Tài liệu Adaptive WCDMA (P5) docx

5
Modulation and demodulation
5.1 MAXIMUM LIKELIHOOD ESTIMATION
We start again with the ML principle defined in Section 3.1 of Chapter 3. After the
signal despreading, vector of parameters θ to be estimated includes timing of the received
symbols τ
0
, phase of the received carrier θ
0
, frequency offset of the received signal ν
0
,
amplitude of the signal A
0
and data symbols a
n
θ(τ
0
,θ
0
,ν
0
,A
0
,a
n
)(5.1)
After despreading, the narrowband signal can be represented as
r(t) = s(t, θ) + w(t) (5.2)
The likelihood becomes
L(

˜
θ) = C
1
exp

−
C
2
N
0

T
0
|(r(t) − s(t,
˜
θ)|
2
dt

(5.3)
In the sequel, we will use a linear-modulated complex-signal format given by
s(t,
˜
θ) = A
0
exp(j
˜
θ
0
)


˜a
0
h(t − nT −˜τ
0
) + j
˜
b
n
h(t − nT − εT −˜τ
0
) (5.4)
where h( ) is the pulse shape and for ε = 0 or 1/2 we have quadrature phase shift keying
(QPSK) or offset QPSK (OQPSK) signals, respectively. The likelihood function defined
by equation (3.5) now becomes
λ(θ) = R(T
0
,
˜
θ) = Re


T
0
0
r(t)s
∗
(t,
˜
θ) dt


(5.5)
If we define the filters matched to the pulse shape in I and Q channel as
p(n, ˜τ)=

∞
−∞
r(t)h(t − nT −˜τ)dt
q(n, ˜τ,ε)=

∞
−∞
r(t)h(t − nT − εT −˜τ)dt (5.6)
Adaptive WCDMA: Theory And Practice.
Savo G. Glisic
Copyright
¶
2003 John Wiley & Sons, Ltd.
ISBN: 0-470-84825-1
124
MODULATION AND DEMODULATION
then equation (5.5) becomes
R(N,
˜
θ) = Re

exp(−j
˜
θ)
N


n=1
˜a
n
p(n, ˜τ)

+ Im

exp(−j
˜
θ)
N

n=1
˜
b
n
q(n, ˜τ,ε)

(5.7)
In the special, important case of nonstaggered signals (ε = 0), we find q = p.Ifwedefine
c
n
= a
n
+ jb
n
, the correlation integral becomes
R(N,
˜

θ) = Re

exp(−j
˜
θ)
N

n=1
˜c
∗
n
p(n, ˜τ)

(5.8)
5.1.1 Phase and frequency correction: phase rotations and NCOs
For a given phase error θ (n), the complex signal sample (sampling index n) z
in
(n)iscor-
rected by multiplying the sample by a complex correlation factor exp(jθ(n)) as follows:
z
in
(n) = x
in
(n) + jy
in
(n)
z
0
(n) = z
in

(n) × exp(j θ (n)) (5.9)
By using exp(j θ ) = cos θ + j sin θ ,weget
z
0
= x
in
cos θ − y
in
sin θ + j(x
in
sin θ + y
in
sin θ) (5.10)
The operation is known as phase rotation and the block diagram for the realization of
equation (5.10) is shown in Figure 5.1.
Frequency corrections (translations) can be performed by the same circuitry but now
the phase correction will change in time. For the frequency error ν, the correction becomes
z
0
= z
in
exp(j2πnvT
s
)(5.11)


+
_
+
+

x
0
(
n
)
x
in
(
n
)
q(
n
)
y
0
(
n
)
sin q
cos q
y
in
(
n
)
Sine/
Cosine
ROM
Figure 5.1 Phase rotation.
FREQUENCY-ERROR DETECTION

125
A simultaneous phase rotation and frequency translation is performed as
exp[j(2πnvT
s
+ θ)] (5.12)
In the next section we will focus on the problem of detecting phase and frequency error.
The circuit from Figure 5.1 will be used for error corrections, given an error value of
phase or frequency.
5.2 FREQUENCY-ERROR DETECTION
We start again with the likelihood function in the following form:
L(
˜
θ) = exp

