M A T C H E D F I L T E R BOUNDS F O R F A S T F A D I N G RICIAN C H A N N E L S by VAIBHAV DINESH B. E. (Electronics and Communications), Guru Jambheshwar University, India, 1998 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

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7 1

Appendix C : Solving the Integral

74

Appendix D: Calculating the Errors in Amplitude and Phase estimation

77

Bibliography

79

N

iv

L i s t of Figures Figure 1.1

Received signal corresponding to slow and fast fading channels

2

Figure 2.1

Continuously Dispersive and Discrete Path channel outputs

9

Figure 2.2

Linear System model

Figure 2.3

3-D view of the continuous-time impulse response, \z(t, x)|

14

Figure 2.4

3-D view of the discrete-time impulse response, \z(t, x)\

17

Figure 2.5

Power Spectrum of an unmodulated CW carrier

19

Figure 3.1

System Model

22

Figure 3.2

The Transformed System

23

Figure 4.1

BER as a function of f

Figure 4.2

BER as a function of the Rician factor and f

47

Figure 4.3

Effect of the Rician factor for f

47

Figure 4.4

Effect of pulse shape in fast fading Rician channel, K

Figure 4.5

Channel averaging ability of different pulse shapes

Figure 4.6

Effect of the roll-off factor on the system performance, K

Figure 4.7

Two-beam frequency selective fading, K

Figure 4.8

Three-beam frequency selective fast fading, f

Figure 4.9

Three-beam frequency selective very-slow fading, a = 0

53

Figure 4.10

Three-beam frequency selective fast fading, f

54

Figure 4.11

Three-beam frequency selective very-slow fading, a > 0

55

Figure 4.12

Three-beam fast fading, K

56

Figure 4.13

Three-beam slow fading, K

= 0 dB

56

Figure 4.14

Three-beam fast fading as a function of K; [a;p] :[0.0;0.0],[1.0;1.0]

57

12

N

for the Rician channel, K

= 0 dB

dB

N

N

= 1.28 dB

= 0 dB

N

dB

dB

= 0 dB

= 0 dB N

= 0 dB and f

N

48 49

dB

dB

46

= 0.64 and a = 0

= 0.64 and a > 0

= 0.64

50 51 52

Figure 4.15

Three-beam fast fading as a function of K; [a;p] :[0.0;1.0],[1.0;0.0]

Figure 4.16

Three-beam slow fading as a function of K; [a;p] :[0.0;0.0],[1.0;1.0] ....58

Figure 4.17

Three-beam slow fading as a function of K; [a;p] :[0.0;1.0],[1.0;0.0] ....59

Figure 4.18

Performance degradation in a flat fading Rician channel

60

Figure 4.19

Performance degradation in a flat fading Rayleigh channel

61

Figure 4.20

Effect of mis-match in phase on the BER in a Rician channel

62

Figure 4.21

Effect of mis-match in phase on the BER in a Rayleigh channel

62

Figure 4.22'

Effect of mis-match in amplitude on the BER in a Rician channel

63

Figure 4.23

Effect of mis-match in amplitude on the BER in a Rayleigh channel

64

Figure D. 1

Resolution of the fading sample with/without error

77

vi

58

Glossary Acronyms

ASK AWGN

Amplitude Shift Keying Additive White Gaussian Noise

BER

Bit Error Rate

BFSK

Binary Frequency Shift Keying

BPSK

Binary Phase Shift Keying

BS

Base Station

CD

Continuously Dispersive

CW

Continuous Wave

DP

Discrete Path

EFD

Intrinsic Frequency Diversity

ISI

Inter-Symbol Interference

LOS

Line Of Sight

MFB

Matched Filter Bound

MS

Mobile Station

MSE

Mean Square Error

OFDM

Orthogonal Frequency Division Multiplexing

PAM

Pulse Amplitude Modulation

PSK

Phase Shift Keying

RRC

Root Raised Cosine

SNR

Signal to Noise Ratio

vii

Symbols

a

roll-off factor of the root raised cosine pulse

A

transmitted signal amplitude

c

speed of light

D

rms

rms delay spread

E

b

bit energy

f

carrier frequency

c

Af

Coherence bandwidth

f

d

maximum Doppler frequency

f

N

maximum normalized Doppler rate

f(t)

received pulse shape corresponding to the diffused channel

f (y)

pdf of Y

h(t)

impulse response of the receive filter

J (.)

zeroth order Bessel function of the first kind

K

Rician factor of the channel

c

Y

0

K

Rician factor (in dB)

N

total number of paths in the discrete channel model

N

noise component at the matched filter output

dB

N

power spectral density of AWGN

n(t)

AWGN noise

P

probability of bit error

p(t)

transmitted pulse shape

Q(v, w)

Marcum's (2-ftmction

r(t)

received (faded and noisy) signal

r {t)

received signal (for a CD channel)

r (t)

received signal (for a DP channel)

