UNIT 2: DIGITAL COMMUNICATION - EXAM-FOCUSED SHORT NOTES

1. SAMPLING & ANALOG-TO-DIGITAL CONVERSION FOUNDATIONS

1.1 Sampling Theorem (Nyquist-Shannon)

• Lowpass Signal Sampling Theorem:

  A bandlimited signal with no frequency components higher than $$\displaystyle f_m $$ Hz can be completely reconstructed from its samples if sampled at a rate $$\displaystyle f_s > 2f_m $$ samples/sec.

  • Minimum Sampling Rate (Nyquist Rate): $$\displaystyle f_N = 2f_m $$.

  • Sampling Frequency: $$\displaystyle f_s = \frac{1}{T_s} $$.

  • Reconstruction: Using an ideal lowpass filter (sinc interpolation).

• Bandpass Signal Sampling Theorem:

  For a signal occupying bandwidth $B$ Hz centered at $$\displaystyle f_c $$ (where $$\displaystyle f_c >> B $$), the minimum sampling rate is $$\displaystyle f_s = 2B $$, provided $$\displaystyle f_c $$ is an integer multiple of $B$.

  • Key Insight: Bandpass signals can be sampled at a rate much lower than $$\displaystyle 2f_{\text{max}} $$ due to spectral gaps.

[!TIP] Exam Trap: Distinguish clearly between lowpass (baseband) and bandpass sampling conditions. Bandpass sampling is also called undersampling or harmonic sampling.

1.2 Types of Sampling

Aperture Effect: In flat-top sampling, the effective sampling instant is the centroid of the pulse, causing a $\text{sinc}(f\tau)$ attenuation in the spectrum.

1.3 Aliasing & Anti-Aliasing

• Aliasing: Overlapping of spectral replicas due to $$\displaystyle f_s < 2f_m $$. Causes irreversible distortion; high frequencies masquerade as low frequencies.

• Anti-Aliasing Filter: A lowpass filter (with cutoff $$\displaystyle f_c \leq f_s/2 $$) placed before the sampler to bandlimit $x(t)$ and prevent aliasing.

• Aliasing Error: The error between the original and reconstructed signal due to spectral overlap.

[!TIP] Exam Must-Know: Aliasing is preventable (via anti-aliasing filter) but not correctable after sampling.

1.4 Quantization

• Need: To convert continuous amplitude samples into discrete digital values (PCM requirement).

• Uniform Quantization:

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

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

  • Assumption: $$\displaystyle e_q $$ is uniformly distributed in $(-\Delta/2, \Delta/2)$.

• Quantization Noise Power ($$\displaystyle \sigma_q^2 $$):

$$\sigma_q^2 = \frac{\Delta^2}{12} = \frac{(x_{\text{max}} - x_{\text{min}})^2}{12 \cdot 2^{2n}}$$

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

  For a full-scale sinusoidal signal ($$\displaystyle x(t) = A \sin(2\pi f t) $$), $$\displaystyle P_x \approx A^2/2 $$.

$$\text{SQNR} = \frac{P_x}{\sigma_q^2} \approx \frac{3 \cdot 2^{2n}}{2} \left(\frac{A}{x_{\text{max}}}\right)^2 \ \text{(linear)}$$

$$\text{SQNR}_{\text{dB}} \approx 6n + 1.76 + 20\log_{10}\left(\frac{A}{x_{\text{max}}}\right) \ \text{dB}$$

> For a full-scale sine wave ($$\displaystyle A = x_{\text{max}} $$), $$\displaystyle \text{SQNR}_{\text{dB}} \approx 6n + 1.76 \ \boxed{\text{dB}} $$.

• Non-Uniform Quantization & Companding:

  • Purpose: Improve SQNR for low-amplitude signals (where uniform quantization noise is relatively high).

  • Principle: Compress signal at transmitter (using a compressor), uniform quantize, expand at receiver.

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

  • A-law (Europe, ITU): Piecewise linear approximation.

1.5 Pulse Modulation Techniques (ADC Chain)

PCM Block Diagram: Sampler → Quantizer → Encoder → Transmitter → Channel → Receiver → Decoder → Reconstructor.

Delta Modulation: Encodes difference between current and previous sample. Output is 1-bit stream.

2. BASEBAND TRANSMISSION & LINE CODING

2.1 Line Encoding / Baseband Signaling

Manchester Rule: Bit value determined by transition direction at the bit center.

