Signal & RF Formulas
RF formulas connect frequency, wavelength, and impedance for transmission lines and antennas. These calculators help you size matching networks, estimate antenna dimensions, and reason about high-frequency layout. They are essential for wireless, transmission-line, and microwave work.
Signal & RF Formula Calculators
- Wavelength λ = c / f Frequency 100 MHz → Wavelength = 2.998 m
- Free Space Path Loss FSPL = 20log(d) + 20log(f) + const Distance 1 mm, Frequency 100 MHz → FSPL = -47.55 dB
- Signal-to-Noise Ratio SNR = 10 × log₁₀(Ps/Pn) Signal Power 10 mW, Noise Power 0.1 µW → SNR = 50 dB
- Nyquist Sampling Rate fs_min = 2 × fmax Max Signal Frequency 0.22 kHz → Min Sampling Frequency = 0.44 kHz
- Link Budget Prx = Ptx + Gtx − Ltx − FSPL + Grx − Lrx Tx Power (dBm) -2 dBm, Tx Gain (dBi) 0.2 dBi, Tx Loss (dB) 0.3 dB, Path Loss (dB) 0.66 dB, Rx Gain (dBi) 0.44 dBi, Rx Loss (dB) 1.41 dB → Prx = -3.73 dBm
- Antenna Gain G = 10 × log₁₀(4πAe/λ²) Effective Aperture 10 mm², Wavelength 1 mm → G = 20.99 dBi
- Skin Depth δ = 1 / √(πfμσ) Frequency 100 kHz, Permeability (H/m) 1 × 10⁻⁷ H/m, Conductivity (S/m) 1000 S/m → δ = 0.1784 m
- Friis Noise Figure (3-stage) NFtotal = NF1 + (NF2−1)/G1 + (NF3−1)/(G1×G2) Stage 1 Noise Factor 0.025, Stage 1 Gain (linear) 0.055, Stage 2 Noise Factor 0.118, Stage 2 Gain (linear) 0.0118, Stage 3 Noise Factor 0.25 → NFtotal = -1172 linear
- Thermal Noise Power Pn = k × T × BW Temperature (K) 0.025 K, Bandwidth 100 kHz → Noise Power = 3.45 × 10⁻¹⁴ µW
- VSWR VSWR = (1 + |Γ|) / (1 − |Γ|) Reflection Coefficient |Γ| 0.055 → VSWR = 1.116
- Return Loss RL = −20 × log10(|Γ|) Reflection Coefficient |Γ| 0.025 → Return Loss = 32.04 dB
- Effective Radiated Power ERP = Ptx × Gt Transmitter Power 22 mW, Antenna Gain (linear) 0.025 → Effective Radiated Power = 0.55 mW
- Third-Order Intercept Point OIP3 = Pout + (IIP3 − Pin) Output Power (dBm) -4.4 dBm, Input IP3 (dBm) -9.4 dBm, Input Power (dBm) -0.94 dBm → Output IP3 (dBm) = -12.86 dBm
- Equivalent Noise Temperature Te = T0 × (NF − 1) Reference Temperature 55 K, Noise Factor (linear) 118 → Noise Temperature = 6435 K
- Group Delay (Approximation) τg ≈ 1 / (2π × BW) Bandwidth 100 kHz → Group Delay = 1.592 µs
- Reflection Coefficient Γ = (ZL − Z0) / (ZL + Z0) Load Impedance 1 Ω, Characteristic Impedance 2.2 Ω → Reflection Coeff. = -0.375
- Insertion Loss from S21 IL = −20 × log10(|S21|) |S21| Magnitude 0.025 → Insertion Loss = 32.04 dB
- Doppler Frequency Shift Δf = fc × v / c Carrier Frequency 100 MHz, Relative Velocity 0.025 → Doppler Shift = 0.008339 Hz
- Radar Range Equation Rmax = (Pt×Gt×Gr×λ²×σ / ((4π)³×Smin))^0.25 Transmit Power 10 mW, Tx Antenna Gain 0.025, Rx Antenna Gain 0.055, Wavelength 1 mm, Radar Cross Section 10 mm², Min Detectable Signal 22 mW → Rmax = 4.21 × 10⁻⁵
- Carson's Rule FM Bandwidth BW = 2 × (Δf + fm) Peak Frequency Deviation 0.22 kHz, Max Modulating Frequency 0.47 kHz → Occupied Bandwidth = 1.38 kHz
- Spurious-Free Dynamic Range SFDR = 20 · log10(Vsignal / Vspur) Signal Amplitude 0.33 V, Worst Spur Amplitude 10 mV → SFDR = 30.37 dB
