Contents
Chapter 14 .......................................................................................................................... 2
14.1 Characterization of fading multipath channels ......................................................... 2
14.1.1 Channel correlation functions and power spectra ......................................... 4
14.1.1.1 The effect of multipath of the channel ............................................... 4
14.1.1.2. Effects of time variations of the channel ........................................... 6
1.4.1.2 Statistical models for fading channels ................................................ 11
14.2 The effect of signal characteristics on the choice of a channel model ................... 12
14.3 Frequency-nonselective slowly fading channel ..................................................... 14
14.4 Diversity techniques for fading multipath channels ............................................... 18
14.4.1 Binary signals ...................................................................................... 19
14.4.3 M-ary orthogonal signals..................................................................... 28
14.5 Digital signaling over a frequency-selective, slowly fading channel ..................... 30
14.5.1 A tapped-delayed-line channel model ......................................................... 30
14.5.2 RAKE demodulator ..................................................................................... 33
14.5.3 Performance of RAKE demodulator ........................................................... 37
Appendix 1. Matched filters ........................................................................................... 41
1
Chapter 14
14.1 Characterization of fading multipath channels
A channel is a medium between transmitter and receiver and may have some properties, for
example:
(1). It may offer not only a single path between transmitter and receiver; instead, it consists
of a lot of paths. Such a medium is called as multipath channel. These paths cause the time
spread in the transmitted signal due to the variation of path length.
(2). The nature of the multipath (the number of path, the strength of each path, the total delay
of multipath, etc.) varies with time because of time variations in the structure of the medium.
This leads to a time-varying multipath channel.
(3). Moreover, the time variations of the medium appear to be unpredictable to the user of
the channel. As a result, it is reasonable to characterize the time-variant multipath channel in
statistics.
The transmitted signal is expressed as
s(t )  Re  sl (t )e j 2 f ct 
(14.1)
The received signal experienced such a time-variant multipath channel can be expressed by
x(t )   n (t ) s[t   n (t )]
(14.2)
n
where n (t ) is the attenuation factor for the signal received on the n-th path with the
propagation delay,  n (t ) .1 This channel is called “multipath” because it is assumed n ³ 2 ,
here and is “time-variant 2” because the amplitude  n (t ) and/or time delay  n (t ) in
individual path are time variant.
Substituting (14.1) into (14.2), we obtain