0

b

cd

dp

Rf(tj, t )

autocorrelation of the process f(t)

fad^

faded signal at the channel output

r (t)

received signal without noise (for the DP channel)

S(f)

power spectrum density of the Rayleigh fading process

2

r

s

viii

s(t)

transmitted signal

SfadW

r e c e i v e d pulse shape

T

r e c e i v e d s i g n a l duration

Ar

s a m p l i n g interval

Af„

Coherence time

T

delay spread o f the c h a n n e l

T

pulse duration

v

m a x i m u m velocity of M S

c

M

p

v

(z)

Whittaker function

X

m a t c h e d filter output after s a m p l i n g

x(t)

m a t c h e d filter output

Y

s i g n a l c o m p o n e n t at the m a t c h e d filter output

z(t, x)

i m p u l s e response o f a linear c h a n n e l ( g e n e r a l i z e d notation)

cd(t> ) x

z

i m p u l s e response o f a C D channel

ray(t' )

i m p u l s e response o f the d i f f u s e d c h a n n e l

rice(t> )

i m p u l s e response o f the R i c i a n c h a n n e l

z {t)

c o m p l e x f a d i n g process c o r r e s p o n d i n g to the «th b e a m

x

z

x

z

n

ix

Greek Symbols

diffused beam strength

a

2

a

attenuation corresponding to the direct/specular path

0

strength of the nth faded beam

«„

nth path attenuation characteristic function 8(r)

delta function

r(z)

Gamma function

„C) • e~ ° m

• s(t-x transmitted at time t , may arrive in con2

2

junction with another component of phase (pi transmitted at time tj, resulting in constructive or

Chapter 2 The Wireless Channel

10

destructive interference between the two. We can characterize signal dispersion in terms of Delay Viewed in the frequency domain, signal dispersion is expressed in terms of Coher-

Spread (T ). M

ence Bandwidth (Af

less than Af

c

c

) which implies that the signal components having frequency separation

are treated alike by the channel. It can be shown [12] that Af = c

M

. Based on

this, we can further classify fading as: • Flat or frequency non-selective fading: When A f

c

is large relative to the signal bandwidth,

the channel is said to be flat. Viewed in the time domain, the channel ideally causes no delay spread in the transmitted signal, i.e. T

M

= 0.

• Frequency Selective fading: If the coherence bandwidth is small compared to the signal bandwidth, the channel is said to be frequency selective. The attendant delay spreading causes inter-symbol interference (ISI). Frequency-spreading of the signal is a direct consequence of the spatial changes of the receiver relative to its surroundings, accounting for propagation path changes. As a result, the channel offers a time-variant response. This is characterized by the Coherence Time At , which is c

defined as the duration over which the channel's impulse response can be considered as timeinvariant. Coherence time is inversely proportional to the Doppler Spread, At

c

«*= 4- [12]. The id

Doppler spread is a function of the MS velocity, v, and signal wavelength, X, and is a measure of the spectral broadening of the signal / „ = /Vcos(e„)

where f

(2.2.3)

is the Doppler shift [15] for the n wave arriving at an angle 0 with respect to the th

n

n

Chapter 2 The Wireless Channel

11

direction of motion of the MS and / . (f

d

= ^) is the maximum Doppler frequency.

Depending on the channel's coherence time with respect to the signal duration, the Doppler spreading results in: • Fast Fading: The channel exhibits a rapidly time-varying impulse response relative to the symbol duration and the signal experiences spectral broadening. Fast fading often causes the occurrence of frequent deep fades in the received signal. • Slow Fading: When the channel's coherence time is long compared to the symbol duration, the channel is considered as slowly fading. Equivalently, the Doppler spread is small. It is important to understand that the time-spreading and frequency-spreading are independent phenomenons. A wireless channel may exhibit any combinations of Flat/Frequency Selective and Fast/Slow fading. 2.3 Channel Model and its Statistical Characterization 2.3.1 The Linear Channel In terms of the complex lowpass equivalent form of a bandpass system [12], we can express the impulse response of the channel as z(t, x), which represents the response of the channel at time t to an impulse applied at time t-x. In this section an expression for z(t, x) applicable for both flat and frequency selective fading channels is obtained. In doing so, we follow an approach similar to that of [12]. The general expression of the impulse response is given as z(t, x) = X « „ ( 0 • e~

J2nfMt)

(2.3.1)

• 6(T-T„(0)

n

where f is the carrier frequency of the bandpass signal; oc„(0 and x (t) are the attenuation and c

n

12

Chapter 2 The Wireless Channel

delay of the n ray at time t, respectively. th

Equation (2.3.1) is a general expression from the point of view that it accounts for both Rician and Rayleigh fading channels and accommodates for all types of fading discussed in Section 2.2.2. Since the specular path is a non-fading component of the channel, we can implement the impulse response of a Rician fading channel, z {t, x), by taking a constant attenuation, a , and rice

0

associated delay, x , for thefirstarriving ray 0

z (t, x) = 0 C Q • erice

j 2 n f

^ • 8(t - 1 ) 0

+ Xa„(0 • e-

]lKfMt)

• 8(t - x (t)) n

(2.3.2)

Figure 2.2 shows the block diagram representation of the response r(f) of a fading channel z(t, x), corrupted with noise n(t), to an input s{t).