Differential Encoding: b_i' = b_i ⊕ b_{i-1}. Advantage: If signal inverted, decoded bits remain same.

2.2 Transmission Impairments

• Inter-Symbol Interference (ISI):

  • Cause: Bandwidth limitation → pulse spreading → overlap with adjacent symbols.

  • Mitigation:

    1. Nyquist Criterion (Zero-ISI): $H(f)$ must satisfy $$\displaystyle H(f) \cdot \sum_{k=-\infty}^{\infty} \text{sinc}(fT - k) = \text{constant} $$. (e.g., raised-cosine).

    2. Equalization: Compensate channel distortion.

• Crosstalk: Unwanted coupling between adjacent channels/cores.

• Eye Pattern:

  • Use: Visualize effects of ISI, noise, jitter.

  • Interpretation:

    • Eye Opening: Indicates margin for sampling (larger = less ISI).

    • Slope of Eye: Indicates sensitivity to timing jitter.

    • Vertical Height: Indicates noise margin.

3. PASSBAND MODULATION SCHEMES (DIGITAL MODULATION)

3.1 Binary Modulation

3.2 Quadrature Modulation: QPSK

• Principle: Two orthogonal BPSK signals on I (In-phase) and Q (Quadrature) channels.

$$s(t) = I(t)\cos(2\pi f_c t) - Q(t)\sin(2\pi f_c t)$$

where $I(t), Q(t) \in \{\pm A/\sqrt{2}\}$.

• Signal Constellation: 4 points at phases 0°, 90°, 180°, 270°.

  • Mapping: 00 → 0°, 01 → 90°, 11 → 180°, 10 → 270°.

• Generation & Detection:

  • Tx: Serial-to-Parallel → 2 BPSK modulators (on $\cos$ & $\sin$) → Sum.

  • Rx: Coherent detection. Separate I & Q channels using $\cos$ & $\sin$ mixers → LPFs → Samplers → Decision → Parallel-to-Serial.

• Spectral Properties:

  • Bandwidth Efficiency: $$\displaystyle \eta = \frac{R_b}{B} = 2 \ \text{bits/sec/Hz} $$ (twice BPSK).

  • Null-to-Null BW: $$\displaystyle \approx 2R_b/T_b = 2R_b $$? Wait, for QPSK, symbol rate $$\displaystyle R_s = R_b/2 $$, so $$\displaystyle B \approx R_s = R_b/2 $$? Correction: Main lobe BW $$\displaystyle \approx 2R_s = R_b $$. So $$\displaystyle \eta = R_b / R_b = 1 $$? Let's clarify: For rectangular pulse shaping, first null BW = $$\displaystyle 2R_s $$. For QPSK, $$\displaystyle R_s = R_b/2 $$, so BW = $$\displaystyle 2 \times (R_b/2) = R_b $$. Hence $$\displaystyle \eta = R_b / R_b = 1 \ \text{bit/s/Hz} $$? No, QPSK transmits 2 bits per symbol, so $$\displaystyle \eta = (2R_s)/ (2R_s) = 1 $$? Actually, bandwidth efficiency is bits per second per Hz. $$\displaystyle R_b = 2R_s $$, BW $$\displaystyle \approx 2R_s $$, so $$\displaystyle \eta = 2R_s / (2R_s) = 1 \ \text{bit/s/Hz} $$. But commonly stated as 2 b/s/Hz because they compare to BPSK which uses same BW for half the rate. Standard result: QPSK has same bandwidth as BPSK but double data rate, so $$\displaystyle \eta_{\text{QPSK}} = 2 \times \eta_{\text{BPSK}} $$. If BPSK $$\displaystyle \eta=1 $$, QPSK $$\displaystyle \eta=2 $$.

  • Spectral Shape: Similar to BPSK but power is split between carriers.

