Timing control for Modem receivers

For a modem receiver using an adaptive equalizer with fractional tap spacing, method and apparatus are disclosed for controlling the sample-timing phase. By evaluating bandedge components of the received signal in a particular way, a timing-phase vector signal is derived which is independent of the signal energy at the bandedges and of the quality of frequency separation of the filters for the bandedge signals. After an initial period, the current timing-phase vector signal is captured and stored as a reference. Thereafter, the sampling phase of the receiver is kept at its initial random value, represented by the stored reference timing-phase vector. The necessity to initially change the sampling phase in the receiver to a value which is forced by the received signal is avoided.

FIELD OF INVENTION 
Present invention is concerned with timing control in modem receivers, and 
in particular with control of the sampling phase in a receiver comprising 
an adaptive equalizer whose tap spacing is a fraction of the modulation 
interval T. 
BACKGROUND 
Several methods and systems are known for controlling the sampling time in 
modem receivers which utilize equalizers with a tap spacing equal to the 
modulation interval T, or a fraction thereof. The known methods and 
systems were disclosed, e.g., in the following publications and patent: 
(a) D.L. Lyon: "Timing Recovery in Synchronous Equalized Data 
Communication", IEEE Transactions on Communications, Vol. COM-23 (1975) 
pp. 269-274. 
(b) F. G. Caron et al.: U.S. Pat. No. 4,039,748 "Method and Device for 
Synchronizing the Receiver Clock in a Data Transmission System". 
(c) D. Godard: "Passband Timing Recovery in an All-Digital Modem Receiver", 
IEEE Transactions on Communications, Vol. COM-26 (1978) pp. 517-523. 
(d) G. Ungerboeck: "Fractional Tap-Spacing Equalizer and Consequences for 
Clock Recovery in Data Modems", IEEE Transactions on Communications, Vol. 
COM-24 (1976) pp. 856-864. 
(e) P. R. Chevillat, D. Maiwald, G. Ungerboeck: "Rapid Training of a 
Voice-Band Data-Modem Receiver Employing an Equalizer with Fractional-T 
Spaced Coefficients", IEEE Transactions on Communications, Vol. COM-35 
(1987) pp. 869-876. 
Three of these disclosures (a, b and c) describe schemes which employ 
bandpass filters to extract signal components from the bandedges of the 
received signal for timing control ("bandedge timing"). The average energy 
of the sum of the bandedge signals is a periodic function, with period T, 
of the sampling phase at which the bandedge signals are observed. This 
dependency is exploited for timing-phase control. In particular, it is 
argued that the sampling phase should be adjusted to a value for which the 
maximum bandedge energy is obtained. Equalizers with T-spaced taps achieve 
optimum performance for this phase. 
The known schemes adjust the sampling phase only to this phase, and later 
maintain it at this value. The selectivity of the bandpass filters plays 
an important role. Furthermore, the signal power in the bandedge regions, 
which depends on the a priori unknown attenuation characteristics of the 
currently used transmission channel, acts as a multiplicative factor in 
the obtained phase-error measurements, and thus influences the dynamic 
behavior of the timing control scheme. It is desireable to have a timing 
control scheme which is independent of these conditions. For receivers 
with an equalizer, whose taps are spaced by a fraction of T (FTS 
equalizer), it is furthermore advantageous to have a timing control scheme 
which only measures the initial random sampling phase and then maintains 
the sampling phase at this value. 
OBJECTS OF THE INVENTION 
It is an object of the invention to devise a timing control scheme that is 
insensitive to the frequency separation of bandpass filters and whose 
dynamic behavior is independent of the signal power in the bandedge 
regions. 
It is another object to provide a timing control scheme for receivers with 
an FTS equalizer that allows to capture an inititial random sampling phase 
and then to maintain the sampling phase at this value. 
SUMMARY OF THE INVENTION 
These objects are achieved by a timing control method and apparatus as 
defined in claims 1 and 4. Preferred particular embodiments of this method 
and of the apparatus are defined in the dependent claims. 
The invention achieves the objects mainly by forming a timing-phase vector 
tpv, whose angle represents the current sampling phase, based on lowpass 
filtering the difference of the products of two complex bandpass-filter 
outputs obtained at T/2-spaced time instants, whereby the result becomes 
bias free and independent of the frequency separation of bandpass filters; 
and by normalizing the magnitude of tpv by an appropriate gain control for 
the lowpass filter such that the magnitude of tpv becomes independent of 
the power of the bandedge signals. 
An advantage of present invention is that it allows to use simple bandpass 
filters with non-critical frequency separation. 
A further advantage is that dynamic behavior of the disclosed timing 
control scheme does not depend on the attenuation characteristics of the 
currently used transmission channel. 
Another advantage of the invention is that it allows to maintain the 
receiver sampling-time phase at its initial random value. 
These and other advantages will become more apparent from the following 
description of a preferred embodiment of the invention with reference to 
the drawings.

