An Aliasing{Free Receiver with Variable Sample
Rate Digital Feedback M=T NDA Timing
Synchronization
Uwe Lambrette, Klaus Langhammer and Heinrich Meyr
Lehrstuhl fur Integrierte Systeme der Signalverarbeitung
Aachen Univ. of. Technology, D-52056 Aachen
email [email protected]
Keywords: Digital Receiver, NDA Timing Synchronization
Aliasing, Variable Sample Rate, DVB
Abstract - The evolving digital television broadcasting standard does not standardize
the data rate 1=T of the transmitted data but instead leaves it completely unspecied.
We propose two digital receiver algorithms for the processing of an extended range of
variable sample rates 1=Ts based on a timing synchronization algorithm producing synchronized samples at rate M=T where M is variable. The algorithms are compared in
terms of their complexity and performance. One of the algorithms is based on ltering
the received samples prior to timing synchronization and choosing M = 2, whereas the
second algorithm also uses the sample rate 3=T in timing recovery loop and matched
lter depending on T=Ts and thus avoids the necessity of preltering. Both algorithms
can be implemented causing negligible loss in a DVB receiver.
This paper was published in part at Globecom 1996.
In this paper we propose a digital feedback (FB) non{data{aided (NDA) timing synchronization algorithm that is capable of processing variable symbol rates relative to
the sample rate T=Ts. Modems for variable data rates have thus far been discussed in
1,2] with an emphasis on interpolator control and interpolation and their construction
has been outlined in 3]. In this paper, we discuss NDA timing synchronization for
variable{rate modems in a detailed manner with an emphasis on the aliasing problems
and the timing synchronizer structure.
In section 2, we dene the signal and noise model used, section 3 covers a review
on optimum receiver structures leading to an implementable synchronization structure
along with the denition of the notation. Section 4 covers the concept of digital timing
synchronization using NDAFB algorithms. Section 5 discusses the eects of aliasing
and two possible remedies. Section 6 describes some aspects of the discrete{time nite{
range implementation. In section 7, simulation results conrm the predictions of the
previous sections.
2 Signal and Transmission Model
We consider a transmission of root{raised cosine (Nyquist) pulses with a variable symbol
duration T and symbol rate R = 1=T . Before sampling and A/D conversion, the received
signal is passed through an anti{aliasing lter (AAF) with transfer function A(f ). The
AAF usually has a bandwidth much smaller than 2=T in order to prevent noise from
being sampled with the A/D converter as this would require a larger dynamic range
of the A/D converter. In this model, the received noise is coloured due to the small
bandwidth 2fg of the analog prelter.
The (well{known) model of the received signal at the receiver is
1
X
rf (mTs ) =
alg(mTs ; lT ; "T )ej ejTsm +nf (mTs )
(1)
l|=;1
{z
}
sf
The ltered gaussian noise nf () has the power spectral density Nf (f ) and the signal sf
has the power spectral density S (f ). The timing oset is always measured with respect
2
H(f)
A(f)
S(f)
2∆f
Nf
IF
amplif.
AGC1
(1+α)/2T a
fg
sampling
A->D
A(f)
f
AGC2
Figure 1: Analog Receiver frontend
to T . The bandwidth used by the signal transmission is 2a, where
a = 1 2+T + f:
(2)
Here f is the maximum allowable frequency oset (in Hz) and is the rollo factor
of the root raised cosine pulses used. We assume that A(f ) = const: for jf j < a. We
also assume that the various AGCs manage to keep the signal power at the A/D input
at a constant value. See gure 1 for the spectra at the analog frontend.
In the receiver, we are faced with multirate signal processing where xl denotes samples
with an average sample rate of R = 1=T and xk denotes samples with an average rate
of RM = M=T = 1=TI , such that the samples xk=lM:::lM +(M ;1) correspond to xl . The
processing and sampling rate Rs = 1=Ts is always larger than M=T but is generated by
a free running oscillator. Samples at this rate are denoted by xm . Figure 2 highlights
the relation of xm, xk and xl. For the implementation discussed in here, sampling rates
of 2=T and 3=T (ie M = 2 3) proved suciency.
