statistical multipathmodel time varying impulse response

16
1 ! " # " $ ! % ! ! ! " & " # &" $ " φ ! ( ! ! )

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1

!

"# "$ !

%

!

!

!

" & "# & "$

' " φ

!

(

!! )

2

!

α φ

)

*+ ,-.

τ/

!

τ

τ !

01!

))(()(),(1

)( tettc n

N

n

tjn

n ττδατ ϕ −==

3

*

Time-invariant case

Time-varying case

2 /

2

1

-

2 ! !

!

4

! ! 2

≈ τ !

3

0 % !4)

- (! δ

ℜ=

=

)(

0

)(2 )()()(tN

n

tjn

tfj nc etetutr φπ α

""##$% $%

.+ /

5

2 /2

6 . 7 8 9 )

(! )

.φ φ :+( )

2

; 2 2 2

( )

0

( ) ( )cos( ( ))N t

I n nn

r t t tα φ=

=( )

0

( ) ( ) sin( ( ))N t

Q n nn

r t t tα φ=

=

5

% %

2φ<= >?$π@

θ

= +

. !

% % 2φ<= >?$π@

θ

2 :+

-- :+

2

λθτπττ θ /cos],2[cos5.)()( nDDprr vffEAAnnnQI

=Ω==

)(]2[sin5.)( ,, ττπτ θ QInnQI rrDprr AfEA −=Ω=

)2sin()()2cos()()( , τπττπττ crrcrr fAfAAQII

−=

)(]2[sin5.)( ,, ττπτ θ QInnQI rrDprr AfEA −=Ω=

)2sin()()2cos()()( , τπττπττ crrcrr fAfAAQII

−=

=0

6

& & %'%%'% = +

1

)2(5.)()( 0 τπττ Dprr fJAAQI

Ω==

Decorrelates over roughly half a wavelength

)]2(5[.)(

)]()([5.)(

0 τπ Dpr

crcrr

fJfS

ffSffSfS

I

II

Ω=

++−=

F

Used to generate simulation values fc+fDfc

Sr(f)

fc-fD

() ()

A

7

" *! " *! -%) ! %

6 9 !!

(+ $

5 !

034 !

!B C

!A*%

!) !)

3 !

8

))

! !

++ ))

0 ! !8

)

! 34 !

D 9

)

9

$ $

% )

! + 8 ! E FF )

6 γ?

6

%)* %)* G ! 3γ

6

ρ

. +

2

( 1) /( 2 )Z Dt e fρ ρ π= −

Zt1 t2 t3

10

%)* %)*

!! !!)

! )

!! !

; 4%

!

!+! !

! )

; !+!

! 4)

" " 3 ! +)(

9

2 2 2

= 4 + A

- ( :+ )

6 ! 34

2

11

,!,! . !

! % !

uB/1<<∆τ uB/1>>∆τ

ττττ ττττ1τ∆ 2τ∆

Narrowband Wideband

,!) ,!) 0

12

) ) τρ E6 τ,

( τ

9 (?

2 /*>τ# τ$H∆ @E2τ∆

τρττττρρρρ

,& ,&

5 5

= =

6 2τI

(

τ ρ)

13

" " 2τ∆E? E2τ 20 στ

2τ 2% % ))

-! AE#:0

0% + 2∆ E6>2τ @J? 2∆ E? +! !

τ

ττττTM

∆∆∆∆BC

F

" "

2 "τ

2

3

#

14

(, (, 5 $ !!σ % !..)

6 $ σ# $σ##? !!σ !σ

6+- 6

+ "%K& ' ?K& ! )

6 +

Frequency correlation function

* " * " ρ E6>2τ=0∆ @E6>2∆ @

A

0%

ρ ≥?)

- E#:A 0% 2∆ J?

2∆ E?

!∆ !

∆∆∆∆TC

ρ

ρρρρBD

F

15

5 ( !

- 6 "%K % ()

% )

)-* /

% * )%* ( !)

A6 ! "-K )-* ' #). + , +- E&)

!

.

Tc

Channel appears static for transmitted digital symbols

16

) )

DoubleFourier

Transform

∆t=0

Fourier Transform

Fourier Transform

∆f=0

" "

6 !

34

2

5 ! (! /

0)1) % !

% (