x(t )  Re  n (t )e  j 2 fc n ( t ) s[t   n (t )] e j 2 fct 

 n

1
(14.3)
As a comparison, we consider a simplest channel: a single path situation. The received signal can be
expressed by x (t ) = a s(t - t ) . The impulse response of this single path channel is a d(t - t ) because
s(t ) * d(t - t ) = a ò
¥
- ¥
2
s(x )d((t - x ) - t )dx = a s(t - t ) .
Remember that the system is called time invariant if y (t ) = T [s(t )], then y (t - to ) = T [s(t - to )] , where
y (t ) is the output of the system and s(t ) is the input, respectively.
2
The corresponding equivalent low pass complex-valued signal rl (t ) is expressed by
rl (t )  n (t )e  j 2 fc n ( t ) s[t   n (t )]
(14.4)
n
Accordingly, the equivalent low pass complex impulse response of the time varying multi
path channel is written as1
c( ; t )  n (t )e  j 2 fc n ( t ) [   n (t )]
(14.5)
n
For some channels, for example, the tropospheric scatter channels, because of the
property of continuum of multipath components, the received signal may be more suitable to
be described in the integral form instead of the summation form as (comparing with (14.2)
for discrete channels)
x (t ) =
¥
ò- ¥
(14.6)
a (t ; t )s(t - t )d t
where a ( t ; t ) is the attenuation of the signal components at delay  and at time instant t.
By replacing s (t ) in (14.6) with(14.1), we obtain the received signal as
¥
x (t ) = Re éêò a (t ; t )e ë -¥
{
j 2p fc t
sl (t - t )d t ù
e j 2p fct
ú
û
}
(14.7)
The equivalent low pass impulse response of the channel is
c( t ; t ) = a ( t ; t )e -
j 2p fc t
(14.8)
where c ( ; t ) represents the response of the channel at time t due to an impulse applied at
time t   . (t represents the time instant of observation and the channel delay is t , so the
emitted epoch of the signal can be retroacted as t - t .)
Note that Equation (14.5) is suitable for discrete multipath channel, and Equation (14.8) is
suitable for continuous multipath channel.
Consider the transmission of unmodulated carrier at frequency f c . Then sl (t )  1 for all t
and the received signal for the case of discrete multipath can be expressed by
rl (t )  n (t )e  j 2 fc n ( t )  n (t )e  jn ( t )
n
1
(14.9)
n
The impulse response of a linear time varying channel (system) can be written as h(t , t ) . The channel output
y (t ) responded to the channel input s(t ) , is y (t ) =
¥
ò- ¥ h(t , t )s(t )d t
. For time invariant channel,
h(t + l , t + l ) = h(t , t ) .
3
The amplitude variations in the received signal represented by(14.9), termed signal fading,
are due to the time variant multipath characteristics of the channel.
Some typical fading channels are:
(1). Rayleigh fading: c ( ; t ) is modeled as a zero-mean complex-valued Gaussian process.
The envelope c( ; t ) at any time instant t obeys Rayleigh distribution. Examples are VHF,
UHF and the HF ionospheric channels.
(2). Rice fading: c ( ; t ) is modeled as a non-zero-mean complex-valued Gaussian process.
The envelope c( ; t ) at any time instant t obeys Rice distribution. Rician fading is usually
considered to be due to a specular component in the received path, for example, a fixed
nonrandom scatterer in the transmitter-receiver path. In this case a deterministic component
may be added in the received signal as
x (t ) = A s(t ) +
¥
ò- ¥
a (t ; t )s(t - t )d t = Re éêA sl (t ) +
ë
{
¥
ò- ¥
a (t ; t )e -
j 2p fc t
sl (t - t )d t ù
e j 2p fct
ú
û
}
(14.10)
where A is the transmission coefficient associated with the specular path.
(3). Nakagami fading: the envelope c( ; t ) at any time instant t obeys Nakagami
distribution.
14.1.1 Channel correlation functions and power spectra
Assume the equivalent low-pass impulse response c ( ; t ) , which is characterized as a
complex-valued random process in the t variable, is wide sense stationary (WSS).
14.1.1.1 The effect of multipath of the channel
The behavior of the multipath channel can be analyzed in both time and frequency domains.
(a). Analysis in the time domain
The autocorrelation function of c ( ; t ) is defined as
1
E [c * ( t 1; t )c( t 2 ; t + D t )] = f c ( t 1, t 2 ; D t )
2
(14.11)
Besides, in most radio transmission, the attenuation and phase of the channel associated path
delay  1 is uncorrelated with the attenuation and phase shift with path delay  2 . This is
called uncorrelated scattering (US). Combination of wide sense stationary and uncorrelated
scattering leads to the WSSUS channel, and then (14.11) can be rewritten as
4
1
E [c * ( t 1; t )c( t 2 ; t + D t )] = f c ( t 1; D t )d( t 1 - t 2 )
2
(14.12)
c ( ; t ) can be measured by transmitting very narrow pulses (i.e., wideband signals) and
cross correlating the received signal with a delayed version of itself. The measured function
of c ( ) may appear as
The range of values of  over which f c (t ) is essentially nonzero is called the multipath
spread of the channel, T m . f c (t ; D t ) gives the average power output as a function of the
delay  and the difference t in observation time.
Let D t = 0 , then f c (t ; D t ) = f c (t ; 0) = f c (t ) represents the average power output of the
channel as a function of the time delay t . Hence, f c (t ) is called the multipath intensity
profile (MIP) or the delay power spectrum (DPS) of the channel.
(b). Analysis in the frequency domain
By taking the Fourier transform of c(t ; t ) with respect to t , we obtain the time-variant
transfer function C ( f ; t ) as
C ( f ;t ) =
¥
ò- ¥ c(t ; t )e- j 2p f t d t
(14.13)
Note that C ( f ; t ) has the same statistics of c(t ; t ) , that is, the transformation does not change
the statistical properties.
With the WSS assumption, the autocorrelation function of C ( f ; t ) is
1
E [C * ( f1; t )C ( f2 ; t + D t )]
2
¥
1 ¥
= ò ò E [c * ( t 1; t )c( t 2 ; t + D t )]e j 2p ( f1t 1 2 -¥ -¥
f C ( f1, f2 ; D t ) =
=
¥
¥
ò- ¥ ò- ¥
¥
=
ò- ¥
=
ò- ¥
¥
f c ( t 1; D t )d( t 1 - t 2 )e j 2p ( f1t 1 -
f c ( t 1; D t )e
j 2p ( f1 - f2 ) t 1
f c ( t 1; D t )e -
j 2p D f t 1
f2 t 2 )
d t 1d t 2
f2 t 2 )
d t 1d t 2
(14.14)
dt 1
dt 1
= f C (D f ; D t )
where D f = f2 - f1 . From the last equation, it is clear that f C (D f ; D t ) is the Fourier
5
transform of the MIP, c ( ; t ) . C ( f ; t ) is called the spaced-frequency, spaced-time
correlation function of the channel. C ( f ; t ) can be measured by transmitting a pair of
sinusoids separated by D f = f2 - f1 and cross-correlating the two separately received
signals with a relative delay t .
By letting t  0 in (14.14) and defining f C (D f ; 0) = f C (D f ) and f c (t ; 0) = f c (t ) , the
transform relationship is expressed as
f C (D f ) =
¥
ò- ¥
f c (t )e-
j 2p D f t
dt
(14.15)
Since f C (D f ) is an autocorrelation function in the frequency variable, it provides us a
measure of the frequency coherence of the channel.
As a result of the Fourier transform relationship between f C (D f ) and f c (t ) , the reciprocal of
the multipath spread is a measure of the coherence bandwidth (D f )c of the channel.
(D f )c »
1
Tm
(14.16)
The coherence bandwidth of the fading channel can be employed to indicate that any two
sinusoids with frequency separation greater than (D f )c will be affected differently by the
channel. When an information-bearing signal of bandwidth W is transmitted through the
channel, the fading channel is classified as frequency-selective if (D f )c = W
and
frequency-nonselective if (D f )c ? W .
Finally, it is interesting to indicate that when the channel consists of only a single path, the
multipath spread of the channel, T m approaches zero, therefore, the resulting coherence
frequency approaches infinity; all the frequency components of the transmitted signal
experience the same impact from the channel!
14.1.1.2. Effects of time variations of the channel
The time variation of the channel is measured by t in f C (D f ; D t ) . The Fourier transform
6
of f C (D f ; D t ) with respect to t is defined as
SC ( D f ; l ) =
¥
ò- ¥
f C (D f ; D t )e-
j 2pl D t
(14.17)
dD t
Let D f = 0 and SC (0; l ) = SC (l ) , (14.17) becomes
SC (l ) =
¥
ò- ¥
f C (0; D t )e-
j 2pl D t
(14.18)
dD t
The function SC (l ) is a power spectrum that gives the signal intensity as a function of the
Doppler frequency l . SC (l ) is then called the Doppler power spectrum of the channel.
An interesting situation: if the channel is time invariant, then in the transmission of a pure
frequency tone, f C (0; D t ) = 1 (normalized or a constant, not a function of D t ), and
then SC (l ) = d(l ) . That is, there is no spectral broadening.
The range of values of  over which SC (l ) is essentially nonzero is called the Doppler
spread B d of the channel. Since SC (l ) is related to f C (D t ) by the Fourier transform, the
reciprocal of B d is a measure of the coherence time (D t )c of the channel.
(D t )c »
1
Bd
(14.19)
A slowly changing channel has a large (D t )c (i.e., a small Doppler spread, B d ). When the
channel is time invariant, we may say that the coherence time is infinite; accordingly, the
Doppler spread approaches zero!
Up to the present time, there are two formulae:
(I). f C (D f ; D t ) =
¥
ò- ¥
f c (t ; D t )e -
j 2p D f t
dt
(14.20)
which is due to the delay  caused by the multipath of the channel; the relationship
between the multipath spread T m and the coherence bandwidth (D f )c is T m » 1/ (D f )c .
(II). SC (D f ; l ) =
¥
ò- ¥
f C (D f ; D t )e-
j 2pl D t
dD t
(14.21)
which is due to the Doppler frequency l caused by the time variation of the channel; the
7
relationship between the coherence time (D t )c and Doppler spread B d is (D t )c » 1/ Bd .
The autocorrelation function of c(t ; t ) in WSSUS model f c (t ; D t ) has two variables, and in
(I) the variable t is assumed to be fixed; if it is treated as variable now, we can obtain
another formula as below.
S (t ; l ) =
¥
ò- ¥
f c (t ; D t )e -
j 2pl D t
(14.22)
dD t
Since f C (D f ; D t ) is obtained by taking Fourier transform, if we take inverse Fourier
transform of SC (D f ; l ) in (14.21) with respect to  f , we must obtain (14.22). That is,
S (t ; l ) =
¥
ò- ¥ SC (D f ; l )e j 2pt D f d D f
(14.23)
By substituting (14.21) into (14.23), we obtain
S (t ; l ) =
¥
¥
ò- ¥ ò- ¥
f C (D f ; D t )e -
j 2pl D t j 2pt D f
e
(14.24)
d D td D f
The relationships can be illustrated in terms of the following figure.
c ( ; t )
 ,f
 t, 
 t, 
S(;)
C (f ; t )
 ,f
SC (f ;  )
Where the thin double arrow represents single Fourier transform; however, the heavy double
arrow represents double Fourier transform. S (t ; l ) is called the scattering function of the
channel. It provides us with a measure of the average power output of the channel as a
function of the time delay  and the Doppler frequency  .
Why is it called scattering function? Because it is this function indicates the extent of
scattering in time delay (not “time”) due to the multipath of the channel, and the extent of
scattering in Doppler frequency (not “frequency”) due to the time variation of the channel.
8
Example 14.1-1. Scattering function of a tropospheric scatter channel
(a). The scattering function S (t ; l ) measured on a 150-mile tropospheric scatter link.
(b). The signal used to probe the channel had a time resolution of 0.1 ms . The time-delay axis,
hence, is quantized in increments of 0.1 ms .
(c). Observe that the multipath spread Tm = 0.7ms .
(d). The Doppler spread, defined as the 3-dB bandwidth of the power spectrum for each
signal path, appears to vary with each signal path, from less than 1 Hz to several Hz.
(e). The largest of these 3-dB bandwidths of the various paths is called the Doppler spread.
9
Example 14.1-2. Multipath intensity profile of mobile radio channels
(a). In urban and suburban areas, typical values of multipath spreads range from 1 to 10 ms .
(b). In rural mountainous areas, the multipath spreads are much greater about 10-30 ms .
Two models for the MIP are
Example 14.1-3. Doppler power spectrum of mobile radio channels.
Jakes’ model is one of widely used model for the Doppler power spectrum of a mobile radio
channel. The autocorrelation function of the time-variant transfer function C ( f ; t ) is given
as
f C (D t ) =
1
E [C * ( f ; t )C ( f ; t + D t )] = J 0(2p fm D t )
2
where J (×) is the zero-order Bessel function of the first kind and fm = vf0 / c is the
10
maximum Doppler frequency, where v is the vehicle speed in meters per second (m/s), f 0
is the carrier frequency, and c is the speed of light ( 3 ´ 108 m/s).
The Doppler power spectrum is
S C (l ) =
¥
ò- ¥
f C (D t )e
- j 2pl D t
dD t =
¥
ò- ¥ J 0(2p fm D t )e
- j 2pl D t
1
ìï
, | f |£ fm
ïï
2
ï
d D t = í p fm 1 - ( f / fm )
ïï
| f |> fm
ïïî 0,
1.4.1.2 Statistical models for fading channels
Several probability density functions are used to describe the statistical properties of the
fading channel. They include Rayleigh, Nakagami, and Rician.
(1). Rayleigh: pR (r ) =
2r - r 2 / W
e
, r ³ 0,
W
with signal parameter, W = E (R 2 )
(2). Nakagami: see Equation 2.1-147, with two parameters, m and W = E (R 2 )
(3). Rician: see Equation 2.1-141, with two parameters, s and  2 . An airplane to ground
communication link, for example, has statistics of Rician distribution.
Propagation models for mobile radio channels
There exist two problems in describing mobile radio channels. The first problem is the mean
path loss. The mean path loss encountered in mobile radio channels may be characterized as
being inversely proportional to d p , where 2  p  4 (remember p  2 in free space).
They are lots of factors affecting the path loss in mobile radio communications, including
base station antenna height, mobile antenna height, operating frequency, atmospheric
conditions, and presence or absence of buildings and trees. One of model for a large city in
an urban area is the Hata model, in which the mean path loss is expressed as
Loss in dB=69.55  26.16log10 f -13.82log10 ht - a( hr )  (44.9 - 6.55log10 ht )log10 d
(14.25)
a (hr )  3.2(log10 11.75hr ) 2 - 4.97, f  400MHz
(14.26)
11
The second problem is the effect of shadowing of the signal due to large obstructions, such
as large buildings, trees, and hilly terrain between the transmitter and the receiver.
Shadowing is usually modeled as a multiplicative and slowly time varying random process.
That is, the received signal may be written by
(14.27)
r (t )= A0g(t )s(t )
where A0 accounts for the antenna gains and the mean path loss between the transmitting
antenna and the receiving antenna, s(t ) is the transmitted signal and g(t ) is a random
process that represents the shadowing effect.
At any time instant, the shadowing process is modeled as lognormally distributed:
1
ìï
e - (ln g ïï
2
ï
2
ps
g
p(g) = í
ïï
ïïî 0
m)2 / 2s 2
(g ³ 0)
(14.28)
(g < 0)
If we define a new random variable X = ln g , then X is a Gaussian random variable as
1
p(x ) =
2ps
2
e - (x -
m)2 / 2s 2
,
- ¥ < x < ¥
(14.29)
Random variable X represents the path loss measured in dB,  is the mean path loss in dB,
and  is the standard deviation of the path loss in dB. For typical cellular and
microcellular environments,  is in the range of 5-12 dB.
14.2 The effect of signal characteristics on the choice of a
channel model
Without noise, the equivalent low pass received signal may be expressed in terms of c ( ; t )
and s l (t ) in the time domain
rl (t ) =
¥
ò- ¥ c(t ;t )s l (t -
t )d t
(14.30)
or correspondingly in the frequency domain as
rl (t ) =
¥
ò- ¥ C ( f ;t )S l ( f )e j 2p ft df
(14.31)
If S l ( f ) has a bandwidth greater than the coherence bandwidth ( f )c of the channel,
S l ( f ) is subjected to different gains and phase shifts across the band. The channel in this
case is said to be frequency-selective; otherwise it is said to be frequency-nonselective.
Additional distortion is caused by the time variation in C ( f ; t ) and leads to a variation in the
received signal strength, termed as fading.
12
Note: Frequency-selectivity is caused by multipath of the channel (coherence frequency of
the channel relative to the transmitted signal bandwidth, W); however, fading is caused by
the time variations of the channel.
If the signaling interval of the transmitted signal is T, the corresponding bandwidth is about
W  1/ T . If T Tm or equivalently,
W
1
 ( f ) c
Tm
(14.32)
then all the frequency components in S l ( f ) undergo the same attenuation and phase shift in
the transmission through the channel; the channel is then called frequency nonselective. This
implies that C ( f ; t ) is a complex-valued constant in the frequency variable. Since S l ( f ) has
its frequency content concentrated in the vicinity of f = 0 , C ( f ; t ) = C (0; t ) . The received
signal is then expressed as
rl (t ) = C (0; t ) ò
¥
- ¥
S l ( f )e j 2p ft df =C (0; t )sl (t )
(14.33)
Thus, when W = (D f )c , the received signal is simply the transmitted signal multiplied by a
complex-valued random process C (0; t ) . In this case, the multipath components in the
received signal are not resolvable.
The transfer function C (0; t ) for a frequency nonselective channel may be expressed by
C (0; t ) = a (t )e-
j f (t )
(14.34)
In addition, the rapidity of the fading on the frequency nonselective channel can be
determined either from the correlation function f C (D t ) or from the Doppler power
spectrum SC (l ) , or equivalently, the coherence time (D t )c or Doppler spread B d .
If T = (D t )c » 1/ Bd , the channel attenuation and phase shift are essentially fixed for the
duration of at least one signaling interval. In this case, the channel is called slowly fading. As
a result, if Tm Bd  1 , the channel is a frequency nonselective slowly fading channel.
Tm Bd is called the spread factor. When Tm Bd  1 , the channel is said to be underspread;
otherwise, it is overspread.
13
Remember that the frequency selectivity is caused by the multipath of the channel (delay
spread) and the fading rapidity is from the time variations of the channel (Doppler spread);
accordingly, there are four possibilities as below.
You may also draw these four situations on the T-T plane or W-W plane.
On the other hand, when W ? (D f )c , the channel is said to be frequency selective. In
frequency selective channel the multipath components in the received signal are resolvable
with a resolution in time delay of 1/W. Since W ? (D f )c , we may write as
W = k ´ (D f )c , k Î I
in frequency domain and T m = k ´ T , k Î I in the time domain. In this
case, we can observe k multipath components in total. It is important to indicate that a
frequency selective channel can be modeled as a tapped delay line (transversal) filter with
time-variant tap coefficients (k taps, for example).
14.3 Frequency-nonselective slowly fading channel
Frequency nonselective channel results in multiplicative distortion of the transmitted signal
sl (t ) , and slowly fading implies that the multiplicative process may be regarded as a constant
during at least one signaling interval. Accordingly, the received signal may be expressed by
14
rl (t ) = a e -
jf
sl (t ) + z (t ),
0£ t £ T
(14.35)
Depending on whether the received signal phase is to be estimated or not, the detector can be
classified as coherent or noncoherent.
(I). Coherent detection:
Assume the channel fading is sufficiently slow that the phase shift f can be estimated from
the received signal precisely (without error), and then coherent detection can be applied.
rl (t )   e  j sl (t )  z (t )
Ä
T
ò0
(×)dt
e j sl * (t )
For a fixed attenuation  (i.e., conditioned on a ), at the output of the sampler, the signal
component is
sO =
T
ò0
a | sl (t ) |2 dt = 2a E b
and the noise component is
nO =
T
ò0
e j f sl * (t )z (t )dt
The SNR at the output of sampler (correlator) is
gb =
| sO |2
4a 2E b2
2 =
E [| nO | ] E [| nO |2 ]
where
T
T
T
0
0
0
E [| nO |2 ] = E [ò e j f sl * (t )z (t )dt ò e - j f sl (l )z (l )d l ] = E [ò
= 2N 0 ò
T
0
T
ò0
T
ò0
sl * (t )z (t )sl (l )z (l )d l dt ]
sl * (t )sl (l )d(t - l )d l dt = 4N 0E b
Hence
gb =
4a 2E b2
a 2E b
=
4N 0E b
N0
The bit error probability conditioned on a can be expressed as a function of the received
SNR gb . Here we already have error probabilities of BPSK and BFSK over AWGN (See
Chapter 5).
BPSK: P2 ( gb ) = Q ( 2gb )
(14.36)
BFSK: P2 ( gb ) = Q( gb )
(14.37)
15
To make probability of bit error meaningful, the random variable  in gb must be
averaged out. If the distribution of gb is known, the average probability of bit error can be
evaluated as
P2 =
¥
ò0
(14.38)
P2 ( gb )p( gb )d gb
where p( gb ) is the probability density function of gb when a is random.
(a). Rayleigh fading: when a is Rayleigh distributed, a 2 is a chi-square distributed so is
gb . The probability density function of gb is
1 e
gb
p( gb ) =
gb / gb
gb ³ 0
,
(14.39)
where gb is the average signal-to-noise ratio.
gb =
Eb
E (a 2 )
N0
BPSK: P2 =
(14.40)
1æ
ç1 2 ççè
gb
1 + gb
ö
÷
÷
÷
ø
(14.41)
(14.41) can be derived below:
P2 =
¥
ò0
P2 ( gb )p( gb )d gb =
= Q ( 2gb )(- e ¥
¥
ò0
ò0
gb (1+ 1/ gb )
gb-
1
1
2 2 p
=
æ
ö1
1
1÷
G( 21) çç1 +
÷
÷
è
2 2 p
gb ø
e-
¥
¥
)0 -
gb / gb
=
ò0
Q ( 2gb )
e-
1 e
gb
gb (1+ 1/ gb )
2 p
1/ 2
d gb =
1/ 2
=
1æ
ç1 2 çèç
gb / gb
gb-
d gb
1/ 2
d gb
¥
1
1
e2 2 p ò0
gb ö
÷
÷
÷
1 + gb ø
gb (1+ 1/ gb )
gb-
1/ 2
d gb
Here we have used the Gamma function1:
G(z ) =
¥
ò0
t z - 1e - t dt
¥
= k z ò t z - 1e - kt dt
0
BFSK: P2 
(Re(z ) > 0)
(Re(z ) > 0, Re(k ) > 0)
1
b 
1 

