How is channel coherence time calculated?

Channel coherence time is derived from the maximum Doppler shift, which depends on the relative velocity and carrier frequency. The maximum Doppler shift is fm = v * fc / c, where v is the speed of the mobile (m/s), fc is the carrier frequency (Hz), and c = 3e8 m/s. The coherence time is then Tc = 0.423 / fm (for 50% correlation of the autocorrelation function) or Tc = 9 / (16 * pi * fm) for the geometric mean definition. For a pedestrian at 3 km/h at 2 GHz: fm = (3/3.6) * 2e9 / 3e8 = 5.6 Hz, Tc = 0.423/5.6 = 76 ms. For a vehicle at 120 km/h at 2 GHz: fm = (120/3.6) * 2e9 / 3e8 = 222 Hz, Tc = 0.423/222 = 1.9 ms. For a high-speed train at 350 km/h at 3.5 GHz: fm = (350/3.6) * 3.5e9 / 3e8 = 1,134 Hz, Tc = 0.423/1134 = 0.37 ms. At mmWave (28 GHz), even pedestrian speeds produce significant Doppler: fm = (3/3.6) * 28e9 / 3e8 = 78 Hz, Tc = 5.4 ms.

How does coherence time affect pilot design?

Channel estimation in OFDM systems uses pilot symbols (known reference signals) placed at regular intervals in time and frequency. The Nyquist sampling theorem requires pilots to be spaced no further apart than Tc/2 in time and Bc/2 in frequency to accurately track the channel variations. In 5G NR, the DMRS (Demodulation Reference Signal) density adapts to mobility. For low mobility (Tc >> slot duration), DMRS is placed in 1 to 2 OFDM symbols per slot (14 symbols, 1 ms at 15 kHz SCS). For high mobility (Tc approaching slot duration), additional DMRS symbols are configured (up to 4 per slot), reducing data capacity but maintaining channel estimation accuracy. The CSI-RS (Channel State Information Reference Signal) for codebook-based feedback must also be transmitted within the coherence time of the intended PMI use: if CSI-RS is sent at time t and the PMI is applied at time t + delta_t, delta_t must be much less than Tc for the PMI to be accurate. At 120 km/h and 3.5 GHz (Tc = 1 ms), CSI feedback latency of 4 to 10 ms means the channel has changed significantly, degrading beamforming accuracy.

How does coherence time interact with OFDM symbol duration?

For OFDM to work effectively, the channel should remain approximately constant during one OFDM symbol (slow fading condition): T_sym much less than Tc. The OFDM symbol duration is T_sym = 1/delta_f + T_CP, where delta_f is the subcarrier spacing and T_CP is the cyclic prefix duration. At 15 kHz SCS: T_sym = 66.7 + 4.7 = 71.4 microseconds. At 120 kHz SCS: T_sym = 8.33 + 0.59 = 8.92 microseconds. For a vehicle at 120 km/h at 3.5 GHz: Tc = 0.36 ms = 360 microseconds. The ratio Tc/T_sym = 360/71.4 = 5 for 15 kHz SCS, which is marginal (some inter-carrier interference from time variation within the symbol). At 120 kHz SCS: Tc/T_sym = 360/8.92 = 40, very comfortable. This is another reason why wider subcarrier spacing is preferred at higher carrier frequencies and mobility: it keeps the symbol duration well within the coherence time, preventing Doppler-induced ICI. The 5G NR numerology design (SCS scaling with carrier frequency) inherently matches this requirement.

Electromagnetic Theory

Coherence Time Channel

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Channel coherence time T_c = 0.423/f_m is the duration over which a wireless channel remains approximately constant. f_m = v×f_c/c is the maximum Doppler shift. Pedestrian (3 km/h, 2 GHz): T_c = 76 ms. Vehicle (120 km/h, 3.5 GHz): T_c = 0.36 ms. Determines pilot spacing (Nyquist: ≤ T_c/2), CSI feedback rate, and OFDM SCS selection (T_sym << T_c).