Worked result for every Signal & RF formula
| Formula | Expression | Worked result |
|---|---|---|
| Wavelength | λ = c / f | Frequency 100 MHz → Wavelength = 2.998 m |
| Free Space Path Loss | FSPL = 20log(d) + 20log(f) + const | Distance 1 mm, Frequency 100 MHz → FSPL = -47.55 dB |
| Signal-to-Noise Ratio | SNR = 10 × log₁₀(Ps/Pn) | Signal Power 10 mW, Noise Power 0.1 µW → SNR = 50 dB |
| Nyquist Sampling Rate | fs_min = 2 × fmax | Max Signal Frequency 0.22 kHz → Min Sampling Frequency = 0.44 kHz |
| Link Budget | Prx = Ptx + Gtx − Ltx − FSPL + Grx − Lrx | Tx Power (dBm) -2 dBm, Tx Gain (dBi) 0.2 dBi, Tx Loss (dB) 0.3 dB, Path Loss (dB) 0.66 dB, Rx Gain (dBi) 0.44 dBi, Rx Loss (dB) 1.41 dB → Prx = -3.73 dBm |
| Antenna Gain | G = 10 × log₁₀(4πAe/λ²) | Effective Aperture 10 mm², Wavelength 1 mm → G = 20.99 dBi |
| Skin Depth | δ = 1 / √(πfμσ) | Frequency 100 kHz, Permeability (H/m) 1 × 10⁻⁷ H/m, Conductivity (S/m) 1000 S/m → δ = 0.1784 m |
| Friis Noise Figure (3-stage) | NFtotal = NF1 + (NF2−1)/G1 + (NF3−1)/(G1×G2) | Stage 1 Noise Factor 0.025, Stage 1 Gain (linear) 0.055, Stage 2 Noise Factor 0.118, Stage 2 Gain (linear) 0.0118, Stage 3 Noise Factor 0.25 → NFtotal = -1172 linear |
| Thermal Noise Power | Pn = k × T × BW | Temperature (K) 0.025 K, Bandwidth 100 kHz → Noise Power = 3.45 × 10⁻¹⁴ µW |
| VSWR | VSWR = (1 + |Γ|) / (1 − |Γ|) | Reflection Coefficient |Γ| 0.055 → VSWR = 1.116 |
| Return Loss | RL = −20 × log10(|Γ|) | Reflection Coefficient |Γ| 0.025 → Return Loss = 32.04 dB |
| Effective Radiated Power | ERP = Ptx × Gt | Transmitter Power 22 mW, Antenna Gain (linear) 0.025 → Effective Radiated Power = 0.55 mW |
| Third-Order Intercept Point | OIP3 = Pout + (IIP3 − Pin) | Output Power (dBm) -4.4 dBm, Input IP3 (dBm) -9.4 dBm, Input Power (dBm) -0.94 dBm → Output IP3 (dBm) = -12.86 dBm |
| Equivalent Noise Temperature | Te = T0 × (NF − 1) | Reference Temperature 55 K, Noise Factor (linear) 118 → Noise Temperature = 6435 K |
| Group Delay (Approximation) | τg ≈ 1 / (2π × BW) | Bandwidth 100 kHz → Group Delay = 1.592 µs |
| Reflection Coefficient | Γ = (ZL − Z0) / (ZL + Z0) | Load Impedance 1 Ω, Characteristic Impedance 2.2 Ω → Reflection Coeff. = -0.375 |
| Insertion Loss from S21 | IL = −20 × log10(|S21|) | |S21| Magnitude 0.025 → Insertion Loss = 32.04 dB |
| Doppler Frequency Shift | Δf = fc × v / c | Carrier Frequency 100 MHz, Relative Velocity 0.025 → Doppler Shift = 0.008339 Hz |
| Radar Range Equation | Rmax = (Pt×Gt×Gr×λ²×σ / ((4π)³×Smin))^0.25 | Transmit Power 10 mW, Tx Antenna Gain 0.025, Rx Antenna Gain 0.055, Wavelength 1 mm, Radar Cross Section 10 mm², Min Detectable Signal 22 mW → Rmax = 4.21 × 10⁻⁵ |
| Carson's Rule FM Bandwidth | BW = 2 × (Δf + fm) | Peak Frequency Deviation 0.22 kHz, Max Modulating Frequency 0.47 kHz → Occupied Bandwidth = 1.38 kHz |
| Spurious-Free Dynamic Range | SFDR = 20 · log10(Vsignal / Vspur) | Signal Amplitude 0.33 V, Worst Spur Amplitude 10 mV → SFDR = 30.37 dB |
Variables used across Signal & RF formulas