2
2 b 
(14.42)
If the fading is sufficiently rapid to preclude the estimation of a stable phase reference by
1
See Abrammowitz, M and Stegun, I. A., “Handbook of mathematical functions with formulas, graph, and
mathematical tables,” Dover, 1970, p. 255.
16
averaging the received signal phase over many signaling intervals, DPSK is an alternative
signaling method. DPSK requires phase stability over two consecutive signaling intervals; it
is quite robust in the presence of signal fading.
For a nonfading channel, the error probability of DPSK is expressed as
P2 ( gb ) =
1 e
2
gb
(14.43)
The average error probability by using (14.39) is
P2 =
1
2(1 + gb )
(14.44)
(II). Noncoherent detection:
Assume we don’t estimate the phase shift and use noncoherent detection for orthogonal FSK.
The nonfading error probability is
P2 ( gb ) =
1 e
2
gb / 2
(14.45)
The average error probability is then
P2 =
1
2 + gb
(14.46)
For large SNR,  b
ìï 1/ 4gb
ïï
ïï 1/ 2g
b
ï
P2 » í
ïï 1/ 2gb
ïï
ïï 1/ gb
ïî
1
for coherent P SK
for coherent , ort hogonal FSK
for DP SK
(14.47)
for noncoherent , ort hogonal FSK
17
(b). Nakagami fading: when a is Nakagami distributed, g = a 2E / N 0 has the probability
density function
p( g ) =
mm
g m - 1e - m g / g
G(m )g m
(14.48)
where g = E [a 2 ]E / N 0 .
14.4 Diversity techniques for fading multipath channels
Deep fade (large attenuation) in the channel will induce significant errors in reception.
Diversity techniques are used to overcome the problem of deep fade in the channel. If we
can supply the receiver several replicas of the same information signal transmitted over
independently fading channels, the probability that all the signal components will fade
simultaneously is reduced considerably. That is, if p is the probability that any one signal
will fade below some critical value, then p L is the probability that all L independently
fading replicas of the same signal will fade below the critical value.
(A). Frequency diversity: The same information-bearing signal is transmitted on L carriers,
where the separation between successive carriers equals or exceeds the coherence bandwidth
( f )c of the channel.
(B). Time diversity: The same information-bearing signal is transmitted in L different time
slots, where the separation between successive time slots equals or exceeds the coherence
time ( t )c of the channel.
The fading channel fits the model of a bursty error channel; frequency diversity or/and time
18
diversity can be viewed as a repetition coding through frequencies or/and time slots.
(C). Space diversity: The same information-bearing signal is transmitted and received over
multiple antennas, where the space separation between antennas must sufficiently far apart
that the multipath components in the signal have significantly different propagation delays at
the antennas. Usually a separation of a few wavelengths is required between two antennas in
order to obtain signals that fade independently.
(D). RAKE technique: Use of a wideband signal, whose bandwidth W is much greater than
the coherence bandwidth ( f )c of the fading channel (therefore, the channel is treated as a
frequency selective channel from the viewpoint of the wideband signal). Such a signal
bandwidth will resolves the multipath components and, thus, provide the receiver with
several independently fading signal paths (path diversity in the time domain). The time
resolution is 1/W, with a multipath spread of Tm seconds; there are Tm /(1/ W )  TmW
resolvable signal components. Besides, since Tm  1/( f )c , the number of resolvable signal
components may also be expressed as W /( f )c . Thus, the use of a wideband signal may be
viewed as another method for obtaining frequency diversity (in the frequency domain) of
order L  W /( f )c .
14.4.1 Binary signals
Assume that
(1). There are L diversity channels.
(2). Each channel is frequency nonselective and slowly fading.
(3). The envelope of signal in each channel is Rayleigh distributed.
(4). The fading processes among the L diversity channels are statistically independent.
(5). The signal in each channel is corrupted by a zero-mean AWGN process.
(6). The noise processes in the L channels are mutually statistically independent, with
identical autocorrelation functions.
The equivalent low pass received signals for the L channels can be expressed as
rlk (t ) = a k e -
jf k
skm (t ) + z k (t ),
k = 1, 2, L , L , m = 1, 2
(14.49)
where {a ke - j f k } represent the attenuation factors and phase shifts for the L channels, the
equivalent low pass signal, skm (t ) (note: it is better to use slkm (t ) ) denotes the m-th signal
19
transmitted on the k-th channel, and zk (t ) denotes the AWGN on the k-th channel. All
signals in the set {skm (t )} have the same energy.
A block diagram illustrating the binary digital communication system model is as below.
The optimum demodulator for the signal received from the k-th channel consists of two
matched filters with impulse responses as
bk 1(t ) = s *k 1 (T - t )
(14.50)
bk 2 (t ) = s *k 2 (T - t )
(14.51)
As a result, 2L matched filters are required. However, if BPSK is employed,
then sk 1 (t )   sk 2 (t ) , so only a single matched filter is needed for each path. Following the
matched filters is a combiner that forms the two decision variables.
The combiner that achieves the best performance is the one on which each matched filter
output is multiplied by the corresponding complex-valued (conjugate) channel gain a ke j f k .
Here assume channel attenuations {  k } and phase shifts {k } are estimated without error.
The channel gain a ke j f k is treated as a weighting function; consequently, a strong signal
carries a larger weight than a weak signal.
After the complex-valued weighting operation is performed, two sums are formed. One is
the real part of the weighted outputs from the matched filters corresponding to a transmitted
signal “0”; the other is the real part of the weighted outputs from the matched filters
corresponding to a transmitted signal “1”. This optimum combiner is called a maximal ratio
combiner (MRC).
(a). Rayleigh fading channel
(I). BPSK with Lth-order diversity
Although in general 2L matched filters (correlators) is required, only L matched filters is
needed for BPSK. The signals are set to be s11 * (t ) = s21 * (t ) = L = sL 1 * (t ) and
20
s12 * (t ) = s22 * (t ) = L = sL 2 * (t ) = - s11 * (t ) .

rl1 (t )
T 1e j1
s *11 (t )


 ()dt
 ()dt
1e
s *12 (t )  s *11 (t )


rlL (t )
Re[ ]

 ()dt
T Le jL
s *L1 (t )



j1


Re[ ]
2L matched filters
2006-05-21

 ()dt
Le j
s *L2 (t )  s *L1 (t )
L
The receiver structure consisting of 2L correlators.
rl1 (t )