• Error Probability:

$$P_e(\text{QPSK}) = 2Q\left(\sqrt{\frac{E_b}{N_0}}\right) - Q^2\left(\sqrt{\frac{E_b}{N_0}}\right) \approx 2Q\left(\sqrt{\frac{2E_b}{N_0}}\right) \text{ for high SNR}$$

> **Relation to BPSK:** $$\displaystyle P_e(\text{QPSK}) \approx 2 P_e(\text{BPSK}) $$? Actually BPSK: $$\displaystyle P_e = Q(\sqrt{2E_b/N_0}) $$. QPSK: $$\displaystyle P_e = 2Q(\sqrt{E_b/N_0}) - Q^2(...) $$. For same $$\displaystyle E_b/N_0 $$, QPSK has slightly higher error. But for same **bandwidth**, QPSK uses half the $$\displaystyle E_b $$? Wait, careful: For same **bit rate** and **bandwidth**, QPSK uses half the power per bit? Actually, if total power $P$ is same, BPSK: $$\displaystyle E_b = P/R_b $$. QPSK: $$\displaystyle E_b = P/(2R_s) = P/R_b $$? Because $$\displaystyle R_b = 2R_s $$, so $$\displaystyle E_b = P/R_b $$ same. So comparison at same $$\displaystyle E_b/N_0 $$: QPSK has $\approx 2 \times$ BPSK error? Let's derive: BPSK: $$\displaystyle P_e = Q(\sqrt{2E_b/N_0}) $$. QPSK: $$\displaystyle P_e = Q(\sqrt{E_b/N_0}) + Q(\sqrt{E_b/N_0}) - ... \approx 2Q(\sqrt{E_b/N_0}) $$ for large SNR. Since $$\displaystyle Q(\sqrt{2E_b/N_0}) < Q(\sqrt{E_b/N_0}) $$, QPSK error is higher. But often they say "QPSK has same error performance as BPSK" because they compare at same **$$\displaystyle E_s/N_0 $$** (symbol energy). For QPSK, $$\displaystyle E_s = 2E_b $$, so $$\displaystyle P_e(\text{QPSK}) \approx 2Q(\sqrt{E_s/(2N_0)}) = 2Q(\sqrt{E_b/N_0}) $$. While BPSK: $$\displaystyle P_e = Q(\sqrt{2E_b/N_0}) = Q(\sqrt{E_s/N_0}) $$. So at same $$\displaystyle E_s/N_0 $$, BPSK is better. **Standard result:** For coherent detection, $$\displaystyle P_e(\text{QPSK}) = Q\left(\sqrt{\frac{2E_b}{N_0}}\right) $$? No, that's BPSK. I'll use the common approximation: $$\displaystyle P_e(\text{QPSK}) \approx 2Q\left(\sqrt{\frac{E_b}{N_0}}\right) $$.

*   **Advantages over BPSK:** **Double bandwidth efficiency** (same BW, double data rate). Same hardware complexity.

3.3 Differential Phase Shift Keying (DPSK)

• Principle: Encode bits as phase change relative to previous symbol.

  • Generation: $$\displaystyle s(t) = A \cos(2\pi f_c t + \theta_n) $$, where $$\displaystyle \theta_n = \theta_{n-1} + \Delta\phi_n $$, $$\displaystyle \Delta\phi_n = 0 $$ for '0', $\pi$ for '1' (for DPSK).

  • Detection (Differential Coherent): Compare phase of current and previous received symbols using a delay-and-multiply circuit.

• Comparison with PSK:

  • Advantage: No need for precise carrier phase synchronization (phase ambiguity resolved).

  • Disadvantage: $\approx 1 \ \text{dB}$ performance loss (higher $$\displaystyle P_e $$) compared to coherent PSK.

3.4 M-ary Modulation

• M-ary PSK:

  • Constellation: $M$ points equally spaced on a circle.

  • Phase Shift: $$\displaystyle \phi_k = \frac{2\pi k}{M}, k=0,1,...,M-1 $$.

  • Bandwidth Efficiency: $$\displaystyle \eta = \log_2 M \ \text{bits/s/Hz} $$ (same as QAM).

  • Trade-off: As $M$ ↑, $\eta$ ↑ but $$\displaystyle P_e $$ ↑ (minimum distance between points ↓).

• M-ary FSK:

  • Orthogonal FSK: Frequencies separated by $$\displaystyle f_m = \frac{k}{T_s} $$, $k$ integer. Minimum distance max → best $$\displaystyle P_e $$.

  • Bandwidth: $$\displaystyle B \approx M \Delta f + 2R_s $$ (much larger than PSK/QAM).

  • Non-orthogonal: Frequencies closer → reduced BW but higher $$\displaystyle P_e $$.

• Quadrature Amplitude Modulation (QAM):

  • Principle: Independent amplitude modulation on I and Q carriers.

  • Constellation: Grid of points (e.g., 16-QAM: $4 \times 4$ grid).

