1Homework 1 of EE226
For radio wireless channel, we can model its input-output relationship as
y(t) =
∫
ht(t− τ)x(τ)dτ (1)
where
ht(τ) =
∑
i
αi
ri(t)
δ(τ − ri(t)
c
) (2)
Assume that at t = 0 there are two significant reflecting/scattering objects in the environment,
both of them are located in the same direction from the transmitter to the receiver but farther
away from the receiver. One object is moving towards the receiver at speed v1, and the other is
moving away from the receiver at speed v2. Assuming x(t) = cos(2pifct), we have
y(t) =
α0
r0
cos(2pifc(t− r0
c
)) +
α1
r1(t)
cos(2pifc(t− r1(t)
c
)) +
α2
r2(t)
cos(2pifc(t− r2(t)
c
)) (3)
where r0 is a constant distance between the transmitter and the receiver, r1(t) = r0+2d1(t) and
r2(t) = r0 + 2d2(t) where d1(t) = d1 − v1t is the distance between the receiver and the first
object and d2(t) = d2 + v2t is the distance between the receiver and the second object.
1) What is the Doppler spread Ds in the received signal at t = 0?
2) What is the delay spread Td of the channel at t = 0?
3) What is the coherence time Tc of the channel at t = 0?
4) What is the coherence bandwidth Wc of the channel at t = 0?
5) If we want the channel to be a slow fading channel for a coding delay equal to 1ms at
t = 0, what is the upper limit for v1 and v2 (assuming fc = 1GHz and v1 = v2)?
6) If we want the channel at t = 0 to be a flat fading channel for a bandwidth W of interest
equal to 20MHz, what is the constraint on d1 and d2?
April 1, 2021 DRAFT

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