DIGITAL COMMUNICATION (EC-502) - Unit 1 Short Notes

UNIT 1: DIGITAL COMMUNICATION - COMPREHENSIVE NOTES

Based on rigorous analysis of RGPV past examination papers (Jun 2025, Dec 2024, May 2024, Nov 2023, Nov 2022), these notes are engineered for maximum exam impact. All topics are presented in the exact sequence and depth as per the approved blueprint.

I. FOUNDATIONS: ANALOG-TO-DIGITAL CONVERSION

A. Sampling Theory

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

  Key Formula: $$\displaystyle f_s \geq 2f_m $$

• Sampling Theorem for Bandpass Signals: For a signal with bandwidth $B$ and highest frequency $$\displaystyle f_H $$, the minimum sampling rate is $$\displaystyle f_s \geq 2B $$. However, if the signal's spectrum consists of non-overlapping bands, $$\displaystyle f_s $$ can be as low as $2B$ even if $$\displaystyle f_H >> 2B $$.

• Types of Sampling:

  • Ideal (Impulse) Sampling: Signal multiplied by an impulse train $\delta(t)$. Spectrum is periodic with period $$\displaystyle f_s $$. Reconstruction uses an ideal low-pass filter.

  • Natural Sampling: Top of sample follows analog signal during pulse width. Requires a sample-and-hold circuit.

  • Flat-Top Sampling: Sample is held constant (flat) for duration $$\displaystyle T_s $$. Uses a sample-and-hold. Easier for practical circuits but introduces aperture effect.

• Aliasing: Occurs when $$\displaystyle f_s < 2f_m $$. High-frequency components "fold back" into lower frequencies, causing irreversible distortion.

  Anti-aliasing Filter: A low-pass filter (with cutoff $$\displaystyle f_c \leq f_s/2 $$) placed before the sampler to bandlimit the signal and prevent aliasing.

• Aperture Effect: In flat-top sampling, the finite pulse width causes the effective sampled value to be an average of the signal over the pulse duration, introducing a $$\displaystyle \text{sinc}(fT_s) $$ attenuation in the spectrum.

B. Quantization

• Need: To convert continuous-amplitude samples into discrete digital values (finite number of levels).

• Uniform Quantizer: Step size $\Delta$ is constant.

  • Mid-tread: Zero reconstruction level. Contains origin.

  • Mid-riser: Zero threshold. Does not contain origin.

• Non-Uniform Quantization & Companding: Non-linear quantization where step size is smaller for small amplitudes (reduces quantization noise for low-level signals). Implemented via companding:

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

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

• Quantization Noise & SQNR Derivation:

  • For a uniform quantizer with $L$ levels, step size $$\displaystyle \Delta = \frac{V_{max}-V_{min}}{L} $$.

  • Quantization error $$\displaystyle e_q(t) $$ modeled as uniform noise over $[-\Delta/2, \Delta/2]$ with power $$\displaystyle P_e = \frac{\Delta^2}{12} $$.

  • Signal power for a full-scale sinusoid: $$\displaystyle P_s = \frac{(V_{max}-V_{min})^2}{8} $$.

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

$$SQNR = \frac{P_s}{P_e} = \frac{3 \cdot 2^{2n}}{2} \approx 6.02n + 1.76 \ \text{dB}$$

    where $$\displaystyle n = \log_2 L $$ is number of bits.

    \boxed{SQNR_{\text{dB}} \approx 6.02n + 1.76}

> **Exam Tip:** Derivation of SQNR is a **very high-frequency** question. Be prepared to derive from first principles.

C. Pulse Modulation Techniques

• Pulse Code Modulation (PCM):

  1. Sampling: (Ideal) at $$\displaystyle f_s \geq 2f_m $$.

  2. Quantization: Uniform or non-uniform.

  3. Encoding: Binary representation of quantized sample.

  Block Diagram:

  DiagramCANVAS: PCM Transmitter: Anti-aliasing Filter -> Sampler -> Quantizer -> Encoder (Binary). Receiver: Decoder -> Reconstruction Filter (Low-pass)

  • Advantages: Robust to noise, easy regeneration, supports multiplexing.

  • Disadvantages: High bandwidth requirement (bit rate $$\displaystyle R_b = n f_s $$).

• Delta Modulation (DM):

  • Principle: Encodes the difference (Δ) between current and previous sample into 1 bit.

  • Generation: Comparator output (1 if input > previous, 0 otherwise) integrated to form staircase approximation.

  • Detection: Integrator in receiver.

  • Errors:

    • Granular Noise: Occurs when signal slope is small and Δ is too large.

