $$\displaystyle f_s \geq 2f_m $$ samples/sec. The minimum rate $$\displaystyle 2f_m $$ is the Nyquist Rate.

  Aliasing: If $$\displaystyle f_s < 2f_m $$, high-frequency components fold back into lower frequencies, causing irreversible distortion. Prevented by anti-aliasing filter (low-pass) before sampling.

• Bandpass Sampling Theorem: For a signal occupying $$\displaystyle f_1 $$ to $$\displaystyle f_2 $$ Hz ($$\displaystyle f_2 > f_1 $$), the minimum sampling rate is $$\displaystyle f_s = 2(f_2 - f_1) $$ provided $$\displaystyle f_2 $$ is an integer multiple of $$\displaystyle (f_2 - f_1) $$.

• Aperture Effect: In practical sampling (e.g., sample-and-hold), each sample is held for a finite duration $T$. This acts as a low-pass filter with sinc response, attenuating high frequencies. Compensated by an equalizer.

B. Types of Sampling

C. Pulse Modulation Techniques (PAM, PWM, PPM)

Generation & Detection of PWM/PPM (Brief):

• PWM Gen: Comparator compares $x(t)$ with a high-frequency sawtooth. Output high when $$\displaystyle x(t) > $$ sawtooth.

• PWM Det: RC integrator converts PWM to PAM, then sample & hold.

• PPM Gen: Use PWM pulse to trigger a monostable that generates a fixed-width pulse after a delay proportional to $x(t)$.

• PPM Det: Correlator with a locally generated clock. Measure time difference between received pulse and slot start.

D. Time Division Multiplexing (TDM)

• Principle: Multiple signals share a single channel by allocating non-overlapping time slots in a recurring frame. Each signal is sampled, and samples are transmitted sequentially.

• Process: Sampling → Pulse Modulation (e.g., PAM) → Multiplexer (assigns slot) → Transmission → Demultiplexer → Demodulation → Reconstruction.

• Requires precise synchronization between multiplexer and demultiplexer.

II. QUANTIZATION & PULSE CODE MODULATION (PCM)

A. Quantization

• Need: Convert continuous-amplitude samples (from sampling) into a finite set of discrete levels for digital transmission.

• Uniform Quantization:

  • Step Size: $$\displaystyle \Delta = \frac{V_{max} - V_{min}}{L} $$, where $$\displaystyle L = 2^n $$ levels, $n$ = bits/sample.

  • Quantizer: Maps input $x$ to nearest quantization level $$\displaystyle x_q = k\Delta $$, $k \in \mathbb{Z}$.

• Quantization Noise (Mean Square Error):

  The error $$\displaystyle e = x - x_q $$ is uniformly distributed in $[-\Delta/2, \Delta/2]$.

  Variance (Power) of Quantization Noise:

$$\sigma_q^2 = \frac{\Delta^2}{12}$$

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

  For a uniform quantizer with $L$ levels and input signal power $$\displaystyle \sigma_x^2 $$:

$$\text{SQNR (dB)} = 10 \log_{10} \left( \frac{\sigma_x^2}{\sigma_q^2} \right) \approx 10 \log_{10} \left( \frac{3 \sigma_x^2 L^2}{2} \right) \approx 6n + 10 \log_{10} \left( \frac{3 \sigma_x^2}{2} \right) \text{ dB}$$

> **Key Insight:** SQNR improves by ~6 dB per additional bit.

B. Pulse Code Modulation (PCM)

• Complete PCM System Block Diagram:

  Input Analog Signal → Anti-aliasing Filter → Sampler → Quantizer → Encoder → Digital Channel → Decoder → Reconstruction Filter → Output Analog Signal.

• Companding (Compressing/Expanding):

  • Need: Non-uniform quantization to improve SQNR for low-amplitude signals (like speech) where probability is high.

  • μ-law (North America, Japan): $$\displaystyle y = \frac{\ln(1+\mu|x|)}{\ln(1+\mu)} \cdot \text{sgn}(x) $$, $$\displaystyle \mu=255 $$.

  • A-law (Europe, ITU): Piecewise linear with $$\displaystyle A=87.6 $$.

  • Application: Used in G.711 standard for telephony (8-bit PCM at 8 kHz).

C. Delta Modulation (DM) & Adaptive Delta Modulation (ADM)

• DM Principle: Encodes the difference (delta) between current and previous sample. Uses 1-bit quantizer.