Category: Electromagnetic Theory

Key relation: T_c = 0.423/f_m

120 km/h @ 3.5 GHz: 0.36 ms

Understanding Channel Coherence Time

A wireless channel changes over time because the transmitter, receiver, or scatterers are in relative motion. Each multipath component experiences a Doppler shift proportional to the cosine of its arrival angle relative to the direction of motion, creating a spread of frequency shifts across the multipath components. This Doppler spread causes the channel's complex gain to fluctuate over time, with a fluctuation rate determined by the maximum Doppler shift. The coherence time is the time-domain dual of the Doppler spread, just as coherence bandwidth is the frequency-domain dual of the delay spread.

Channel coherence time drives many fundamental design choices in wireless systems. It determines how often channel estimates must be updated (pilot density), how quickly CSI feedback becomes stale (limiting adaptive beamforming at high speeds), how long codes can be interleaved (to average over fading dips), and the maximum useful coherent integration time (for radar or positioning). The 5G NR numerology, with its scalable subcarrier spacing, is designed so that the OFDM symbol duration remains well within the coherence time at each carrier frequency and mobility scenario.

Coherence Time Formulas

Maximum Doppler Shift:
f_m = v × f_c / c

Coherence Time (50% correlation):
T_c = 0.423 / f_m

Pilot Spacing Constraint:
Δt_pilot ≤ T_c / 2 (Nyquist in time)

Where v = velocity (m/s), f_c = carrier (Hz), c = 3×10⁸ m/s. 120 km/h at 3.5 GHz: f_m = 1,167 Hz, T_c = 0.36 ms, pilots every ≤0.18 ms. At 28 GHz: f_m = 9,333 Hz, T_c = 45 μs, SCS must be ≥120 kHz.

Mobility and Coherence Time

Scenario	Speed	f_c	f_m	T_c
Pedestrian	3 km/h	2 GHz	5.6 Hz	76 ms
Urban vehicle	60 km/h	3.5 GHz	194 Hz	2.2 ms
Highway vehicle	120 km/h	3.5 GHz	1,167 Hz	0.36 ms
HST	350 km/h	3.5 GHz	1,134 Hz	0.37 ms
Pedestrian mmWave	3 km/h	28 GHz	78 Hz	5.4 ms

Common Questions

Frequently Asked Questions

How is T_c calculated?

f_m = v×f_c/c. T_c = 0.423/f_m. Pedestrian 3 km/h at 2 GHz: f_m = 5.6 Hz, T_c = 76 ms. 120 km/h at 3.5 GHz: f_m = 1,167 Hz, T_c = 0.36 ms. HST 350 km/h at 3.5 GHz: T_c = 0.37 ms. mmWave pedestrian: f_m = 78 Hz, T_c = 5.4 ms.

How does T_c affect pilot design?

Nyquist: pilots every ≤ T_c/2. 5G NR DMRS: 1 to 4 symbols per slot (1 ms). Low mobility: 1 to 2 DMRS sufficient. High mobility (T_c < 1 ms): 4 DMRS symbols, reducing data capacity. CSI feedback latency (4 to 10 ms) exceeds T_c at vehicle speeds, degrading beamforming accuracy.

T_c and OFDM symbol duration?

Need T_sym << T_c to avoid Doppler-induced ICI. 15 kHz SCS: T_sym = 71 μs. At 120 km/h, 3.5 GHz (T_c = 0.36 ms): ratio = 5, marginal. 120 kHz SCS: T_sym = 8.9 μs, ratio = 40, comfortable. 5G numerology inherently matches SCS to frequency/mobility.

Related Terms

← Coherence Time Coherency Matrix →

Coherence Time Channel

Understanding Channel Coherence Time

Coherence Time Formulas

Mobility and Coherence Time

Frequently Asked Questions

How is Tc calculated?

How does Tc affect pilot design?

Tc and OFDM symbol duration?

Related Terms

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How is T_c calculated?

How does T_c affect pilot design?

T_c and OFDM symbol duration?