Small-Scale Fading

Small-Scale Fading

PROF. MICHAEL TSAI

2017/10/30

Multipath Propagation

2

Multipath Channel Impulse Response

t₀

t

t

t

t

t

t

t

t

0)

h

(t,t)

An example of the time-varying discrete-time impulse response for a multipath radio channel

The channel impulse response when 𝑡 = 𝑡_# (what you receive at the receiver when you send an impulse at time 𝑡_#)

𝜏_# = 0, and represents the time the first signal arrives at the receiver.

Summed signal of all multipath components arriving at 𝜏_'~𝜏_')*.

Excess delay: the delay with respect to the first arriving signal (𝜏)

Maximum excess delay: the delay of latest arriving signal

Time-Variant Multipath Channel Impulse Response

4

t

t₀

t

t

t

t

t

t

t

0) t(t₁) t₁

t₂

t(t₂) t₃

t(t₃)

h

(t,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

( ) å

( ) [ { ( ) } ] ( )

=

- +

-

=

0

) ( ,

) ( 2

exp ,

,

i

i i

i c i

b

t a t j f t t t t

h t t p t f t d t

( ) å

[ ] ( )

=

- -

=

0

exp

N i

i i

i

b

a j

h t q d t t

Phase change due to different arriving time Additional phase change due to reflections

Amplitude change (mainly path loss) Summation over all MPC

6

t

t₀

t

t

t

t

t

t

t

0) t(t₁) t₁

t₂

t(t₂) t₃

t(t₃)

h

(t,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 the wireless

channel

7

t

t₀

t

t

t

t

t

t

t

0) t(t₁) t₁

t₂

t(t₂) t₃

t(t₃)

h

(t,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 the wireless

channel

Power delay profile

• To predict h

(t) 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 |h

(t;t)|

over a local area.

8

𝑝 𝑡 ≈ 𝛿(𝑡 − 𝜏)

𝑃 𝑡; 𝜏 ≈ 𝑘 ℎ

𝑡; 𝜏

TX 𝑝(𝑡) 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

10

Inside 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

11

Already talked about this

Time dispersion parameters

• Mean Excess Delay

• RMS Delay Spread

12

å å å

å =

=

k

k k

k k

k k k

k k

P P a

a

) (

) (

2 2 __

t t t t

t

First moment of the power delay profile

å å å

å =

=

k

k k

k k

k k k

k k

P P a

a

) (

)

(

2 2 2 __2

t t t t

t

__ 2 __

2

( t ) t

s

= -

Second moment of the power delay profile Square root of the second central 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

13

Example: Time dispersion parameters

14

Coherence Bandwidth

• Coherence bandwidth is a statistical measure of the range of frequencies over which the channel can be considered “flat”

à a channel passes all spectral components with

approximately equal gain and linear phase.

Coherence Bandwidth

• Bandwidth over which Frequency Correlation function is above 0.9

• Bandwidth over which Frequency Correlation function is above 0.5

16

s

50

» 1 B

s

5

» 1 B

Those two are approximations derived from empirical results.

Typical RMS delay spread values

17

Signal Bandwidth &

Coherence Bandwidth

18

f t

Transmitted Signal

𝑇₈

𝑇₈: symbol period

𝐵₈ 𝐵₈: signal bandwidth

t₀

t

t

t

t

t

t

t

𝑃(𝑡; 𝜏)

𝑇₈ ≈ 1 𝐵₈

𝐵_<

𝜎_>

Frequency-selective fading channel

19

𝐵_<

𝐵₈ f

f

𝐵

> 𝐵

TX signal

Channel

RX signal

×

= =∗

t₀

t

t

t

t

t

t

t

𝑃(𝑡; 𝜏)

𝜎_>

𝑇₈ t

𝑇

< 𝜎

These will become inter- symbol interference!

𝑇₈

Flat fading channel

𝐵_<

f 𝐵₈

f

𝐵

< 𝐵

TX signal

Channel

RX signal

×

= =∗

𝑇

> 𝜎

t₀

t

t

t

t

t

t

t

𝑃(𝑡; 𝜏)

𝜎_>

t 𝑇₈

𝑇₈ 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.

21

Example

0 1 2 3 4 5 -30dB

-20dB -10dB 0dB

t P(t)

Would this channel be suitable for AMPS or GSM without the use of an equalizer?

P s P

k

k k

k

k

µ

t t t

t 4 . 38

01 . 0 1 . 0 1 . 0 1

) 01 . 0 ( 0 ) 1 . 0 ( 1 ) 1 . 0 ( 2 ) 1 ( 5 )

( ) ( Delay

Excess

Mean

=

+ +

+

+ +

= +

=

= å

å

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

µ

t t t

t =

+ +

+

+ +

= +

= å

å

Example

• Therefore:

• Since B

> 30KHz, AMPS would work without an equalizer.

• GSM requires 200 KHz BW > B

à An equalizer would be needed.

µ s t

t

s

( ) 21 . 07 ( 4 . 38 ) 1 . 37 Spread

Delay

RMS

__2

- = - =

=

=

s KHz

B

146

) 37 . 1 ( 5

1 5

Bandwidth 1

Coherence = = = =

µ

s