Channel

n{t)

AWGN Figure 2.2 Linear System model The signal r(t) can be written as

r(t) = J z(t,x)s(t-x)dx

+ n(t)

(2.3.3)

Substituting (2.3.1) in (2.3.3) and neglecting the noise term for the moment, the received signal,

Chapter 2 The Wireless Channel

13

r {t), is given as s

•',(*) = 5>«C) • e~

J2KfMt)

.

• s(t-x (t)) n

(2.3.4)

n

T h e a u t o c o r r e l a t i o n f u n c t i o n o f z(t, x) i s defined as [12]

SR (Af z

JCCJ, x )) = | • E[z*(t, Tj) • z(f + A / , x ) ] . 2

(2.3.5)

2

A s s u m i n g u n c o r r e l a t e d s c a t t e r i n g , i.e. i n d e p e n d e n c e o f attenuation a n d phase s h i f t s a s s o c i a t e d w i t h different path delays, a n d letting

At - 0, (2.3.5) c a n be re-written as

SR (0;(T!, x )) = 9t(x,) • 8(T! - x ) . z

2

(2.3.6)

2

T h e f u n c t i o n SR(x), k n o w n as the d e l a y p o w e r s p e c t r u m o f the c h a n n e l , represents the average p o w e r output o f the c h a n n e l as a f u n c t i o n o f the t i m e d e l a y x . I n u r b a n e n v i r o n m e n t s , 9t(x) c a n be a p p r o x i m a t e d b y the c o n t i n u o u s one-sided e x p o n e n t i a l f u n c t i o n [13], i.e.

SR(x) = ^ - • e x p f - - M rms

where

D

rms

^ rms'

x >0

is the r m s d e l a y spread. In some p r e v i o u s w o r k s

(2.3.7)

[6], the class o f t w o - s i d e d c o n t i n u -

ous spectra i s c o n s i d e r e d b y a s s u m i n g the G a u s s i a n s p e c t r u m . H o w e v e r , f o r s i m p l i c i t y , w e c o n f i n e o u r a n a l y s i s t o o n e a n d t w o b e a m cases ( D o u b l e - S p i k e s p e c t r u m

(2.3.8)) w h i c h

can easily be

extended t o a s y s t e m w i t h N , ( N > 2), beams b y a p p l y i n g a s i m i l a r p r o c e d u r e .

= ^(8(x) + 8 ( x - a ) )

(2.3.8)

A snapshot o f the i m p u l s e response w i t h c o n t i n u o u s o n e - s i d e d e x p o n e n t i a l d e l a y p o w e r s p e c t r u m is s h o w n i n F i g u r e

2.3.

T h e f i g u r e is p l o t t e d a s s u m i n g

D

rms

= 0.1 sec a n d

a Doppler

14

Chapter 2 The Wireless Channel

frequency of 10 Hz. One needs to be careful while comparing Figure 2.1, pertaining to the response of the fading channel to an impulse, with the three-dimensional plot of the impulse response, \z(t, x)|; since z(t, x) is a function of two time variables, t and x, whereas the response of the channel is a function of t only. It is, however, possible to trace the channel's response to an impulse from the three-dimensional plot of it's impulse response.

0.2 x [sec]

0.3 0.4

0

Figure 2.3 3-D view of the continuous-time impulse response, \z(t, x)| It should be noted that the terms 'one beam' and 'two beam' models do not simply refer to the arrival of one and two rays at the mobile; rather each beam may be interpreted as a collection of several rays with varying path-delays closely distributed about the mean delays x, and x , 2

respectively [14]. Thus, on splitting the second term in (2.3.2) into two parts (summation series) each reflecting the case of a single Rayleigh faded beam and denoting their respective indices by n and n , {

2

Chapter 2 The Wireless Channel

15

with each ray in the group having an associated delay of x„ (0 and ^(0, respectively, where x (0 Bi

=

x,+Ax (0,

AT

Bi

B i

(o«r

p

(2.3.9)

T is the transmitted signal block/pulse duration, we obtain p

z {t,i) = a -e~

-Hi-* )+z (t,T)

+ z (t,%)

J2nfcXo

rice

0

0

l

(2.3.10)

2

where z {t,x) = e

J

/c

x

'X n,(0^ a

-/27C/-T,

z (r,t) = e

;

/ c 2

1

1

-7'2TC/ AT C

X « „ *2W ~

g

n

n |

(2.3.11)

