Quick Revision Notes — Communication Systems

Quick Revision Notes for Communication Systems

These revision notes are designed for the night before your exam or the final week before GATE. Each topic is compressed into its most essential points — the formulas you must remember and the common mistakes students make. Read these after you've studied the full material; they're reminders, not replacements for deep learning.

Module 1: Signals and Systems Fundamentals

Key relationships to remember:

• Time domain multiplication ↔ Frequency domain convolution (and vice versa)
• Wider in time → narrower in frequency (time-bandwidth uncertainty principle)
• Real signal → symmetric magnitude spectrum, antisymmetric phase spectrum
• Bandwidth × Duration ≥ 1/2 (fundamental limit on simultaneous localization)
• Modulation by cos(2πfct) shifts spectrum to ±fc, doubling baseband bandwidth for DSB
• Energy signal: finite total energy, zero average power (e.g., single pulse)
• Power signal: finite average power, infinite energy (e.g., periodic signals)
• Parseval's theorem: energy computed in time domain equals energy in frequency domain

Module 2: Analog Modulation

AM (DSB-FC) — Most tested points:

• Modulated signal: s(t) = Ac[1 + μ·m(t)]cos(2πfct)
• Total power: Pt = Pc(1 + μ²/2), efficiency η = μ²/(2 + μ²), max 33.3% at μ = 1
• Overmodulation (μ > 1) → envelope distortion → envelope detector fails
• At μ = 1: carrier has 2/3 power, sidebands share remaining 1/3
• Bandwidth = 2fm always (upper sideband + lower sideband)

DSB-SC vs SSB vs VSB:

• DSB-SC: No carrier → 100% efficient, needs coherent detection, BW = 2fm
• SSB: BW = fm, half DSB-SC power, most bandwidth-efficient analog modulation
• VSB: BW ≈ fm + vestige, used in analog TV broadcasting, compromise between DSB and SSB

FM critical points:

• Instantaneous frequency: fi = fc + kf·m(t), deviation Δf = kf·Am
• Carson's rule: BW = 2(Δf + fm) = 2fm(1 + β) — memorize this formula
• Narrowband FM (β << 1): BW ≈ 2fm, no SNR advantage over AM
• Wideband FM (β >> 1): BW ≈ 2Δf = 2βfm, significant SNR improvement
• FM SNR output: SNRo = 3β²(β+1) × SNRc (above threshold only!)
• FM threshold effect: below ~10 dB channel SNR, performance collapses with noise spikes
• Pre-emphasis/de-emphasis: boosts highs before Tx, cuts after Rx → 13 dB improvement
• FM capture effect: stronger signal suppresses weaker (unlike AM where they add)

Common exam mistakes: Confusing μ (AM, dimensionless ratio ≤ 1) with β (FM, can be much greater than 1). Forgetting that FM bandwidth depends on message amplitude through Δf. Not knowing PM of m(t) equals FM of dm(t)/dt.

Module 3: Noise Performance

Essential noise formulas:

• Thermal noise: N = kTB where k = 1.38×10⁻²³ J/K. Room temp: N₀ = kT = -174 dBm/Hz
• Noise figure: F = 1 + Te/T₀ (T₀ = 290K standard reference temperature)
• Cascaded stages: Ftotal = F₁ + (F₂-1)/G₁ + (F₃-1)/(G₁G₂) + ...
• First stage dominates! A noisy first amplifier ruins everything downstream regardless of later stages

Receiver output SNR comparison:

• AM envelope detection: SNRo ≈ μ²·SNRc (for high channel SNR)
• DSB-SC coherent: SNRo = SNRc (no threshold, linear relationship)
• SSB coherent: SNRo = SNRc (same as DSB-SC)
• FM: SNRo = 3β²(β+1)·SNRc (massive improvement but only above threshold)

Module 4: Sampling and Quantization

Sampling theorem: fs ≥ 2fm (Nyquist rate). Below this → aliasing (irreversible frequency folding). Practical systems use anti-aliasing filter and oversample at 2.2-2.5 × fm.

Quantization formulas:

• Uniform quantizer step: Δ = (Vmax - Vmin)/2ⁿ, levels L = 2ⁿ
• SQNR = 6.02n + 1.76 dB — each extra bit improves quality by exactly 6 dB
• Non-uniform (μ-law): y = ln(1+μ|x|)/ln(1+μ)·sgn(x), μ = 255 (North America), improves speech dynamic range by ~24 dB
• PCM bit rate: Rb = n × fs. Telephone standard: 8 kHz × 8 bits = 64 kbps
• PCM bandwidth: BW ≥ Rb/2 = nfs/2 (Nyquist minimum for baseband)

Delta Modulation:

• Transmits 1 bit per sample (step up or down by Δ)
• Slope overload condition: Δ × fs ≥ 2πfm × Am (signal changes too fast)
• Granular noise: signal flat but output oscillates around true value
• Trade-off: increasing Δ reduces slope overload but increases granular noise

Module 5: Digital Modulation

BER formulas — these WILL be on your exam:

