Formula sheet

Telecommunications Link Design Formula Sheet

Telecom formulas for dB units, bandwidth, symbol rate, sampling, thermal noise, SNR, noise figure, RF and optical link budgets, latency, jitter, and radar range.

Branch: Telecommunications Engineering
Content: Formula sheet
Updated: May 29, 2026
Revision: v1.0.8 · reviewed

This formula sheet collects first-pass relationships for telecommunications link design, receiver sensitivity, sampling, optical fiber, latency, jitter, and radar range checks. Always state references, units, bandwidth definitions, noise assumptions, modulation, coding, environment, and required margin before comparing results.

Decibel units

Power ratio in decibels:

\displaystyle L_{dB}=10\log_{10}\left(\frac{P_2}{P_1}\right)

Voltage ratio with equal impedance:

\displaystyle G_{dB}=20\log_{10}\left(\frac{V_2}{V_1}\right)

Power in dBW:

\displaystyle P_{dBW}=10\log_{10}\left(\frac{P_W}{1\,W}\right)

Power in dBm:

\displaystyle P_{dBm}=10\log_{10}\left(\frac{P_{mW}}{1\,mW}\right)

Conversion:

P_{dBm}=P_{dBW}+30

Use dB for ratios, dBW or dBm for absolute power, dBi for antenna gain relative to isotropic gain, and dB-Hz for carrier-to-noise density quantities.

Bandwidth and capacity

Idealized Shannon capacity:

C=B\log_2(1+SNR)

where $C$ is channel capacity, $B$ is bandwidth, and $SNR$ is linear signal-to-noise ratio.

Spectral efficiency:

\displaystyle \eta=\frac{R_b}{B}

where $R_b$ is bit rate.

Noise-equivalent bandwidth should be distinguished from occupied bandwidth, filter 3 dB bandwidth, and allocated regulatory bandwidth.

Symbol Rate and Occupied Bandwidth

Symbol rate:

\displaystyle R_s=\frac{R_b}{k}

where $k=\log_2(M)$ bits per symbol for an ideal $M$ -ary modulation before coding and overhead.

Gross bit rate with coding overhead:

\displaystyle R_{gross}=\frac{R_{payload}}{R_c}(1+\alpha_{oh})

where $R_c$ is code rate and $\alpha_{oh}$ represents framing, pilots, guards, and protocol overhead.

Raised-cosine occupied bandwidth approximation:

B\approx R_s(1+\alpha)

where $\alpha$ is roll-off factor.

Spectral efficiency including overhead:

\displaystyle \eta_{net}=\frac{R_{payload}}{B}

State whether rates are payload, coded, line, symbol, or occupied-bandwidth values before comparing systems.

Sampling

Nyquist sampling condition for an ideal band-limited signal:

f_s>2B

Nyquist frequency:

\displaystyle f_N=\frac{f_s}{2}

Sampling period:

\displaystyle T_s=\frac{1}{f_s}

Aliasing occurs when energy above $f_N$ folds into the sampled band. Practical designs require anti-alias filtering and margin above the theoretical minimum sample rate.

Quantization

Number of quantization levels for an ideal $N$ -bit converter:

L=2^N

Quantization step:

\displaystyle \Delta=\frac{V_{FS}}{2^N}

Ideal full-scale sinusoidal quantization SNR:

SNR_q\approx 6.02N+1.76\ \text{dB}

Quantization is separate from sampling. A system can satisfy the sampling theorem and still have poor amplitude resolution or timing noise.

Thermal noise

Thermal noise power:

N=kTB

where $k$ is Boltzmann’s constant, $T$ is absolute temperature, and $B$ is noise bandwidth.

At approximately 290 K:

N_{dBm}\approx -174+10\log_{10}(B_{Hz})

Receiver noise floor:

N_{floor,dBm}\approx -174+10\log_{10}(B_{Hz})+NF_{dB}

The result depends on bandwidth and reference temperature.

Signal-to-noise ratio

Linear signal-to-noise ratio:

\displaystyle SNR=\frac{P_s}{P_n}

SNR in decibels:

SNR_{dB}=P_{s,dB}-P_{n,dB}

Carrier-to-noise ratio:

\displaystyle C/N=\frac{C}{N}

Carrier-to-noise density in dB-Hz:

\left(C/N_0\right)_{dBHz}=\left(C/N\right)_{dB}+10\log_{10}(B_{Hz})

Energy per bit to noise density in dB:

\left(E_b/N_0\right)_{dB}=\left(C/N_0\right)_{dBHz}-10\log_{10}(R_{b,Hz})

Use consistent units: $C/N_0$ is commonly in dB-Hz, while $E_b/N_0$ is in dB.

Noise figure

Noise factor:

\displaystyle F=\frac{SNR_{in}}{SNR_{out}}

Noise figure:

NF_{dB}=10\log_{10}(F)

Equivalent noise temperature:

T_e=(F-1)T_0

where $T_0$ is commonly 290 K.

Friis cascade noise factor:

\displaystyle F_{total}=F_1+\frac{F_2-1}{G_1}+\frac{F_3-1}{G_1G_2}+\cdots

Use linear gains and linear noise factors in the Friis equation.