2C
2
N
0

T
0
Re[r(t)s
∗
(t,
˜
θ)]dt

(5.13)
To emphasize the existence of frequency error, the signal defined by equation (5.4) is
rewritten as

s
∗
(t,
˜
θ) =
∞

n=−∞
[˜a
n
h(t − nT −˜τ)− j
˜
b
n
h(t − nT − εT −˜τ)] exp[−j(
˜
θ + 2π ˜vt)]
(5.14)
In this case equation (5.13) becomes
L(
˜
θ)
∼
=
N−1

m=0
exp

2C

2
N
0
˜a
m
Re


∞
−∞
r(t)h(t − mT −˜τ)e
−j(
˜
θ+2π ˜vt)
dt

× exp

2C
2
N
0
(−
˜
b
m
) Im


∞

−∞
r(t)h(t − mT − εT −˜τ)e
−j(
˜
θ+2π ˜vt)
dt

(5.15)
L(
˜
θ)
∼
=
N−1

m=0
exp

2C
2
N
0
˜a
m
Re[p(m)]

× exp

2C
2

N
0
(−
˜
b
m
) Im[q(m)]

(5.16)
Joint maximization of equation (5.16) with respect to all the parameters would be rather
complex for practical implementation. To remove data from equation (5.16), we use aver-
aging of the function. For M-ary modulation this can be represented as
L
a,b
=
N−1

m=0

m

i=1
1
M
exp

2C
2
N
0

a
i
Re[p(m)]


×

m

i=1
1
M
exp

−
2C
2
N
0
b
i
Im[q(m)]


(5.17)
126
MODULATION AND DEMODULATION
In the simple case of binary modulation we have
L
a,b

=
N−1

m=0
cosh

2C
2
N
0
Re[p(m)]

· cosh

2C
2
N
0
Im[q(m)]

(5.18)
For nonoffset QPSK modulation, q(m) = p(m) and equation (5.18) becomes
L
a,b
=
N−1

m=0
cosh


2C
2
N
0
Re[p(m)]

· cosh

2C
2
N
0
Im[p(m)]

(5.19)
By taking the logarithm of equation (5.19), we have

a,b
 ln[L
a,b
] =
N−1

m=0

N−1

m=0
ln cosh


2C
2
N
0
Re[p(m)]

+ ln cosh

2C
2
N
0
Im[p(m)]


(5.20)
The following approximations are used at this point:
ln cosh(x)
∼
=
x
2
2
,|x|1
∼
=
|x|,|x|1 (5.21)
For the small value of the argument we have

a,b

∼
=
C
3
N−1

m=0
({Re[p(m)]}
2
+{Im[p(m)]}
2
)
= C
3
N−1

m=0
|p(m)|
2
(5.22)
Equation (5.22) can be maximized by changing ˜ν in p(m) in the open loop search. By
taking the derivative of equation (5.20), we get the tracker for the QPSK signal.
∂
a,b
∂˜v
=
N−1

m=0
2C

2
N
0
Re[p
v
(m)]tanh

2C
2
N
0
Re[p(m)]

+
N−1

m=0
2C
2
N
0
Im[p
v
(m)]tanh

2C
2
N
0
Im[p(m)]


(5.23)
where p
v
(m)∂p(m)/∂ ˜v. A sample of frequency-error detector control signal is
u
v
(n) = Re[p
v
(n)]tanh

2C
2
N
0
Re[p(n)]

+ Im[p
v
(n)]tanh

2C
2
N
0
Im[p(n)]

(5.24)
FREQUENCY-ERROR DETECTION
127

Re
Im
Re
Im
v
~
Signal matched filter
r
(
t
)
Frequency
rotator
Filter strobes
p
and
q
are complex
Non offset signals:
p
(
n
) =
q
(
n
)
Offset signal:
q

(
n
) is time-offset from
p
(
n
) by e
T

h
(−
t
)
−
j
2p
th
(−
t
)
Frequency
loop
filter
Frequency
error
detector
{
p
v
(

n
)}, {
q
v
(
n
)}
u
v
(
n
)
{
p
(
n
)}, {
q
(
n
)}
Figure 5.2 ML-derived frequency detector.
and its time average U
v
= E
n
[u
v
(n)] is called the detector characteristic or S-curve. The
block diagram realizing equation (5.24) is shown in Figure 5.2.

