# Ece562 Channel Modeling Slides

Embed Size (px)

### Text of Ece562 Channel Modeling Slides

• 8/13/2019 Ece562 Channel Modeling Slides

1/40

Multipath Channel between pair of Tx & Rx Antennas

v

d

X

Y

Mobile

BS Antenna

Multipath channel seen at location (d,)

2

• 8/13/2019 Ece562 Channel Modeling Slides

2/40

Distance between Tx & Rx Antennas

C h a n n e

l G

a i n ( d B )

3

• 8/13/2019 Ece562 Channel Modeling Slides

3/40

Channel Model for Point-to-Point Communications

s(t) h(t) y(t)

w(t)

r (t)f c f c f

|S (f )|

Real bandpass channel model

s(t) h(t) y(t)

w(t)

r (t) 0 f |S (f )|

Complex baseband channel model 4

• 8/13/2019 Ece562 Channel Modeling Slides

4/40

Passband-Baseband relationships (Proakis)

Passband to Baseband Baseband to Passband

S (f ) = S + (f + f c ) = 2u(f + f c )S (f + f c ) S (f ) = S ( f f c )+ S (f f c ) 2s(t) = s+ (t)ej 2f c t = 1 2 [s(t) + j s(t)]ej 2f

c t s(t) = Re[ 2s(t)ej 2f c t ]

h(t) = 1 2 h+ (t)ej 2f c t h(t) = 2Re[ h(t)ej 2f c t ]

w(t) = w+ (t)ej 2f c t = 1 2 [ w(t) + j w(t)]ej 2f c t w(t) = Re[ 2w(t)ej 2f c t ]

5

• 8/13/2019 Ece562 Channel Modeling Slides

5/40

Properties of Complex Baseband Additive Noise w (t )

If w(t) is zero mean, Gaussian, then w(t) is zero mean, complex Gaussian

Let w(t) = wI (t) + jwQ (t). If w(t) is wide sense stationary (WSS), thenwI (t) and wQ (t) are jointly WSS, and

Rw I ( ) = Rw Q ( ) , and Rw I w Q ( ) = Rw Q w I ( )A complex process with this property is called proper complex Dene ACF of w(t) by

Rw ( ) = E[ w(t + )w(t)]

Then from proper complex property

Rw ( ) = 2 Rw I ( ) + j 2Rw Q w I ( )

6

• 8/13/2019 Ece562 Channel Modeling Slides

6/40

Proper Complex Processes (Neeser & Massey )

Let Y = Y I + j Y Q be a complex random vector with

Y I = E[( Y I m Y I )(Y I m Y I )] , Y Q = E[( Y Q m Y Q )(Y Q m Y Q )]

Y I Y Q = E[( Y I m Y I )(Y Q m Y Q )] , Y Q Y I = E[( Y Q m Y Q )(Y I m Y I )]

Complex covariance Y = E (Y m Y )(Y m Y ) = Y I + Y Q + j Y Q Y I Y I Y Q

Complex pseudo-covariance

Y = E (Y m Y )(Y m Y ) = Y I Y Q + j Y Q Y I + Y I Y Q

7

• 8/13/2019 Ece562 Channel Modeling Slides

7/40

Denition 1. Y is a proper complex vector if Y = 0 , i.e.

Y I = Y Q and Y Q Y I = Y I Y Q

For proper complex Y ,

Y = 2 Y I + j 2 Y Q Y I

The scalar case: If Y is proper complex scalar, then Y I and Y Q areuncorrelated and

2Y = E[

|Y

mY

|2] = 22Y I = 2

2Y Q

If Y is proper and Gaussian it is said to be proper complex Gaussian (PCG) or

circularly complex Gaussian8

• 8/13/2019 Ece562 Channel Modeling Slides

8/40

Useful results on PCG random vectors

Result 1. If Y is a PCG vector, the pdf of Y is given by

pY (y ) := pY I Y Q (y I , y Q )

= 1n | Y |

exp (y m Y ) 1Y (y m Y )

The pdf of Y has circular symmetry.

Notation: Y CN (m Y , )Result 2. If Y is PCG, then Z = AY + b is also PCG.The circular property is preserved under linear transformations.

