Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If...
Transcript of Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If...
Small-Scale Fading I
PROF. MICHAEL TSAI
2011/10/27
Multipath Propagation
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RX just sums up all Multi Path Component (MPC).
Multipath Channel
Impulse Response
t0
t
τ0 τ1 τ3τ2 τ4 τ5 τ6 τ(t0)
hb(t,τ)
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 ��)
�� = 0, 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
Channel Impulse Response
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t
t0
τ0 τ1 τ3τ2 τ4 τ5 τ6 τ(t0)
τ(t1)t1
t2 τ(t2)
t3
τ(t3)
hb(t,τ)
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
( ) ( ) ( ){ }[ ] ( )∑−
=
−+−=1
0
)(,)(2exp,,N
iiiicib ttttfjtath τδτφτπττ
( ) [ ] ( )∑−
=
−−=1
0
expN
iiiib jah ττδθτ
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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t
t0
τ0 τ1 τ3τ2 τ4 τ5 τ6 τ(t0)
τ(t1)t1
t2 τ(t2)
t3
τ(t3)
hb(t,τ)
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 the wireless
channell
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t
t0
τ0 τ1 τ3τ2 τ4 τ5 τ6 τ(t0)
τ(t1)t1
t2 τ(t2)
t3
τ(t3)
hb(t,τ)
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 the wireless
channell
Power delay profile
• To predict hB(ττττ) a probing pulse p(t) is sent s.t.
• Therefore, for small-scale channel modeling, POWER
DELAY PROFILE is found by computing the spatial
average of |hB(t;ττττ)|2 over a local area.
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� � ≈ (� − �)
� �; � ≈ � ℎ� �; ��
TXRX
�(�)
Average over
several measurements
in a local area
Example: power delay profile
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From a 900 MHz cellular system in San Francisco
Example: power delay profile
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In side a grocery store at 4 GHz
Time dispersion parameters
• Power delay profile is a good representation of the
average “geometry” of the transmitter, the receiver,
and the reflectors.
• To quantify “how spread-out” the arriving signals are,
we use time dispersion parameters:
• Maximum excess delay: the excess delay of the latest arriving MPC
• Mean excess delay: the “mean” excess delay of all arriving MPC
• RMS delay spread: the “standard deviation” of the excess delay of
all arriving MPC
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Already talked about this
Time dispersion parameters
• Mean Excess Delay
• RMS Delay Spread
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∑
∑
∑
∑==
kk
kkk
kk
kkk
P
P
a
a
)(
)(
2
2
__
τ
ττττ
First moment of the power delay profile
∑
∑
∑
∑==
kk
kkk
kk
kkk
P
P
a
a
)(
)( 2
2
22__
2
τ
ττττ
2____
2 )(ττστ −=
Second moment of the power delay profile
Square root of the second moment of
the power delay profile
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 noise
• No way to be aware of the “latest”
• Maximum Excess Delay (practical version):
• The time delay during which multipath energy falls to X dB below the maximum.
• This X dB threshold could affect the values of the time-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
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Example: Time dispersion
parameters
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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.
Recall this:
Transfer function
Coherence Bandwidth
• Bandwidth over which Frequency Correlation
function is above 0.9
• Bandwidth over which Frequency Correlation
function is above 0.5
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τσ50
1≈cB
τσ5
1≈cB
Those two are approximations derived from empirical results.
Typical RMS delay
spread values
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Signal Bandwidth &
Coherence Bandwidth
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tf
Transmitted Signal
��
��: symbol period
��
��: signal bandwidth
t0
τ0 τ1 τ3τ2 τ4 τ5 τ6
�(�; ��
�� �1
��
����
Frequency-selective fading channel
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��
f��
f
�� � ��
TX signal
Channel
RX signal
� �∗
t0
τ0 τ1 τ3τ2 τ4 τ5 τ6
���; ��
��
t��
�� " ��
These will become inter-
symbol interference!
��
Flat fading channel
��
f
��
f
�� " ��
TX signal
Channel
RX signal
� �∗
�� � ��
t0
τ0 τ1 τ3τ2 τ4 τ5 τ6
���; ��
��
t
��
��
No significant ISI
Equalizer 101
• An equalizer is usually used in a frequency-selective 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 from the estimated one, return to step 1.
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Example
0 1 2 3 4 5
-30dB
-20dB
-10dB
0dB
τ
P(τ)
Would this channel be suitable for
AMPS or GSM without the use of
an equalizer?
sP
P
kk
kkk
µτ
τττ 38.4
01.01.01.01
)01.0(0)1.0(1)1.0(2)1(5
)(
)(DelayExcessMean
__
=+++
+++===∑
∑
22222
2__
2 07.2101.01.01.01
0)01.0(1)1.0(2)1.0(5)1(
)(
)(s
P
P
kk
kkk
µτ
τττ =
++++++==
∑
∑
Example
• Therefore:
• Since BC > 30KHz, AMPS would work without an
equalizer.
• GSM requires 200 KHz BW > BC ����An equalizer would be needed.
sµττστ 37.1)38.4(07.21)(SpreadDelayRMS 22____
2 =−=−==
KHzBC 146)37.1(5
1
5
1BandwidthCoherence ====
µστ
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t
t0
τ0 τ1 τ3τ2 τ4 τ5 τ6 τ(t0)
τ(t1)t1
t2 τ(t2)
t3
τ(t3)
hb(t,τ)
Two main aspects
of the wireless
channell
Doppler Effect
• Difference in path lengths #$ � %&'() � *#+,-.)
• Phase change #/ �01#$
2�01*#+
2,-.)
• Frequency change, or Doppler shift,
3% �4
01#/
#+�*
2&'()
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Example
• Consider a transmitter which radiates a sinusoidal carrier frequency of 1850 MHz. For a vehicle moving 60 mph, compute the received carrier frequency if the mobile is moving
1. directly toward the transmitter.
2. directly away from the transmitter
3. in a direction which is perpendicular to the direction of arrival of the transmitted signal.
• Ans:
• Wavelength=5 ��
67�
8 �9
:;� �<� 0.162(@)
• Vehicle speed A = 60@�ℎ = 26.82C
�
1. DE =�F.:�
�.F�cos 0 = 160 JK
2. DE =�F.:�
�.F�cos L = −160(JK)
3. Since cosM
�= 0, there is no Doppler shift!
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3% =4
01#/
#+=*
2&'()
Doppler Effect
• If the car (mobile) is moving toward the direction of
the arriving wave, the Doppler shift is positive
• Different Doppler shifts if different ) (incoming angle)
• Multi-path: all different angles
• Many Doppler shifts ���� Doppler spread
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