Important Formulas — Communication Systems

Important Formulas in Communication Systems

This is your comprehensive formula reference for communication systems. Every formula here has been carefully organized with explanations of when and how to use it. During exam preparation, understanding what each variable represents and when a formula applies is just as important as memorizing the expression itself.

Fundamental Signal and Noise Formulas

Thermal Noise Power: N = kTB

Where k = 1.38 × 10⁻²³ J/K (Boltzmann's constant), T = temperature in Kelvin, B = bandwidth in Hz. At room temperature (T = 290K), noise power density N₀ = kT = -174 dBm/Hz. This means every Hz of bandwidth adds -174 dBm of irreducible noise. A 1 MHz bandwidth receiver sees N = -174 + 60 = -114 dBm of thermal noise.

Signal-to-Noise Ratio: SNR = P_signal / P_noise (linear) = 10·log₁₀(P_signal/P_noise) dB

Noise Figure: NF = SNR_input / SNR_output = 1 + (Te/T₀)

Where Te is effective noise temperature and T₀ = 290K. A receiver with NF = 3 dB doubles the noise (adds as much noise as the input thermal noise). For cascaded stages, use Friis' formula for noise:

F_total = F₁ + (F₂ - 1)/G₁ + (F₃ - 1)/(G₁·G₂) + ...

This shows why the first amplifier's noise figure dominates — subsequent stages' noise contributions are divided by preceding gain.

Decibel Conversions:

• Power: dBm = 10·log₁₀(P_watts/0.001)
• Voltage: dBV = 20·log₁₀(V/1)
• Useful reference: 0 dBm = 1 mW, 30 dBm = 1 W, -30 dBm = 1 μW

Analog Modulation Formulas

Standard AM (DSB-FC):

• Modulated signal: s(t) = Ac[1 + μ·m(t)]cos(2πfct)
• Modulation index: μ = Am/Ac (must be ≤ 1 to avoid overmodulation)
• Total power: Pt = Pc(1 + μ²/2) for single-tone modulation
• Transmission efficiency: η = μ²/(2 + μ²) — maximum 33.3% at μ = 1
• Bandwidth: BW = 2fm (twice the highest message frequency)

DSB-SC (Suppressed Carrier):

• Signal: s(t) = Ac·m(t)·cos(2πfct)
• Power: Pt = Pc·μ²/2 (all power in sidebands — 100% efficient)
• Bandwidth: BW = 2fm (same as AM)
• Requires coherent detection (carrier recovery needed)

SSB (Single Sideband):

• Bandwidth: BW = fm (half of DSB)
• Power: Pt = Pc·μ²/4 (half of DSB-SC)
• Most bandwidth-efficient analog modulation

Frequency Modulation:

• Instantaneous frequency: fi = fc + kf·m(t)
• Frequency deviation: Δf = kf·Am (maximum frequency shift)
• Modulation index: β = Δf/fm
• Carson's bandwidth rule: BW = 2(Δf + fm) = 2fm(β + 1)
• Narrowband FM (β << 1): BW ≈ 2fm (similar to AM)
• Wideband FM (β >> 1): BW ≈ 2Δf
• FM SNR advantage: SNRo = 3β²(β + 1)·SNRc (output SNR improves with β²)
• FM threshold: SNRc ≈ 10 dB (below this, noise spikes appear)
• Pre-emphasis/de-emphasis improvement: additional 13 dB for audio

Phase Modulation:

• Signal: s(t) = Ac·cos(2πfct + kp·m(t))
• Phase deviation: Δφ = kp·Am
• Relationship to FM: PM with m(t) is equivalent to FM with dm(t)/dt
• Bandwidth: Same Carson's rule applies with β = Δφ for sinusoidal modulation

Digital Communication Formulas

Sampling and Quantization:

• Nyquist rate: fs ≥ 2fm (minimum sampling frequency)
• Quantization levels: L = 2ⁿ (n = number of bits)
• Quantization step size: Δ = (Vmax - Vmin)/L
• Uniform PCM SQNR: SQNR = 6.02n + 1.76 dB
• Bit rate: Rb = n·fs bits/second
• Non-uniform (μ-law) SQNR improvement: approximately 24 dB for speech signals

Bit Error Rate (BER) for Various Modulations:

The Q-function: Q(x) = (1/2)erfc(x/√2) ≈ (1/(x√(2π)))·e^(-x²/2) for x > 3

OFDM Parameters:

• Subcarrier spacing: Δf = 1/Tu (Tu = useful symbol duration)
• Total bandwidth: B = N·Δf (N = number of subcarriers)
• Guard interval (cyclic prefix): Tg > τmax (maximum delay spread)
• Spectral efficiency: η = N/(N + Ng) × log₂M × R (where R = code rate, Ng = guard samples)
• Peak-to-Average Power Ratio: PAPR ≈ 10·log₁₀(N) dB for large N

Information Theory Formulas

Shannon Channel Capacity: C = B·log₂(1 + S/N) bits/second

This is the most important formula in communication systems. It tells you the absolute maximum error-free data rate achievable over a channel with bandwidth B and signal-to-noise ratio S/N.