Result 3. (Central Limit Theorem). If {Y k } is a sequence of independentproper complex random vectors (not necessarily Gaussian), then the sum

k Y k (after appropriate normalization) converges to a PCG vector.

9

• 8/13/2019 Ece562 Channel Modeling Slides

9/40

Proper Complex and PCG processes

Covariance and pseudo-covariance functions of Y (t) = Y I (t) + jY Q (t):C Y (t + , t ) = E [( Y (t + ) mY (t + ))( Y (t) mY (t))]C Y (t + , t ) = E [( Y (t + ) mY (t + ))( Y (t) mY (t))]

Denition 2. {Y (t)} is proper complex if C Y Y (t + , t ) = 0 , i.e.C Y I (t + , t ) = C Y Q (t + , t ) and C Y Q Y I (t + , t ) = C Y I Y Q (t + , t )

For proper complex {Y (t)},C Y (t + , t ) = 2 C Y I (t + , t ) + j 2C Y Q Y I (t + , t)

Denition 3. A proper complex process {Y (t)} is PCG if, for all n, and allt1 , t 2 , . . . , t n , the samples Y (t1), Y (t2), . . . , Y (tn ) are jointly PCG .

Result 4. If a PCG process

{Y (t)

} is passed through a linear system, the

output is PCG as well.10

• 8/13/2019 Ece562 Channel Modeling Slides

10/40

Part II: From Point-to-Point CommunicationsModel to Mobile Communications Channel Model

11

• 8/13/2019 Ece562 Channel Modeling Slides

11/40

Complex Baseband Model for Point-to-Point

Communications

s(t) h(t) y(t)

w(t)

r (t)

If bandpass noise {w(t)} is AWGN with PSD N 0 / 2, then baseband noise{w(t)} is a PCG white process with

Rw ( ) = E[ w(t + )w(t)] = N 0 ( )

Rw I w Q ( ) = Rw Q w I ( ) = 0 = {wI (t)}; {wQ (t)} independent Rw I ( ) = Rw Q ( ) = 12 Rw ( ) = N 02 ( )

S w I (f ) = S w Q (f ) = 12 S w (f ) =

N 02 for all f

12

• 8/13/2019 Ece562 Channel Modeling Slides

12/40

Mobile Comm. Channel Model for link between Tx/Rx Pair

v

d

X

Y Mobile

BS Antenna

Causal LTI system corresponding to multipath prole at (d,)

s(t) hd, ( ) y(t)

w(t)

r (t)

13

• 8/13/2019 Ece562 Channel Modeling Slides

13/40

Multipath Prole and Channel Impulse Response

At location (d,), n-th path connecting Tx and Rx antennas has

amplitude gain of n (d,) delay of n (d,) carrier phase shift of n (d,) = 2f c n (d,) + constant.

Thus y(t) =n

n (d,) ej n (d, ) s(t n (d,))

= hd, ( ) = n n (d,) e

j n (d, )

( n (d,))

As MS moves, (d,) varies with time = Time varying channel

14

• 8/13/2019 Ece562 Channel Modeling Slides

14/40

Two scales of variation

Small scale variations movements of the order of few carrier wavelengths multipath prole roughly constant # paths, strengths, delays channel variations due to phase differences in paths average power gain in vicinity of (d,) is G(d,) =

n

2n

typical values: f c = 1 G Hz = c = 0 .3 m.

Large scale variations variations in G(d,) that result from changing multipath prole scale of distance between objects in environment typical values: 10s of meters (outdoor)

15

• 8/13/2019 Ece562 Channel Modeling Slides

15/40

Separation of scales

Small and large scale variations separated s(t) y(t)

g(d,)

hd, ( )x(t)

hd, ( ) is hd, ( ) normalized to have average power gain of 1, i.e.hd, ( ) =

n n (d,) ej n (d, ) ( n (d,))

with

n

2n (d,) = 1

amplitude gain (real) g(d,)G(d,) = g2(d,)