  • Generation/Detection: Same as QPSK but with multi-level I/Q signals.

  • Error Probability: $$\displaystyle P_e \approx 4\left(1 - \frac{1}{\sqrt{M}}\right) Q\left(\sqrt{\frac{3E_b}{N_0 (M-1)}}\right) $$ for square M-QAM.

  • Trade-off: Higher $\eta$ than PSK for same $M$, but more sensitive to noise and nonlinearities.

3.5 Special Modulation: Minimum Shift Keying (MSK)

• Principle: A special case of Offset QPSK (OQPSK) with continuous phase.

  • OQPSK: Offset Q-channel by $$\displaystyle T_s/2 $$ relative to I-channel → reduces envelope variations.

  • MSK: Further, modulation index $$\displaystyle h = 0.5 $$. Frequency separation $$\displaystyle \Delta f = \frac{1}{4T_s} $$.

• Spectral Properties:

  • Narrower main lobe than QPSK/OQPSK (more power in main lobe).

  • Lower out-of-band sidelobes → better spectral containment.

• Comparison:

  | Feature | QPSK | OQPSK | MSK | | :--- | :--- | :--- | :--- | | Phase Shifts | 0°, 90°, 180°, 270° | Same | Continuous | | Envelope Variations | High (π/2, π shifts) | Reduced | Constant envelope | | Bandwidth (main lobe) | $$\displaystyle 2R_s $$ | $$\displaystyle \approx 1.5R_s $$ | $$\displaystyle \approx 1.5R_s $$? Actually MSK main lobe BW = $$\displaystyle 1.5R_s $$? Standard: MSK BW ≈ $$\displaystyle 1.5 R_s $$ (null-to-null). | | Spectral Efficiency | Moderate | Moderate | High (narrow main lobe). |

MSK = OQPSK with sinusoidal pulse shaping → continuous phase → constant envelope → robust to nonlinearities.

3.6 Comparison of Modulation Techniques

4. SIGNAL SPACE REPRESENTATION & DETECTION

4.1 Gram-Schmidt Orthogonalization Procedure

• Purpose: Transform a set of $n$ linearly independent signals $$\displaystyle \{s_1(t), s_2(t), ..., s_n(t)\} $$ into an orthogonal basis $$\displaystyle \{\phi_1(t), \phi_2(t), ..., \phi_n(t)\} $$ spanning the same signal space.

• Steps:

  1. $$\displaystyle \phi_1(t) = s_1(t) $$

  2. $$\displaystyle \phi_2(t) = s_2(t) - \frac{\langle s_2, \phi_1 \rangle}{\|\phi_1\|^2} \phi_1(t) $$

  3. $$\displaystyle \phi_3(t) = s_3(t) - \frac{\langle s_3, \phi_1 \rangle}{\|\phi_1\|^2} \phi_1(t) - \frac{\langle s_3, \phi_2 \rangle}{\|\phi_2\|^2} \phi_2(t) $$

  4. Continue...

• Normalization: $$\displaystyle \psi_i(t) = \frac{\phi_i(t)}{\|\phi_i\|} $$, where $$\displaystyle \|\phi_i\| = \sqrt{\langle \phi_i, \phi_i \rangle} $$.

• Signal Representation: Any signal $$\displaystyle s_k(t) $$ can be expressed as $$\displaystyle s_k(t) = \sum_{i=1}^{n} a_{ki} \psi_i(t) $$, where $$\displaystyle a_{ki} $$ are coefficients.

• Application in Digital Comm: Represent $M$ modulated signals as vectors in $n$-dimensional space. Detection reduces to choosing the closest signal point (minimum distance rule).

4.2 Optimal Detection

• Matched Filter (MF):

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

  • Transfer Function: $$\displaystyle H(f) = K S^*(f) e^{-j2\pi f T_b} $$, where $S(f)$ is FT of $s(t)$.

  • Impulse Response: $$\displaystyle h(t) = K s(T_b - t) $$ (time-reversed, shifted version of signal).

  • Block Diagram: RF/IF → Mixer with local carrier (coherent) → LPF → MF → Sampler → Decision.

• Correlator Detector:

  • Principle: Computes inner product $$\displaystyle \int_0^{T_b} r(t) \cdot s_i(t) dt $$ with each possible signal $$\displaystyle s_i(t) $$.

  • Relationship: Matched filter output at $$\displaystyle T_b $$ equals correlator output for that signal.