 ()dt
1e j
s *11 (t )
1
T
rlL (t )

s *L1 (t )
 ()dt


 Le
jL
Re[ ]
L matched filters
2006-05-21
The receiver structure consisting of L correlators.
Note that the signals used for basis functions are not “normalized” yet. By following the
receiver structure of L matched filters, the output of the MRC can be expressed as a single
decision variable in the form
21
L
æ
U = Re ççç2E b å a k2 +
çè
k= 1
L
ö
÷
2
÷
a
N
=
2
E
b å ak +
å k k ÷÷ø
k= 1
k= 1
L
L
å
(14.52)
a k N kr
k= 1
where N kr denotes the real part of the complex-valued Gaussian noise variable
N kr
ïì
= Re ïí e j f k
ïï
îï
ïü
ï
z
(
t
)
s
*
(
t
)
dt
ý
k
lkm
ò
ïï
0
ï
þ
T
(14.53)
To derive above equation, we consider the 1st channel. The input to the combiner, the random
variable R1(T ) , is
T
T
R1(T ) = a 1e j f 1 ò rl 1 (t )s11 * (t )dt = a 1e j f 1 ò [a 1e 0
=
s11(t ) + z 1(t )]s11 * (t )dt
0
T
a 12
jf 1
(14.54)
T
ò s11(t )s11 * (t )dt + a 1 e ò z1(t )s11 * (t )dt = a 12 (2E b ) + a 1N 1
0
0
1444444442
444444443
jf 1
N1
At the output of the combiner, the decision variable U is
L
U = Re å
k= 1
L
{2Eba k 2 + a k N k } = 2Eb å a k 2 +
k= 1
L
å
(14.55)
a k N kr
k= 1
U is a Gaussian random variable with conditional mean and conditional variance conditioned
on ak
L
E (U ) = 2Eb å a k 2
(14.56)
k= 1
L
L
ö÷2
÷
=
å å a k a m E (N kr N mr ) =
÷
÷
ø
k= 1m = 1
T
L
ìï
ü
ï
= å a k 2E {Re ïí e j f k ò z k (t )s *k (t )dt ïý}2
ïï
ïï
k= 1
0
ï
îï
þ
æL
s U 2 = E ççç å a k N kr
çèk = 1
L
å
a k 2E (N kr 2 )
k= 1
T
T
æ1
ö÷2
1 - jf k
çç j f k
= å a k E ç e ò z k (t )s *k (t )dt + e
ò z k * (t )sk (t )dt ø÷÷÷÷
ççè 2
2
k= 1
0
0
T T
L
æ
1 ç
= å a k 2 E ççe j 2f k ò ò z k (t )z k ( e)sk * (t )sk * ( e)dtd e
4 ççè
k= 1
0 0
L
2
T T
ö
÷
÷
z
*
(
t
)
z
*
(
e
)
s
(
t
)
s
(
e
)
dtd
e
÷
k
k
k
k
òò
÷
÷
ø
0 0
T T
+ 2 ò ò z k (t )z k * ( e)sk * (t )sk ( e)dtd e + e 0 0
T T
L
=
(14.57)
j 2f k
L
T
1
1
a k 2E [ò ò z k (t )z k * ( e)sk * (t )sk ( e)dtd e ] = å a k 2E [ò 2N 0 | sk (t ) |2 dt ]
å
2 k= 1
2 k= 1
0 0
0
L
= 2E bN 0 å a k 2
k= 1
22
Here we have used E (N kr N mr ) = E (N kr ) ´ E (N mr ) = 0, k ¹ m because of the statistical
independence property for different paths.
The SNR at the output of combiner is the ratio of the power of signal component to the
power of noise component as
L
L
é
ù2
2
ê2E b å a k 2 ú
2
E
b å ak
ê
ú
[E (U )]2
k
=
1
û =
k= 1
SNR o =
= ë
L
N0
sU 2
2E bN 0 å a k 2
(14.58)
k= 1
By defining  b , the SNR per bit, is
L
Eb å a k 2
k= 1
gb =
N0
L
=
å
gk
(14.59)
k=1
the probability of error is that U is less than zero and is expressed by
P2( gb ) = Q ( SNR o ) = Q ( 2gb )
(14.60)
Note that SNRo = [E (U )]2 / s U 2 is the signal-to-noise (power) ratio and is also the sole
parameter on which the error probability of receiver depends. On the other hand, gb is a
signal energy-to-noise power spectral density per bit, or simply signal-to-noise per bit
( gb = E b / N 0 can be seen in AWGN environment). Because of the attenuation,
gk = E a k 2 / N 0 is the instantaneous SNR on the kth channel. To obtain the average error
probability of (14.60), we need to find the probability density, p( b ) .
Here we use the characteristic function of  b to find the probability density. Since  k is
Rayleigh distributed,  k 2 is chi-square distributed, so is gk = E ba k 2 / N 0 . For L=1, gb = g1 ,
the characteristic function of g1 is
y g1 ( jv ) = E (e jv g 1 ) =
1
1 - jv gc
(14.61)
where  c is the average SNR per channel, which is assumed to be identical for all channels.
That is
gc =
Eb
E (a k 2 )
N0
(14.62)
independent of k. Since the fading on the L channels is assumed to be mutually statistically
independent, the {gk } are statistically independent, and hence, the characteristic function
23
for the sum  b is simply the result in (14.61) raised to the Lth power.
y gb ( jv ) =
1
(1 - jv gc )L
(14.63)
But this is the characteristic function of a chi-square distributed random variable with 2L
degrees of freedom (see Chapter 2). The probability density is
p( gb ) =
1
gbL - 1e (L - 1) ! gc L
gb / gc
(14.64)
The average error probability is
¥
P2 =
L - 1 æL - 1 + k ö
L
1
P
(
g
)
p
(
g
)
d
g
=
[
(1
m
)]
å ççççè k ÷÷÷÷ø[12 (1 + m)]k
ò 2 b b b 2
k= 0
0
(14.65)
where
gc
1 + gc
m=
(14.66)
(14.65) can be derived as follows.
P2 =
¥
¥
ò P2 ( gb )p( gb )d gb =
ò Q(
0
2gb )
0
1
gbL - 1e (L - 1) ! gc L
gb / gc
d gb
ég L - 1 L - 1
ù¥
k (L - 1)(L - 2) L (L - k )
( L - 1- k ) ú
ê b
gb
ê - 1 + å (- 1)
ú
( -gc1)k + 1
k= 1
êë gc
ú
û0
¥
L- 1
- gb
L- 1
é
1
(L - 1)(L - 2) L (L - k ) (L - 1- k ) ù
1
e
g
úd gb
+ò
g - 2 e - gb / gc êê b- 1 + å (- 1)k
gb
L 2 p b
- 1)k + 1
ú
(
L
1)
!
g
(
c
g
k
=
1
ê
ú
g
c
c
ë
û
0
¥
L
1
é
1
1
1
= e - gb (1+ 1/ gc ) êêgbL - 1- 2 + å (L - 1)(L - 2) L (L - k )( gc )k gb(L - 1- k ò
L
1
2 (L - 1) ! gc 2 p
k= 1
ë
0
¥
ì
ïï
1
1
1
= e - gb (1+ 1/ gc ) gbL - 1- 2d gb
í
2 (L - 1) ! gc L - 1 2 p ïï ò
î 0
¥
L- 1
ü
ï
1
+ å (L - 1)(L - 2) L (L - k )( gc )k ò e - gb (1+ 1/ gc ) gb(L - 1- k - 2 )d gb ïý
ïï
k= 1
0
þ
1
L- 2
ì
ïï
æ gc ÷
ö
1
1
= í G(L - 21) çç
÷
è 1 + gc ÷
ø
2 (L - 1) ! gc L - 1 2 p ïïî
1
L- 1
ïï
æ g öL - k - 2 ü
+ å (L - 1)(L - 2) L (L - k )( gc )k G(L - k - 21) çç c ÷
ý
÷
÷
è 1 + gc ø
ïï
k= 1
þ
1
=
Q ( 2gb ) e (L - 1) ! gc L
gb / gc
If the average SNR per channel  c
1 (10 dB) , then
1
2
(1   )  1 and
1
2
1
2)
ù
úd gb
ú
û
(1   )  1/ 4 c .
Besides,
L - 1 æL
çç
ç
è
k= 0ç
å
- 1+ kö
÷
÷
=
÷
÷
k
ø
æ2L - 1 ö
÷
çç
÷
çç L ÷
÷
è
ø
(14.67)
24
L æ2L - 1 ö
æ1 ö
÷
ç
÷
÷
P2 » ççç
÷ çç
÷
÷
è 4gc ÷
ø çè L ø
(14.68)
Thus, the error probability has been improved with diversity techniques because it has been
reduced from 1/ gc to 1/ gc raised to the Lth power.
(2). Coherent BFSK with Lth-order diversity
The two decision variables at the output of the maximal ratio combiner may be expressed as
L
æ
U 1 = Re ççç 2E b å a k 2 +
è
k=1
ö
÷
a kN k1 ÷
÷
÷
ø
k=1
L
å
æL
ö
÷
U 2 = Re çç å a k N k 2 ÷
÷
÷
çè k = 1
ø
(14.69)
(14.70)
where sk 1 (t ) was assumed to be transmitted, N k 1 and N k 2 are noise components at the
output of the matched filters, respectively. The computation of error probability is similar to
that for BPSK. The conditional error probability is
P2( gb ) = Q( gb )
(14.71)
Due to the same fading situation, (14.64) and (14.65) can still be used. For coherent
detection, the parameter  is defined as
m=
gc
2 + gc
(14.72)
Finally, for large values of  c , the error probability can be approximated as
L æ2L - 1 ö
æ1 ö
÷
ç
÷
÷
P2 » ççç
÷
÷
÷ ççèç L ø
÷
è 2gc ø
(14.73)
(3). DPSK with Lth-order diversity
(4). Noncoherent BFSK with Lth-order diversity
The noncoherent FSK system under AWGN was derived in another file. The results can be
directly applied here. Assume s1(t ) was transmitted, the two decision variables at the output
of maximum combiner may be expressed as
25
L
å
U1 =
| 2E ba ke -
jf k
+ N k 1 |2
k=1
L
å
U2 =
(14.74)
| N k2 |
2
k=1
where the complex-valued Gaussian noise variables of k-th channel are
Tb
N k1 =
ò zk (t )sk 1 * (t )dt
0
Tb
Nk2 =
ò zk (t )sk 2 * (t )dt
0
The probability of error in (14.65) also applies to square-law-combined FSK with parameter
 defined as
m=
gc
2 + gc
(14.75)
An alternative derivation (Pierce, 1958) is to decide the probability densities of U1 and U 2
first. Since  k e  jk , N k 1 and N k 2 are zero-mean Gaussian distributed, the decision
variables U1 and U 2 are distributed according to a chi-square probability distribution with
2L degrees of freedom.
p(U 1 ) =
1
(2s 12 )L (L
æ U1 ö
÷
U 1L - 1 exp ççç÷
÷
è 2s 12 ø
- 1)!
(14.76)
where
s 12 =
1
E {| 2E s a ke 2
jf k
+ N k 1 |2 } = 2E s N 0(1 + gc )
(14.77)
Similarly,
p(U 2 ) =
1
(2s 22 )L (L
æ U2 ö
÷
U 2L - 1 exp ççç÷
÷
è 2s 22 ø
- 1)!
(14.78)
with
s 22 = 2E s N 0
(14.79)
It can be proved that the resulting average probability of error is (14.65) with  defined in
(14.75).
Summary of received signal model and error probabilities
(I). Single channel over Rayleigh fading without channel gain estimation: (Section 14.3)
26
Received signal: rl (t ) = a e - j f sl (t ) + z (t ),
0£ t £ T
Average SNR: gb = E ( a 2 )E b / N 0
Coherent reception
Noncoherent reception
P2,BPSK,Coherent =
1æ
ç1 2 ççè
gb ö
÷
÷ (14.35)
ø
1 + gb ÷
P2,DPSK =
P2,BFSK,Coherent =
1æ
ç1 2 ççè
ö
gb ÷
((14.40)
÷
÷
2 + gb ø
P2,BFSK ,NonCoherent =
1
2(1 + gb )
(14.44)
1
2 + gb
(14.46)
(II). Multiple channels over Rayleigh fading with channel gain estimation: (Section 14.4)
Received signal: rlk (t ) = a ke -
jf k
L
Average SNR: gb =
gc =
skm (t ) + z k (t ),
Eb
a 2 =
N 0 kå= 1 k
k = 1, 2, L , L , m = 1, 2
L
å
gk = L gk , ( E ( a 12 ) = E ( a 22 ) = L = E ( a L 2 ) )
k=1
Eb
E ( a k 2 ) = gk
N0
Coherent reception
L - 1 æL
P2,BPSK ,Coh = [12 (1 - m)]L å ççç
è
k= 0ç
m=
- 1+ kö
÷
÷
[12 (1 + m)]k
÷
÷
k
ø
gc
1 + gc
L - 1 æL - 1 + k ö
÷
÷
P2,DPSK = [12 (1 - m)]L å ççç
[12 (1 + m)]k
÷
÷
k
ç
ø
k= 0è
m=
L - 1 æL - 1 + k ö
÷
÷
P2,BFSK ,Coh = [12 (1 - m)]L å ççç
[12 (1 + m)]k
÷
÷
k
ç
ø
k= 0è
m=
Noncoherent reception
gc
2 + gc
gc
1 + gc
L - 1 æL - 1 + k ö
÷
÷
P2,BFSK ,NC = [12 (1 - m)]L å ççç
[12 (1 + m)]k
÷
÷
k
ç
ø
k= 0è
m=
gc
2 + gc
(b). Nakagami fading
All the derivations for Nakagami fading are the same as those for Rayleigh fading but
with the different probability density function.
The Nakagami probability density function for the single-channel SNR parameter
 b   2 Eb / N 0 is given as
p( gb ) =
1
g m - 1e G(m )( gb / m )m b
gb / ( gb / m )
(14.80)
On the other hand, the Rayleigh probability density function for the L-channel SNR
with gc = gb / L
27
p( gb ) =
1
gbL - 1e (L - 1) ! gc L
gb / gc
(14.81)
Comparing the two formulae, we observe that when L=m=integer, the two formulae are
identical, which means that the performance of single channel system under Nakagami
fading at the fading figure m is the same as that of L-channel diversity system under
Rayleigh fading.
How about the formula for the Kth order diversity system in a Nakagami fading channel?
Because of the similarity in the above equations, the characteristic function of a Nakagami
random variable must be in the form
1
(1 - jv gb / m )m
y gb ( jv ) =
(14.82)
which is consistent with (14.63) for the characteristic function of the combined signal in a
system with Lth order diversity in a Rayleigh fading channel. Consequently, a K-channel
system in a Nakagami fading channel with independent fading is equivalent to an L=Km
channel diversity in a Rayleigh fading channel.
14.4.3 M-ary orthogonal signals
The performance of M-ary orthogonal signals transmitted over a Rayleigh fading channel is
assessed. The orthogonal signals may be viewed as M-ary FSK with a minimum frequency
separation of an integer multiple of 1/T, where T is the signaling interval.
 The same information-bearing signal is transmitted on L diversity channels.

