Small-Scale Fading
PROF. MICHAEL TSAI
2016/04/15
Multipath Propagation
2
RX just sums up all Multi Path Component (MPC).Multipath Channel Impulse Response
t0
t
0
1
2
3
4
5
6 (t0)h
b(t,)
An example of the time-varying discrete-time impulse response for a multipath radio channel
The channel impulse
response when (what you receive at the receiver
when you send an impulse at time )
, and represents the time the first signal arrives at the receiver.
Summed signal of all multipath components arriving at .
Excess delay: the delay with respect to the first arriving signal ()
Maximum excess delay:
the delay of latest arriving signal
Time-Variant Multipath Ch annel Impulse Response
4
t
t0
0
1
2
3
4
5
6 (t0)(t1) t1
t2
(t2) t3
(t3)
h
b(t,)
Because the transmitter, the receiver, or the
reflectors are moving, the impulse response is time- variant.
• The channels impulse response is given by:
• If assumed time-invariant (over a small- scale time or distance):
Multipath Channel Impulse Response
Phase change due to different arriving time Additional phase change due to reflections
Amplitude change (mainly path loss) Summation over all MPC
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) ( ,
) ( 2
exp ,
,
Ni
i i
i c i
b
t a t j f t t t t
h
10
exp
N i
i i
i
b
a j
h
6
t
t0
0
1
2
3
4
5
6 (t0)(t1) t1
t2
(t2) t3
(t3)
h
b(t,)
Following this axis, we study how “spread-out” the impulse response are.
(related to the physical layout of the TX, the RX, and the reflectors at a single time point)
Following this axis, we study how fast the channel changes over time.
(related to the moving speed of the TX, the RX, and the reflectors)
Two main aspects of t
he wireless channel
7
t
t0
0
1
2
3
4
5
6 (t0)(t1) t1
t2
(t2) t3
(t3)
h
b(t,)
Following this axis, we study how “spread-out” the impulse response are.
(related to the physical layout of the TX, the RX, and the reflectors at a single time point)
Two main aspects of t
he wireless channel
Power delay profile
• To predict h
b( ) a probing pulse p(t) is sent s .t.
• Therefore, for small-scale channel modeling, P OWER DELAY PROFILE is found by computing the s patial average of |h
B(t; )|
2over a local area .
8
� ( � ) ≈ �(� − �)
� ( � ; � ) ≈ � | h
�( � ; � ) |
2
TX RX
�(�)
Average over several
measurements in a local area
Example: power delay profile
9
From a 900 MHz cellular system in San Francisco
Example: power delay profile
1 0
Inside a grocery store at 4 GHz
Time dispersion parameters
• Power delay profile is a good representation o f the average “geometry” of the transmitter, the receiver, and the reflectors.
• To quantify “how spread-out” the arriving si gnals are, we use time dispersion parameters:
• Maximum excess delay: the excess delay of the latest a rriving MPC
• Mean excess delay: the “mean” excess delay of all ar riving MPC
• RMS delay spread: the “standard deviation” of the ex cess delay of all arriving MPC
1 1
Already talked about this
Time dispersion parameters
• Mean Excess Delay
• RMS Delay Spread
1 2
First moment of the power delay profile
Second moment of the power delay profile Square root of the second
central moment of the power delay profile
k
k k
k k
k k k
k k
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) (
2 2 __
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22 2 2 __2
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Time dispersion parameters
• Maximum Excess Delay:
• Original version: the excess delay of the latest arriving MPC
• In practice: the latest arriving could be smaller than the nois e
• No way to be aware of the “latest”
• Maximum Excess Delay (practical version):
• The time delay during which multipath energy falls to X dB belo w the maximum.
• This X dB threshold could affect the values of the ti me-dispersion parameters
• Used to differentiate the noise and the MPC
• Too low: noise is considered to be the MPC
• Too high: Some MPC is not detected
1 3
Example: Time dispersion parameter s
1 4
Coherence Bandwidth
• Coherence bandwidth is a statistical measure of the range of frequencies over which the channel can be considered “flat”
a channel passes all spectral components with
approximately equal gain and linear phase.
Coherence Bandwidth
• Bandwidth over which Frequency Correlation fu nction is above 0.9
• Bandwidth over which Frequency Correlation fu nction is above 0.5
1 6
Those two are approximations derived from empirical results.
50
1 B
c
5
1
B
cTypical RMS delay spread values
1 7
Signal Bandwidth & Cohere nce Bandwidth
1 8
f t
Transmitted Signal
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�symbol period
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�signal bandwidth
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TX signal
Channel
RX signal
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2
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These will become inter-symbol
interference!
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�Flat fading channel
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TX signal
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RX signal
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t0
0
1
2
3
4
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�No significant ISI
Equalizer 101
• An equalizer is usually used in a frequency-selecti ve fading channel
• When the coherence bandwidth is low, but we need to use high data rate (high signal bandwidth)
• Channel is unknown and time-variant
• Step 1: TX sends a known signal to the receiver
• Step 2: the RX uses the TX signal and RX signal to estimate the channel
• Step 3: TX sends the real data (unknown to the receiver)
• Step 4: the RX uses the estimated channel to process the RX signal
• Step 5: once the channel becomes significantly different fro m the estimated one, return to step 1.
2 1
Example
0 1 2 3 4 5 -30dB
-20dB -10dB 0dB
P()
Would this channel be suitable for AMPS or GSM without the use of an equalizer?
P s P
k
k k
k
k
4.38
01 . 0 1 . 0 1 . 0 1
) 01 . 0 ( 0 ) 1 . 0 ( 1 ) 1 . 0 ( 2 ) 1 ( 5 )
( ) ( Delay
Excess
Mean __
2 2
2 2
2 2
__
2
21 . 07
01 . 0 1 . 0 1 . 0 1
0 ) 01 . 0 ( 1
) 1 . 0 ( 2
) 1 . 0 ( 5
) 1 ( )
( ) ( P s P
k
k k
k
k
Example
• Therefore:
• Since B
C> 30KHz, AMPS would work without an equ alizer.
• GSM requires 200 KHz BW > B
C An equalizer wou ld be needed.
s
( ) 21 . 07 ( 4 . 38 ) 1 . 37 Spread
Delay
RMS
__ 2 2__
2