  • Implementation: Often easier to implement as a correlator in digital domain after ADC.

[!TIP] Key Insight: Matched filter and correlator are equivalent for optimal detection in AWGN. Both implement the same inner product operation.

5. INFORMATION THEORY & CHANNEL CAPACITY

5.1 Fundamental Concepts

• Entropy ($H(X)$): Average information content/uncertainty of a discrete random variable $X$.

$$H(X) = -\sum_{i=1}^{m} p_i \log_2 p_i \ \text{bits}$$

*   Maximum when all $$\displaystyle p_i $$ equal (uniform distribution).

• Information Rate ($R$): Entropy per unit time. For a source emitting symbols every $$\displaystyle T_s $$ sec, $$\displaystyle R = \frac{H(X)}{T_s} \ \text{bits/sec} $$.

5.2 Shannon-Hartley Theorem

• Statement: The channel capacity $C$ (maximum error-free data rate) of a bandlimited AWGN channel is:

$$\boxed{C = B \log_2(1 + \text{SNR}) \ \text{bits/sec}}$$

where $B$ = channel bandwidth (Hz), $$\displaystyle \text{SNR} = \frac{P}{N_0 B} $$ (power signal-to-noise ratio).

• Implications:

  1. $C$ increases with both $B$ and SNR.

  2. Trade-off: Can increase $C$ by increasing $B$ or SNR (or both).

  3. For fixed $B$, $C$ increases logarithmically with SNR.

  4. For fixed SNR, $C$ increases linearly with $B$.

5.3 Channel Capacity & Shannon's Theorem

• Channel Capacity ($C$): The supremum of all information rates $R$ that can be achieved with arbitrarily low error probability.

• Shannon's Channel Coding Theorem: Reliable communication (error probability → 0) is possible if and only if $$\displaystyle R < C $$.

  If $$\displaystyle R > C $$, no coding scheme can achieve arbitrarily low error.

• Mutual Information ($I(X;Y)$): Information gained about $X$ from observing $Y$. $$\displaystyle C = \max_{p(x)} I(X;Y) $$.

• Binary Symmetric Channel (BSC):

  • Crossover probability $P$ (bit flip probability).

  • Capacity: $$\displaystyle C = 1 - H(P) \ \text{bits/channel use} $$, where $$\displaystyle H(P) = -P\log_2 P - (1-P)\log_2(1-P) $$.

  • Example: For $$\displaystyle P=0.1 $$, $$\displaystyle C = 1 - H(0.1) \approx 1 - 0.469 = 0.531 \ \text{bits/use} $$.

5.4 Bandwidth Efficiency (η)

$$\eta = \frac{R}{B} \ \frac{\text{bits/sec}}{\text{Hz}}$$

• Relation to Modulation: $$\displaystyle \eta_{\text{max}} $$ for a modulation scheme is $$\displaystyle \log_2 M $$ (for M-ary signaling) if ideal Nyquist pulses are used.

  • BPSK/QPSK: $$\displaystyle \eta = 1 \ \text{bit/s/Hz} $$? Actually QPSK: $$\displaystyle R_b = 2R_s $$, $$\displaystyle B \approx R_s $$, so $$\displaystyle \eta = 2R_s / R_s = 2 $$. But often they say QPSK has same BW as BPSK but double rate, so $$\displaystyle \eta_{\text{QPSK}} = 2 \ \text{bits/s/Hz} $$. BPSK: $$\displaystyle \eta = 1 \ \text{bit/s/Hz} $$.

  • 16-QAM: $$\displaystyle \eta = 4 \ \text{bits/s/Hz} $$.

• Shannon Limit: $$\displaystyle \eta \leq \log_2(1 + \text{SNR}) $$. For high SNR, $$\displaystyle \eta \approx \log_2(\text{SNR}) $$.

6. ERROR CONTROL CODING (ECC)

6.1 Introduction

• Need: To combat channel noise, achieve reliable communication at lower $$\displaystyle E_b/N_0 $$ or within fixed bandwidth.

• Code Rate ($$\displaystyle R_c $$): $$\displaystyle R_c = \frac{k}{n} $$, where $k$ = message bits, $n$ = codeword length. Redundancy $$\displaystyle = 1 - R_c $$.

• Trade-off: More redundancy → better error correction → lower net data rate.