5.2.1 QPSK tracking algorithm: practical version
Further simplification is obtained if we use
tanh(x)
∼
=
x,|x|1
∼
=
sgn(x),|x|1 (5.25)
which for nonoffset QPSK results in
u
v
(n) = Re[p(n)]Re[p
v
(n)] + Im[p(n)]Im[p
v
(n)]
= Re[p(n)p
∗
v
(n)] (5.26)
5.2.2 Time-domain example: rectangular pulse
After considerable labor, the S-curve defined as u
v
(n) = E
n
[U
v
(n)] for a unit-amplitude
rectangular pulse in time domain and random data is found to be [1]

U
v
(v, τ ) = C
c
A
0

τ
sin πvτ
πv
2

cos πvτ −
sin πvτ
πvτ

+ (T − τ )
sin πv(T − τ )
πv
2
×

cos πv(T − τ ) −
sin πv(T − τ )
πv(T − τ )

(5.27)
For random data the S-curve is shown in Figure 5.3.
128
MODULATION AND DEMODULATION

If the data pattern is assumed to rotate by 90
◦
from one symbol to the next, that is,
c
m+1
= jc
m
, the S-curve is as given in Figure 5.4.
For binary phase shift keying (BPSK) dotting signal where c
m+1
=−c
m
,theS-curve
is as given in Figure 5.5.
One should notice that for random data, S-curve demonstrates regular shape with the
slope decreasing with the timing error. Even for nonsynchronized systems with τ /T =
0.5, the system would operate. For rotating data the impact of timing is larger. For dotting
signal, if the timing error becomes large, the S-curve not only has a reduced slope, which
is equivalent to reducing the signal-to-noise ratio in the loop, but also changes the sign
resulting in the devastating effects of generating the control signal that would cause loss
of synchronization.
0
1
−1
−2
0.2
0.3
0.4
0.5
Frequency error, ∆

vT

U
v
∆t
/
T
= 0
0.1
2
Figure 5.3 Frequency detector S-curves for rectangular pulses (nonoffset signal).
0 1
−1
0.125
0.25
Frequency error, ∆
vT

U
v

∆τ/
T
= 0
Figure 5.4 S-curve for rotating pattern: square pulses.
CARRIER PHASE MEASUREMENT: NONOFFSET SIGNALS
129
0.125
0.25
0.5

0
−1
Frequency error, ∆
vT

∆τ /
T
= 0
U
v
1
Figure 5.5 S-curve for BPSK dotting pattern; square pulses.
5.3 CARRIER PHASE MEASUREMENT:
NONOFFSET SIGNALS
In this case possible solutions will depend very much on a number of parameters.
Regarding the signal format, there will be differences for single amplitude [M-ary phase-
shift keying (MPSK)] versus multiamplitude [M-ary quadrature amplitude modulation
(MQAM)] or offset versus nonoffset signal. Different representations such as rectangular
versus polar representation of phase error or parallel versus serial representation of signal
(offset only) will result in different solutions. Additional knowledge such as clock timing
(clock-aided) or data or decisions [data-aided (DA) or decision-directed (DD)] will also
be of great importance. Configurations such as feedforward (FF) versus feedback (FB)
will also offer different advantages and drawbacks.
5.3.1 Data-aided (DA) operation
In this case a preamble c
n
and timing τ
0
are available and equation (5.8) becomes
R(N,

˜
θ)= Re

exp(−j
˜
θ)
N

n=1
c
∗
n
p(n)