    • Slope Overload: Occurs when signal slope is too high for Δ to follow.

• Adaptive Delta Modulation (ADM): Step size Δ is adaptively changed based on recent signal changes to reduce both granular noise and slope overload.

• Comparison (PAM, PWM, PPM, PCM):

  | Feature | PAM | PWM | PPM | PCM | | :--- | :--- | :--- | :--- | :--- | | Information | Sample amplitude | Pulse width | Pulse position | Quantized sample code | | Bandwidth | High | Very High | Very High | Highest | | Noise Immunity | Poor | Moderate | Good | Excellent | | Complexity | Low | High | High | Moderate |

D. Time-Division Multiplexing (TDM)

• Basic Principle: Multiple signals share the same channel by allocating non-overlapping time slots in a repeating frame.

• Synchronization: Critical for correct slot assignment.

  • Frame Synchronization: Special pattern (e.g., 1's and 0's) at start of frame.

  • Bit Synchronization: Clock recovery from the data stream.

II. BASEBAND TRANSMISSION & LINE CODING

A. Digital Baseband Signals & Signaling

Common Pitfall: Confusing Manchester (transition at center defines bit) with Differential Manchester (transition at start defines bit, center transition always present).

B. Spectral Properties of Baseband Signals

• Power Spectral Density (PSD): Describes power distribution vs. frequency.

  • Unipolar NRZ: Significant low-frequency (DC) component.

  • Bipolar (AMI) & Manchester: Zero DC component, main lobe at $$\displaystyle f = 1/T_b $$ (Manchester has stronger high-frequency components).

  • RZ: Wider main lobe than NRZ due to shorter pulses.

C. Inter-Symbol Interference (ISI)

• Cause: Bandlimiting (channel or filter) causes pulse spreading. Past pulses interfere with current pulse detection.

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

$$H(f) = \frac{1}{f_s} \sum_{k=-\infty}^{\infty} X(f - kf_s)$$

where $X(f)$ is any function with $$\displaystyle X(0)=f_s $$.

*   **First Criterion (Time Domain):** $$\displaystyle h(kT_s) = \begin{cases} 1, & k=0 \\ 0, & k \neq 0 \end{cases} $$

*   **Second Criterion (Frequency Domain):** Above equation.

• Raised Cosine Filter: Practical filter satisfying Nyquist criterion with roll-off factor $\alpha$ ($0 \leq \alpha \leq 1$).

  Bandwidth: $$\displaystyle B = \frac{f_s}{2}(1+\alpha) = \frac{R_b}{2}(1+\alpha) $$.

  \boxed{B = \frac{R_b}{2}(1+\alpha)}

D. Eye Pattern

• Concept: Oscilloscope display of superimposed received signal over multiple bit intervals.

• Use:

  • ISI: Eye closure vertically.

  • Timing Jitter: Horizontal eye closure.

  • Optimum Sampling Instant: Center of open eye.

  • Noise Margin: Vertical opening at sampling instant.

E. Differential Encoding

• Principle: Encode based on transition/no-transition relative to previous bit, not absolute level.

  • $$\displaystyle d_k = b_k \oplus d_{k-1} $$ (for DPSK/Differential Manchester).

• Advantages:

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

  2. Simple receiver (only needs previous bit).

• Disadvantages: ~3 dB performance penalty compared to coherent detection.

F. Crosstalk

• Definition: Unwanted coupling between adjacent channels/wires.

• Types:

  • NEXT (Near-End Crosstalk): Interference measured at the source end of the interfering channel.

  • FEXT (Far-End Crosstalk): Interference measured at the destination end.

III. DIGITAL PASSBAND MODULATION SCHEMES

A. Binary Modulation

• BPSK: Carrier phase shifted by 0° (for 1) or 180° (for 0).

  • Generation: Multiply binary data (+1/-1) with carrier $$\displaystyle \cos(2\pi f_c t) $$.

  • Detection: Coherent (correlator/matched filter). $$\displaystyle s_1(t) = \sqrt{\frac{2E_b}{T_b}}\cos(2\pi f_c t) $$, $$\displaystyle s_0(t) = -\sqrt{\frac{2E_b}{T_b}}\cos(2\pi f_c t) $$.

  • Bandwidth: $$\displaystyle \approx 2R_b $$ (for practical pulses).

  • Pe (Coherent): $$\displaystyle \boxed{P_e = Q\left(\sqrt{\frac{2E_b}{N_0}}\right)} $$

• BFSK:

  • Coherent: Two orthogonal carriers ($$\displaystyle f_1, f_2 $$ with $$\displaystyle |f_1-f_2| = 1/T_b $$). $$\displaystyle P_e = Q\left(\sqrt{\frac{E_b}{N_0}}\right) $$.

