CH:\PTER
6:
CARRIE:!cfw_ ANO S\'41101. SYNClilWNI/ A TIO'-
359
receiver. The process of extracting such a clock signal at the receiver is usually
called symbol synchronization or timing recovery.
Timing recovery is one of the most critical functions that
CHAPTER ti: CARRIER ANO SYMBOL SYNC'HRO'IZATIO~
347
6-2-4 Decision-Directed Loops
A problem arises in maximizing either (6-2-9) or (6-2-10) when the signal
sit; </) carries the information sequence cfw_In. In this case we can adopt one of
two approaches:
C"HAPTER
6:
CARRIER
AND SYMBOL
SYNCHRONl7
AllO!'
345
4)(1)
FIGURE 6-2-6
vco
Equivalent PLL model with additive noise.
where, by definition, tl<I> = <P - cf> is the phase error. Thus, we have the
equivalent model for theP.PLL with additive noise as shown i
CHAPTER 6: CARRIER AND SYMBOL. SYNCHRONIZATION
Sampler
349
~-.
Delay
T
Received
signal
Loop
filter
Delay
T
sine
c
Ji< )dt
Phase
y
x
estimator
FIGURE 6-2-10
Carrier recovery for M-ary PSK using a decision-feedback PLL.
In the case of M-ary PSK, the DFPLL h
348
DIGITAL COMMUNICATIONS
I1
< )dt
Decision
u
cos(2Jl:f. 1 + > Sampler
Time
90"
Received
signal
phase shift
sync.
sin(21t/,1+
t)
Loop
di)
filter
Delay
T
1'1GURE 6-2-9
Carrier recovery with a decision-feedback PLL.
with respect to c/> and setting the deri
CHAPTER
The double-frequency
signal c(t)s(t) through
bearing signal
6:
CARRIER
ANO SYMBOL
SYNCHRONIZATION
3J9
component may be removed by passing the product
a lowpass filter. This filtering yields the information-
y(t) = ~A(t) cos( - (J,)
(6-2-3)
Note th
('HAPrF.R
ll:
CARRIER
AND SYMBOL S'NO!RONILATION
343
where the factor of ~ has been absorbed into the gain parameter K. By
substituting from (6-2-14) for G(s) into (6-2-18), we obtain
(6-2-19)
H(s) = l + (r2 + l/K)s-+ (r1/K)s2
Hence. the closed-loop syste
CHAPTER
6:
CARRIER
AND
SYMBOL
341
SYNCHRONIZATION
-f
FIGURE 6-2-1
()dt
A PLL for obtaining the ML estimate of the phase of an
vco
sin(:?~f. r + ~Mt I '-'
unmodulated carrier.
FIGURE 6-2-2
r.
A (one-shot) ML estimate of the phase of an
unmodulated carrier.
342
DIGITAL COMMUNICATIONS
The loop filter is a lowpass filter that responds only to the low-frequency
component !sin ( tb - <(>) and removes the component at 2fc. This filter is
usually selected to have the relatively simpJe transfer function
G(s )
1
+'t
CHAPH:R
where A : : :
ti
CARRIER
AND SYMBOL
SYNCHRONIZATION
35)
1 with equal probability. Clearly, the pdf of A is given as
p(A)
=
!o(A - 1) + ~S(A + 1)
Now, the likelihood function A( <I>) given by (6-2-9) is conditional on a given value
of A and must b
JS0
DIGITAL COMMUNICATIONS
The two signals are added to generate the error signal
e(t) =-!A
sin (cl> - (b) + ~nc(I) sin (cl> - 4> - 8m)
+ ~n. (t) cos (cl> - 4> - 8,) +double-frequency terms
(6-2-42)
This error signal is the input to the loop filter that p
CHAPTER
!: BLOCK ANO CONVOLUTIONAL CHANNEL CODES
451
The table lookup decoding method using the syndrome is practical only when
n - k is small, e.g., n - k < 10. This method is impractical for many interesting
and powerful codes. For example, if n - k = 2
l"HAPrt'R
CARRIER AND SYMBOL SYNCHRO!l:ll.ArtO~
n;
357
BandpJ"
Rl,1.-i' .,1
.\tth~r-. \ c-r
k,
fihl'r tuned to
.W/
h.'\.~
vco
Frequency divider
~M
Outpot
FlGVRE 62-14
Carrier recoverv with .\1th power law device for M-a.ry PSK.
demodulation.
