Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If...

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Small-Scale Fading I PROF. MICHAEL TSAI 2011/10/27

Transcript of Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If...

Page 1: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

Small-Scale Fading I

PROF. MICHAEL TSAI

2011/10/27

Page 2: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

Multipath Propagation

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RX just sums up all Multi Path Component (MPC).

Page 3: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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

Page 4: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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.

Page 5: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

• 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

Page 6: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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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

Page 7: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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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

Page 8: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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

Page 9: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

Example: power delay profile

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From a 900 MHz cellular system in San Francisco

Page 10: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

Example: power delay profile

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In side a grocery store at 4 GHz

Page 11: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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

Page 12: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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

Page 13: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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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Page 14: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

Example: Time dispersion

parameters

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Page 15: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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

Page 16: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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.

Page 17: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

Typical RMS delay

spread values

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Page 18: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

Signal Bandwidth &

Coherence Bandwidth

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tf

Transmitted Signal

��

��: symbol period

��

��: signal bandwidth

t0

τ0 τ1 τ3τ2 τ4 τ5 τ6

�(�; ��

�� �1

��

����

Page 19: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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!

��

Page 20: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

Flat fading channel

��

f

��

f

�� " ��

TX signal

Channel

RX signal

� �∗

�� � ��

t0

τ0 τ1 τ3τ2 τ4 τ5 τ6

���; ��

��

t

��

��

No significant ISI

Page 21: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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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Page 22: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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 =

++++++==

Page 23: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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

Page 24: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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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

Page 25: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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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Page 26: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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&'()

Page 27: Small-Scale Fading I - 國立臺灣大學•The channels impulse response is given by: • If assumed time-invariant (over a small-scale time or distance): Multipath Channel Impulse

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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