(0

•8(T-(f +

2

2

•8(T-(f +AT (0))

2

AT (0)). ll2

2

An upper-bound on At (r) can be obtained by realizing [14] that the greatest change in n

the path length occurs when the mobile is directly in line with the direction of arrival of the signal. Over the duration T , this would correspond to a change in path-length of v • T for a mobile p

moving at velocity v. Therefore the change in delay (Ax (t)) n

associated with the change in

path-length would be v • T /c, where c is the speed of light. Since (Ax (t)) n

max

is much smaller

than the pulse duration, we can drop the Ax (t) factor from the delta functions in (2.3.11). n

However, the same cannot be neglected in the exponential terms due to the large value of f . c

Assuming that each of the two beams consists of a large number of independent rays and

Chapter 2 The Wireless Channel

16

the phases of the rays are uniformly distributed from 0 to 2n [15], the Central Limit Theorem [16] can be applied to (2.3.11). Taking Xj = 0 and x = a, (2.3.10) simplifies to 2

t r i c e d * )

=

OL -e~

-Hx-x )

+ (t)-d(x) + e~

i2nfeXo

0

z (t)-^-o)

i2lifea

0

Zl

2

(2.3.12)

where z (t) and z (t) are complex Gaussian random processes accounting for the attenuation x

2

and phase of the two independent beams. Since the direct/specular path is assumed to have the shortest path-length and the first Rayleigh faded component arrives with negligible delay, we have x = 0. Normalizing the two 0

random processes such that i^z^r)! ] = £[|z (0| ] = 1 and explicitly specifying the 2

2

2

strengths of the two Rayleigh faded beams as 0Cj and oc respectively, we can re-write (2.3.12) as 2

z {t,x) rice

=

a -5(x) + a z (0-5(x) + a z (0-6(x-a) 0

1

1

2

2

(2.3.13)

Equation (2.3.13) is a general expression for the impulse response of a time-varying, frequency selective channel. The impulse response corresponding to a flat fading channel is z (t,x) rice

= a -8(x) + a z (r)-5(x) 0

1

1

(2.3.14)

Figure 2.4 illustrates the impulse response corresponding to (2.3.13) assuming a unity gain channel with a Rician factor of 1. The figure is plotted with a Doppler frequency of 20 Hz and a delay(a) of 0.2 sec.

17

Chapter 2 The Wireless Channel

0.20 0

Figure 2.4 3-D view of the discrete-time impulse response, \z{t, %)\ The Rician factor K is defined as the power of the specular component relative to the power in the Rayleigh faded or diffused component and is often expressed in dB. Thus («o)

(

K

dB

where ^f d (^) a

e

= 10-log (£) = 10 log 10

2

1

10

(2.3.15)

is the average delay power spectrum of the fading component of the channel,

( a ) is the power of the specular path and the power of the diffused component is obtained by 2

0

integrating the average delay power spectrum i.e. J'iKf j (t)dx. ac e

2.3.2 RF Spectrum and Autocorrelation of the Fading Process The present work is based on Clarke's model [17] which assumes a fixed transmitter with

18

Chapter 2 The Wireless Channel

a v e r t i c a l l y p o l a r i z e d antenna. T h e statistical characteristics o f the r a n d o m process z (t) t

are based

o n i s o t r o p i c s c a t t e r i n g o f the c o m p o n e n t s / r a y s r a n d o m l y a r r i v i n g at the r e c e i v e r at u n i f o r m l y distributed angles a n d e x p e r i e n c i n g s i m i l a r attenuation over s m a l l - s c a l e distances.

F i g u r e 2.5 d e p i c t s the s p e c t r a l b r o a d e n i n g o f an u n m o d u l a t e d c o n t i n u o u s w a v e ( C W ) c a r r i e r at f r e q u e n c y f

c

due to D o p p l e r shift c a u s e d b y the R a y l e i g h c h a n n e l . T h u s e a c h o f the

rays arrives at the M S i n its o w n frequency, offset f r o m f

c

s p e c t r u m d e n s i t y S(f)

w i t h i n ±f .

a n d a u t o c o r r e l a t i o n f u n c t i o n §(t)

d

The corresponding power

o f the R a y l e i g h f a d i n g p r o c e s s z-(r)

f o r the case o f a v e r t i c a l w h i p antenna o n the M S are g i v e n b y [15]

S(f)

=

3b

(2.3.16)

w h e r e b is the average p o w e r r e c e i v e d b y an i s o t r o p i c antenna a n d (O

d

= 2nf , d

f

d

b e i n g the

m a x i m u m D o p p l e r frequency, and

We accept this thesis as conforming to the required standard

THE UNIVERSITY OF BRITISH COLUMBIA March 2002 © Vaibhav Dinesh, 2002

In

presenting

degree freely

at

this

the

available

copying

of

department publication

of

in

partial

fulfilment

of

the

University

of

British

Columbia,

I

agree

for

this or

thesis

reference

thesis by

this

for

his thesis

and

study.

scholarly

or for

her

I further

purposes

gain

that

agree

may

be

It

is

representatives.

financial

requirements

shall

not

that

the

Library

permission

granted

by

understood

be

for

allowed

the

The University of British Columbia Vancouver, Canada

Date

DE-6

(2/88)

O S M A R C H 200Z .