3
TS
x(mT)≡xm
m
x(kTI )≡x(kT/M) ≡xk
T/M
k
T
x(lT)≡xl
t
asynchronous
synchronous
l
Figure 2: Relation of time axes
3 Receiver Structure
3.1 ML-based Receiver Structure
ML{based reception intends to eliminate the unwanted parameters from the likelihood
function L() in order to obtain a likelihood function solely dependent on the symbol
vector a. Several approaches to do so are known (Non{Data{Aided (NDA), Data{Aided
(DA), Decision{Directed (DD) Synchronization) and all lead to realizations with small
performance degradation compared to the theoretical optimum.
We recall from 4] that the likelihood function may be written as
L(a " T ) = 2IRaH ej z ; aH Ha
(3)
assuming a negligible frequency oset where z = : : : z;1 z0 z1 : : :] is the output of the
digital matched lter
1
X
z(lT + "T ) =
rf (mTs)gMF (;mTs + lT ; "T )
(4)
m=;1
The digital matched lter is time{variant except when Ts =T is an integer4]. It may
be separated into a time{variant variable{rate interpolation and decimation lter and
a xed rate time{invariant matched lter. For odd M , we have to align the samples in
the matched lter by Ti=2. Let
8
>
>
<
0
M even
>
: 1=(2M ) M odd
odd = >
(5)
and k = kTI + (" + odd)T . Then
1
rf (mTs )h(k m " + odd M Ts=T )
m=;1
1
X
z(lT + "T ) =
rf ( k )gMF (;kT=M + (l ; odd)T ):
k=;1
rf ( k ) =
X
4
(6)
(7)
rf (mTs)
Prefilter
rf (kT/M+εT/M)
1/Ts
Interpolator
selection / bypass
of prefilter
depending on Ts/T
µ k m=mk
0 m=mk
}
M/T
m=mk
e
1/Ts
l
f,l
=z(lT+εT)
M/T
1/T
Viterbi
Decoder
^
θk
NCO
M/T
ek
wl 1/T
el
Loop
Filter
TNCO
Matched
Filter
^
-jθk
M/T
TED
1/Ts
z(kT/M+εT/M)e
1/T
Phase
Recovery
M/T
decreasing sample rate by discarding
n-1 out of n samples
increasing sample rate by
repeating sample
decreasing sample rate by
accepting only samples
marked by control signal
increasing sample rate by
controlled repetition of sample
1/T,... average
sample rate
Figure 3: Receiver. The prelter is discussed in section 5.2.
Later we will also use k0 = k ; oddT .
3.2 Implementation{Friendly Receiver Structure
The receiver based upon the joint estimation of data and sync parameters cannot be
realized due to complexity constraints. For the application of receiving a continuous
data stream, we now suggest a specic receiver structure. Synchronized samples at the
rate RM are generated before applying the matched lter. The timing recovery loop
is closed before the matched lter in order to remove unwanted interactions between
timing and carrier recovery. The carrier loop is closed 'around' the matched lter. The
resulting receiver structure is depicted in gure 3. The prelters will be discussed later.
Acquisition may now be separated by acquiring rst timing and then carrier. From now
on, we will focus on the timing recovery scheme used.
4 Timing Synchronization
We describe the NDA timing feedback synchronization technique.
5
Timing synchronization for continuous data streams is performed by a feedback loop
consisting of a timing error detector, a loop lter and either a controllable VCO or a
digital interpolator. The latter solution has several advantages: It allows to reduce the
interaction between analog and digital circuitry (and hence diminishes design time and
test complexity) it allows to use cheaper analog components and its design is easier to
handle as no joint analog and digital modeling technique has to be employed.
