What is the relationship between average fade duration and level crossing rate?

Average fade duration and level crossing rate are complementary second-order fading statistics connected by the probability of being in a fade. The fraction of time the signal is below the threshold equals the product of the average fade duration and the level crossing rate: P(r < R) = AFD times N_R, where N_R is the level crossing rate at threshold R. For Rayleigh fading, N_R = sqrt(2*pi) * f_m * rho * exp(-rho^2). Combining these, the AFD can also be expressed as AFD = P(r < R) / N_R = (1 - exp(-rho^2)) / N_R. At deep fade levels (rho much less than 1), the fades are infrequent (low N_R) but long; at shallow fade levels (rho near 1), fades are frequent (high N_R) but short.

How does average fade duration affect interleaver design?

The interleaver depth in a coded communication system must be large enough to spread burst errors across multiple codewords so that the error correcting code sees approximately random errors rather than bursts. The minimum interleaver depth is typically set to 5 to 10 times the average burst error length, which is the product of the AFD and the data rate. For an AFD of 1 ms at a data rate of 10 Mbps, the burst length is 10,000 bits, requiring an interleaver depth of 50,000 to 100,000 bits. In LTE, the turbo code interleaver sizes range from 40 to 6,144 bits, and additional channel interleaving across the TTI (1 ms) and frequency diversity from OFDM across 1,200 subcarriers provide the necessary burst error randomization. At mmWave frequencies, the Doppler rate is much higher, producing shorter but more frequent fades, which actually relaxes the interleaver depth requirement.

Propagation & Fading

Average Fade Duration

Q: How is average fade duration calculated for a Rayleigh fading channel?

For a Rayleigh fading channel, the average fade duration is calculated as AFD = (e^(rho^2) - 1) / (rho * f_m * sqrt(2*pi)), where rho is the threshold level normalized to the RMS signal level and f_m is the maximum Doppler frequency. The maximum Doppler frequency f_m = v/lambda, where v is the vehicle speed and lambda is the wavelength. For example, at 1.9 GHz (lambda = 15.8 cm) with a vehicle traveling at 60 km/h (16.7 m/s), f_m = 105.7 Hz. At a threshold 10 dB below the RMS level (rho = 0.316), the AFD is approximately 0.74 milliseconds. This means each fade event lasts less than 1 ms on average, but at a data rate of 1 Mbps, this corresponds to approximately 740 corrupted bits per fade event.

Q: How does average fade duration affect interleaver design?

The interleaver depth in a coded communication system must be large enough to spread burst errors across multiple codewords so that the error correcting code sees approximately random errors rather than bursts. The minimum interleaver depth is typically set to 5 to 10 times the average burst error length, which is the product of the AFD and the data rate. For an AFD of 1 ms at a data rate of 10 Mbps, the burst length is 10,000 bits, requiring an interleaver depth of 50,000 to 100,000 bits. In LTE, the turbo code interleaver sizes range from 40 to 6,144 bits, and additional channel interleaving across the TTI (1 ms) and frequency diversity from OFDM across 1,200 subcarriers provide the necessary burst error randomization. At mmWave frequencies, the Doppler rate is much higher, producing shorter but more frequent fades, which actually relaxes the interleaver depth requirement.

/av-rij fayd doo-ray-shuhn/

The mean time duration that a Rayleigh or Rician fading signal envelope remains continuously below a specified threshold level. Average fade duration (AFD) is a second-order fading statistic (along with level crossing rate) that characterizes the temporal behavior of multipath fading and directly determines the burst error length in digital communication, driving the design of interleaver depth, ARQ timeout, and handover hysteresis in mobile systems.

Category: Propagation & Fading

Symbol: τ̄ (tau-bar)

Typical Range: 0.1 to 10 ms (mobile at 1-2 GHz)

Understanding Average Fade Duration

In a multipath fading environment, the received signal envelope fluctuates rapidly as the mobile receiver moves through constructive and destructive interference patterns. When the envelope drops below a threshold (typically set at the receiver sensitivity or the FEC coding threshold), a burst of errors occurs until the signal recovers. The AFD tells the designer how long, on average, each of these error bursts lasts. Combined with the level crossing rate (how often the signal crosses the threshold), AFD completely characterizes the second-order temporal statistics of the fading process.

AFD is inversely proportional to the maximum Doppler frequency (f_m = v/λ), which means faster-moving vehicles experience shorter but more frequent fades. At 1.9 GHz and 120 km/h, each fade at the -10 dB threshold lasts only about 0.37 ms. At walking speed (5 km/h), the same fade lasts approximately 8.9 ms. This speed dependence has critical implications: slow-moving users experience long burst errors that challenge interleaver design, while fast-moving users experience short fades that are easier to code through but complicate channel estimation.

AFD Formulas

Average Fade Duration (Rayleigh):
τ̄ = (e^ρ² - 1) / (ρ × f_m × √(2π))

Where:
ρ = R / R_rms (threshold normalized to RMS level)
f_m = v / λ (maximum Doppler frequency, Hz)
v = mobile velocity (m/s), λ = wavelength (m)

Level Crossing Rate (Rayleigh):
N_R = √(2π) × f_m × ρ × e^-ρ²

Relationship:
P(r < R) = N_R × τ̄ = 1 - e^-ρ²

Burst Error Length:
L_burst = τ̄ × R_data (bits)

AFD at 1.9 GHz by Speed and Threshold

Speed	f_m (Hz)	AFD at -10 dB	AFD at -20 dB	Burst Length at 1 Mbps (-10 dB)
5 km/h (walk)	8.8	8.9 ms	1.0 ms	8,900 bits
30 km/h (urban)	52.8	1.49 ms	0.17 ms	1,490 bits
60 km/h (suburban)	105.7	0.74 ms	0.084 ms	740 bits
120 km/h (highway)	211.4	0.37 ms	0.042 ms	370 bits

Common Questions

Frequently Asked Questions

How is average fade duration calculated for a Rayleigh fading channel?

For Rayleigh fading, AFD equals (e^(rho^2) - 1) / (rho times f_m times sqrt(2*pi)), where rho is the threshold normalized to the RMS level and f_m is the maximum Doppler frequency (v/lambda). At 1.9 GHz with a vehicle at 60 km/h (f_m = 105.7 Hz) and a threshold 10 dB below the RMS level (rho = 0.316), the AFD is approximately 0.74 ms. At 1 Mbps, this corresponds to about 740 corrupted bits per fade event. Deeper thresholds produce shorter fades because the signal spends less time at extreme levels.

What is the relationship between AFD and level crossing rate?

AFD and level crossing rate (LCR) are complementary second-order statistics connected by the fade probability: P(r less than R) = AFD times N_R. For Rayleigh fading, N_R = sqrt(2*pi) * f_m * rho * exp(-rho^2). At deep fade levels (small rho), fades are infrequent but long. At shallow levels (rho near 1), fades are frequent but short. Together, AFD and LCR completely describe the temporal fading behavior needed for system design.

How does AFD affect interleaver design?

The interleaver depth must be 5 to 10 times the average burst error length (AFD times data rate) to spread burst errors across multiple codewords. For an AFD of 1 ms at 10 Mbps, the burst length is 10,000 bits, requiring 50,000 to 100,000 bits of interleaving. In LTE, turbo code interleavers (up to 6,144 bits) combine with channel interleaving across the 1 ms TTI and OFDM frequency diversity across 1,200 subcarriers. At mmWave frequencies, higher Doppler rates produce shorter fades, which actually relaxes the interleaver requirement.

Related Terms

← Average Detector (EMC) Average Power Rating →