COMPUTER

for

make

it

extensive

head

without

EN6/NEE.RiN°°)

7 1

Appendix C : Solving the Integral

74

Appendix D: Calculating the Errors in Amplitude and Phase estimation

77

Bibliography

79

N

iv

L i s t of Figures Figure 1.1

Received signal corresponding to slow and fast fading channels

2

Figure 2.1

Continuously Dispersive and Discrete Path channel outputs

9

Figure 2.2

Linear System model

Figure 2.3

3-D view of the continuous-time impulse response, \z(t, x)|

14

Figure 2.4

3-D view of the discrete-time impulse response, \z(t, x)\

17

Figure 2.5

Power Spectrum of an unmodulated CW carrier

19

Figure 3.1

System Model

22

Figure 3.2

The Transformed System

23

Figure 4.1

BER as a function of f

Figure 4.2

BER as a function of the Rician factor and f

47

Figure 4.3

Effect of the Rician factor for f

47

Figure 4.4

Effect of pulse shape in fast fading Rician channel, K

Figure 4.5

Channel averaging ability of different pulse shapes

Figure 4.6

Effect of the roll-off factor on the system performance, K

Figure 4.7

Two-beam frequency selective fading, K

Figure 4.8

Three-beam frequency selective fast fading, f

Figure 4.9

Three-beam frequency selective very-slow fading, a = 0

53

Figure 4.10

Three-beam frequency selective fast fading, f

54

Figure 4.11

Three-beam frequency selective very-slow fading, a > 0

55

Figure 4.12

Three-beam fast fading, K

56

Figure 4.13

Three-beam slow fading, K

= 0 dB

56

Figure 4.14

Three-beam fast fading as a function of K; [a;p] :[0.0;0.0],[1.0;1.0]

57

12

N

for the Rician channel, K

= 0 dB

dB

N

N

= 1.28 dB

= 0 dB

N

dB

dB

= 0 dB

= 0 dB N

= 0 dB and f

N

48 49

dB

dB

46

= 0.64 and a = 0

= 0.64 and a > 0

= 0.64

50 51 52

Figure 4.15

Three-beam fast fading as a function of K; [a;p] :[0.0;1.0],[1.0;0.0]

Figure 4.16

Three-beam slow fading as a function of K; [a;p] :[0.0;0.0],[1.0;1.0] ....58

Figure 4.17

Three-beam slow fading as a function of K; [a;p] :[0.0;1.0],[1.0;0.0] ....59

Figure 4.18

Performance degradation in a flat fading Rician channel

60

Figure 4.19

Performance degradation in a flat fading Rayleigh channel

61

Figure 4.20

Effect of mis-match in phase on the BER in a Rician channel

62

Figure 4.21

Effect of mis-match in phase on the BER in a Rayleigh channel

62

Figure 4.22'

Effect of mis-match in amplitude on the BER in a Rician channel

63

Figure 4.23

Effect of mis-match in amplitude on the BER in a Rayleigh channel

64

Figure D. 1

Resolution of the fading sample with/without error

77

vi

58

Glossary Acronyms

ASK AWGN

Amplitude Shift Keying Additive White Gaussian Noise

BER

Bit Error Rate

BFSK

Binary Frequency Shift Keying

BPSK

Binary Phase Shift Keying

BS

Base Station

CD

Continuously Dispersive

CW

Continuous Wave

DP

Discrete Path

EFD

Intrinsic Frequency Diversity

ISI

Inter-Symbol Interference

LOS

Line Of Sight

MFB

Matched Filter Bound

MS

Mobile Station

MSE

Mean Square Error

OFDM

Orthogonal Frequency Division Multiplexing

PAM

Pulse Amplitude Modulation

PSK

Phase Shift Keying

RRC

Root Raised Cosine

SNR

Signal to Noise Ratio

vii

Symbols

a

roll-off factor of the root raised cosine pulse

A

transmitted signal amplitude

c

speed of light

D

rms

rms delay spread

E

b

bit energy

f

carrier frequency

c

Af

Coherence bandwidth

f

d

maximum Doppler frequency

f

N

maximum normalized Doppler rate

f(t)

received pulse shape corresponding to the diffused channel

f (y)

pdf of Y

h(t)

impulse response of the receive filter

J (.)

zeroth order Bessel function of the first kind

K

Rician factor of the channel

c

Y

0

K

Rician factor (in dB)

N

total number of paths in the discrete channel model

N

noise component at the matched filter output

dB

N

power spectral density of AWGN

n(t)