4.2 Timing Error Detector and Loop Filter
We use the timing error detector (TED) devised by Gardner 5] which is known to
produce an error estimate that will lead to timing estimates approaching the CramerRao bound (CRB) 6] independent of the carrier. The TED algorithm is obtained
by setting @L=@" = 0 and removing the data dependency. Secondly, the derivative
@gMF ()=@" according to (4) is simplied until @z(lT + "T )=@" const:(rf (k+1 ) ;
rf (k;1 )) is obtained, as we want to apply the TED before the matched lter. This
approximation does not only hold for RM = 2=T (as suggested in 5]), but also for
higher M . We obtain
;
ek = rf ( k0 ) rf ( k0 + T=M ) ; rf ( k0 ; T=M ) :
When computing the S{curve, we closely follow the derivation in 5] and nally obtain
(for root {raised cosine pulses, rollo )
E el (") = ; (M4M
2 ; 42 ) sin(2
") 1
+
1
;
sin M + sin M
2
(8)
The S{curve is displayed in gure 4 for M = 2 4 and is derived in the appendix.
The loop lter is described by the equations
xl+1 = xl + el
wl = Kixl + Kp el
(9)
(10)
and hence the timing recovery loop employed is a standard second order type 2 loop
running at an average sample rate R.
6
Figure 4: S-curve of the timing error detector for dierent sampling rates
4.3 Interpolator Control (TNCO)
Let us rst assume that both timing oset " and TI =Ts are known. We have to reconstruct the synchronized sampling instants of the transmitted sequence at an intermediate
rate RM in units of Ts .
kTI + "T = mk Ts + k Ts
(11)
The task of the interpolator control is to compute the quantities mk and k from the
loop lter output and the NCO control word. Dene
(m k) = kTI =Ts + "T=Ts ; m
(12)
Obviously for m = mk , (mk k) = k . Reasonable k lie between 0 and 1, thus we are
interested in the values mk for which 0 < k < 1 holds. The mk increase monotonically,
ie. mk+1 mk as k increases monotonically. In the case of practical interest TI =Ts is
always greater than one and in this case mk+1 > mk . If (m k) > 1 we can be sure
that (m + 1 k) is a better candidate for k . If (m + 1 k) < 0 we can be sure that
(m k) = k and m = mk , thus < 0 is the criterion for computing a new output
7
the procedure.
Compared to the approach chosen by Gardner and Erup 2,1] we nd our 7] () to
be normalized to the sampling time instead of the time TI which allows to obtain k
without dividing by TI . Compared to the 'alternative approach' from 2] we do not need
to compute the dierence between mk and mk+1.
4.4 Interpolator Design
If the complexity of the interpolator remains unconstrained, the interpolator computes
the exact value of the (bandlimited) rf (k ). However, an estimate of the interpolated
value based on few samples 2,7] is sucient, we can optimize the lters h() from eqn.
(6) using an exhaustive search. Ie. for any the MSE of all possible lter vectors h (constrained by quantization and lter length) is computed and the vector with minimum
MSE is selected. A very good overview of fractional delay lter design with extensive
references is given in 8].
5 Aliasing Eects
5.1 Constraints on T
s
After interpolation, a controlled decimation is carried out. Therefore it may occur that
signal and noise spectra start aliasing which do not interfere before the decimation
process to the rate RM < Rs.
As shown in gure 5 (b), aliasing occurs in the case where the bandwidth of the
antialiasing prelter fg ceases to satisfy
fg (M=T ; a):
(13)
The aliasing of the noise components Nf only, as shown in gure 5 (a) is not crucial as
this noise will be removed by subsequent matched ltering 4].
In order to remove unwanted aliasing, two strategies will be discussed here. The
rst deals with preltering the received signal whereas the second removes aliasing
by increasing the intermediate sample rate RM . In both cases we assume that the
8
Sf(f)
fg,a
∆f=0
2∆f
Nf(f)
0
a
1/Ts-fg,b
fg,b
1/Ts-fg,a 1/Ts-a
1/Ts
f
a=((1+α)/2T+∆f)
a) fg,a<(M/T-a)
0
b) fg,b=(M/T-a)
0
M/T-fg,a
M/T-fg,b=a
M/T
M/T
f
f
Figure 5: Aliasing after Decimation. Top: The Power Spectrum of rf sampled at rate Rs.
a) M=T ; fg > a. b) M=T ; fg = a.
bandwidth of the analog prelter is slaved to the sample rate by fg = c=Ts. In order to
avoid any eect from the prelter on the useful signal, we have to lower bound c by
fg a
(14)
The tightest bound is achieved when T=Ts = 2, such that
cmin = aT2
9
(15)
transition band
transition band
stop band
stop band
∆f t
∆f t
pass band
f
-f g
-fs
-fp
fp
fs
fg
"don’t care" region
Figure 6: Frequency bands for optimum lter design.