6.2 Linear Block Codes

• Structure: $(n, k)$ linear block code. Generator matrix $$\displaystyle \mathbf{G}_{k \times n} $$, Parity-check matrix $$\displaystyle \mathbf{H}_{(n-k) \times n} $$ such that $$\displaystyle \mathbf{G} \mathbf{H}^T = \mathbf{0} $$.

• Encoding: $$\displaystyle \mathbf{c} = \mathbf{u} \mathbf{G} $$, where $\mathbf{u}$ is $k$-bit message vector.

• Syndrome Decoding:

  1. Compute syndrome: $$\displaystyle \mathbf{s} = \mathbf{r} \mathbf{H}^T $$.

  2. If $$\displaystyle \mathbf{s} = \mathbf{0} $$, assume no error (or undetectable).

  3. If $\mathbf{s} \neq \mathbf{0}$, look up in standard array or use syndrome table to find error pattern $\mathbf{e}$.

  4. Correct: $$\displaystyle \mathbf{c} = \mathbf{r} - \mathbf{e} $$.

• Hamming Codes:

  • Property: Single-error-correcting (SEC).

  • Relation: $$\displaystyle n = 2^m - 1 $$, $$\displaystyle k = n - m $$, where $m$ = parity bits.

  • Example (7,4): $$\displaystyle m=3 $$, $$\displaystyle n=7 $$, $$\displaystyle k=4 $$, $$\displaystyle R_c=4/7 $$.

  • Parity-check matrix $\mathbf{H}$: All non-zero $m$-bit columns.

$$\mathbf{H} = \begin{bmatrix} 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 & 1 & 1 \end{bmatrix}$$

*   **Syndrome:** 3-bit, directly points to error position (if single error).

6.3 Cyclic Codes

• Property: Any cyclic shift of a codeword is another codeword.

• Encoding using Generator Polynomial $g(x)$:

  • Codeword polynomial $$\displaystyle c(x) = m(x) \cdot x^{n-k} + r(x) $$, where $r(x)$ is remainder of $$\displaystyle m(x)x^{n-k} $$ divided by $g(x)$.

  • Implementation: Shift register circuit with feedback based on $g(x)$.

• Decoding: Syndrome calculation using $$\displaystyle s(x) = r(x) \mod g(x) $$. If $s(x) \neq 0$, error detected. For single-error-correcting cyclic codes (like Hamming), syndrome directly gives error location.

6.4 Convolutional Codes

• Concept: Encode $k$ input bits into $n$ output bits per time unit, with memory (constraint length $K$).

• Parameters: $(n, k, K)$ or $(n, k, L)$ where $$\displaystyle L = K-1 $$ is memory length.

• Representation:

  • State Diagram: $$\displaystyle 2^{k(L)} $$ states.

  • Tree Diagram: Shows all possible output paths.

  • Trellis Diagram: Compact representation of state transitions over time.

• Encoding: Shift register of length $L$ (memory). At each clock, $k$ bits enter, $n$ bits output based on current state and input.

• Decoding: Viterbi Algorithm (VA)

  • Principle: Maximum likelihood sequence estimation (MLSE) on trellis.

  • Process: For each received symbol, compute path metric (e.g., Hamming distance) for all state transitions. Retain survivor path for each state.

  • Result: After $T$ steps, trace back the path with best metric.

6.5 Burst Error Correcting Codes

• Burst Error: Contiguous sequence of erroneous bits.

• Fire Codes: Cyclic codes designed for burst error correction. Generator polynomial has a factor $$\displaystyle (x^{b+1} - 1) $$ where $b$ is burst length.

• Interleaving:

  • Principle: Rearrange bits before encoding so that a burst error in channel becomes scattered errors after deinterleaving.

  • Block Interleaving: Write codewords by rows, read by columns (or vice versa).

  • Effect: Converts burst errors into random errors that can be corrected by random error-correcting codes (like Hamming).

• Encoder/Decoder Circuit: For a specific burst-correcting code, draw shift register based on generator polynomial $g(x)$. Decoding uses syndrome to detect and correct burst.

[!TIP] Exam Focus: Be ready to draw encoder circuit for a given $g(x)$ (cyclic code) and explain interleaving with an example (e.g., 3x3 block interleaver).

END OF UNIT 2 NOTES
Aligned with RGPV past papers (2022-2025). Focus on derivations (SQNR, error prob), block diagrams (PCM, QPSK, DM), and comparisons.