AWGN noise

P

probability of bit error

p(t)

transmitted pulse shape

Q(v, w)

Marcum's (2-ftmction

r(t)

received (faded and noisy) signal

r {t)

received signal (for a CD channel)

r (t)

received signal (for a DP channel)

0

b

cd

dp

Rf(tj, t )

autocorrelation of the process f(t)

fad^

faded signal at the channel output

r (t)

received signal without noise (for the DP channel)

S(f)

power spectrum density of the Rayleigh fading process

2

r

s

viii

s(t)

transmitted signal

SfadW

r e c e i v e d pulse shape

T

r e c e i v e d s i g n a l duration

Ar

s a m p l i n g interval

Af„

Coherence time

T

delay spread o f the c h a n n e l

T

pulse duration

v

m a x i m u m velocity of M S

c

M

p

v

(z)

Whittaker function

X

m a t c h e d filter output after s a m p l i n g

x(t)

m a t c h e d filter output

Y

s i g n a l c o m p o n e n t at the m a t c h e d filter output

z(t, x)

i m p u l s e response o f a linear c h a n n e l ( g e n e r a l i z e d notation)

cd(t> ) x

z

i m p u l s e response o f a C D channel

ray(t' )

i m p u l s e response o f the d i f f u s e d c h a n n e l

rice(t> )

i m p u l s e response o f the R i c i a n c h a n n e l

z {t)

c o m p l e x f a d i n g process c o r r e s p o n d i n g to the «th b e a m

x

z

x

z

n

ix

Greek Symbols

diffused beam strength

a

2

a

attenuation corresponding to the direct/specular path

0

strength of the nth faded beam

«„

nth path attenuation characteristic function 8(r)

delta function

r(z)

Gamma function

„C) • e~ ° m

• s(t-x transmitted at time t , may arrive in con2

2

junction with another component of phase (pi transmitted at time tj, resulting in constructive or

Chapter 2 The Wireless Channel

10

destructive interference between the two. We can characterize signal dispersion in terms of Delay Viewed in the frequency domain, signal dispersion is expressed in terms of Coher-

Spread (T ). M

ence Bandwidth (Af

less than Af

c

c

) which implies that the signal components having frequency separation

are treated alike by the channel. It can be shown [12] that Af = c

M

. Based on

this, we can further classify fading as: • Flat or frequency non-selective fading: When A f

c

is large relative to the signal bandwidth,

the channel is said to be flat. Viewed in the time domain, the channel ideally causes no delay spread in the transmitted signal, i.e. T

M

= 0.

• Frequency Selective fading: If the coherence bandwidth is small compared to the signal bandwidth, the channel is said to be frequency selective. The attendant delay spreading causes inter-symbol interference (ISI). Frequency-spreading of the signal is a direct consequence of the spatial changes of the receiver relative to its surroundings, accounting for propagation path changes. As a result, the channel offers a time-variant response. This is characterized by the Coherence Time At , which is c

defined as the duration over which the channel's impulse response can be considered as timeinvariant. Coherence time is inversely proportional to the Doppler Spread, At

c

«*= 4- [12]. The id

Doppler spread is a function of the MS velocity, v, and signal wavelength, X, and is a measure of the spectral broadening of the signal / „ = /Vcos(e„)

where f

(2.2.3)

is the Doppler shift [15] for the n wave arriving at an angle 0 with respect to the th

n

n

Chapter 2 The Wireless Channel

11

direction of motion of the MS and / . (f

d

= ^) is the maximum Doppler frequency.

Depending on the channel's coherence time with respect to the signal duration, the Doppler spreading results in: • Fast Fading: The channel exhibits a rapidly time-varying impulse response relative to the symbol duration and the signal experiences spectral broadening. Fast fading often causes the occurrence of frequent deep fades in the received signal. • Slow Fading: When the channel's coherence time is long compared to the symbol duration, the channel is considered as slowly fading. Equivalently, the Doppler spread is small. It is important to understand that the time-spreading and frequency-spreading are independent phenomenons. A wireless channel may exhibit any combinations of Flat/Frequency Selective and Fast/Slow fading. 2.3 Channel Model and its Statistical Characterization 2.3.1 The Linear Channel In terms of the complex lowpass equivalent form of a bandpass system [12], we can express the impulse response of the channel as z(t, x), which represents the response of the channel at time t to an impulse applied at time t-x. In this section an expression for z(t, x) applicable for both flat and frequency selective fading channels is obtained. In doing so, we follow an approach similar to that of [12]. The general expression of the impulse response is given as z(t, x) = X « „ ( 0 • e~

J2nfMt)

(2.3.1)