5.2 Constraining Received Signal Bandwidth
In this section M = 2 is xed for all cases of T=Ts. We introduce a lowpass lter
preceeding the timing recovery loop (gure 3, 'frontend'). The lter shall reduce fg to
fs < fg such that no more aliasing occurs.
Since the signal is band limited due to A(f ) the frequencies beyond fg can be regarded
as dont{care regions. No optimization is carried out in the transition band, the transition bandwidth ft is left as an optimization parameter, instead. The bands where
the approximation error has to be minimized is the pass- and the stopband as shown in
gure 6. Because of this the MSE of the symmetric ideal spectra and an approximation
H~ i (f ) can be dened by
hR
i
R
fp
2
jfT
2
jfT
s )j2 df + fg jH
s )j2 df
~
~
j
1
;
H
(e
(e
i
fs i
MSEi = 0
:
(16)
fp + fg ; fs
The usage of several digital lowpass lters allows us to increase the bandwidth of
the transition. Thus a reduction of implementation complexity is possible, because the
dont{care regions are enlarged.
It can be seen from gure 5 that a digital lter i with frequencies fpi and fsi does
not inuence the signal and avoids aliasing within the signal spectra in the range of
Ts (14)
fpiTs :
fsi Ts (13)
(17)
M ; aT
T
aT
It follows that the edge-frequencies of the next lter i + 1 have to be chosen to
!
aT
f
si Ts
fpi+1 = T M ; aT + V and fsi+1 = fpi+1 + ft
(18)
s
10
ft is the transition bandwidth which is assumed constant for all lters. Solving these
equations for all fpi and fsi and eliminating ft under the assumptions Ts=T jmin = 0:24
for given imax and fp0 = fs0 = fg gives the information needed to design9] the imax
lters.
MSE results for a dierent number of lters and varying lter lengths are displayed
in gure 7: Shown are the maximum MSE values for a bank of 1,2 and 4 lters. A
coecient quantization of 5 and 6 bit was chosen. For a given lter length of the
prelters, the value of the worst{case MSE, being valid for only one Ts =T is shown. Ie.
MSE is maximized for a given number of coecients Np and lters over Ts=T and i
(which selected according to T=Ts) according to a MINMAX strategy 10].
MSE (Np wordlength) = max
i
!
max MSEi(lterbank wordlength)
T=Ts2Range i
(19)
Thus in the diagram MSE values for dierent Ts =T are plotted and compared explaining
the jagged shape of the graphs. In general it is observed that with increasing lter length
the quantization becomes the limiting factor on minimizing the MSE whereas for small
lter lengths, the number of lters imax is dominant for the performance. Not much can
be gained by using more than 2 lters. The resulting prelter must be selected according
to the actual normalized sampling rate Ts=T . For the case that is investigated later using
simulation, we used 2 lters which were selected according to table 1. In this case, no
additional degradation (in terms of the BER) is attained from the prelter, whereas
using one lter this is not the case.
Settings M = 2 V = 0:01 c = 0:38
Filter
Valid Range i
none
0:5
0:3167
1
0:3267 > Ts =T 0:2707
2
0:2807
0:24
Table 1: Selection of Prelter depending on T=Ts
11
maximum MSE for filter bank
MSE should
be at least 15dB
below σ2
1 Filter (5 bit)
0.03
1 Filter (6 bit)
0.025
0.02
selected
value
0.015
2 Filters (5 bit)
2 Filters (6 bit)
[selected]
0.01
4 Filters (5 bit)
0.005
4 Filters (6 bit)
0
10
12
14
15
16
18
20
22
Number of Filter Taps
Figure 7: Results for quantized lter coecient MSE
5.3 Increasing the MF sample rate
A second technique employs the adaptive increase of the MF and TED sample rate (ie.