(5.28)
At the maximum point, ∂R/∂θ vanishes and we have
Im

exp(−j
ˆ
θ)

n
c
∗
n
p(n)

= 0 ⇔


n
Im[exp(−j
ˆ
θ)c
∗
n
p(n)] = 0 (5.29)
130
MODULATION AND DEMODULATION
and we define again the sample of the S-curve as
u
θ
(n) = Im

exp(−j
˜
θ)

n
c
∗
n
p(n)

(5.30)
Under the ideal conditions the matched filter output pulse becomes
p(n) = A
0
c
n

exp(jθ
0
) + v(n) (5.31)
where v(n) is a sample of noise (assumed zero-mean Gaussian) and A
0
is signal amplitude.
Averaging gives the S-curve as
U
0
= EIm{exp(−j
˜
θ)c
∗
n
[A
0
c
n
exp(jθ
0
) + v(n)]}
= A
0
E|c
n
|
2
Im{exp[j(θ
0
−

˜
θ)]}=A
0
E|c
n
|
2
sin θ (5.32)
where θ = θ
0
−
˜
θ.
5.3.2 Decision-directed (DD) operation
If preamble is not available, detected data can be used instead, resulting in a DD solution.
Implementation of equation (5.30) for such a case is shown in Figure 5.6.
Specific DD algorithm
If we represent the output of the phase rotator p(n)e
−j
˜
θ
as the complex sequence
{x(n), y(n)}, the output of the decision algorithm as ˆc
n
=ˆa
n
+ j
ˆ
b
n

= sgn(x
n
) + jsgn(y
n
)
*
*
~
Timing
Phase
rotator
Decisions
Data out
Integrator
Complex signal
Real signal
Complex conjugate
Matched
filter
r
(
t
)
p
(
n
)
Sample
Loop
filter

Im( )
h
(−
t
)
u
q
(
n
)
c
n
e
_
j
q
~
ˆ
c
n
*
t
ˆ
q
Figure 5.6 Decision-directed carrier tracking.
CARRIER PHASE MEASUREMENT: NONOFFSET SIGNALS
131
where sgn(v)=+1(−1) if v is greater than 0 (v<0), then equation (5.30) is given by
u
n

(n) = Im(x
n
+ jy
n
)(ˆa
n
+ j
ˆ
b
n
)=y
n
sgn(x
n
) − x
n
sgn(y
n
)(5.33)
This is known as four-phase hard-limiting Costas detector that is so widely used in
QPSK systems.
Rectangular representation
If we use the following steps:
• exp(−j
˜
θ)→ a rectangular representation in equation (5.28)
λ(
˜
θ)= Re


(cos
˜
θ − j sin
˜
θ)
N

n=1
ˆc
∗
n
p(n)

(5.34)
differentiate with respect to
˜
θ
• bring all the expressions into the summation sign
• take the real part of the derivative
• the ML estimate
ˆ
θ occurs for the value of
˜
θ at which the derivative goes to zero
N

n=1
sin
ˆ
θ Re[ˆc

∗
n
p(n)] −
N

n=1
cos
ˆ
θ Im[ˆc
∗
n
p(n)] = 0
Solving for the angle gives
ˆ
θ(n) = arctan













n−1


i=n−M
Im[ˆc
∗
i
p(i)]
n−1

i=n−M
Re[ˆc
∗
i
p(i)]













(5.35)
Implementation of equation (5.35) is shown in Figure 5.7.
5.3.3 Nondecision-aided measurements
Why might one choose to avoid DD measurements? First of all there are some circum-
stances, such as acquisition intervals or low signal-to-noise ratios, for which data decisions

are of poor quality and should not be used. One can show that in BPSK the equivalent
signal-to-noise ratio in such systems will be reduced by factor (1–2P
e
)
2
where P
e
is the
bit error rate. Omitting the decision operation might reduce equipment complexity. (Not
likely to be a good reason in a digital implementation. Indeed, we find digital DD meth-
ods are often simpler than non-DD methods.) In this case the likelihood function will be
averaged out with respect to data.