The received
J58
DIGITAL COMMUNICATIONS
The variance of the phase error in the PLL resulting from the additive noise
may be expressed in the simple form
2
-1
ML
(6-2-59)
u- ;:;"'
'IL
where 'IL is the loop SNR and _S;,:.L is the M-phase power loss. SML has been
evaluat
J56
DIGITAL COMMUNIC'ATIONS
called tht Costas loop. The received signal is multiplied by cos (21Cfct + </J) and sin
(21Cfct + 4> ), which are outputs from the VCO. The two products are
Yc(i)
= (s(t) + n(t)] COS (2efct + J>)
=
Ys(I)
!IA(t) + nc(t)] cos A</
CllAPIER
~:
("ARRll'R
AND SY:-.1801. 'WNOIRONll.A
1"101\
355
easilv show that both components have spectral power in the frequency band
centered at 2.f,. Consequently. the bandpass filter with bandwidth Bhr centered at 2f.
which produces the desired sinus
CHAPTER Ii:
CARRIF.R
AND SYMBOL SYNOIRO!'llZATION
353
Sampler
j I I JOI
ntI
t=11T
l: (I
k
vco
11"'1
Sampler
j I I tdt
FIGURE 6211
t= 11T
Non-decision-directed PLL. for carrier phase estimations of PAM signals.
Although this equation can be manipulated fur
352
DIGITAL COMMUNIC-ATIO~S
Eumple 6-2-3
Let us consider the same signal as in Example 6-2-2, but now we assume that
the amplitude A is zero-mean gaussian with unit' variance. Thus,
V2i
= _l_e-A212
p(A)
If we average A( <P) over the assumed pdf of A, we o
.J40
DIGITAL COMMl!N!CATIONS
Note that the first term of the exponential factor does not involve the signal
parameter <J>. The the signal energy over the observation interval2(t; </> ), is a
constant equal to third term, which contains the integral of s T
Ifweassumethat theeciency factoris0.5,thenwith:
3 108 =0.3m D = 3 0.3048m
c
f
= = 109
weobtain:
G =G =45.8458=16.61dB
R
T
(b) Theeective radiated poweris:
EIRP =PT GT =GT =16.61dB
(c) Thereceived power is:
PT GT GR
P =
4d
=2.995 109 =85.23 dB =55.23 dBm
(d) MSK isbasically binaryFSK withfrequency separation off =1/2T. For thisfrequency
separation the binary signals are orthogonal with coherent detection. Consequently, the error
probability forsymbolbysymbol detection oftheMSKsignal yields anerrorprobabil
If s1 (t) is transmitted, then the received signal is :
r(t) = 2E
b
cos(2f t + ) + n(t)
c
b
2
Crosscorrelating r(t) by
rc =
c
0 r(t)
T
T
cos(2fc t) and sampling the output at t = T , results in
2 cos(2f t)dt
E
2 cos(2f t)dt
b cos(2f t + ) cos(2f t)dt +
=
which is valid for large x, that is for high SNR. In this case :
VT
2
1
r
Eb
r
r
+E
b
b
r E
2
I0
22
2 e
0
2
dr 2
e(r
2 2 E
0
Eb ) /22 dr
b
This integral is further simplied if we observe that for high SNR, the integrand is dominant
in the vicinity of Eb a
1
where, e.g: p(n1c )= 4No exp(n2 /4N E ).
0
E
Problem 5.41 :
Therstmatched lteroutputis:
r =
Tr
0
1
T
( )h (T )d =
l
0
1
T
r ( )s (T (T )d =
l
0
r ( )s ( )d
l
Similarly:
r =
T
0
2
r ( )h (T )d =
l
T
0
2
r ( )s (T (T )d =
l
T
0
rl ( )s ( )d
which arethesa
Problem 5.38 :
(a) The optimal ML detector (see 5-1-41) selects the sequence C i that minimizes the
quantity:
D(r, C ) n & (r
=
i
k
C )2
ik
k
=
1
The metrics of the two possible transmitted sequences are
n
D(r, C ) =w& (r 2
& (r
) +
k
1
k
k=
1
an
d
k=w
+
are:
E [n1c n1c ] = E Re
T
z(t)s (t)dt Re
0
=
l1
T
E
=2 0
4
= 12
0
l1
T
z(t)s (t)dt +
2
z (t)sl1 (t)dt
0
[z(a)z (t)] s (t)s (a)dtda
T
l1
0
l1
2N0 T sl1 (t)s (t)dt = 2N0 E
0
E
T
l1
0
4
1
where we have used the identity Re [z] =
that
z(t)s (t)dt
1
4
1
T
Similarly we nd that P (e|C 2 ) = P (e|C 1 ) and since the two sequences are equiprobable
P (e) = Q
Eb (n w)
2
(c) The probability of error P (e) is minimized when
Eb
(nw)
is maximized, that is for w = 0.
This implies that C 1 = C 2 and thus the distance
where the equality (a) follows from the fact that (1)i is 1 for i even and 1 when i is
odd. Solving the system
P
+P
= 1
even
odd
Peven Podd = (1 2p)n
we
obtain
1
Pn = Podd = (1
2p) 2
)
n
(1
(b) Expanding the quantity (1 2p)n , we obtain
n(n 1)
(1 2p)n =