depending on T=Ts). Solving (13) for M leads to
(20)
M aT + cT
Ts
where M T=Ts and we nd that
T = M
T = M ; aT :
(21)
Ts min
Ts max
c
Figure 8 displays achievable sampling rates for various M and c. The -shaped region
contains all values of (c T=Ts) that can be sampled with a sampling rate M=T where
M = 2 3 4. In table 2, the boundaries on T=Ts (Ts =T ) are given for c = 0:38. (It turns
out that there are large areas where eg. both M = 2 and M = 3 can be used, which
is the case for 0:3067 < Ts=T < 0:3333.) In doing so, we have to adopt the matched
lter to the dierent sample rates M=T . As the samples produced by the interpolator
are symmetric to the times lT + T=2, we had to introduce the fractional delay odd
that needs to be compensated by sampling the matched lter for eg. M = 3 at instants
12
s
2 2
3 3
4 4
s
3.26
5.89
8.53
0.3067 0.5
0.3333 0.1698
0.1172 0.25
Table 2: Switching boundaries for M=T {timing synchronization
3.26
5.89
8.53
0.8
M=2
0.6
M=3
M=4
c
0.4
lowest limit
0.25
2
4
6
T/Ts
8
10
12
14
Figure 8: Achievable T=Ts for varying c = cmin and M , When c > 0:4 and T=Ts 2 2 4] is
desired, preltering has to be used.
kTI ; oddT .
6 Simulation Results
In a rst step, we compute the timing error variance and compare it to the CRB. We
use a fully quantized digital receiver. The CRB 11,12,7] lower bounds the timing error
R1
;1
2
2
E
s
V ar("T ; "^T ) N 2BlT R 1;1!j2GjG(!(!)j)jd!
(22)
2 d! :
0
;1
Inserting the root{raised signal spectra (see appendix 2) and integration leads to
T "2 22EBL=N
S
0
2
:
23 (1 + 32 ) ; 16
2
3
(23)
Note that the loop noise bandwidth is a function of Ts=T and M . In our simulations,
BLT = MTs =TbL0 was chosen. We used MbL0 = 2 10;3 in our simulations. Secondly,
the synchronization performance is lower bounded by the quantization of k to 8 levels
("=2 = Ts=8T is the maximum timing error) to "2 min = (Ts =8T )2=12.
13
Figure 9: Simulated and calculated timing error variance
Variance computation and timing error variance simulation result13] are compared
in gure 9. It is seen that the performance loss due to nite wordlength is negligible.
Note that the plain addition of the CRB and the quantization variance oor is too
pessimistic. The variance oor "2 min is somewhat increased by quantization eects of
all signals in the synchronizer.
Secondly, we investigate the BER performance: The DVB standard 14] has the remarkable property of not dening the data rate for digital transmission (Properties of
the DVB standard are outlined in table 3).
We assess the performance of the two schemes by inserting the respective components
into a fully{quantized, bittrue DVB prototype receiver. In our simulations13] we always
focused on the bit error rate for the R = 7=8 convolutional code instead of the MSE
as clipping eects clearly degraded the BER while improving the MSE. The BER thus
proved a more precise means of assessing the performance, results are shown in gure
10. The operating point was selected to be Eb=N0 = 5:5dB with a frequency oset
of fT = 0:125. We varied the normalized sampling rate Ts =T between 0.25 and
14
Figure 10: Simulated BER performance of complete quantized DVB receiver: At
Eb=N0 = 5:5dB, the BER of the RCC = 7=8 punctured convolutional code is 6 10;5,
at Eb=N0 = 5:15dB it is 2 10;4 and at Eb=N0 = 5dB it is 4 10;4.
0.49. For c = 0:38 and V = 0:01 the prelters (selected according to gure 7) have
to be switched according to table 1. If, however, the M=T synchronizer is used, below
Ts =T = 1=3:26 = 0:3067 we have to use M = 3. We simulated the eect of frequency
osets in the lter and the phase recovery loop was also simulated. The matched lter
for the 2=T mode had length 9 taps, whereas the matched lter for the 3=T {mode had
13 taps.
It is seen that the degradation of the receiver is almost independent of the sampling
rate. The degradation of the 3/T scheme is slightly higher. For other SNR and other
convolutional code rates, a similar degradation can be expected.