  • Non-Coherent: No phase tracking. $$\displaystyle P_e = \frac{1}{2} e^{-\frac{E_b}{2N_0}} $$.

• ASK/OOK: 1=carrier on, 0=carrier off. Poor noise immunity.

B. Quadrature & M-ary Modulation

• QPSK (Quadrature PSK):

  • Principle: Two bits per symbol. Four phases: 45° (11), 135° (01), 225° (10), 315° (00). Can be seen as two orthogonal BPSK streams (I & Q).

  • Generation: Split data into I and Q streams, BPSK modulate $$\displaystyle \cos(2\pi f_c t) $$ and $$\displaystyle \sin(2\pi f_c t) $$, then sum.

    Block Diagram:

    DiagramCANVAS: QPSK Transmitter: Serial-to-Parallel -> 2 BPSK Modulators (I on cos, Q on sin) -> Adder. Receiver: Coherent demod with two correlators (cos, sin) -> Decision -> Parallel-to-Serial

  • Constellation: 4 points on unit circle at 45° intervals.

  • Spectral Properties: Same minimum bandwidth as BPSK ($$\displaystyle \approx R_b/2 $$ Hz for same bit rate) but bandwidth efficiency doubles.

  • Error Probability (Coherent): $$\displaystyle \boxed{P_e = Q\left(\sqrt{\frac{2E_b}{N_0}}\right)} $$ (same as BPSK for same $$\displaystyle E_b $$).

  • Advantages over BPSK:

    1. Same Bandwidth, Double Bit Rate (or same bit rate, half bandwidth).

    2. Same $$\displaystyle P_e $$ for given $$\displaystyle E_b/N_0 $$.

• DPSK & DQPSK: Differential encoding applied to PSK. DQPSK uses differential encoding on I and Q bits. ~3 dB loss vs coherent QPSK.

• M-ary PSK/FSK:

  • M-ary PSK: $M$ phases equally spaced. Bandwidth $$\displaystyle \approx \frac{R_b}{\log_2 M} $$. $$\displaystyle P_e $$ increases with $M$.

  • M-ary FSK: $M$ orthogonal frequencies. Bandwidth $$\displaystyle \approx M \cdot \frac{R_b}{\log_2 M} $$. Better $$\displaystyle P_e $$ than M-PSK for same $M$.

• QAM (Quadrature Amplitude Modulation):

  • Combines amplitude and phase modulation (e.g., 16-QAM: 4 amplitudes × 4 phases).

  • Square M-QAM (M=4,16,64,...): Constellation points on $\sqrt{M} \times \sqrt{M}$ grid.

  • Error Probability (Approx for Square M-QAM):

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

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

*   **Trade-off:** Higher $M$ → higher bandwidth efficiency, lower power efficiency (higher $$\displaystyle P_e $$).

C. Continuous Phase Modulation (CPM)

• MSK (Minimum Shift Keying):

  • Special case of CPFSK with modulation index $$\displaystyle h = 0.5 $$.

  • Phase changes continuously by $\pm \pi/2$ per symbol.

  • Spectral Properties: Very compact spectrum, lower sidelobes than QPSK (roll-off ~0.3). Main lobe width $$\displaystyle \approx 1.5 R_b $$.

  • Generation: Can be viewed as Offset QPSK (OQPSK) with sinusoidal pulse shaping (half-sinusoid).

IV. SIGNAL SPACE ANALYSIS & RECEIVER DESIGN

A. Gram-Schmidt Orthogonalization

• Concept: Transform a set of $M$ linearly independent signals $$\displaystyle \{s_i(t)\} $$ into an orthogonal set $$\displaystyle \{\phi_i(t)\} $$ spanning the same space.

• Procedure (for $M$ signals over $[0,T]$):

  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. Continue: $$\displaystyle \phi_k(t) = s_k(t) - \sum_{i=1}^{k-1} \frac{\langle s_k, \phi_i \rangle}{\|\phi_i\|^2} \phi_i(t) $$

• Application: Any signal $$\displaystyle s_i(t) $$ can be represented as $$\displaystyle s_i(t) = \sum_{k=1}^{N} a_{ik} \phi_k(t) $$, where $N \leq M$ is the signal space dimension. Coefficients $$\displaystyle a_{ik} $$ are projections.