  • Block Diagram: Input → Comparator (with feedback from Integrator) → 1-bit Encoder → Bit Stream. Receiver: Bit Stream → Integrator → Output.

  • Problems:

    1. Granular Noise: Small step size $\Delta$ causes noise when signal is flat.

    2. Slope Overload: Large step size $\Delta$ cannot track steep signal slopes.

• Adaptive Delta Modulation (ADM): Step size $\Delta$ is adaptively changed based on recent bit pattern (e.g., increase $\Delta$ after consecutive 1s or 0s). Reduces both granular noise and slope overload.

III. LINE CODING & BASEBAND TRANSMISSION

A. Digital Data Representation & Line Encoding

B. Baseband Transmission Issues

• Inter-Symbol Interference (ISI): Smearing of pulses due to channel bandwidth limitation or multipath, causing overlap with adjacent bits.

• Nyquist Criterion for Zero ISI: The overall system frequency response $H(f)$ must satisfy:

$$\sum_{k=-\infty}^{\infty} H\left(f + \frac{k}{T_b}\right) = \text{Constant}$$

where $$\displaystyle T_b $$ = bit period. This leads to **Nyquist pulses** (e.g., sinc, raised-cosine).

• Eye Pattern: Oscilloscope display of overlayed received bits.

  • Eye Opening: Vertical height → noise margin; Horizontal width → timing jitter tolerance.

  • Best Sampling Instant: Center of eye.

• Crosstalk: Unwanted coupling between adjacent transmission lines (near-end, far-end).

C. Differential Encoding

• Concept: Encode data based on change from previous bit, not absolute level.

  • Differential Manchester and DPSK are examples.

  • Encoding: $$\displaystyle d_k = b_k \oplus d_{k-1} $$.

• Advantages:

  1. Immunity to phase ambiguity (e.g., 180° inversion in BPSK).

  2. Simpler receiver for non-coherent detection (DPSK).

• Disadvantages:

  1. Error propagation: 1 bit error corrupts current and next decoded bit.

  2. Requires differential encoder/decoder.

IV. DIGITAL MODULATION TECHNIQUES (Passband Transmission)

A. Binary Modulation

B. Quadrature Modulation

• QPSK (Quadrature PSK):

  • Two bits per symbol (00, 01, 11, 10).

  • Maps to 4 phases: $$\displaystyle 0^\circ, 90^\circ, 180^\circ, 270^\circ $$.

  • Generation: Split data into I (even bits) and Q (odd bits). Modulate two BPSK carriers ($\cos$ and $\sin$) with 90° phase shift, then sum.

  • Detection: Coherent demodulation with two matched filters (I & Q channels), then combine.

  • Bandwidth: Same as BPSK but double the bit rate (2 bits/symbol).

• Offset QPSK (OQPSK): Offset Q-channel bits by $$\displaystyle T_b/2 $$ relative to I-channel. Reduces maximum phase transition from 180° to 90° → lower spectral sidelobes.

• π/4-QPSK: Alternates between QPSK constellations rotated by 45°. Constant envelope, used in IS-136.

C. Differential Modulation

• DPSK (Differential PSK):

  • Principle: Encode data differentially first: $$\displaystyle d_k = b_k \oplus d_{k-1} $$. Then transmit using PSK (phase change $$\displaystyle \Delta \phi = 0^\circ $$ for $$\displaystyle d_k=0 $$, $$\displaystyle 180^\circ $$ for $$\displaystyle d_k=1 $$).

  • Detection: Compare phase of current and previous received symbols (non-coherent).

  • Comparison with Coherent PSK: ~3 dB worse BER for same $$\displaystyle E_b/N_0 $$, but avoids carrier phase recovery.

D. Minimum Shift Keying (MSK)

• Principle: Special case of Continuous-Phase FSK (CPFSK) with modulation index $$\displaystyle h = 0.5 $$. Frequency separation $$\displaystyle \Delta f = \frac{1}{2T_b} $$ ensures phase continuity.

• Spectral Properties: More compact spectrum than QPSK (lower sidelobes), constant envelope.

• Advantages over QPSK: Better spectral efficiency, robust to non-linearities (constant envelope). Used in GSM, Bluetooth.

E. M-ary Modulation

• M-ary PSK: $M$ phases equally spaced on a circle. Bandwidth efficiency = $$\displaystyle \log_2 M $$ bits/symbol/Hz. Error probability increases with $M$ for same $$\displaystyle E_b/N_0 $$.