Practical implications:

• Doubling bandwidth doubles capacity (linear relationship)
• Doubling SNR adds only 1 bit/s/Hz (logarithmic relationship)
• At SNR = 0 dB: C = B bit/s (1 bit/s/Hz)
• Shannon limit: minimum Eb/N₀ = ln(2) = -1.59 dB

Entropy: H(X) = -Σᵢ p(xᵢ)·log₂(p(xᵢ)) bits/symbol

Maximum entropy occurs when all symbols are equally probable: H_max = log₂(M) for M symbols. For a binary source with probability p: H(X) = -p·log₂(p) - (1-p)·log₂(1-p).

Mutual Information: I(X;Y) = H(X) - H(X|Y) = H(Y) - H(Y|X)

Channel capacity C = max_{p(x)} I(X;Y) — capacity is the maximum mutual information optimized over all possible input distributions.

Source Coding Theorem: Average code length L ≥ H(X)/log₂(D) for D-ary codes. Huffman coding achieves L < H(X) + 1.

Wireless Channel and Propagation Formulas

Free-Space Path Loss: FSPL = (4πd/λ)² = (4πdf/c)²

In dB: FSPL(dB) = 20·log₁₀(d) + 20·log₁₀(f) - 147.55 (d in meters, f in Hz)

Friis Transmission Equation: Pr = Pt·Gt·Gr·(λ/4πd)²

In dB: Pr(dBm) = Pt(dBm) + Gt(dBi) + Gr(dBi) - FSPL(dB)

Link Budget: Pr = Pt + Gt - Lft + Gr - Lfr - FSPL - Lmisc

Where Lft, Lfr = feeder losses, Lmisc = rain attenuation, body loss, etc.

Cellular Reuse:

• Cluster size: N = i² + ij + j² (valid: 1, 3, 4, 7, 12, 13, 19, ...)
• Co-channel reuse distance: D = R√(3N)
• Signal-to-Interference Ratio: SIR = (1/6)(D/R)ⁿ = (1/6)(3N)^(n/2)
• With path loss exponent n = 4: SIR = (3N)²/6
• For N = 7: SIR = (21)²/6 = 73.5 = 18.7 dB

Error Control Coding Formulas

Hamming Code Parameters:

• Code length: n = 2ᵐ - 1
• Data bits: k = 2ᵐ - 1 - m
• Parity bits: m (minimum 3)
• Code rate: R = k/n
• Minimum distance: dmin = 3 (corrects 1 error, detects 2)
• Example: (7,4) Hamming — 7-bit codeword, 4 data bits, corrects single errors

Error Correction Capability:

• Errors correctable: t = ⌊(dmin - 1)/2⌋
• Errors detectable: dmin - 1
• For BCH codes: t-error-correcting BCH over GF(2ᵐ) has n = 2ᵐ - 1, n - k ≤ mt

Coding Gain: G_coding = 10·log₁₀(R·dmin) dB (approximate, at high SNR)

Where R = code rate and dmin = minimum distance. A rate-1/2 code with dmin = 10 provides approximately 10·log₁₀(5) = 7 dB coding gain.

Shannon Limit for Coded Systems: Eb/N₀(required) ≥ (2^(2R) - 1)/(2R) where R = information rate / bandwidth

Optical Communication Formulas

Fiber Attenuation: P_out = P_in · 10^(-αL/10) where α = attenuation coefficient (dB/km), L = length (km)

Typical values: α = 0.2 dB/km at 1550 nm, α = 0.35 dB/km at 1310 nm

Numerical Aperture: NA = √(n₁² - n₂²) = n₁·√(2Δ) where Δ = (n₁-n₂)/n₁

Acceptance angle: θmax = sin⁻¹(NA)

Bandwidth-Distance Product:

• Multimode fiber: B·L ≈ 500 MHz·km (graded-index)
• Single-mode fiber: Limited by chromatic dispersion: B·L ∝ 1/(D·Δλ) where D = dispersion coefficient (ps/(nm·km))

Key Takeaways

1. The thermal noise floor (-174 dBm/Hz at room temperature) sets the ultimate sensitivity limit — every receiver design starts from this number.
2. Shannon's capacity formula C = B·log₂(1+SNR) defines the theoretical maximum — practical systems operate 3-10 dB from this limit.
3. FM trades bandwidth for noise immunity (SNR improves as β²), while AM is bandwidth-efficient but noise-sensitive — this fundamental trade-off recurs throughout communications.
4. BER formulas all involve Eb/N₀ (energy per bit to noise density ratio) — this is the universal metric for comparing modulation schemes regardless of bandwidth or data rate.
5. Friis' cascade formula shows that front-end noise figure dominates system sensitivity — invest in a good LNA before anything else.
6. Coding gain provides "free" SNR improvement by adding structured redundancy — modern codes (LDPC, Polar) achieve within 0.1 dB of Shannon's limit.