16

• 8/13/2019 Ece562 Channel Modeling Slides

16/40

Part III: Small Scale Variations in Gain Basics

17

• 8/13/2019 Ece562 Channel Modeling Slides

17/40

Small Scale Variations

Small scale variations are captured in hd,hd, ( ) =

n

n (d,) ej n (d, ) ( n (d,))

where { n (d,)} are normalized so that n 2n (d,) = 1 As (d,) changes with t, channel becomes time varying:h(t; ) := hd ( t ) , ( t ) ( ) =

n

n (t) ej n ( t ) (

n (t))

Assume g(d,) is constant over small scales(t) y(t)

g

h(t; )x(t)

y(t) =

0

h(t; )x(t )d

18

• 8/13/2019 Ece562 Channel Modeling Slides

18/40

Doppler shifts in phase

For movements of the order a few wavelengths { n (t)} and { n (t)} are roughly constant

BUT n (t) = 2f c n (t) + const. changes signicantly Doppler shift is function of angle n of path w.r.t. velocity vector

vv tn

v t cos n

n (t + t ) n (t) 2f c v t cosn

c

= 2v t cosn

c

= 2 f max t cosn

f max = v/ c is the maximum Doppler frequency

19

• 8/13/2019 Ece562 Channel Modeling Slides

19/40

Delay Prole of Channel

n and n are roughly independent of t, i.e. time variations are mainly dueto changes in n= h(t; ) =

n

n ej n ( t ) ( n )

0 DS

|h(t; )|

0 (LOS)

n

n20

• 8/13/2019 Ece562 Channel Modeling Slides

20/40

21

• 8/13/2019 Ece562 Channel Modeling Slides

21/40

Denition 4. The quantity DS = max n min n is called the delay spread of the channel.

w.l.o.g. assume min n = 0 . Then max n = DS and

y(t) = DS

0h(t; )x(t )d

If passband bandwidth of s(t), W 1 DS , then x(t) is roughly constantover time intervals of order of DS

= y(t) x(t) DS

0h(t; )d = x(t)

n

n ej n ( t ) = x(t)E (t)

= h(t; ) E (t) ( )22

• 8/13/2019 Ece562 Channel Modeling Slides

22/40

s(t) y(t)x(t)

g E (t)

E (t) = DS

0h(t; )d =

n

n ej n ( t )

23

• 8/13/2019 Ece562 Channel Modeling Slides

23/40

Purely Diffuse (no LOS) Scattering Rayleigh fading

v

X

Y MS

BS

No LOS path or no single path dominates all other paths If we model {n } as independent random Unif[0, 2], then {E (t)} is zero

mean process

The process { n ej n ( t )} is proper complex If number of paths is large, by CLT (Result 3),

{E (t)

} is a Proper Complex Gaussian (PCG) random process

24

• 8/13/2019 Ece562 Channel Modeling Slides

24/40

First Order Statistics for Purely Diffuse Scattering

For xed t, E (t) = E I (t) + jE Q (t) is PCG r.v. withE |E (t)|2 =

n

2n = 1

= E I (t) and E Q (t) are independent N (0, 1/ 2) r.v.s

Envelope and phase(t) =

|E (t)

| =

E 2I (t) + E 2Q (t) , and (t) = tan 1

E Q (t)

E I (t)

For xed t, (t) and (t) are independent, (t) has a Rayleigh pdf and(t) is Unif[0, 2].

0 1 2 30

0.2

0.4

0.6

0.8

1

p (x) = 2 xex2

u(x)

25

• 8/13/2019 Ece562 Channel Modeling Slides

25/40

Autocorrelation function of {E (t )}RE (t, t + ) = E [ E

(t)E (t + )]

= En

n ej n ( t )i

i ej i ( t + )

=n

2n E ej [ n ( t + ) n ( t )]

n 2n e

j 2f max cos n

= RE ( )

{E (t)} is approximately stationary

In-phase and Quadrature components have correlation functions

RE I ( ) = RE Q ( ) = 12

Re {RE ( )} = 12

n

2n cos(2f max cosn )

RE Q E I ( ) =

RE I E Q ( ) =

1

2Im

{RE ( )

} =

1

2 n 2n sin(2f max cosn )