Each diversity channel is assumed to be frequency-nonselective and slowly fading.

The fading processes on the L channels are assumed to be mutually statistically
independent.

An AWGN process corrupts the signal on each diversity channel and the processes are
mutually statistically independent.
In this section, only noncoherent detection with square-law combining is discussed.
The output of the combiner containing the signal is
L
U1 =
å
| 2E s a ke -
jf k
+ N k 1 |2
(14.83)
k=1
while the output of the remaining M-1 combiners are
L
Un =
å
| N kn |2, n = 2, 3, L , M
(14.84)
k=1
28
The probability of error is 1-probability of correct detection. That is,
1 - Pr (U 1 > U n , n = 2, 3, L , M ) = 1 - Pr (U 1 > U 2, U 1 > U 3, L ,U 1 > U M )
The random variables U 1,U 2, L ,U M are mutually independent because of the orthogonality
of signals and mutually statistically independent AWGN. Consequently, the probability
density of U1 is in (14.76). On the other hand, U 2, L ,U M are identically distributed and
described in (14.78).
When U1 is fixed, the joint probability Pr (U 1 > U 2, U 1 > U 3, L ,U 1 > U M ) is equal to
Pr (U 1 > U 2 ) raised to the M-1 power.
U1
P (U 2 < U 1 ) =
ò p(U 2 )dU 2 =
0
k
æ U öL - 1 1 æ U 1 ö
÷
çç1 - exp ççç- 12 ÷
÷
÷
å
2
÷ k ! èç 2s 2 ø
÷
è 2s 2 ø
k= 0
(14.85)
where  22  2 Es N 0 . The average probability of error is obtained by averaging over p(U1 ) .
The form of the average probability of error is (Hahn, 1962)
¥
pM = 1 -
M- 1
é
öL - 1 1 æ U 1 ö÷k ù
ê1 - exp æ
ú
çç- U 12 ÷
ç
dU 1
÷
÷
÷å k ! èçç 2s 22 ø÷ ú
ê
èç 2s 2 ø
k= 0
ë
û
L- 1
ö
ö æ
U1 ÷
U 1k ÷
çç1 - e - U 1
÷
´
dU 1
÷
å
÷
ø çè
1 + gc ÷
k! ÷
ø
æ U1 ö
1
ò (2s 12 )L (L - 1) !U 1L - 1 exp çççè- 2s 12 ø÷÷÷´
0
¥
æ
1
= 1- ò
U 1L - 1 exp çççL
è
(1
+
g
)
(
L
1)
!
c
0
(14.86)
k= 0
where gc is the average SNR per diversity channel. The average SNR per bit is
gb = L gc / log2 M = L gc / k . Because
æL - 1 U 1k
çç
çè å k !
k= 0
n
ö
÷
÷
=
÷
÷
ø
n ( L - 1)
å
bknU 1k
(14.87)
k= 0
where b kn is the set of coefficients in the above expansion. As a result,
æM - 1 ÷
ö
÷
1)n + 1 çç
÷ n ( L - 1)
çè n ÷
æ 1+ g
ø
å bkn (L - 1 + k ) !çççè1 + n + cn gc
(1 + n + n gc )L k = 0
M - 1 (-
PM =

1
(L - 1) ! nå= 1
(14.88)
When there is no diversity (L=1), the error probability becomes
M - 1 (-
PM =
å
n=1

k
ö
÷
÷
÷
ø
æM - 1 ÷
ö
÷
1)n + 1 çç
÷
÷
çè n ø
1 + n + n gc
The bit error probability can be obtained from PM as Pb = PM
(14.89)
2k - 1
.
2k - 1
29
14.5 Digital signaling over a frequency-selective, slowly
fading channel
If the spread factor of the channel is Tm Bd
satisfy W ( f )c and the signal duration T
1, we may confine the signal bandwidth to
( t )c . Hence, the channel is frequency
nonselective, slowly fading. The techniques used to overcome the severe fading are diversity.
On the other hand, if a channel with bandwidth W much wider than ( f )c is available now,
and if we would like to fully utilize such a wide bandwidth but keep the channel frequency
nonselective slowly fading, what can we do? One method is to partition the full bandwidth
into several sub-channels (FDM), each of which has a mutual separation at least ( f )c .
Then the same signal can be transmitted on the FDM sub-channels, thus, frequency diversity
is obtained. The other approach is to use RAKE technique, which will discuss in the
following section.
14.5.1 A tapped-delayed-line channel model
Instead of using FDM, a wideband signal s (t ) 1 is employed to cover the available
bandwidth, W (bandpass frequency band). The channel is still assumed to be slowly fading
by virtue of the assumption that T = (D t )c . The bandwidth of the equivalent lowpass signal2
sl (t ) is then confined to | f |£ 12W , and just such band limitation, the equivalent lowpass
signal can be decomposed into a series expression by using the sampling theorem.
¥
sl (t ) =
å
n=- ¥
n sin[pW (t - n / W )]
sl ( )
W
pW (t - n / W )
(14.90)
The Fourier transform of sl (t ) is
ìï 1
ïï
S l ( f ) = ïí W
ïï
ïïî
¥
å
n=- ¥
n
sl ( ) e W
0
j 2p fn / W
, f £
1
W
2
(14.91)
, elsewhere
The noiseless received signal from a frequency-selective channel (because here we already
assume W ? (D f )c ) was previously expressed in the form (See Equation 14.2-2)
1
Such a wideband signal can be achieved by using some spectrum-spreading signals, for example, the
so-called direct-sequence spreading signal, if the occupation of the wideband spectrum is permitted.
2
Usually, transmitted signals may be in direct sequence BPSK (DS/BPSK), for example, s(t ) = c(t )d (t ) ,
where c(t ) is the spectrum-spreading waveform and d (t ) is the information signal.
30
¥
ò C ( f ; t )S l ( f )e j 2p ft df
r l (t ) =
(14.92)
- ¥
where C ( f ; t ) is the time-variant transform function of c ( ; t ) , the time-variant impulse
response. Substituting (14.91) into (14.92) yields
¥
ò- ¥ c(t ; t )sl (t -
r l (t ) =
=
=
¥
1
W
å
n=- ¥
n
sl ( )
W
t )d t =
ò
f £
C ( f ; t )e j 2p f (t -
df »
n /W )
1
2W
¥
1
W
¥
ò- ¥ C ( f ; t )S l ( f )e j 2p ft df
n
1
sl ( )c( t14442
- n 4443
/ W ;t ) =
W
W
{ t - t at t = n / W
n=- ¥
t
14444444444442
4444444444443
å
¥
1
W
å
n=- ¥
¥
å
sl (t -
n=- ¥
¥
n
sl ( ) ò C ( f ; t )e j 2p f (t W -¥
n /W )
df
(14.93)
n
)c(n / W ; t )
W
A convolut ion-sum form
Here we have used the assumption that the available channel bandwidth W (also the
bandwidth of the wideband signal) is also the bandwidth of the time-variant transfer
function C ( f ; t ) . Thus we may release the integration bounds from  12 W to  . Observing
the above equation, we know that the received lowpass signal can be expressed as a
convolution sum.
If we define cn (t ) =
1 n
1
c( ; t ) =
c( t ; t ) | t = n / W
W W
W
as a set of time-variable channel coefficients,
the received lowpass signal can be expressed as
¥
r l (t ) =
å
(14.94)
cn (t )sl (t - n / W )
n=- ¥
That is, the time-variant frequency-selective channel can be modeled or represented as a
tapped delay line (TDL) with tap spacing 1/W (or Tc ) and tap weight coefficients {cn (t )} .
As a matter of fact, we deduce from(14.94) the low-pass impulse response for such a
TDL-based channel is1
¥
c( t ; t ) =
å
(14.95)
cn (t )d( t - n / W )
n=- ¥
¥
1
You may prove it by using r l (t ) = c( t ; t ) * sl (t ) =
ò c( t ; t )sl (t - t )d t =
å
cn (t )sl (t - n / W ) , and
n=- ¥
compare with Equation (14.5).
31
The corresponding time-variant transfer function is1
¥
C ( f ; t ) = F [c( t ; t )] =
å
cn (t ) e -
j 2p fn / W
(14.96)
n=- ¥
That is, with an equivalent low pass signal bandwidth 21W andW ? (D f )c , the resulting
delay resolution in the MIP (multipath intensity profile) or MDP (multipath delay profile)
is 1/ W . Since the multipath spread is T m , then the useful (truncated) total taps in the TDL
model for frequency-selective fading channel is
êT
L = ê m
êë1/ W
ú
ú+ 1 = ëT mW û+ 1
ú
û
(14.97)
Henceforth the noiseless received signal can be expressed as
L
rl (t ) =
å
cn (t )sl (t - n / W ) 2
(14.98)
n=1
Indeed, this channel model is a kind of finite length impulse response (FIR) filter. Note that
from WSS assumption, cn (t ) =
1 n
1
c( ; t ) =
c( t ; t ) | t = n / W
W W
W
are complex-valued stationary
processes and from the US assumption {cn (t )} are mutually uncorrelated, henceforth, if
{cn (t )} are Gaussian random processes, they are statistically independent. The statistical
distribution of magnitude of taps | cn (t ) |= a n (t ) may be Rayleigh, Nakagami, etc., and that
of the phases n (t ) are usually treated as uniform.
1
Equation (14.96) in some sense can be treated as a Fourier series representation of a periodic signal C ( f ; t )
in the frequency domain, f, where the period of the time-variant transfer function is 1/ W = T c 1/ W = T c .
The important thing here is the corresponding time domain or, precisely speaking, the time-delay domain, t .
L
2
The corresponding (multipath) channel impulse response is c( t ; t ) =
å
cn (t )d( t - n / W ) , which is a
n=1
specific example of the generic linearly time-varying discrete multipath model of (14.5). On the other hand, the
frequency response of the channel is the Fourier transform as shown in (14.96), where the number of branches
L
is L, C ( f ; t ) = F [c( t ; t )] =
å
cn (t ) e -
j 2p fn / W
. If the channel is slowly fading, the coefficient can be viewed
n=1
as a constant (precisely a random variable) in a symbol time, that is, cn (t ) = cn . For a general wireless mobile
channel model also see: Rappaport T. S., Wireless Communications: Principles and Practice, 2nd ed., Section
5.2.
32
sl (t )
1/W
c1 (t )
1/W