• 6(T-T„(0)

n

where f is the carrier frequency of the bandpass signal; oc„(0 and x (t) are the attenuation and c

n

12

Chapter 2 The Wireless Channel

delay of the n ray at time t, respectively. th

Equation (2.3.1) is a general expression from the point of view that it accounts for both Rician and Rayleigh fading channels and accommodates for all types of fading discussed in Section 2.2.2. Since the specular path is a non-fading component of the channel, we can implement the impulse response of a Rician fading channel, z {t, x), by taking a constant attenuation, a , and rice

0

associated delay, x , for thefirstarriving ray 0

z (t, x) = 0 C Q • erice

j 2 n f

^ • 8(t - 1 ) 0

+ Xa„(0 • e-

]lKfMt)

• 8(t - x (t)) n

(2.3.2)

Figure 2.2 shows the block diagram representation of the response r(f) of a fading channel z(t, x), corrupted with noise n(t), to an input s{t).

Channel

n{t)

AWGN Figure 2.2 Linear System model The signal r(t) can be written as

r(t) = J z(t,x)s(t-x)dx

+ n(t)

(2.3.3)

Substituting (2.3.1) in (2.3.3) and neglecting the noise term for the moment, the received signal,

Chapter 2 The Wireless Channel

13

r {t), is given as s

•',(*) = 5>«C) • e~

J2KfMt)

.

• s(t-x (t)) n

(2.3.4)

n

T h e a u t o c o r r e l a t i o n f u n c t i o n o f z(t, x) i s defined as [12]

SR (Af z

JCCJ, x )) = | • E[z*(t, Tj) • z(f + A / , x ) ] . 2

(2.3.5)

2

A s s u m i n g u n c o r r e l a t e d s c a t t e r i n g , i.e. i n d e p e n d e n c e o f attenuation a n d phase s h i f t s a s s o c i a t e d w i t h different path delays, a n d letting

At - 0, (2.3.5) c a n be re-written as

SR (0;(T!, x )) = 9t(x,) • 8(T! - x ) . z

2

(2.3.6)

2

T h e f u n c t i o n SR(x), k n o w n as the d e l a y p o w e r s p e c t r u m o f the c h a n n e l , represents the average p o w e r output o f the c h a n n e l as a f u n c t i o n o f the t i m e d e l a y x . I n u r b a n e n v i r o n m e n t s , 9t(x) c a n be a p p r o x i m a t e d b y the c o n t i n u o u s one-sided e x p o n e n t i a l f u n c t i o n [13], i.e.

SR(x) = ^ - • e x p f - - M rms

where

D

rms

^ rms'

x >0

is the r m s d e l a y spread. In some p r e v i o u s w o r k s

(2.3.7)

[6], the class o f t w o - s i d e d c o n t i n u -

ous spectra i s c o n s i d e r e d b y a s s u m i n g the G a u s s i a n s p e c t r u m . H o w e v e r , f o r s i m p l i c i t y , w e c o n f i n e o u r a n a l y s i s t o o n e a n d t w o b e a m cases ( D o u b l e - S p i k e s p e c t r u m

(2.3.8)) w h i c h

can easily be

extended t o a s y s t e m w i t h N , ( N > 2), beams b y a p p l y i n g a s i m i l a r p r o c e d u r e .

= ^(8(x) + 8 ( x - a ) )

(2.3.8)

A snapshot o f the i m p u l s e response w i t h c o n t i n u o u s o n e - s i d e d e x p o n e n t i a l d e l a y p o w e r s p e c t r u m is s h o w n i n F i g u r e

2.3.

T h e f i g u r e is p l o t t e d a s s u m i n g

D

rms

= 0.1 sec a n d

a Doppler

14

Chapter 2 The Wireless Channel

frequency of 10 Hz. One needs to be careful while comparing Figure 2.1, pertaining to the response of the fading channel to an impulse, with the three-dimensional plot of the impulse response, \z(t, x)|; since z(t, x) is a function of two time variables, t and x, whereas the response of the channel is a function of t only. It is, however, possible to trace the channel's response to an impulse from the three-dimensional plot of it's impulse response.

0.2 x [sec]

0.3 0.4

0

Figure 2.3 3-D view of the continuous-time impulse response, \z(t, x)| It should be noted that the terms 'one beam' and 'two beam' models do not simply refer to the arrival of one and two rays at the mobile; rather each beam may be interpreted as a collection of several rays with varying path-delays closely distributed about the mean delays x, and x , 2

respectively [14]. Thus, on splitting the second term in (2.3.2) into two parts (summation series) each reflecting the case of a single Rayleigh faded beam and denoting their respective indices by n and n , {

2

Chapter 2 The Wireless Channel

15

with each ray in the group having an associated delay of x„ (0 and ^(0, respectively, where x (0 Bi

=

x,+Ax (0,

AT

Bi

B i

(o«r

p

(2.3.9)

T is the transmitted signal block/pulse duration, we obtain p

z {t,i) = a -e~

-Hi-* )+z (t,T)

+ z (t,%)