7 Conclusions
We presented two techniques for designing wide{range variable symbol rate fully digital
receivers based on a free running oscillator clock. One of the approaches is based on
15
increases the intermediate synchronized sampling rate. In terms of implementation
eciency, the second approach probably will prove superior, as only a second matched
lter running at rate 3=T is needed, whereas the rst technique requires two additional
lters running at a sampling rate 1=Ts and using more coecients. Both techniques
do not produce signicant additional loss in a fully quantized bittrue receiver model
suitable for DVB reception.
Further work has to detail the acquisition performance, since false locks may occur at
multiples of the symbol rate.
8 Appendix
8.1 Detector S-Curve
We assume that the symbol duration T is acquired correctly already and compute the
shape and the slope of the detector curve depending on the time lag of the synchronized
samples with respect to the correct sampling instant. In this case,
ek = rf ( k ) (rf ( k + T=M ) ; rf ( k ; T=M )) :
is the instantaneous error detector output (at rate 1=T ) and
E ek =
X
q
g( ; (q ; 1=M
)T )g( + qT )} +
{z
|
(q)
X
q
g| ( ; (q ; 1=M
)T )g( + qT )}
{z
(q )
(24)
is the average of the detector output in case that the noise is stationary (noise terms
cancel in this case) and the symbols are iid with symbol energy one, g is the modulation
pulse. We closely follow 5] in the notation, (basically the dierence in the computations
is that 1=2 has to be replaced by 1=M ) and rstly compute one using the Fourier
transform where
Z
A(f ) = ( )e;j2f d
B(f ) =
Z
( )e;j2f d
We will see that in the innite series P and P only few terms survive. The quantity
A(f ) results to
Z
A(f ) = g( + (q ; 1=M )T )g( + qT )e;j2f d
(25)
16
A(f ) =
Z
G( )ej2qT
Z
g( + (q ; 1=M )T )e;j2 (f ;) dd
(26)
and using 0 = + (q ; 1=M )T we nally obtain
A(f ) =
Z
G( )ej2qT
Z
Z
g( 0)e;j2( ;(q;1=M )T )(f ;)d 0d
0
A(f ) = ej2f (q;1=M )T G( )G(f ; )ejT d
(27)
Inserting this in Pq (q ) and using the time shift relation of the fourier transform we
obtain
X
q
Z
Z
(q ) = G( )ejT G(f ; )ej2f e;j2fT=M
X
q
ej2fqT dfd:
(28)
and using the fact that 5]
T
1 j2fqT X
1
e
= (f ; m=T )
;1
;1
X
(29)
we obtain
Z
X
(q ) = T1 G( )ejT G(m=T ; )ej2m=T e;j2m=M d
q
m
Z 1=T
m
=1
X
(;1)m ej2m=T
G( )G(m=T ; )ej2T=M d (30)
= T1
;
1
=T
m=;1
in the case that G(f ) = 0 for jf j > 1=T (which is usually the case). Performing the
same steps for (and B) we obtain
X
X
q
Z
X
( q) = T1 (;1)m ej2m=T G( )G(m=T ; )e;j2T=M d
m
(31)
The quantity of interest was
(;1)m ej2m=T G(f )G(m=T ; f ) sin 2
fT
M df
q
m=;1
(32)
For a real modulation pulse g, (G is symmetric) the summand for m = 0 vanishes due
to the odd symmetry and the surviving terms are
U ( ) =
X
(q ) ; (q ) = 2Tj
1
X
Z
Z 1=T
2
j
j
2
=T
G
(f )G(1=T ; f ) sin 2
fT
U ( ) = ; T e
df
M
0
Z 0
+ej2=T ;1=T G(f )G(;1=T ; f ) sin 2
fT
df
:
M
17
(33)
Computation of the S- curve
M =2
M =3
M =4
sin
√
2 fT
M
1 1
− sin
2 2
1 1
− sin
2 2
⎛
⎜
⎝
⎛
⎜
⎝
fT
fT
−
−
1
2
1
2
⎞
⎟
⎠
⎞
⎟
⎠
α=0.35
0.3
0.4
0.5
0.6
fT
0.7
nonzero Overlap
and Integration Region
Figure 11: Integrand factors of the S{curve gain
We admit that there is not much dierence to the original result of 5] where only M = 2
was investigated, but the extension to other oversampling factors is dicult to establish.