• M-ary FSK: $M$ orthogonal frequencies. Bandwidth $$\displaystyle \approx M \cdot \frac{1}{T_b} $$. Orthogonality condition: $$\displaystyle \int_0^{T_b} \cos(2\pi f_i t) \cos(2\pi f_j t) dt = 0 $$ for $i \neq j$.

• M-ary QAM: Combination of ASK & PSK. $$\displaystyle M = 2^k $$ points in a square grid (e.g., 16-QAM, 64-QAM). Higher bandwidth efficiency but more susceptible to noise and non-linearities.

F. Spectral Properties Comparison

G. Performance Comparison & Applications

V. SIGNAL SPACE, DETECTION & ERROR PROBABILITY

A. Signal Space Representation

• Concept: Represent any set of $M$ signals $$\displaystyle \{s_i(t)\} $$, $0 \leq t \leq T$, as vectors in an $N$-dimensional space using an orthogonal basis $$\displaystyle \{\phi_1(t), ..., \phi_N(t)\} $$.

$$s_i(t) = \sum_{j=1}^{N} s_{ij} \phi_j(t)$$

where $$\displaystyle s_{ij} = \int_0^T s_i(t) \phi_j(t) dt $$ are the **coordinates** (vector components).

• Gram-Schmidt Orthogonalization Procedure:

  1. Start with first basis vector: $$\displaystyle \phi_1(t) = \frac{s_1(t)}{\|s_1(t)\|} $$.

  2. For $$\displaystyle k = 2 $$ to $M$:

     • Projection: $$\displaystyle p_k(t) = s_k(t) - \sum_{j=1}^{k-1} \left[ \int_0^T s_k(t) \phi_j(t) dt \right] \phi_j(t) $$.

     • Normalize: $$\displaystyle \phi_k(t) = \frac{p_k(t)}{\|p_k(t)\|} $$.

  3. Result: $$\displaystyle \{\phi_1, ..., \phi_M\} $$ is an orthonormal basis for the signal set.

• Use: Reduces continuous-time detection problem to vector detection in Euclidean space. Optimum receiver correlates with basis functions.

B. Optimum Receivers

• Matched Filter:

  • Principle: Maximizes SNR at sampling instant $$\displaystyle t=T $$ for known signal $s(t)$ in AWGN.

  • Impulse Response: $$\displaystyle h(t) = s(T-t) $$, $0 \leq t \leq T$.

  • Block Diagram: Input r(t) → Matched Filter → Sampler at t=T → Decision Device.

  • Advantage: Equivalent to a correlator (integrates $r(t)$ with $s(t)$ over $[0,T]$). Optimal in sense of maximum likelihood detection.

• Correlator Detector:

  • Principle: Computes $$\displaystyle z = \int_0^T r(t) s(t) dt $$.

  • Comparison with Matched Filter: For baseband signals, matched filter output at $$\displaystyle t=T $$ equals correlator output. Often implemented as matched filter followed by sampler.

C. Error Probability Analysis

• BPSK (Coherent):

  • Signals: $$\displaystyle s_1(t) = \sqrt{E_b} \cos(2\pi f_c t) $$, $$\displaystyle s_0(t) = -\sqrt{E_b} \cos(2\pi f_c t) $$.

  • Probability of Bit Error:

$$P_b = Q\left( \sqrt{\frac{2E_b}{N_0}} \right)$$

where $$\displaystyle Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty e^{-t^2/2} dt $$.

• QPSK (Coherent):

  • Equivalent to two orthogonal BPSK channels (I & Q). Each bit has same $$\displaystyle E_b/N_0 $$ as BPSK.

  • Probability of Symbol Error:

$$P_s = 2Q\left( \sqrt{\frac{E_b}{N_0}} \right) - Q^2\left( \sqrt{\frac{E_b}{N_0}} \right) \approx 2Q\left( \sqrt{\frac{E_b}{N_0}} \right)$$

*   **Probability of Bit Error (Gray coding):** $$\displaystyle P_b \approx \frac{1}{2} P_s = Q\left( \sqrt{\frac{E_b}{N_0}} \right) $$.

> **Key:** QPSK BER is **same as BPSK** for same $$\displaystyle E_b/N_0 $$, but transmits 2 bits/symbol.

• M-ary Systems (General Concept): As $M$ increases (for fixed $$\displaystyle E_b $$), minimum distance between constellation points decreases → higher $$\displaystyle P_e $$. Trade-off: Bandwidth efficiency ↑, Power efficiency ↓.