• Use: Simplifies optimum receiver design (correlator/matched filter bank) and $$\displaystyle P_e $$ calculation via Euclidean distance between signal points.

B. Optimum Receivers

• Matched Filter:

  • Principle: Maximizes output SNR at a specific sampling time $T$.

  • Impulse Response: $$\displaystyle h(t) = s(T-t) $$ for signal $s(t)$ over $[0,T]$.

  • Transfer Function: $$\displaystyle H(f) = S^*(f) e^{-j2\pi f T} $$.

  • Output at $$\displaystyle t=T $$: $$\displaystyle z(T) = \int_0^T r(t) s(T-t) dt = \int_0^T r(t) s(t) dt $$ (if $s(t)$ real & even).

• Correlator Detector:

  • Integrates $r(t) \cdot s(t)$ over $[0,T]$.

  • Equivalence: Matched filter and correlator give identical performance for white noise. Correlator is often easier to implement for baseband signals.

C. Probability of Error Analysis (Signal Space Approach)

1. Represent all signals $$\displaystyle s_i(t) $$ in orthogonal basis $$\displaystyle \{\phi_k(t)\} $$ → signal vectors $$\displaystyle \mathbf{s}_i = (a_{i1}, a_{i2}, ..., a_{iN}) $$.

2. Received vector $$\displaystyle \mathbf{r} = \mathbf{s}_i + \mathbf{n} $$, where $\mathbf{n}$ is Gaussian noise vector with covariance $$\displaystyle \sigma_n^2 \mathbf{I} $$.

3. Optimum receiver (ML) chooses signal with minimum Euclidean distance $$\displaystyle \|\mathbf{r} - \mathbf{s}_i\|^2 $$.

4. $$\displaystyle P_e $$ for a signal pair depends on their Euclidean distance $$\displaystyle d_{ij} = \|\mathbf{s}_i - \mathbf{s}_j\| $$:

$$P(e|s_i \text{ sent}) = Q\left(\frac{d_{ij}}{2\sigma_n}\right)$$

For equiprobable signals, average $$\displaystyle P_e $$ is sum over nearest neighbors.

V. INFORMATION THEORY & CHANNEL CAPACITY

A. Fundamental Concepts

• Entropy (H(X)): Average uncertainty of discrete random variable X.

$$H(X) = -\sum_{i} p(x_i) \log_2 p(x_i) \ \text{bits}$$

• Joint Entropy (H(X,Y)) & Conditional Entropy (H(Y|X)):

$$H(X,Y) = -\sum_{i,j} p(x_i,y_j) \log_2 p(x_i,y_j)$$

$$H(Y|X) = H(X,Y) - H(X)$$

• Mutual Information (I(X;Y)): Information about X gained from Y.

$$I(X;Y) = H(X) - H(X|Y) = H(Y) - H(Y|X)$$

• Information Rate: For a discrete memoryless source with entropy $H(X)$, rate $$\displaystyle R = H(X) $$ bits/symbol. For a channel, capacity $C$ is max mutual information.

B. Channel Capacity

• Definition: Maximum achievable reliable (arbitrarily low $$\displaystyle P_e $$) information rate over a channel.

$$C = \max_{p(x)} I(X;Y) \ \text{bits/channel use}$$

• Shannon's Channel Capacity Theorem (AWGN Channel):

  \boxed{C = B \log_2\left(1 + \frac{S}{N}\right) = B \log_2\left(1 + \frac{E_b R_b}{N_0 B}\right)}

  where $B$ = bandwidth (Hz), $S/N$ = signal-to-noise ratio (linear).

• Shannon-Hartley Theorem: Specific form for AWGN channel (above).

• Bandwidth Efficiency (η): $$\displaystyle \eta = \frac{R_b}{C} $$ bits/sec/Hz. Measures how efficiently bandwidth is used.

• Trade-off: For fixed $C$, increasing $B$ allows decreasing $S/N$ (and vice versa). There is no trade-off without increasing $$\displaystyle P_e $$ if $$\displaystyle R_b > C $$.

C. Binary Symmetric Channel (BSC)

• Model: Input $X \in \{0,1\}$, Output $$\displaystyle Y = X \oplus e $$, where $$\displaystyle e=1 $$ (error) with prob $P$, $$\displaystyle e=0 $$ with prob $1-P$.

• Capacity:

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

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

\boxed{C_{\text{BSC}} = 1 - H(P)}

• Example: For $$\displaystyle P=0.1 $$, $H(0.1) \approx 0.469$, $C \approx 0.531$ bits/channel use.

VI. ERROR CONTROL CODING

A. Introduction

• Adds redundancy to transmitted data to detect/correct errors.

