Exercise E1
A mobile station receives from a transmitter 2 waves: a direct wave and a reflected wave coming from opposite directions. (see figure).
R = 150 m
The mobile station moves toward the reflecting wall with a constant speed of 6 m/sec. Carrier frequency is 1.8 GHz.
Assuming power attenuation due to reflection equal to 6 dBs and that excess attenuation due to path length difference be negligible, determine:
1. The deterministic RMS Delay spread when the receiver is at 200 m from the wall.
2. The deterministic RMS Doppler spread in the same conditions as above.
3. Considering now a successive time instant when the mobile is closer to the wall. Will the RMS delay spread increase or decrease?
SOLUTION
Be P0 the direct ray power and Pr the reflected ray power. It is:
Pr = P0 / A
Attenuation A according to the assumptions, is only due to reflection, therefore:
0 tot
0 10 r
6
4 P 5
P 4
4
10 P P
A= ≈ ⇒ = ⇒ =
Then, assuming the direct-ray relative delay equal to zero, we have:
TM 0= t0⋅ P0
Ptot + tr⋅ Pr
Ptot = 0⋅4
5+ 400 3⋅108⋅1
5= 0.26 µsec
DS= TM 02⋅ P0
Ptot + t
(
r − TM 0)
2⋅PPrtot
= 0.26⋅10
(
−6)
2⋅45+ 1.33⋅10(
−6− 0.26⋅10−6)
2⋅15 == 0.05408⋅10−12+ 0.229⋅10−12 = 0.53⋅10-6sec
Doppler spread calculation is very similar.
By definition, the Doppler shift for a generic wave is : fd = − f0⋅
v⋅ ˆki
c
where f0 is the nominal carrier frequency, v mobile speed vector and ˆki the direction of arrival versor.
R = 200 m
Therefore:
c Hz f v f
c Hz f v f
dr d
10 36 3 10 6 8 . 1
10 36 3 10 6 8 . 1
8 9
0
8 9
0 0
+
⋅ =
⋅
⋅ +
=
⋅ +
=
−
⋅ =
⋅
⋅
−
=
⋅
−
=
Hz 6 . 5 21
108 5
36 1 5 36 4
0
0 = 0 ⋅ + ⋅ =− ⋅ + ⋅ =− =−
tot r d tot
d P
f P P f P
W r
(
0 0)
2 0(
0)
2( )
2( )
24 1
36 21.6 36 21.6
5 5
28.8 Hz
d d r r
tot tot
P P
W f W f W
P P
= − ⋅ + − ⋅ = − + ⋅ + + ⋅ =
=
If the mobile comes closer to the wall then while powers do not change (assumption) the relative delay tr decreases, therefore, the DS value decreases. Of course DS would be equal to zero on the wall due to the null relative delay tr .
On the contrary, there is no Doppler spread change since Doppler shifts do not change.