DETAILED DESCRIPTION 
1. Principles of receivers using equalizers with "fractionally-spaced" taps 
and of their timing control 
The disclosed timing control method is suited for modem receivers which 
incorporate an adaptive equalizer whose tap spacing is a fraction of one 
modulation interval T (FTS equalizer). Usually, the spacing is chosen 
equal to T/2. An FTS equalizer permits the achievement of a low 
mean-square error essentially independently from the sampling phase (cf. 
above-cited article by Ungerboeck). Hence, it is sufficient to "capture" a 
random sampling phase at the beginning of receiver training, and then to 
maintain this phase during subsequent receiver operations. The invention 
constitutes an efficient solution for performing these functions. 
The disclosed method is particularly useful in connection with the fast 
start-up technique described in above-cited article by Chevillat et al., 
which benefits from the absence of a timing-preamble sequence and adjusts 
the equalizer by spectral division from a cyclic pseudo-random sequence. 
However, the method is equally well suited for modem receivers which are 
trained more conventionally by first receiving a timing-preamble sequence 
and then adjusting the equalizer by the slower and simpler least 
mean-square gradient algorithm from an equalizer training sequence. 
Earlier equalizers with T-spaced taps achieve a low mean-square error only 
for certain sample-timing phases. Hence, it is necessary to establish a 
suitable sample-timing phase before the equalizer can be trained. For this 
purpose, usually a timing-preamble sequence is sent prior to an equalizer 
training sequence. From the timing-preamble sequence, the receiver 
recognizes the beginning of signal reception and determines the particular 
sampling phase at which the T-spaced equalizer is able to operate. The 
sampling phase must be changed to this phase before equalizer training can 
start. This procedure is also used in modem receivers which employ FTS 
equalizers to achieve better equalization, but do not fully exploit the 
sampling-phase independence of these equalizers. 
2. Prior solution for control of the timing-phase 
The disclosed timing control method of present invention is related to the 
scheme described in U.S. Pat. No. 4,039,748 and in the article by Godard 
cited above. 
FIG. 3 of the Godard article illustrates a timing control scheme in which 
the received carrier-modulated signal is converted to a complex passband 
signal by a phase splitter, also referred to as a receive Hilbert filter 
(the obtained complex-valued "analytic" signal contains only signal 
components at positive frequencies). From this signal the components 
around the upper and lower bandedge frequencies, f.sub.0 +1/2T and f.sub.0 
-1/2T Hz, where f.sub.0 denotes the carrier frequency, are extracted by 
two complex bandpass filters. For consistency with later descriptions, let 
the complex output signals of these two filters be x.sup.U (t) and x.sup.L 
(t) (U=upper, L=lower); in the Godard article these signals are denoted 
g.sub.2 (t) and g.sub.1 (t), respectively. The imaginary part of the 
complex correlation product x.sup.U (t).x.sup.L (t), where the overbar 
designates a conjugate-complex value value, is formed and sampled once per 
modulation interval at time nT+.tau., where .tau. denotes the sampling 
phase. The quantity is used as a phase-error signal in the arrangement of 
a second-order loop which adjusts .tau. such that the imaginary part of 
the correlation product vanishes in the mean. It can be shown that this 
phase corresponds approximately to the sampling phase at which the maximum 
average energy of the sum of the bandpass filter outputs is obtained, and 
hence constitutes a suitable sampling phase for an equalizer with T-spaced 
taps. 
Note that the correlation product is sampled only once per modulation 
interval. As will be shown later, the expectation of the correlation 
product takes on the form E{x.sup.U (nT+.tau.).x.sup.L 
(nT+.tau.)}=A'+C'exp(j2.pi..tau./T), where A' and C' are generally 
complex-valued quantities which depend only on the received signal 
spectrum and the characteristics of the bandpass filters. The exponential 
term indicates the dependence on the sampling phase .tau.. 
The Godard article suggests that A' be zero (see Eq. (27)). However, it was 
found experimentally and confirmed mathematically that the quantity A' 
does not vanish, unless ideal bandpass filters with no spectral overlap 
are used. If simple first- or second-order bandpass filters are employed, 
the value of A' cannot always be neglected compared to the magnitude of 
C'. The latter depends critically on the signal power in the bandedge 
regions. If, owing to severe channel attenuation at the bandedges, the 
magnitude of C' becomes small, a non-zero value of A' can lead to biased 
phase-errors estimates and in extreme cases to completely wrong 
sampling-phase adjustment. 
Note further that considering only the imaginary part of the above 
correlation product as a phase-error signal restricts the adjustment of 
the sampling phase to one particular value. The scheme does not allow to 
measure an arbitrary sampling phase. Also, since the power of the 
correlation products is not controlled, the effective bandwidth of the 
employed phase-locked loop depends on the signal power in the bandedge 
regions. The dynamic behavior of the phase-locked loop is thus influenced 
in an undesirable manner by the spectral shape of the received signal. 
Finally, it should be noted that the timing control scheme described in the 
Godard article works equally well on a complex baseband signal obtained by 
shifting the passband signal in frequency to baseband by multiplication 
with exp(-j2.pi.f.sub.0 t). The complex bandpass filters must then have 
their center frequencies at +1/2T and -1/2T, respectively. 