RF link budget

Effective isotropic radiated power:

EIRP=P_t+G_t-L_t

Received power:

P_r=P_t+G_t+G_r-L_{path}-L_{misc}

Free-space path loss:

\displaystyle L_{fs}=20\log_{10}\left(\frac{4\pi d}{\lambda}\right)

Using distance in kilometers and frequency in MHz:

L_{fs,dB}=32.44+20\log_{10}(d_{km})+20\log_{10}(f_{MHz})

Link margin:

M=P_{available}-P_{required}

Fade margin:

M_{fade}=P_r-P_{sens}-M_{impl}

Interference margin:

M_I=C/I_{available}-C/I_{required}

Include feeder loss, polarization loss, pointing loss, atmospheric loss, rain fade, implementation loss, interference margin, and required availability.

Uplink carrier-to-noise density

Simplified uplink dB-Hz expression:

\left(C/N_0\right)_{dBHz}=EIRP_{dBW}-L_{path,dB}+\left(G/T\right)_{dB/K}-k_{\text{dBW/K/Hz}}

where $EIRP$ is in dBW, path losses are in dB, $G/T$ is in dB/K, and:

k\approx -228.6\ \text{dBW/K/Hz}

Required bit-energy condition:

M_{dB}=\left(E_b/N_0\right)_{available,dB}-\left(E_b/N_0\right)_{required,dB}

The required value depends on modulation, coding, bit error rate, implementation loss, and service availability.

Antenna and wavelength checks

Wavelength:

\displaystyle \lambda=\frac{c}{f}

Approximate far-field distance:

\displaystyle R_{ff}\approx \frac{2D^2}{\lambda}

where $D$ is the largest antenna dimension.

Approximate antenna effective aperture:

\displaystyle A_e=\frac{G\lambda^2}{4\pi}

Higher antenna gain improves link margin but usually narrows beamwidth and increases pointing sensitivity.

Optical fiber link budget

Received optical power before design margin:

P_{rx}=P_{tx}-L_{fiber}-L_{conn}-L_{splice}-L_{split}

Fiber attenuation:

L_{fiber}=\alpha L

where $\alpha$ is attenuation per unit length and $L$ is fiber length.

Available power margin after required design allowance:

M_{available}=P_{rx}-P_{sens}-M_{design}

where $P_{sens}$ is receiver sensitivity and $M_{design}$ is the reserved design margin for aging, repairs, dirty connectors, and environmental variation.

Optical loss budgets should include connector cleanliness, bend loss, repair margin, wavelength, aging, launch conditions, and receiver overload limits.

Optical Dispersion Screening

Chromatic dispersion broadening:

\Delta t_D=|D|\Delta\lambda L

where $D$ is dispersion coefficient, $\Delta\lambda$ is source spectral width, and $L$ is fiber length.

Modal dispersion bandwidth-distance screening:

B\cdot L\le (B\cdot L)_{rated}

Pulse-spreading margin:

M_t=T_{bit}-\Delta t_{total}

where:

\displaystyle T_{bit}=\frac{1}{R_b}

Dispersion checks should include wavelength, fiber type, source linewidth, receiver bandwidth, connector quality, launch condition, temperature, and target bit error rate.

Latency

Propagation delay:

\displaystyle t_p=\frac{d}{v}

Serialization delay:

\displaystyle t_s=\frac{N_{bits}}{R_b}

Total one-way latency estimate:

t_{oneway}=t_p+t_s+t_{queue}+t_{proc}+t_{switch}+t_{protocol}

Round-trip time:

RTT\approx 2t_{oneway}

For fiber, propagation speed is roughly $c/n$ , where $n$ is refractive index.

Jitter

Peak-to-peak jitter:

J_{pp}=t_{max}-t_{min}

Root-mean-square jitter:

\displaystyle J_{rms}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(t_i-\bar{t})^2}

Unit interval:

\displaystyle UI=\frac{1}{R_s}

where $R_s$ is symbol rate.

Timing margin is often evaluated as a fraction of the unit interval. Clock recovery, sampling aperture, phase noise, and packet queueing can all contribute to jitter.

Radar range scaling

Monostatic radar received power:

\displaystyle P_r=\frac{P_tG^2\lambda^2\sigma}{(4\pi)^3R^4L}

where $P_t$ is transmit power, $G$ is antenna gain, $\lambda$ is wavelength, $\sigma$ is radar cross section, $R$ is range, and $L$ represents losses.

The fourth-power range dependence means radar links are highly sensitive to range. Doubling range can reduce received echo power by approximately 12 dB before other effects are considered.

Measurement checks

Bit error rate estimate:

\displaystyle BER=\frac{N_{errors}}{N_{bits}}

Packet loss ratio:

\displaystyle PLR=\frac{N_{lost}}{N_{sent}}

Mean latency:

\displaystyle \bar{t}=\frac{1}{N}\sum_{i=1}^{N}t_i

Percentile latency:

P_q=\text{the value below which }q\text{ percent of measured latencies fall}

Measurements should state resolution bandwidth, sample rate, averaging, test duration, traffic load, packet size, clock reference, calibration state, environmental condition, and acceptance threshold.

REF

Disciplines