c2 (t )
1/W

1/W
c3 (t )
cL (t )




L
rl (t )  cn (t )sl (t  n / W )
n1
Additive noise z(t )
The tapped delay line model for frequency selective fading channel.
14.5.2 RAKE demodulator
A frequency selective channel is modeled as a TDL with statistically independent
time-variant tap weights cn (t ) . The statistical independent tap weights provide with L replicas
of the same transmitted signal at the receiver. Such a receiver achieves the same performance
as the equivalent Lth-order diversity communication system.
Assume binary signaling with “equal-energy” is emitted over the frequency selective
channel sl1 (t ) and sl 2 (t ) , which may be “antipodal or orthogonal” and the signaling
interval T ? T m . Because of T ? T m , inter-symbol interference (ISI) due to multipath is
very small and can be neglected. The bandwidth of the signal exceeds the coherence
bandwidth of the channel; therefore, the received signal can be expressed as
L
rl (t ) =
å
ck (t )sli (t - k / W ) + z (t ) = vi (t ) + z (t ), 0 £ t £ T , i = 1, 2
(14.99)
k=1
where z (t ) is a complex-valued zero-mean WGN process, and {ck (t ) } are assumed to be
known or can be precisely estimated. At the receiver, the optimum demodulator consists of
two filters matched to v1 (t ) and v2 (t ) , respectively. (Or cross correlators instead; also note
that two filters are not matched to slm (t ) ). For coherent detection of the binary signals, the
outputs for the nth individual multipath signal are1
Tb + n / W
rn =
ò
rl (t )cn * (t )sli * (t - n / W )dt , n = 1, 2, L , L
(14.100)
n /W
Here sli * (t - n / W ) (corresponding v1 (t ) or v2 (t ) ) is the matched filter matched to the
nth delay, and the tap weight cn (t ) is obtained from the estimation. By the assumption of
T = Tb ? T m
1
, the above equation may be approximately expressed as
Simon, M. K, et al, Spread Spectrum Communications Handbook, Revised ed., 1994, pp. 440-451.
33
Tb + n / W
ò
rn =
Tb
rl (t )cn * (t )sli * (t - n / W )dt »
n /W
ò rl (t )cn
* (t )sli * (t - n / W )dt , n = 1, 2, L , L
(14.101)
0
The decision variables are
L
U   rn
n 1
In the textbook, the decision variables when the transmitted signal is sl1(t ) are
Um
T
é
ù
= Re êck * (t ) ò rl (t )slm * (t - k / W )dt ú, m = 1, 2
0
ë
û
L
é T
ù
éL
ù
T
= Re êò rl (t )( å ck * (t ) slm * (t - k / W ))dt ú = Re êå ò rl (t )ck * (t ) slm * (t - k / W )dt ú
ê 0
ú
êk = 1 0
ú
k=1
ë
û
ë
û
L
éL
ù
T
= Re êå ò ( å cn (t ) sl 1(t - n / W ) + z (t ))ck * (t ) slm * (t - k / W )dt ú
êk = 1 0 n = 1
ú
ë
û
L
L
é
T
= Re êå å ò cn (t )ck * (t ) sl 1(t - n / W )slm * (t - k / W )dt
êk = 1 n = 1 0
ë
L
ù
T
+ å ò z (t )ck * (t ) slm * (t - k / W )dt ú
ú
0
k=1
û
L
= Re[å | ck (t ) |2
k=1
T
ò0
L
sl 1(t - n / W )slm * (t - k / W )dt ) +
L
å å
k=1n=1
k¹ n
T
cn (t )ck * (t ) ò sli (t - n / W )
0
éL
ù
T
´ slm * (t - k / W )dt ]+ Re êå ò z (t )ck * (t ) slm * (t - k / W )dt ú
êk = 1 0
ú
ë
û
L
= Re å | ck (t ) |2
+
k=1
L
T
å ò
k=1 0
T
ò0
sl 1(t - n / W )slm * (t - k / W )dt
ù
z (t )ck * (t ) slm * (t - k / W )dt ú, m = 1, 2
ú
û
(14.102)
Here the slowly fading channel assumption is made; hence, the coefficients are constant
during a single bit interval, and the autocorrelation property of sl (t ) is used. sl (t ) is a
spectrum-spreading signal whose autocorrelation approaches zero if the time lag between
sl (t - k / W ) and sl (t - n / W ) has | k - n |³ 1 . Besides, the bounds of integral should be as
those in (14.100) with the assumptionT b ? T m .
Also note again that vm (t ) is not a single function but a sum of slm (t ) and its time-lag
versions. See the following figure for the detail configuration.
34
An alternative realization of the optimum demodulator employs a single delay line through
which is passed the received signal rl (t ) . The signal at each tap is correlated
with ck * (t )slm * (t ), k = 1, 2, L , L, m = 1, 2 . This result for individual multipath signal can be
derived from (14.100) as
Tb + n / W
ò
rn =
rl (t )cn * (t )sli * (t - n / W )dt
n /W
(14.103)
Tb
=
ò rl (t +
n / W )cn * (t + n / W )sli * (t )dt n = 1, 2, L , L
0
L
éL
ù
T
U m = Re å rk = Re êêå ò rl (t + k / W )ck * (t + k / W )slm * (t )dt ú
ú, m = 1, 2
k= 1
ëk = 1 0
û
(14.104)
The benefit of the above equation lies in the simplification of reference signal by using a
single function sl (t ) instead of the compound signal vm (t ) . In the following figure, y ( t ) is
the received signal and Wss is the bandwidth of spreading waveform.
35
Because the system is causal, y(t + L / W ss ) may be treated as the head of the received signal,
the received signal can be rearranged as below. Following the change, the order of the path
has been reversed.
36
14.5.3 Performance of RAKE demodulator
Assume that
(1). The fading is sufficiently slow to allow us to estimate ck (t ) perfectly (without noise).
(2). Within any one signaling interval, ck (t ) is treated as a constant, i.e., ck (t )  ck .
(3). The signal is assumed to have zero autocorrelation at nonzero time offset,
Tb
ò sli (t -
n / W )sli (t - k / W ) » 0, k ¹ n , i = 1, 2
(14.105)
0
The decision variable may be expressed by
éL
ù
T
é T
ù
*
U m = Re êò rl (t )vm* (t )dt ú = Re êêå ck* ò rl (t )slm
(t - k / W )dt ú
ú, m = 1, 2
0
ë 0
û
ëk = 1
û
(14.106)
Suppose s1(t ) was transmitted; then the received signal is
L
rl (t ) =
å
cn sl 1(t - n / W ) + z (t ), 0 £ t £ T
(14.107)
n=1
Substituting (14.107) into (14.106) yields
L
éL
ù
éL
ù
T
T
*
*
êå ck*
U m = Re êêå ck* å cn ò sl 1(t - n / W )slm
(t - k / W )dt ú
+
Re
z (t )slm
(t - k / W )dt ú
ò
ú
ê
ú
0
0
ëk = 1 n = 1
û
ëk = 1
û
éL
ù
T
*
= Re êêå | ck |2 ò sl 1(t - k / W )slm
(t - k / W )dt ú
ú
0
ëk = 1
û
éL
ù
T
*
+ Re êêå ck* ò z (t )slm
(t - k / W )dt ú
ú , m = 1, 2
0
k
=
1
ë
û
(14.108)
When the binary signals are antipodal, a single decision variable suffices. Thus
L
æ
U 1 = Re ççç2E b å a k 2 +
çè
k= 1
L
L
ö
÷
akN k ÷
= 2E b å a k 2 + å a k Re(N k )
÷
÷
ø
k= 1
k= 1
k= 1
L
å
(14.109)
where a k = | ck | and
Tb
N k = e j f k ò z (t )sl 1 * (t - k / W )dt
(14.110)
0
When observing (14.109), we find it is identical to the output of an MRC in a system with
Lth-order diversity, Equation (14.55). Furthermore, if all the tap weights have the same mean
square value, E ( k 2 ) , then Equations (14.64) and (14.65) can apply; otherwise the
37
evaluation must be done again. The average error probability for distinct { k } is derived
below.
The error probability is
ïì Q ( 2gb ) r r = - 1 (antipodal )
P2 ( gb ) = Q ( gb (1 - r r )) = ïí
ïï Q ( gb ) r r = 0 (orthogonal )
î
(14.111)
and
L
gb =
Eb
a 2 =
N 0 kå= 1 k
L
å
gk
(14.112)
k= 1
{ k } follows the chi-squared distribution with two degrees of freedom
p( gk ) =
1 e
gk
gk / gk
(14.113)
where gk is the average SNR for the kth path, defined as
gk =
Eb
E (a k 2 )
N0
(14.114)
The characteristic function of gk is
Y gk ( j n ) =
1
1 - j ngk
(14.115)
But gb is the sum of L statistically independent components {gk } , the characteristic function
of gb is
L
Ygb ( j n ) =
Õ 1-
k= 1
1
j ngk
(14.116)
The error probability of  b can be obtained by taking the inverse Fourier transform of the
characteristic function as
L
p( gb ) =
pk e
k = 1 gk
å
gk / gk
, gb ³ 0
(14.117)
where
38
L
pk =
g
Õ gk -k gi
(14.118)
i= 1
i¹ k
The average error probability can be obtained by averaging over (14.117) as
L
P2 =
1
p
2 kå= 1 k
é
ê1 êë
gk (1 - r r ) ù
ú
2 + gk (1 - r r ) ú
û
(14.119)
and when gk ? 1
æ2L - 1 ÷
ö L
1
÷
P2 = ççç
÷
Õ
÷
çè L ø k = 1 2gk (1 - r r )
(14.120)
We have assumed the tap weights are estimated perfectly. In practice, the relative good
estimates can be obtained if the channel fading is sufficiently slow, e.g. (D t )c / T ³ 100 ,
where T is the signaling interval. The following figure demonstrates a method of estimating
the tap weights for binary orthogonal schemes.
For example, assume sl1 (t ) was transmitted, by using(14.104) we have
éL
ù
T
é T
ù
*
U m = Re êò rl (t )vm* (t )dt ú = Re êêå ò rl (t + k / W )ck* (t )slm
(t )dt ú
ú, m = 1, 2
ë 0
û
ëk = 1 0
û
where the received signal is from (14.99)
L
rl (t ) =
å
ck (t )sli (t - k / W ) + z (t ) = vi (t ) + z (t ), 0 £ t £ T , i = 1, 2
k=1
Thus, at the k-th tap, one of component of received signal is ck (t ) sl1 (t ) . Following the tap,
there have two branches; along the left branch the result is ck (t ) sl1 (t ) sl1 *(t )  z(t ) sl1 *(t )
39
and the right branch the result is ck (t )sl1 (t ) sl 2 *(t )  z (t ) sl 2 *(t ) . Following the definition of
the orthogonal signal (see p.177, Chapter 4),
é 2E b j 2p D mft j 2p fct
2E b
cos[2p fc t + 2p D mft ] = Re ê
e
e
êë T b
Tb
= Re[slm (t )e j 2p fct ], m = 1, 2, L , M , 0 £ t £ T b
sm (t ) =
ù
ú
ú
û
The signals will pass the lowpass filter, so it is better to the response in the frequency
domain.
The left branch:
(a). The signal component: assume the channel gain is constant over a single signaling
interval. That is, ck (t )  ck
The Fourier transform of ck (t ) sl1 (t ) sl1 *(t )  ck sl1 (t ) sl1 *(t ) of the left branch is
The Fourier transform of ck sl1 (t ) sl1 *(t ) is
ck
2 Eb
2 Eb
2E
 ( f  f ) *
 ( f  f )  ck b  ( f )
Tb
Tb
Tb
and the Fourier transform of ck (t ) sl1 (t ) sl 2 *(t )  ck sl1 (t ) sl 2 *(t ) of the right branch is
ck
2 Eb
2 Eb
2E
 ( f  f ) *
 ( f  2f )  ck b  ( f  f )
Tb
Tb
Tb
The sum of two signals pass the lowpass filter, the output of which is ck
ck
2 Eb
 ( f ) or
Tb
2 Eb
in the time domain because it is assumed ( t )c / T  100 , which means
Tb
Bd  1/( t )c  f /100 ( f  1/ Tb ) , hence  ( f  f ) is out of the band Bd and is
rejected.
(b). The rest of signal components and noise part at both branches contribute the uncertainty.
For example, the noise part of the left branch z (t ) sl1 *(t ) is a zero-mean Gaussian
distributed random process. That is, E[ z(t ) sl1 *(t )]  E[ z(t )]sl1 *(t )  0 and the
auto-covariance function
E[ z (t ) sl1 * (t ) z * (t   ) sl1 (t   )]  E[ z (t ) z * (t   )]sl1 (t   ) sl1 * (t )
 2 N 0 ( ) sl1 (t   ) sl1 * (t )  2 N 0 sl1 (t ) sl1 * (t )  2 N 0
2 Eb
Tb
It seems to have some problems. The first is that at the output of the lowpass filter an
additional gain 2 Eb / Tb which must be removed. The second is the output is ck (t ) not
ck * (t ) . Clearly, the function of complex conjugate operation must be embedded in the
lowpass filter.
The method for estimating the channel gains in the orthogonal signaling is not proper for the
antipodal signaling, because the addition of the two correlator outputs results in signal
40
cancellation. The configuration is below. A single correlator is used and its output is fed to
the input of the low pass filter after the information-bearing signal is removed. To
accomplish this, a delay of one signaling interval ( Tb ) is introduced into the channel
estimation procedure. First, the receiver must decide whether the information in the received
signal is +1 or –1, and second, it uses the decision to remove the information from the
correlator output prior to feeding it to the lowpass filter.
For example, at the k-th tap, one of the received signal component is ck (t ) sli (t ) , may be
ck (t ) sl1 (t ) or ck (t ) sl 2 (t )  ck (t ) sl1 (t ) , therefore, the output of the multiplier is
ck (t ) sli (t ) sl1 *(t )  ck (t ) . That the sign must be removed becomes apparent. However, it is
no way to know the “current information” until the detector makes the decision. As an
alternative, the previous (ready) information bit can be employed, which is the same to the
output of the delay (assume the previous decision is correct). In other words, the estimate
value ck * (t ) is the tap weight of the previous signaling interval.
Appendix 1. Matched filters
A received signal r (t ) may be expressed by
r (t )  s (t )  n (t )
where s (t ) is the signal component and n(t ) is the additive noise. Several cases are
discussed.
(1). Known signals in white noise
The matched filter can be expressed as
hm (t )  F 1 kS * ( f )e  j 2 ftm   ks (tm  t )
41
where tm represents the best observation time and k is a constant and can be set to one for
convenience. The impulse response of the optimum filter is the mirror image of the signal
component s (t ) , delayed by an interval tm . That is, the filter is matched to a specific signal
and so called the matched filter.
The phase response is also very important and above equation states that the phase shifts in
s (t ) should be negated in such a way that all frequency components in s (t ) add in phase at
exactly the time tm  t .
s (t ) is assumed to have a finite duration (0,T). The impulse response of the matched filter
s(tm  t ) can be obtained by folding s (t ) about the vertical axis and shifting it to the right
by tm seconds. Restricting consideration to the physically realizable case with minimum
delay, we may choose tm  T .
s(t )
The synthesis of a linear time-invariant system involves the design of a system to attain a
given impulse response and frequency transfer function. One way in which this can be
accomplished for the general case is by noting that the impulse response of an LTI system
can be approximated by a delay line that is tapped at various points and weighted by a set of
gain factors.
Consider a causal system with impulse response h(t ) and a causal input f (t ) . The output
g ( t ) can be expressed in terms of convolution integral as
t
t
0
0
g (t )   f ( )h(t   )d   f (t   )h( )d
The above integration can be approximated by
g (t ) 
t / 
 f (t  k  )h(k  )
k 0
The result can be realized by using a delay line with taps at the delays k  ; the output of
each tap is multiplied by the preset weight h( k  )  .
42
A filter constructed using a tapped delay line, tap weights, and adder in the configuration is
called a transversal filter. Both digital and analog methods can be used to implement
transversal filters. For digital implementation, shift registers or CCD (charge-coupled device)
can be used as delays. On the other hand, for analog implementation, SAW
(surface-acoustic-wave) filter can be used.
43