• BPSK: Pe = Q(√(2Eb/N₀)) — best power efficiency for binary schemes
• QPSK: Pe = Q(√(2Eb/N₀)) — same BER as BPSK but double spectral efficiency
• BFSK coherent: Pe = Q(√(Eb/N₀)) — 3 dB worse than BPSK
• BFSK non-coherent: Pe = (1/2)exp(-Eb/2N₀)
• DPSK: Pe = (1/2)exp(-Eb/N₀) — about 1 dB worse than coherent BPSK
• OOK non-coherent: Pe = (1/2)exp(-Eb/4N₀)

Bandwidth efficiency ladder:

• BPSK: 1 bit/s/Hz | QPSK: 2 | 8-PSK: 3 | 16-QAM: 4 | 64-QAM: 6 bits/s/Hz
• Higher efficiency requires more Eb/N₀: 16-QAM needs ~4.5 dB more than QPSK

Why QPSK is special: It's two independent BPSK signals on I and Q channels. Double spectral efficiency with identical BER performance per Eb/N₀. Every practical wireless system uses at least QPSK.

Module 6: Information Theory

Shannon's Channel Capacity — the most important formula in communication systems: C = B × log₂(1 + SNR) bits/second

• Absolute upper limit: CANNOT exceed regardless of coding complexity
• Achievable: CAN approach with sufficiently complex codes (LDPC, Polar)
• Doubling bandwidth B → doubles capacity (linear)
• Doubling SNR → adds only ~1 bit/s/Hz (logarithmic — diminishing returns)
• Shannon limit: minimum Eb/N₀ for error-free communication = ln(2) = -1.59 dB

Entropy: H(X) = -Σ p(x)·log₂p(x) bits/symbol

• Binary source: H = -p·log₂(p) - (1-p)·log₂(1-p)
• Maximum H = log₂(M) when all M symbols equiprobable
• H = 0 for deterministic source (no uncertainty, no information)

Source coding: Average code length L ≥ H(source). Huffman is optimal instantaneous code: L < H + 1. Efficiency η = H/L̄.

Module 7: Error Control Coding

Hamming code (7,4): 4 data + 3 parity bits, dmin = 3, corrects 1 error, detects 2. General Hamming: (2ᵐ-1, 2ᵐ-1-m), always dmin = 3, rate R = 1 - m/(2ᵐ-1).

Universal relationships:

• Errors correctable: t = ⌊(dmin-1)/2⌋
• Errors detectable: e = dmin - 1
• Singleton bound: dmin ≤ n - k + 1
• Coding gain ≈ 10·log₁₀(R × dmin) dB at high SNR

CRC: Polynomial division, append remainder as check bits. Error detection only. CRC-32 detects: all single errors, all double errors, all odd-count errors, all bursts ≤ 32 bits.

Convolutional codes: Described by (n,k,K). Rate-1/2, K=7 is standard. Decoded by Viterbi algorithm (ML trellis search). Free distance dfree = 10, gain ≈ 5-6 dB.

Module 8: Wireless and Cellular

• Free-space path loss: FSPL(dB) = 20·log₁₀(d) + 20·log₁₀(f) + 32.44 (d in km, f in MHz)
• Path loss exponent: n = 2 (free space), 3-4 (urban), 4-6 (indoor)
• Rayleigh fading: no LOS, deep fades possible (30-40 dB), envelope Rayleigh distributed
• Rician fading: strong LOS + scattered. K-factor = direct/scattered power
• Doppler spread: fd = v·fc/c. Coherence time: Tc ≈ 1/(2fd)
• Coherence bandwidth: Bc ≈ 1/(5στ). If signal BW > Bc → frequency-selective fading → use OFDM
• Cellular: cluster size N, reuse distance D = R√(3N)
• SIR = (1/6)(3N)^(n/2). For N=7, n=4: SIR = (21)²/6 = 73.5 = 18.7 dB

Common GATE Traps

1. Eb/N₀ vs SNR: Eb/N₀ = SNR × (B/Rb). They're different! QPSK BER same as BPSK per Eb/N₀, NOT per SNR
2. Carson's rule: Approximate (98% power). Exact FM BW is infinite (Bessel sidebands)
3. Shannon formula: log₂(1+SNR), NOT log₂(SNR). The "+1" matters at low SNR
4. Noise bandwidth ≠ 3-dB bandwidth: For RC filter: noise BW = (π/2) × f₃dB
5. Hamming distance: Between two different codewords. dmin = smallest across ALL valid codeword pairs

Key Takeaways

1. Communication chain: source → source coding → channel coding → modulation → channel → demodulation → decoding → sink. Every topic fits here.
2. Trade-offs: bandwidth vs power (FM), complexity vs performance (coherent detection), rate vs reliability (coding). Understanding trade-offs is understanding communication systems.
3. Shannon's C = B·log₂(1+SNR) is the single most important equation. Modern codes achieve within 0.1 dB.
4. For GATE: 80% of questions need numerical formula application. Practice speed and accuracy.
5. Always verify units: Hz for bandwidth, bits/s for rate, dB for ratios, dBm for absolute power. Wrong conversions cause wrong answers.
6. If you can trace a signal from source to destination explaining every block, you're exam-ready.