• M-ary QAM: For square $M$-QAM (e.g., 16-QAM, 64-QAM), symbol error probability is complex, but approximate for large $M$:

$$P_s \approx 4 \left(1 - \frac{1}{\sqrt{M}}\right) Q\left( \sqrt{\frac{3 E_b}{(M-1) N_0}} \right)$$

VI. CHANNEL CODING & ERROR CONTROL

A. Information Theory Basics

• Entropy $H(X)$: Average information content per symbol of discrete source $X$.

$$H(X) = -\sum_{i=1}^{m} p(x_i) \log_2 p(x_i) \text{ bits/symbol}$$

• Information Rate $R$: $$\displaystyle R = r \cdot H(X) $$ bits/sec, where $r$ = symbol rate.

• Mutual Information $I(X;Y)$: Average information that $Y$ conveys about $X$. $$\displaystyle I(X;Y) = H(X) - H(X|Y) $$.

B. Channel Capacity & Fundamental Theorems

• Shannon-Hartley Theorem (AWGN Channel):

$$\boxed{C = B \log_2 \left(1 + \frac{S}{N}\right) = B \log_2 \left(1 + \text{SNR}\right)}$$

where $C$ = channel capacity (bits/sec), $B$ = bandwidth (Hz), $S/N$ = signal-to-noise power ratio.

> **Implication:** For fixed $B$, capacity increases with $\log(\text{SNR})$. For fixed SNR, capacity increases linearly with $B$.

• Information Capacity Theorem (General):

$$C = \max_{p(x)} I(X;Y)$$

Capacity is the maximum mutual information over all input distributions.

• Binary Symmetric Channel (BSC) Capacity:

  For crossover probability $p$, capacity is:

$$C = 1 - H(p) \text{ bits/channel use}$$

where $$\displaystyle H(p) = -p \log_2 p - (1-p) \log_2 (1-p) $$.

*Example:* For $$\displaystyle p=0.1 $$, $$\displaystyle C = 1 - H(0.1) \approx 1 - 0.469 = 0.531 $$ bits/use.

C. Block Codes

• Linear Block Codes:

  • Parameters: $(n, k)$ — $k$ info bits → $n$ coded bits ($n-k$ parity).

  • Generator Matrix $G$: $k \times n$ matrix. Codeword $$\displaystyle \mathbf{c} = \mathbf{u} G $$, $\mathbf{u}$ = info vector.

  • Parity-Check Matrix $H$: $(n-k) \times n$ matrix, $$\displaystyle G H^T = 0 $$. Syndrome $$\displaystyle \mathbf{S} = \mathbf{r} H^T $$. $$\displaystyle \mathbf{S}=0 $$ → no error (or undetectable).

  • Syndrome Decoding: Pre-compute syndrome table for all correctable error patterns.

• Cyclic Codes:

  • Subclass of linear codes where cyclic shift of a codeword is another codeword.

  • Generator Polynomial $g(x)$: Degree $n-k$. Codeword polynomial $c(x)$ is divisible by $g(x)$.

  • Encoding (Shift Register): Implement division by $g(x)$.

  • Decoding: Syndrome computed by evaluating $r(x)$ at roots of $g(x)$ (using syndrome polynomial).

• Hamming Codes (e.g., (7,4)):

  • Single-error correcting (SEC). $$\displaystyle n = 2^r - 1 $$, $$\displaystyle k = n - r $$.

  • For (7,4): $$\displaystyle r=3 $$ parity bits. Positions: 1,2,4 are parity (powers of 2). Data in positions 3,5,6,7.

  • Parity Equations:

    $$\displaystyle p_1 = d_3 \oplus d_5 \oplus d_7 $$

    $$\displaystyle p_2 = d_3 \oplus d_6 \oplus d_7 $$

    $$\displaystyle p_4 = d_5 \oplus d_6 \oplus d_7 $$

  • Syndrome bits $$\displaystyle (S_1, S_2, S_4) $$ give binary position of error.

D. Convolutional Codes

• Encoding:

  • Constraint Length $K$: Number of input bits that affect current output.

  • Code Rate $k/n$: $k$ input bits → $n$ output bits per time unit.

  • Represented by generator polynomials or state diagram/trellis.

• Decoding - Viterbi Algorithm:

  • Principle: Maximum Likelihood Sequence Estimation (MLSE). Finds maximum likelihood path through trellis.