By introducing the real Ge and Go, we may write
G(f ) = Ge(f ) + jGo(f )
(34)
where Ge (f ) (Go (f )) obey an even (odd) symmetry in case that g is real. For the general
case (non-symmetric g) we obtain 5]
q
U ( ) = ; T4 G2I + G2R sin( 2T
; )
(35)
where
Z 0
GI = ;1=T (Go(f )Ge (;1=T ; f ) + Ge (f )Go (;1=T ; f )) sin 2
fT
M df
Z 1=T
GR = 0 (Ge (f )Ge (1=T ; f ) ; Go(f )Go (1=T ; f )) sin 2
fT
M df:
= tan;1 GGI
R
(36)
as in 5]. For a real g that also is symmetric, we obtain
Z 1=T
4
2
U ( ) = ; T sin T 0 G(f )G(1=T ; f ) sin 2
fT
M df}
|
{z
X
18
(37)
Firstly, we consider root{raised cosine pulses.
r
1
Gr = p 1 ; sin 2
(2fT ; 1)
(38)
2
holds in the spectral range f 2 (1 ; )=2T (1 + )=2T ] of interest. In this case,
s
s
Z 1+
2T 1
X = 1 2 1 ; sin 2 (2fT ; 1) 1 + sin 2 (2fT ; 1) sin 2
fT
M df (39)
2T
which leads to
Z 1+
2T
X = 1 cos 2 (1 ; 2fT ) sin 2
fT
(40)
M df:
;
;
2T
Solving the integral (using 17]) we obtain
2
1+ + sin 1; 3
1+ + sin 1;
sin
sin
1 4 2 M
2 M
; 2 M2 ;1 2 M 5
(41)
X = 2
T
2 +1
M M that simplies to
2
1
M
X = T sin M (1 + ) + sin M (1 ; ) M 2 ; 42
(42)
and leads to the expression given in the paper previously. For raised{cosine pulses,
we rst have to look at the (normalized) symbol energy which is, after ltering a unit
energy root{raised pulse with a matched lter, reduced to (4 ; )=4T < 1=T . In this
derivation, however, we do not correct the variation of the pulse energy but keep the
spectrum
Z
(43)
Gr = 21 1 ; sin 2
(2fT ; 1) G2r df = 4 4;T of the received pulse for f 2 (1 ; )=2T (1 + )=2T ]. In this case,
Z 1+ 1
1
2T
2
X = 1 4 ; 4 sin 2 (2fT ; 1) sin 2
fT
(44)
M df
2T
which simplies to
Z 1+
1
2
fT
2T
X = 8 1 sin M 1 + cos (2fT ; 1) df
(45)
2T
whose only signicant dierence to the root{raised case is a factor two in the argument
of the cosine. This integral can be solved using 17] and the result is
M
(46)
X = 16
T cos M (1 ; ) ; cos M (1 + ) 1 ; 12=M 2
The maxima of the S{curve can be found from gure 12. The slope of the S{curve is
U ( = 0) = ;8
X=T 2.
;
;
19
Maximum of S-Curve
2XT
0.9
0.8
0.7
root-raised cosine
pulses
0.6
M =2
0.5
M =3
simulated
points
0.4
M =4
0.3
raised cosine pulses
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
α
rolloff
0.7
0.8
0.9
1
Figure 12: Maxima of the S{curve for various M and . Compare gure 4.
8.2 Computation of the CRB
As known, the timing error variance results to 7,11,12]
1
R
#;1
"
j
G(!)j2d!
2
E
s
2BLT 1R;1
Varf0T ; ^T g = ~
:
(47)
N0
!2jG(!)j2d!
;1
The timing error variance strongly depends on the pulse shape (ie. the rollo parameter
). The larger , the smaller the variance.
We have to insert Gr (!) ! = 2
f where Gr corresponds to the root raised cosine
pulse.
1
Z
;1
1
Z
;1
1;
jG(!)j2d!