• Types of Errors: Random (independent), Burst (contiguous).

• Code Parameters: $(n, k)$ block code: $k$ info bits → $n$ transmitted bits. Code rate $$\displaystyle R_c = k/n $$.

B. Linear Block Codes

• Structure: All valid codewords form a linear subspace. Generator matrix $\mathbf{G}$ ($k \times n$), Parity-check matrix $\mathbf{H}$ ($(n-k) \times n$) with $$\displaystyle \mathbf{G}\mathbf{H}^T = \mathbf{0} $$.

• Encoding: $$\displaystyle \mathbf{c} = \mathbf{u}\mathbf{G} $$, where $\mathbf{u}$ is info 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 error pattern from syndrome table (assumes few errors).

• Cyclic Codes (Subclass):

  • Codewords as polynomials $c(x)$. Generator polynomial $g(x)$ divides $$\displaystyle x^n+1 $$.

  • Encoding: $$\displaystyle c(x) = u(x) x^{n-k} + r(x) $$, where $r(x)$ is remainder of $$\displaystyle u(x)x^{n-k} $$ divided by $g(x)$.

  • Implementation: Shift register encoder.

  • Syndrome: $$\displaystyle s(x) = r(x) \mod g(x) $$.

• Hamming Codes:

  • Single-error-correcting (SEC). Satisfies $$\displaystyle 2^m \geq n+1 $$ where $$\displaystyle m = n-k $$ parity bits.

  • (7,4) Hamming Code: $$\displaystyle n=7, k=4, m=3 $$. Can correct $$\displaystyle t=1 $$ error. Generator and parity-check matrices standard.

C. Convolutional Codes

• Description: Encoding with memory. Input data stream processed by a shift register of length $K$ (constraint length). Output $n$ bits per $k$ input bits → rate $k/n$.

• Representations:

  • State Diagram: States = contents of shift register (except inputs). $$\displaystyle 2^{K-1} $$ states for binary.

  • Tree Diagram: Shows all possible output paths.

  • Trellis Diagram: Compact state diagram with time, showing state transitions and outputs.

• Viterbi Algorithm (ML Decoding):

  1. Compute branch metrics (e.g., Hamming distance) for each received sequence segment.

  2. Add-Compare-Select (ACS): For each state at time $t$, accumulate path metric from previous states.

  3. Traceback: Select path with maximum metric (minimum distance) through trellis.

D. Burst Error Correcting Codes

• Concept: Designed to correct errors occurring in bursts (length ≤ $b$).

• Fire Codes: A class of cyclic codes with burst error correction capability. Encoder/decoder use shift registers and syndrome calculation. General idea: Interleaving can also convert burst errors to random errors for block codes.

VII. SYSTEM-LEVEL & COMPARATIVE TOPICS

A. Comparison of Modulation Techniques

B. Applications & Choice Factors

• Factors: Bandwidth constraints, power constraints, channel (fading, Doppler), complexity/cost, required $$\displaystyle P_e $$.

• Typical Applications:

  • BPSK: Deep-space, low-data-rate.

  • QPSK/OQPSK: Satellite, wireless (Wi-Fi, CDMA), digital video broadcasting.

  • MSK: GSM (mobile comms).

  • M-QAM (16/64): Cable modems, DSL, high-density modems (high SNR required).

  • FSK: Low-rate telemetry, caller ID.

C. Bandwidth Calculation

• General: $$\displaystyle B = \frac{R_b}{\eta} $$, where $\eta$ is bandwidth efficiency (bits/sec/Hz).

• Approximate Bandwidths for Raised-Cosine Filter ($\alpha$ roll-off):

  • BPSK: $$\displaystyle B \approx R_b (1+\alpha)/2 $$? Correction: For BPSK, $$\displaystyle B \approx R_b (1+\alpha) $$. For QPSK (same symbol rate $$\displaystyle R_s = R_b/2 $$), $$\displaystyle B \approx R_s (1+\alpha) = R_b(1+\alpha)/2 $$.

  • QPSK: $$\displaystyle B \approx \frac{R_b}{2}(1+\alpha) $$.

  • MSK: $$\displaystyle B \approx 0.8 R_b $$ (for $\alpha \approx 0.3$ equivalent).

  Example from Past Paper: $$\displaystyle R_b = 1 \text{ Mbps}, \eta=0.8 \Rightarrow B = \frac{1 \text{ Mbps}}{0.8 \text{ bits/sec/Hz}} = 1.25 \text{ MHz} $$.

END OF UNIT 1 NOTES