How is coherence bandwidth calculated from delay spread?

The RMS delay spread sigma_tau characterizes the time dispersion of multipath arrivals and is calculated as the square root of the second central moment of the power delay profile (PDP): sigma_tau = sqrt(tau_bar2 - tau_bar^2), where tau_bar = sum(P_k * tau_k) / sum(P_k) is the mean delay and tau_bar2 = sum(P_k * tau_k^2) / sum(P_k) is the mean-square delay, with P_k and tau_k being the power and delay of the k-th multipath component. The coherence bandwidth depends on the correlation threshold. For 0.5 correlation (commonly used): Bc = 1 / (5 * sigma_tau). For 0.9 correlation (stricter): Bc = 1 / (50 * sigma_tau). For example, in a typical urban environment with sigma_tau = 1 microsecond: Bc(0.5) = 1 / (5 * 1e-6) = 200 kHz. This means signals narrower than 200 kHz experience flat fading, while wider signals experience frequency-selective fading. Indoor environments with sigma_tau = 50 ns have Bc(0.5) = 4 MHz, and open rural areas with sigma_tau = 100 ns have Bc(0.5) = 2 MHz.

How does coherence bandwidth affect OFDM design?

OFDM converts a wideband frequency-selective channel into many narrowband flat-fading subchannels. For each subcarrier to experience flat fading, its bandwidth (approximately equal to the subcarrier spacing delta_f) must be much less than the coherence bandwidth: delta_f much less than Bc. In 5G NR, the subcarrier spacing options are 15, 30, 60, 120, and 240 kHz. At sub-6 GHz in urban environments (Bc approximately 200 kHz), 15 kHz subcarrier spacing works well (15 kHz much less than 200 kHz). At mmWave with smaller delay spreads (Bc approximately 5 to 10 MHz), wider subcarrier spacing (120 or 240 kHz) is used, which also reduces the OFDM symbol duration to be less affected by phase noise. The cyclic prefix duration must exceed the maximum delay spread to prevent inter-symbol interference: T_CP must be greater than tau_max. For 15 kHz SCS, the normal CP is 4.7 microseconds, supporting delay spreads up to about 4.7 microseconds. For 120 kHz SCS, the CP is 0.59 microseconds, limiting deployment to environments with delay spread below 0.59 microseconds.

What happens when signal bandwidth exceeds coherence bandwidth?

When the signal bandwidth B exceeds the coherence bandwidth Bc, the channel is frequency-selective: different frequency components of the signal experience different fading (some may be in deep fade while others are strong). This has both negative and positive effects. The negative effect is inter-symbol interference (ISI) in single-carrier systems: multipath replicas of successive symbols overlap in time, causing detection errors. An equalizer (linear MMSE, decision feedback, or Viterbi-based MLSE) is needed to compensate, with complexity growing with the ratio B/Bc. The positive effect is frequency diversity: because different parts of the spectrum fade independently, it is unlikely that the entire signal bandwidth is in a deep fade simultaneously. Coding and interleaving across the bandwidth can exploit this diversity, improving BER performance by a diversity order proportional to the number of independently fading frequency bins (approximately B/Bc). OFDM with channel coding naturally exploits this: a coded OFDM system with bandwidth 10 * Bc has approximately 10x frequency diversity, gaining 5 to 8 dB over a flat-fading channel at BER = 10^-3 with appropriate coding.

Electromagnetic Theory

Coherence Bandwidth

/koh-heer-unts band-width/

Coherence bandwidth B_c is the frequency range over which a multipath channel has correlated fading. B_c = 1/(5σ_τ) for 0.5 correlation, where σ_τ is RMS delay spread. Signal BW < B_c: flat fading. Signal BW > B_c: frequency-selective fading (ISI but also frequency diversity). OFDM subcarrier spacing Δf << B_c ensures flat fading per subcarrier. Urban: σ_τ ≈ 1 μs, B_c ≈ 200 kHz.