26

• 8/13/2019 Ece562 Channel Modeling Slides

26/40

Isotropic Scattering Environment

Isotropic scattering may be approximated by continuum of paths with p() =

12

(uniform)

to getRE ( ) =

12

ej 2f max cos d = J 0(2f max )

where J 0(

) is the zeroth order Bessel function of the rst kind 

J 0(x) = 12

cos(x cos )d

AlsoRE I ( ) = RE Q ( ) =

12

RE ( )

RE Q E I ( ) = RE I E Q ( ) = 0

27

• 8/13/2019 Ece562 Channel Modeling Slides

27/40

Accuracy of Bessel Approximation

0 0.5 1 1.5 2 2.5 30.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5BesselN=8N=16

f max or c

R E

I (

) o r

R E I

( )

Even for a few uniformly distributed discrete paths (N = 8 , 16)we get an ACF that is well approximated by a Bessel function

28

• 8/13/2019 Ece562 Channel Modeling Slides

28/40

Fading process is fundamentally a function of location To get fading as function of time we assumed MS is traveling along

constant velocity vector v.

Distance relative to location at time 0 equals vt Fading process in is given by

E ( ) = E (/v )

Fading ACF over distance variable is given byRE () = E E ( + ) E ( )

= RE

v

= RE

f max c

=

p() e

j 2 cos c d

= J 0

2 c

29

• 8/13/2019 Ece562 Channel Modeling Slides

29/40

Coherence distance c is measure of distance separation over which E remains roughly unchanged

c can be dened more precisely in terms of the ACF as (say): c = largest such that |RE () | > 0.9RE (0) = 0 .9

Coherence time T c = cv For isotropic Rayleigh fading

c 0.1 c , and T c 0.1f max .

Fading is said to be slow if T c T s , where T s is the symbol period

30

• 8/13/2019 Ece562 Channel Modeling Slides

30/40

Scattering with LOS Component

If LOS (specular) path with parameters 0 , 0 and 0(t), thenE (t) = 0 ej 0 ( t ) + 1 20 E (t)

where {E (t)} is zero mean PCG, Rayleigh fading process Note: {E (t)} is zero-mean process, but not Gaussian since LOS

component dominates diffuse components in power

Rice Factor: =

power in the specular componenttotal power in diffuse components

= 201 20

From the denition of it follows that

0 =

+ 1, and 1 20 =

1( + 1)

31

• 8/13/2019 Ece562 Channel Modeling Slides

31/40

For xed t, the envelope has Ricean pdf (Rice ) p (x) = 2 x( + 1) I 0 2x ( + 1) exp x2( + 1) u(x)

where I 0() is zeroth order modied Bessel function of 1st kind 

I 0(y) = 12

exp( y cos )d .

0 1 2 30

0.5

1

1.5

2

RayleighRicean =1Ricean =5Ricean =10

32

• 8/13/2019 Ece562 Channel Modeling Slides

32/40

v = 72 km/hr = 20 m/s; f c = 900 MHz c = 1/3 m f max = 60 Hz

0 50 100 150 200 25020

15

10

5

0

5

10

15

t (ms)

C h a n n e

l G a

i n ( d B )

RayleighRicean =5Ricean =10

33

• 8/13/2019 Ece562 Channel Modeling Slides

33/40

34

• 8/13/2019 Ece562 Channel Modeling Slides

34/40

s(t) y(t)

g

h(t; )x(t)

s(t) has passband bandwidth of W

If W 1 DS , then fading is at

If W > 1 DS , then fading is frequency selective

From Paulraj et al 

Urban 5 s

Mall 0.3 sIndoors 0.1 s

s(t) has baseband bandwidth of W/ 2 = x(t) has bandwidth W/ 2 By Sampling Theorem (sinc interpolation formula)

x(t ) =

=

x (t /W ) sinc[W ( /W )]35

• 8/13/2019 Ece562 Channel Modeling Slides

35/40

y(t) =

DS

0h(t; ) x(t

)d

=

= x (t /W )

DS

0h(t; ) sinc [W ( /W )] d

E (t) = DS

0h(t; ) sinc [W ( /W )] d

E (t) 0 for < 0 and for /W > DS . If L =

DS W then

y(t) L 1

=0

x (t /W ) E (t)

