Numerical: rms delay spread; flat or selective for AMPS/GSM

Step 1 – Convert powers to linear values

P (dB):    0   −10    −20    −23
P (lin):   1   0.1    0.01   0.00501
τ (ns):    0   100    200    400
ΣP  =  1 + 0.1 + 0.01 + 0.00501 = 1.11501

Step 2 – Mean excess delay

τ̄ = Σ P(τ_k) τ_k / Σ P(τ_k)
  = (1×0 + 0.1×100 + 0.01×200 + 0.00501×400) / 1.11501
  = (0 + 10 + 2 + 2.005) / 1.11501
  = 14.005 / 1.11501 = 12.56 ns

Step 3 – Second moment

τ̄² (mean square) = Σ P τ² / Σ P
  = (0.1×100² + 0.01×200² + 0.00501×400²) / 1.11501
  = (1000 + 400 + 801.9) / 1.11501
  = 1974.8 ns²

Step 4 – RMS delay spread

σ_τ = √(1974.8 − 12.56²) = √(1974.8 − 157.8)
    = √1817.0 = 42.63 ns

Step 5 – Coherence bandwidth (90% correlation)

B_c(90%) ≈ 1 / (50 σ_τ) = 1 / (50 × 42.63 × 10⁻⁹)
         = 469.2 kHz

(For reference, the 50% coherence bandwidth is 1/(5σ_τ) ≈ 4.69 MHz.)

Step 6 – Decide the type of fading A channel is flat for a system when its signal bandwidth B_s is smaller than B_c.

Answer: σ_τ = 42.63 ns, B_c(90%) ≈ 469.2 kHz; the channel is flat fading for both AMPS (30 kHz) and GSM (200 kHz). GSM's 200 kHz is about 43% of B_c, so it is closer to the limit; GSM receivers still include an equalizer to handle more dispersive channels.