0
0
P2   P2 ( b ) p( b )d  b   Q ( 2 b )
1
 b L 1e  b /  c d  b
( L  1)! c L

 L 1

1
( L  1)!
 b /  c
k

Q
(
2

)
e
 ( L 1 k ) 
  ( 1)
b
L
1 k 1 b
( L  1)! c
( L  1  k )!(  c )
 k  0
 0

1
e  b  12  b /  c
b e
( L  1)! c L 2 
0


 L 1

( L  1)!
k
 ( L 1 k ) d  b
  ( 1)
1 k 1 b
( L  1  k )!(  c )
 k  0

 L 1 ( L  1)! c k 1 ( L 1 k  12 ) 
e  b (11/  c )  
b
d  b
 k  0 ( L  1  k )!

1
1
 
2 ( L  1)! c L 2 

1
1
 
2 ( L  1)! c L 2 
( L  1)! c k 1

k  0 ( L  1  k )!

1
1

1
2 2  (1  1/  c ) 2


e
  b (11/  c )
b
( L 1 k  12 )
d b
0
( L  k  12 )
1

L  k  12
L  k 1
(1  1/  c )
k  0 ( L  1  k )!  c
1
1

2 2 


L 1


0
L 1
L 1
1
 ( L  1  k )! (
k 0
L 1
1
1

1
2 2(1  1/  c ) 2
 ( L  1  k )!(
1
1

1
2 2(1  1/  c ) 2
 ( L  1  k )!(
1
1

1
2 2(1  1/  c ) 2
 (k ')!(
1 3  5 
1
k 0
L 1
c
 1)
L  k 1
 1)
L  k 1
1
k 0
L 1
1
k '0
1  L 1  2k   1   2 
 
2 2 k  0  k   4 
c
1
1 3  5 
 1) L  k 1
c
 (2( L  k  1)  1) 
2 L 1 k
 (2( L  k  1)  1)
2 L 1 k
[2( L  k  1)]!
2
( L  1  k )!
2( L 1 k )
(2k ')!
 1) 22 k ' ( k ')!
k'
c
k
The result is identical to (A13) in Eng & Milstein, Coherent DS-CDMA performance in
Nakagami multipath fading, IEEE Comm., 1995, pp.1134-1143, where different derivations
was done.
44
14.5.1 Tapped delay line (TDL) model for wireless channel
(1) Signal bandwidth W
f c  frequency selective.
(2) Signal duration T
tc  slowly fading.
Assume the channel is frequency
selective slowly fading
1. The equivalent lowpass signal sl (t ) has a bandwidth
f  12 W
From the sampling theorem
sl (t ) 

n sin[ W (t  n / W )]
 W (t  n / W )
 s (W )
n 
l
By taking the FT both sides, we obtain
1

sl ( f )  W



n
 s (W ) e
n 
l
0
 j 2 fn / W
1
, f  W
2
, o. .
2. At the receiver, the equivalent lowpass signal
45


rl (t )   c( ; t ) sl (t   )d   C ( f ; t ) sl ( f )e j 2 ft df

1
W


1
W
1

W


n
 s (W ) 
n 
f  12W

n
 s (W ) 
n 
C ( f ; t )e j 2 f ( t n / W )df
l
l


C ( f ; t )e j 2 f ( t n / W )df

n
1
sl ( )c( t  n / W ; t ) 