ZT
= 2
= 2T
0
= 2
1+
d! + 2
ZT
1;
T
1 1 + sin 1 ; T ! d!
2
2
(48)
1;
!2jG(!)j2d!
ZT
0
1+
!2d! + 2
ZT
1;
T
20
T
2 1 + sin 2 1 ; ! d!
!2 1
= 23 T
(1 + 32) +
1
;
T
!2 sin 2
1 ; !T
:
(49)
The last integral can easily be solved using the substitution !0 = =T ; !. We obtain
References
2
Var 2BLT~ 2 3
:
2ES =N0 3 T 2 (1 + 32) ; 16
T 2
(50)
1] L. Erup, \Simulation of Interpolators and Their Control," in Proceedings of the IEEE CAMAD-3,
IEEE, 1990.
2] F. M. Gardner and L. Erup, \Interpolation in Digital Modems - Part i," IEEE Transactions on
Communications, vol. 41, no. 3, pp. 501{507, 1993.
3] G. Karam, K. Maalej, V. Paxal, and H. Sari, \Variable Symbol{Rate Modem Design for Cable and
Satellite Broadcasting," in Proc. 7th International Tirrhenian Workshop on Digital Communications
(M. Luise and E. Bighlieri, eds.), (London), Springer, 1996.
4] H.Meyr, M.Oerder, A.Polydoros, \On Sampling Rate, Analog Preltering and Sucient Statistics
for Digital Receivers," IEEE Transactions on Communications, 1994.
5] F. Gardner, \A BPSK/QPSK Timing{Error Detector for Sampled Receivers," IEEE Transactions
on Communications, vol. COM{34, pp. 423{429, May 1986.
6] M. Oerder and H. Meyr, \Derivation of Gardner's timing error detector from the maximumlikelihood
principle," IEEE Transactions on Communications, vol. COM-35, pp. 684{685, June 1987.
7] H. Meyr et al., Synchronization in Digital Communications, vol. 2. John Wiley & Sons, 1997. to be
published.
8] T. I. Laakso, V. Valimaki, M. Karjalainen, and U. Laine, \Splitting the Unit Delay," IEEE Signal
Processing Magazine, pp. 30{60, January 1996.
9] The MathWorks, MATLAB Reference Guide. Cochituate Place, 24 Prime Park Way, Natick, Mass.
01760, 1994. Documentation of MATLAB 4.2.
10] L. Scharf, Statistical Signal Processing. Addison Wesley, 1993.
11] M. Moeneclaey, \A simple lower bound on the linearized performance of practical symbol synchronizers.," IEEE Transactions on Communications, vol. COM{31, pp. 1029{1032, 1983.
12] M. Moeneclaey, \A fundamental lower bound on the performance of joint carrier and bit synchronizers," IEEE Transactions on Communications, vol. COM{32, pp. 1007{1012, 1984.
13] Synopsys, Mountain View, CA, COSSAP stream driven simulator user guide, 1994. version 6.7.
14] European Telecommunications Institute, \Digital Broadcasting System for Television, Sound and
Data Services Framing Structure, Channel Coding and Modulation for 11/12 GHz Satellite Services," Draft DE/JTC-DVB-6, ETSI Secretariat, 06921 Sophia Antipolis { France, August 1994.
15] Y. Yasuda, \High{Rate Punctured Convolutional Codes for Soft Decision Viterbi Decoding," IEEE
Transactions on Communications, vol. COM-32, March 1984.
16] J. P. Odenwalder, Optimum Decoding of Convolutional Codes. PhD thesis, University of Caliornia,
Los Angeles, 1970.
17] I. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Products. Academic Press, 1980. corrected and enlarged Edition.
21
Modulation
QPSK with Root{Raised Cosine Pulses
(rollo = 0:35) and Gray{Encoding
Channel Coding
Convolutional Code (CC)15,16] RCC = 1=2 : : : 7=8
Reed{Solomon Code
(204,188)
Operating Point
EbCC =N0
4.2 : : : 6:15dB
BER behind CC
2 10;4
Example for Symbol Rates
ranging
20: : : 44MS/s
Table 3: Outline of the DVB standard
22