- λ — Wavelength in meters (m)
- f — Frequency in hertz (Hz)
- c — Speed of light ≈ 3 × 10⁸ m/s
- FSPL — Path loss in decibels (dB)
- d — Distance between transmitter and receiver (m)
- SNR — Signal-to-noise ratio (dB)
- Psignal — Signal power (W)
- Pnoise — Noise power (W)
- fs_min — Minimum sampling frequency (Hz)
- fmax — Maximum signal frequency (Hz)
- Prx — Received power (dBm)
- Ptx — Transmitter output power (dBm)
- Gtx — Transmit antenna gain (dBi)
- Ltx — Transmitter cable/connector losses (dB)
- Grx — Receive antenna gain (dBi)
- Lrx — Receiver cable/connector losses (dB)
- G — Antenna gain (dBi)
- Ae — Effective aperture area (m²)
- δ — Skin depth (m)
- μ — Permeability of the conductor (H/m)
- σ — Conductivity of the conductor (S/m)
- NFtotal — System noise factor (linear)
- NF1 — Stage 1 noise factor (linear)
- NF2 — Stage 2 noise factor (linear)
- NF3 — Stage 3 noise factor (linear)
- G1 — Stage 1 gain (linear power ratio)
- G2 — Stage 2 gain (linear power ratio)
- Pn — Available noise power (W)
- k — Boltzmann constant (1.38 × 10⁻²³ J/K)
- T — Temperature in Kelvin (K)
- BW — Measurement bandwidth (Hz)
- VSWR — Voltage Standing Wave Ratio (dimensionless, ≥1)
- |Γ| — Magnitude of reflection coefficient (0 to 1)
- RL — Return loss in positive decibels (dB)
- ERP — Effective radiated power (W)
- Gt — Antenna gain (linear, not dB)
- OIP3 — Output third-order intercept point (dBm)
- Pout — Fundamental output power per tone (dBm)
- IIP3 — Input third-order intercept point (dBm)
- Pin — Input power per tone (dBm)
- Te — Equivalent noise temperature (K)
- T0 — Reference temperature, standard 290K (K)
- NF — Noise factor (linear, not dB)
- τg — Group delay (s)
- Γ — Voltage reflection coefficient (−1 to +1, dimensionless)
- Z_L — Load impedance (Ω)
- Z₀ — Transmission line characteristic impedance (Ω)
- IL — Insertion loss (dB, positive value)
- |S₂₁| — Forward transmission coefficient magnitude (0 to 1)
- Δf — Doppler frequency shift (Hz)
- f_c — Carrier (transmitted) frequency (Hz)
- v — Relative velocity between source and observer (m/s)
- P_t — Transmitter peak power (W)
- G_t — Transmit antenna gain (linear, not dB)
- G_r — Receive antenna gain (linear, not dB)
- S_min — Minimum detectable signal power (W)
- f_m — Maximum modulating frequency (Hz)
- SFDR — Spurious-free dynamic range (dB)
- Vsignal — Amplitude of the fundamental (V)
- Vspur — Amplitude of the worst spurious tone (V)
Questions people ask about Signal & RF
- Link budget calculator — Link Budget
- Return loss formula — Return Loss
- A return loss of — Return Loss
- Effective radiated power chart — Effective Radiated Power
- Group delay formula — Group Delay (Approximation)
- Insertion loss s21 — Insertion Loss from S21
- Radar range equation calculator — Radar Range Equation
- Rmax radar calculator — Radar Range Equation
- Carson bandwidth rule — Carson's Rule FM Bandwidth
- Carson's rule for fm bandwidth — Carson's Rule FM Bandwidth
Related Formula Categories
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