- 2070 Magh · 4+4 marks
Doppler spread fading; delay spread numerical
Short answer
Doppler spread is spectral broadening due to motion: f_c ± f_m. Fast fading if symbol period > coherence time (B_s < B_D); slow fading if symbol period << coherence time. From PDP: τ̄ = 1 µs, σ_τ = 0.976 µs; B_c(90%) = 1/(50σ_τ) = 20.5 kHz, B_c(50%) = 1/(5σ_τ) = 205 kHz.
Define Doppler spread. Describe the types of small scale fading based on Doppler spread. Calculate the mean excess delay and rms delay spread for the multipath profile given below. Estimate the 90% and 50% coherence bandwidth of the channel. [Figure: power delay profile Pr(τ) with impulses at τ = 0 µs (about 0 dB), τ = 1 µs (about −10 dB) and τ = 2 µs (about 0 dB); the vertical axis is marked 0 dB, −10 dB, −20 dB, −30 dB and the time axis is in µs.]
Doppler spread
Doppler spread (B_D) is the range of frequencies over which the received Doppler spectrum is non-zero when a pure sinusoid is sent. Due to motion, the received spectrum occupies f_c − f_m to f_c + f_m, where f_m = v/λ is the maximum Doppler shift. Its time-domain dual is the coherence time T_c ≈ 0.423/f_m.
Types of small scale fading based on Doppler spread
| Point | Fast fading | Slow fading |
|---|---|---|
| Condition | T_s > T_c, B_s < B_D | T_s ≪ T_c, B_s ≫ B_D |
| Channel change | Within one symbol | Over many symbols |
| Effect | Time distortion of the symbol | Channel is static per symbol |
| Typical case | Very low data rate, high speed | Most digital systems |
Numerical (power delay profile)
From the figure: P(0) = 0 dB, P(1 µs) = −10 dB, P(2 µs) = 0 dB.
Linear powers: 1, 0.1, 1 → ΣP = 2.1
Mean excess delay
τ̄ = (1×0 + 0.1×1 + 1×2) / 2.1 = 2.1 / 2.1 = 1 µs
Second moment
τ̄² = (1×0² + 0.1×1² + 1×2²) / 2.1 = 4.1 / 2.1 = 1.952 µs²
RMS delay spread
σ_τ = √(1.952 − 1²) = √0.952 = 0.976 µs
Coherence bandwidth
B_c(90%) ≈ 1/(50 σ_τ) = 1/(50 × 0.976 µs) = 20.49 kHz
B_c(50%) ≈ 1/(5 σ_τ) = 1/(5 × 0.976 µs) = 204.9 kHz
Answer: τ̄ = 1 µs, σ_τ = 0.976 µs, B_c(90%) ≈ 20.5 kHz and B_c(50%) ≈ 205 kHz.
✓ Answer checked by Claude Opus 5.5 (high effort)
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