W t  at  =n/W
W
n 
(using the assumption 1)

n
 s (t  W )c(n / W ; t )
n 
l

A convolution-sum form
However, if W
Let cn (t ) 
rl (t ) 
f c , the approximation does not hold for frequency-nonselective)
1
n
c( ; t ) be a set of time-variable channel coefficients then
W W

 c (t )s (t  n / W )
n 
n
l
The channel can be modeled as a tapped delay line with tap weights {cn (t )} and tap
spacing 1/W
Since
rl (t )  c( ; t ) * sl (t )   c( ; t ) sl (t   )d 

 c (t ) s (t  n / W )
n 
n
l
Therefore
c( ; t ) 

 c (t ) (  n / W )
n 
n
And the corresponding time-variant transfer function
C ( f ; t )  F [c( ; t )] 

 c (t ) e
n 
 j 2 fn / W
n
That is, for a signal bandwidth
1
W (lowpass) or W (bandpass), the resulting resolution in
2
the MIP (multipath intensity profile) or MDP (multipath delay profile) is
1
W
46
3. The total multipath spread is Tm , then the useful (truncated) total taps in the TDL model
is
 T 
L   m   1  TmW   1
1/ W 
L
Henceforth rl (t )   cn (t ) sl (t  n / W ) (without noise)
n 1
sl (t )
1
W
1
W
c1 (t )
c2 (t )
1
W
cL (t )
L
rl (t )   cn (t ) sl (t  n / W )
n 1
 z (t )
z
(
t
)
* This kind of channel model is indeed a finite length impulse response (FIR) filter.
Note：from the US (uncorrelated scattering) assumption, {cn (t )} are mutually uncorrelated,
furthermore if {cn (t )} are Gaussian r.p., they are statistically independent.
47
14-5.2 The RAKE demodulator
1. The frequency selective channel is modeled as a TDL with statistically independent
time-invariant tap weights cn (t ) .
2. The statistical independent tap weights provide with L replicas of the same transmitted
signal at the receiver.
The Lth-order diversity communication system.
Example:
Assume binary signaling with “equal-energy” over the channel Sl1 (t ) and Sl 2 (t ) , maybe
“antipodal or orthogonal”
Note: T Tm ISI can be neglected.
1/ W (" L 
since Tm
1
 1") and T
W
Tm  T
1
W
The received signal
L
rl (t )   ck (t )sli (t  k / W )  z (t )  vi (t )  z (t ), 0  t  T , i  1, 2
k 1
z (t ) : Complex-valued zero-mean WGN process
Assume
ck (t )
are known or can be precisely estimated.
The optimum demodulator consists of two filters matched to v1 (t ) and v2 (t ) ,
respectively (or crosscorrelators instead)
For coherent detection of the binary signals, the decision variables are
T
U m  Re   rl (t )vm* (t )dt 
 0

 L T

*
 Re    rl (t )ck* (t ) slm
(t  k / W )dt  , m  1, 2
 k 1 0

Here we have already used: T
Tm 

T  kw
k
w
T
( )dt   ( )dt
0
To derive an alternative realization of the optimum demodulator, go back to (14.5-7), i.e.,
48
rl (t ) 

 c (t ) s
n 
n
lm
n

t 
 W

, m  1, 2


n
 T
* 
 U m  Re   rl (t )  cn* (t ) slm
t 
0
 W
n 

 
dt 
 
Here we have used the approximation:

T k / w
k/w
k
* 
rl (t )ck* (t ) slm
t 
 W
T
k

* 
dt

rl (t )ck* (t ) slm

t 

0

 W

 dt

 T 
n 
 
U m  Re    rl (t )cn (t )slm
 t   dt 
0
 W 
 n
let t   t 
n
W
n
n *
 T 

U m  Re    rl (t   )cn* (t   )slm
(t )dt 
W
W
 0 n 


1
n
 T

*
 Re    rl (t   )cn (t )slm
(t )dt 
0
W
 n  L

n
 T L

*
 Re    rl (t  )c n (t )slm
(t )dt 
0
W
 n 1

Remember cn ( t ) is a random process and is the nth tap weight (or the nth branch’s
coefficient).
c1 (t ) corresponds the coefficient of the latest branch
c L (t ) corresponds the coefficient of the earliest branch
Hence
n
 T L

*
U m  Re    rl (t  )cLn 1 (t )slm
(t )dt 
W
 0 n 1

From “spread spectrum communications Handbook,” Simon, Omura, Scholtz, and Levilt,
revised edition. P.444-p.448, since
U m,k  Re  
 k / W
T k /W
*
rl (t )ck* (t )slm
(t  k / W )dt  , m  1, 2 ; k  1, 2,..., L

T
*
 Re   rl (t   k / W )ck* (t )slm
(t )dt 
 0

49
 T L

*
 U m  Re    rl (t   k / W )ck* (t )slm
(t )dt  , m  1, 2
0
 n 1

14.5.3 Performance of RAKE demodulator
* Ck (t ) can be perfectly estimated and can be viewed as a constant Ck since the channel
is assumed slowly fading
T
 L

*
U m  Re   Ck*  r (t ) Slm
(t  k w)dt  , m  1, 2
0
 k 1

Suppose Sl1 (t ) was transmitted , them
L
rl ( t )  Cn 1Sl ( t
n )w

(Z ) t, 0  t
T
n 1
The decision variable
L
T
T
 L

 L

U m  Re  Ck*  Cn  Sl1 (t  n w)Slm* (t  k w)dt   Re  Ck * Z (t )Slm (*t  k w)dt  , m  1, 2
0
0
 k 1 n 1

 k 1

T
*
Note that for k  n,  Sl1 (t  n w)Slm
(t  k w)dt , m  1, 2 represents the cross correlation
0
term and is undesired. To take it out ,the signals Sli (t ) must be specifically designed ,for
example “orthogonality” ,

T
0
Sli (t  n w)Slj* (t  k w)dt  0, k  nij , i  1, 2
then
T
 l

 L

2 T
*
*
U m  Re  Ck  Sl1 (t  k w) Slm
(t  k w)dt   Re   Ck*  Z (t )Slm
(t  k w)dt  , m  1, 2
0
0
 k 1

 k 1

assumpe the signals are “antipodal” , then
L
L


U1  Re  2 k2   k N k 
k 1
k 1


Ck   k e jk
50
Channel Model
(I). the simplest case: single path: constant gain, constant delay
If the channel gain = α; the delay=  0
 ; 0
S (t )
r (t )   S (t   0 )
 r (t )  S (t )  C (t )   S (t   )C ( )d
 C (t )   (   0 )  C ( )
* The impulse response of the channel is an impulse itself, and the channel is time-invariant,
why? Note C (t ) is indeed not function of t, or C (t )  C ( ) : function of 
(II). if the transmitted path is more than one, this is the multipath case
C (t )  n (   n ) C  ( )
n
Note that the channel is still T.I. if both  n ,  n are not function of t.
Also note for each path, it has infinite bandwidth.
(III) The gain or delay is/are function of time in each path
C ( ;t )  n t ( ) ( n t ( ) )
To show it is time-variant,
r (t )  C ( t ; S )t ( d )   n t S (t )  n( t
( ))
n
Let’s S '(t )  S (t  t0 )
r '(t )   C ( ; t ) S '(t   )d   C ( ; t ) S (t  t0   )d   n (t ) S (t  t0   n (t ))
51
However
r (t  t0 )  C ( t ; t 0 S )t (t 0  d )  n t  t( 0S t ) t (0 n t  t ( 0
))
 r ' (t ) r (t0 t ) T . V .
In other words, the channel response including the channel gain and channel
propagation delay are affected by the time that the signal is activated.
It should be emphasized the FT of c( ; t )  C ( f ; t ) (if the system is T.I. then t vanished)
IV）for the delay continuous case，it is directly used the formula
r  t    c  ; t  s  t    d   c  t   ; t  s   d
example：（for Haykin ,signals and system ,P.51）
i(t)
L
V(t)
the input signal： v  t   x  t  ;the inductance L is assumed to be constant.
the output signal： i  t   y  t  ;
（1） y  t   i  t  
1 t
x   d
L 
To find the impulse response




（2） y  t   x  t   c  ; t    c  ; t  x  t    d   c  t    x   d
from （1）&（2） c  t   ; t  
Or c  ; t  
1
1
 , t   0
u t      L
L
 0,
o.w.
1
u    independent of t  T.I
L
52
Indeed, let
x '  t   x  t  t0 
y ' t  
1 t
1 t  t0
x   t0  d   x  ' d '  y  t  t0   T .I


L
L 
If now the inductance L  L  t  is function of t
y t  
Then

1 t
x

d




 c t  ; t  x   d
L  t  
 1
,t 

 c t  ; t    L t 
 0
,t 

or c  ; t  
1
u  
L t 
（may let t    ' ）
 c  ; t  is function of time, which leads to be a time-varying system
if we let x '  t   x t  t0  , then
y ' t  
t t0
1 t t0
1
x

'
d

'

x  ' d '  y  t  t0 


L  t  
L  t  t0  
53
Time Invariance vs Time Variance
(I)
S (t )
C (t )
r ( t )  S ( t ) C ( t ) T [ S ( t )]
 r ( t )  C ( t ) S ( t ) C ( t

)S ( )d 

C ( )S (t
)d
let’s S '(t )  S (t  t0 )
r ' (t ) T [ S ' (t ) ] C t( ) S* t' ( ) C
 t ( S) ' d( ) C t  (0  S )  ( t
d)
Set  '    t0  r '(t )   C (t   ' t0 ) S ( ')d '
On the other hand
r (t  t0 )   C (t  t0   ) S ( )d
 r '(t )  T [ S (t  t0 )]  r (t  t0 )
then C(t) is time-invariant ( T.I.)
( II ) Consider another system
S (t )
c( ; t )
r ( t )  C( ; t ) S (t 
)
r ( t )  T [ S ( t )] C( ;t ) *S (t )
C ( ;t )S (t
)d
C(t
)S (  )d
Do as before to examine the system is TI or TV.
let’s S '(t )  S (t  t0 )
r '(t )   C ( ; t ) S '(t   )d   C ( ; t ) S (t  t0   )d
r (t  t0 )   C ( ; t  t 0) S (t  t 0  )d
 r '(t )  T [ S (t  t0 )]  r (t  t0 ) TV
That is the channel C ( ; t ) is time-varying, in fact, the “t” parameter in C ( ; t ) leads
to the T.V. property.
If let t=0 in C ( ; t ) i.e., C ( ) then the channel becomes T.I.
In this case
r ( t )  C( ) S (t 
)d 
C (t
; 0 )S ( or
 d) the same as ( I ).
The key is that the channel gain (response) is only a function of the “age” or “elapsed time”
between transmitter and receiver , no matter when the signal is transmitted ( activated ).
54