= h(t; ) L 1

=0

E (t) ( /W )

36

• 8/13/2019 Ece562 Channel Modeling Slides

36/40

Tapped Delay Line Model

... ...x(t) 1W 2W W L 1W

E 0(t) E

1(t) E

2(t) E (t) E L

1(t)

y(t)

Recall that h(t; ) =n

n ej n ( t ) ( n )

= E (t) = DS

0h(t; ) sinc [W ( /W )] d

=n

n ej n ( t ) sinc[W ( n /W )]

37

• 8/13/2019 Ece562 Channel Modeling Slides

37/40

0 DS

|h(t; )|

0 (LOS)

n

n

38

• 8/13/2019 Ece562 Channel Modeling Slides

38/40

Statistical Model for {E (t )} If {E (t)} includes a dominant LOS component, then it has Riceanenvelope; else it has Rayleigh envelope Autocorrelation function of {E (t)}

RE ( ) = E [( t)E (t + )E ]

n

2n ej 2f max cos n sinc2 [W ( n /W )]

Check: If the fading is at, i.e. DS 1W , E (t) 0 for = 0 , andRE 0 ( )

n

2n ej 2f max cos n sinc2 [W n ]

n

2n ej 2f max cos n RE ( )

Nature of RE ( ) depends on angular location and spread of pathscontributing to tap

39

• 8/13/2019 Ece562 Channel Modeling Slides

39/40

Cross-correlation Between Taps

RE k E ( ) = E [ E k (t + )E

(t)]

n 2n e

j 2f max cos n sinc[W ( n

/W )] sinc[W ( n

k/W )]

Frequency diversity depends on cross-correlation between taps

Fading in neighboring taps can be highly correlated

If tap delays are chosen to match cluster centers in delay prole, then tapswill be less correlated

Cluster model for channel

h(t; ) L c 1

=0

E (t) ( )

where Lc is number of clusters; delay of cluster 40

• 8/13/2019 Ece562 Channel Modeling Slides

40/40

References

 J. G. Proakis. Digital Communications . Mc-Graw Hill, New York, 3rd edition, 1995.

 F. D. Neeser and J. L. Massey. Proper complex random processes with applications to information theory. IEEE Trans. Inform. Th. , 39(4),July 1993.

 S. Rice. Statistical properties of a sine wave plus noise. Bell Syst. Tech. J. , 27(1):109157, January 1948. M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions . Dover, New York, 1964.

 A. J. Paulraj and C. B. Papadias. Space-time processing for wireless communications. IEEE Signal Processing Magazine , pages 4983,November 1997.

 P. A. Bello. Characterization of randomly time-variant linear channels. IEEE Trans. Commun. Systems , pages 360393, December 1963.

 R. Clarke. A statistical theory of mobile radio reception. Bell Syst. Tech. J. , 47(6):9571000, July-August 1968. M. Gudmundson. Correlation model for shadow fading in mobile radio systems. Electron. Lett. , 27(23):21452146, 1991.

 W. C. Jakes, Jr. Microwave Mobile Communications . Wiley, New York, 1974.

 E. Wong and B. Hajek. Stochastic Processes in Engineering Systems . Springer-Verlag, New York, 1985.

 D. Tse and P. Viswanath. Fundamentals of Wireless Communication . Cambridge University Press, 2005.

41 ##### SIMULATION WITH THE CUK TOPOLOGY ECE562: Power Electronics ... · SIMULATION WITH THE CUK TOPOLOGY ECE562: Power Electronics I COLORADO STATE UNIVERSITY Modified in Fall ... Unlike
Documents ##### Redmond Channel Partner Magazine (RCPMag) (Glossary Definition) (Slides)
Technology ##### ECE562 Power Electronics Schedule and Grading · 1 ECE562 Power Electronics Schedule and Grading Class Time: Tuesday and Thursday 5:30 – 6:45 PM in B105 (Engineering B …
Documents ##### SIMULATION OF A SERIES RESONANT CIRCUIT ECE562: Power ... resonant.pdf · SIMULATION OF A SERIES RESONANT CIRCUIT . ECE562: Power Electronics I . COLORADO STATE UNIVERSITY . Modified
Documents Documents