J2nfcXo

rice

0

0

l

(2.3.10)

2

where z {t,x) = e

J

/c

x

'X n,(0^ a

-/27C/-T,

z (r,t) = e

;

/ c 2

1

1

-7'2TC/ AT C

X « „ *2W ~

g

n

n |

(2.3.11)

(0

•8(T-(f +

2

2

•8(T-(f +AT (0))

2

AT (0)). ll2

2

An upper-bound on At (r) can be obtained by realizing [14] that the greatest change in n

the path length occurs when the mobile is directly in line with the direction of arrival of the signal. Over the duration T , this would correspond to a change in path-length of v • T for a mobile p

moving at velocity v. Therefore the change in delay (Ax (t)) n

associated with the change in

path-length would be v • T /c, where c is the speed of light. Since (Ax (t)) n

max

is much smaller

than the pulse duration, we can drop the Ax (t) factor from the delta functions in (2.3.11). n

However, the same cannot be neglected in the exponential terms due to the large value of f . c

Assuming that each of the two beams consists of a large number of independent rays and

Chapter 2 The Wireless Channel

16

the phases of the rays are uniformly distributed from 0 to 2n [15], the Central Limit Theorem [16] can be applied to (2.3.11). Taking Xj = 0 and x = a, (2.3.10) simplifies to 2

t r i c e d * )

=

OL -e~

-Hx-x )

+ (t)-d(x) + e~

i2nfeXo

0

z (t)-^-o)

i2lifea

0

Zl

2

(2.3.12)

where z (t) and z (t) are complex Gaussian random processes accounting for the attenuation x

2

and phase of the two independent beams. Since the direct/specular path is assumed to have the shortest path-length and the first Rayleigh faded component arrives with negligible delay, we have x = 0. Normalizing the two 0

random processes such that i^z^r)! ] = £[|z (0| ] = 1 and explicitly specifying the 2

2

2

strengths of the two Rayleigh faded beams as 0Cj and oc respectively, we can re-write (2.3.12) as 2

z {t,x) rice

=

a -5(x) + a z (0-5(x) + a z (0-6(x-a) 0

1

1

2

2

(2.3.13)

Equation (2.3.13) is a general expression for the impulse response of a time-varying, frequency selective channel. The impulse response corresponding to a flat fading channel is z (t,x) rice

= a -8(x) + a z (r)-5(x) 0

1

1

(2.3.14)

Figure 2.4 illustrates the impulse response corresponding to (2.3.13) assuming a unity gain channel with a Rician factor of 1. The figure is plotted with a Doppler frequency of 20 Hz and a delay(a) of 0.2 sec.

17

Chapter 2 The Wireless Channel

0.20 0

Figure 2.4 3-D view of the discrete-time impulse response, \z{t, %)\ The Rician factor K is defined as the power of the specular component relative to the power in the Rayleigh faded or diffused component and is often expressed in dB. Thus («o)

(

K

dB

where ^f d (^) a

e

= 10-log (£) = 10 log 10

2

1

10

(2.3.15)

is the average delay power spectrum of the fading component of the channel,

( a ) is the power of the specular path and the power of the diffused component is obtained by 2

0

integrating the average delay power spectrum i.e. J'iKf j (t)dx. ac e

2.3.2 RF Spectrum and Autocorrelation of the Fading Process The present work is based on Clarke's model [17] which assumes a fixed transmitter with

18

Chapter 2 The Wireless Channel

a v e r t i c a l l y p o l a r i z e d antenna. T h e statistical characteristics o f the r a n d o m process z (t) t

are based

o n i s o t r o p i c s c a t t e r i n g o f the c o m p o n e n t s / r a y s r a n d o m l y a r r i v i n g at the r e c e i v e r at u n i f o r m l y distributed angles a n d e x p e r i e n c i n g s i m i l a r attenuation over s m a l l - s c a l e distances.

F i g u r e 2.5 d e p i c t s the s p e c t r a l b r o a d e n i n g o f an u n m o d u l a t e d c o n t i n u o u s w a v e ( C W ) c a r r i e r at f r e q u e n c y f

c

due to D o p p l e r shift c a u s e d b y the R a y l e i g h c h a n n e l . T h u s e a c h o f the

rays arrives at the M S i n its o w n frequency, offset f r o m f

c

s p e c t r u m d e n s i t y S(f)

w i t h i n ±f .

a n d a u t o c o r r e l a t i o n f u n c t i o n §(t)

d

The corresponding power

o f the R a y l e i g h f a d i n g p r o c e s s z-(r)

f o r the case o f a v e r t i c a l w h i p antenna o n the M S are g i v e n b y [15]

S(f)

=

3b

(2.3.16)

w h e r e b is the average p o w e r r e c e i v e d b y an i s o t r o p i c antenna a n d (O

d

= 2nf , d

f

d

b e i n g the

m a x i m u m D o p p l e r frequency, and