Category: Electromagnetic Theory

Urban B_c: ≈200 kHz

Indoor B_c: ≈4 MHz

Understanding Coherence Bandwidth

A wireless channel with multipath propagation acts as a frequency-selective filter: the multiple copies of the transmitted signal arrive with different delays and combine constructively at some frequencies and destructively at others, creating a channel frequency response with peaks and nulls. The spacing between adjacent nulls is approximately 1/σ_τ, and the coherence bandwidth describes the frequency range over which the response stays approximately constant. Within this range, the channel is "flat" and can be modeled as a single complex gain; beyond this range, the channel varies and must be equalized.

Coherence bandwidth is the frequency-domain dual of the delay spread, just as coherence time is the time-domain dual of the Doppler spread. Together, these four parameters (delay spread/coherence bandwidth and Doppler spread/coherence time) completely characterize the second-order statistics of a wideband, time-varying wireless channel and drive the design of every modern communication system: subcarrier spacing, cyclic prefix length, pilot density, equalizer complexity, and coding/interleaving depth.

Coherence Bandwidth Formulas

0.5 Correlation:
B_c ≈ 1 / (5 σ_τ)

0.9 Correlation:
B_c ≈ 1 / (50 σ_τ)

RMS Delay Spread:
σ_τ = √(τ̄² - τ̄²) = √(∑P_kτ_k²/∑P_k - (∑P_kτ_k/∑P_k)²)

Where P_k, τ_k = power and delay of k-th multipath. σ_τ = 1 μs (urban): B_c(0.5) = 200 kHz. σ_τ = 50 ns (indoor): B_c(0.5) = 4 MHz. σ_τ = 10 ns (mmWave LOS): B_c = 20 MHz.

Environment and B_c Values

Environment	σ_τ	B_c (0.5)	5G SCS Match	CP Margin
Indoor office	30 to 50 ns	4 to 6.7 MHz	15 to 60 kHz	Large
Suburban	200 to 500 ns	400 kHz to 1 MHz	15 to 30 kHz	Good
Urban	1 to 3 μs	67 to 200 kHz	15 kHz	Tight
Hilly terrain	5 to 20 μs	10 to 40 kHz	15 kHz + ext CP	Insufficient
mmWave LOS	5 to 20 ns	10 to 40 MHz	120 to 240 kHz	Large

Common Questions

Frequently Asked Questions

How is B_c calculated from delay spread?

B_c(0.5) = 1/(5σ_τ). B_c(0.9) = 1/(50σ_τ). σ_τ = RMS delay spread from power delay profile. Urban (σ_τ = 1 μs): 200 kHz. Indoor (50 ns): 4 MHz. mmWave LOS (10 ns): 20 MHz. Wider B_c means less frequency-selective fading and simpler equalization.

How does B_c affect OFDM design?

Subcarrier spacing << B_c for flat fading per subcarrier. 5G NR: 15 kHz (sub-6 urban, B_c ≈ 200 kHz), 120/240 kHz (mmWave, B_c ≈ 10 to 40 MHz). CP > τ_max: 15 kHz normal CP = 4.7 μs (supports up to 4.7 μs delay). 120 kHz CP = 0.59 μs (only short delay spreads).

What happens when BW > B_c?

Frequency-selective fading: ISI requires equalization (MMSE, DFE, MLSE). But also frequency diversity: independent fading across B/B_c subbands. With coding: 10x diversity (B = 10×B_c) yields 5 to 8 dB gain at BER 10^-3 over flat fading. OFDM + coding naturally exploits this.

Related Terms

← Coherence Area Coherence Length →

Coherence Bandwidth

Understanding Coherence Bandwidth

Coherence Bandwidth Formulas

Environment and Bc Values

Frequently Asked Questions

How is Bc calculated from delay spread?

How does Bc affect OFDM design?

What happens when BW > Bc?

Related Terms

Request a Quote

Environment and B_c Values

How is B_c calculated from delay spread?

How does B_c affect OFDM design?

What happens when BW > B_c?