  • Process: Recursively computes path metric (cumulative Hamming distance) for each state at each time step. Retains survivor path per state.

  • Output: Most likely transmitted sequence after traceback depth (typically $5K$).

E. Burst Error Correcting Codes

• Concept: Correct clusters of errors (bursts) common in wireless/fading channels.

• Fire Codes: Use generator polynomial with a factor $$\displaystyle (x^{b+1} - 1) $$ to correct bursts of length $b$.

• Reed-Solomon (RS) Codes:

  • Type: Non-binary BCH codes. Symbols from $$\displaystyle GF(2^m) $$.

  • Parameters: $(n, k)$ with $$\displaystyle n = 2^m - 1 $$. Can correct up to $$\displaystyle t = \frac{n-k}{2} $$ symbol errors.

  • Key: Each symbol = $m$ bits. Excellent for burst errors (a symbol error = $m$ bit errors).

  • Applications: CD/DVD, QR codes, satellite comms (CCSDS), DSL.

F. Applications of Error Correcting Codes (ECC)

• Deep Space Communication: Low-power signals, long distances → strong codes (e.g., concatenated RS+convolutional).

• Wireless (Cellular, Wi-Fi): Fading channels → convolutional, turbo, LDPC.

• Data Storage (HDD, SSD, CD/DVD/Blu-ray): Burst errors from media defects → RS codes.

• DSL (Digital Subscriber Line): Twisted pair noise → trellis-coded modulation (TCM).

• Satellite TV (DVB-S2): LDPC + BCH for near-Shannon performance.

VII. SYSTEM-LEVEL & MISCELLANEOUS TOPICS

A. Bandwidth Efficiency & Trade-offs

• Definition: $$\displaystyle \eta = \frac{\text{Bit Rate (bps)}}{\text{Channel Bandwidth (Hz)}} $$ (units: bits/s/Hz).

• Trade-off: Increasing $\eta$ (e.g., using M-ary modulation) requires higher SNR for same error probability (power efficiency ↓). Conversely, high power efficiency (BPSK) has low $\eta$.

• Calculation: Given bit rate $$\displaystyle R_b $$ and $\eta$, required bandwidth $$\displaystyle B = \frac{R_b}{\eta} $$.

B. Complete System Blocks

• PCM System: [As detailed in Section II.B].

• Delta Modulator:

  Analog Input → Comparator (with ±Δ from Quantizer) → 1-bit Encoder → Bit Stream.

  Bit Stream → Decoder (Integrator) → Reconstructed Analog.

  Feedback: Reconstructed Analog → Quantizer (±Δ) → back to comparator.

C. Key Definitions & Concepts

• Bandwidth: Range of frequencies occupied by signal (e.g., null-to-null, 90% power).

• Aliasing: Overlap of spectral replicas due to undersampling.

• Quantization Noise: Distortion from mapping continuous amplitude to discrete levels; power $$\displaystyle \sigma_q^2 = \Delta^2/12 $$.

• ISI (Inter-Symbol Interference): Overlap of adjacent symbols due to channel dispersion.

• Eye Pattern: Visual tool to assess ISI, noise, timing jitter.

• Differential Encoding: Encoding based on transition; advantage: immunity to phase reversals.

Exam Tips & Common Pitfalls:

1. Aliasing vs. Aperture Effect: Aliasing is frequency-domain folding due to low sampling rate. Aperture effect is time-domain averaging due to finite sample width.

2. SQNR Formula: Remember $\text{SQNR} \approx 6n$ dB for uniform quantizer with full-scale sine wave.

3. QPSK vs. BPSK BER: For same $$\displaystyle E_b/N_0 $$, QPSK BER ≈ BPSK BER (per bit). QPSK has same power efficiency but double bandwidth efficiency.

4. Matched Filter vs. Correlator: Matched filter maximizes SNR at a specific time. Correlator (with integration over symbol period) is a specific implementation for baseband/bandpass with coherent carrier.

5. Shannon-Hartley: Capacity depends on bandwidth and linear in $\log(1+\text{SNR})$, not linear in SNR.

6. Gram-Schmidt: Always start with first signal $$\displaystyle s_1(t) $$, subtract projections of all previous basis functions from $$\displaystyle s_k(t) $$ before normalizing.

7. TDM vs. FDM: TDM shares time, FDM shares frequency.

8. Manchester vs. Differential Manchester: Manchester has transition at bit center; Differential Manchester has transition at bit start always, mid-bit transition indicates data.