3. Receiver front-end functions to obtain a complex baseband signal 
For the disclosed timing control method it is assumed that the receiver 
front-end functions convert a received real-valued carrier-modulated 
signal 
##EQU1## 
to a sampled version of the complex baseband signal 
##EQU2## 
In Eqs (1) and (2), the quantities a.sub.i represent modulation symbols 
from a set of generally complex-valued discrete amplitudes; the signal 
element h(t) describes the overall complex-baseband response of the 
transmission system up to the inputs of the equalizer and the 
timing-control scheme; and w.sub.0 (t) and w(t) denote additive noise 
signals which are neglected in the remaining discussions. The baseband 
signal is sampled at rate T/2 with sampling phase .tau.: 
EQU x.sub.k (.tau.)=x(kT/2+.tau.). (3) 
During the n-th modulation interval, samples x.sub.2n (.tau.) and 
x.sub.2n+1 (.tau.) enter the equalizer delay line, and are also used as 
input to the timing control scheme. 
The sequence of symbols {a.sub.i } is either a random data sequence with 
the property 
EQU E{a.sub.i a.sub.i+m }=E.sub.s .delta..sub.m, (4) 
where E.sub.s represents the average symbol energy and .delta..sub.m 
denotes the Kronecker delta function, or during start-up a suitable 
sequence of training symbols. 
4. Realization of the invention 
FIG. 1 shows a block diagram of a realization of the invention. The 
arrangement comprises the following components: 
receiver front-end elements (11) for converting the received 
carrier-modulated signal to a sampled complex-baseband signal (not part of 
the invention); 
an adaptive equalizer (13) with fractionally-spaced taps (not part of the 
invention); 
two complex bandpass filters BPF-U (15) and BPF-L (17) with center 
frequencies at f=.+-.1/2T. (For an alternative realization of the 
invention with a passband signal, these frequencies should be changed to 
f=f.sub.0 .+-.1/2T.) 
a multiplier element (19) for forming a correlation product of the output 
signals of the two bandpass filters; 
a T/2 delay element (21) and a subtracting element (23) for forming 
differences of subsequent products furnished by the multiplier (19); 
a lowpass filter LPF (25) with a gain control element (27), connected to 
the output of the subtracting element, furnishing at its output a 
timing-phase vector; 
a register or storage unit (29) for holding a timing-phase vector reference 
value; this register is loaded at a particular time (TC-REF) after the 
beginning of signal reception with the then current value of the 
timing-phase vector signal furnished by the LPF (25); 
a phase-error generating element (31) for measuring a phase difference 
between the timing-phase vector values furnished by the low pass filter 
(25) and the reference register (29); 
a timing-phase adjusting element (33) for forming a new timing phase in 
response to a previous timing phase and the current phase error; 
timing means (35) comprising an oscillator or equivalent means, for 
generating a clock signal whose phase is controlled by the adjusting elemt 
(33); and 
a sequencing control unit (37) which furnishes several sequencing control 
signals (TC-E, TC-O, TC-G, TC-P, TC-REF) for determining the sequence in 
which the different units of the arrangement are active. 
The two bandpass filters BPF-U and BPF-L extract from the baseband signal 
the components in the upper and lower roll-off regions, i.e., around the 
Nyquist frequencies +1/2T and -1/2T. The BPF outputs x.sub.2n.sup.U 
(.tau.), x.sub.2n+1.sup.U (.tau.) and x.sub.2n.sup.L (.tau.), 
x.sub.2n+1.sup.L (.tau.) are pairwise correlated and the difference 
betweeen the two T/2-spaced correlation products is formed. The 
expectation of this difference exhibits the desired bias-free form C 
exp(j2.pi..tau./T), even if the bandpass filter exhibit non-negligible 
spectral overlap. Hence, simple single-pole BPF's can be employed. 
Lowpass filtering of the difference of the correlation product greatly 
reduces short-term fluctuations. Hence the timing-phase vector tpv.sub.n 
(.tau.) obtained at the output of the LFP filter will closely resemble the 
expectation of the input signal. 
The magnitude of the timing-phase vector is controlled by an algorithm 
which scales the LPF input gain g.sub.LPF and the output tpv.sub.n (.tau.) 
such that .vertline.tpv.sub.n (.tau.).vertline. remains close to the unit 
radius. 
The phase error .DELTA..tau..sub.n representing the phase difference 
between tpv.sub.n (.tau.) and the reference timing-phase vector 
tpv.sub.REF is computed, and used to control the sampling phase .tau.. 
Since under normal conditions phase differences remain small and the 
magnitudes of the timing-phase vectors are approximately normalized, the 
phase difference arg{tpv.sub.n (.tau.)}-arg{tpv.sub.REF } is well 
approximated by 
EQU .DELTA..tau..sub.n =Im{tpv(.tau.).sub.n tpv.sub.ref }. (5) 
The objective of timing-phase adjustments is the control the sampling phase 
.tau. such as to minimize the phase error .DELTA..tau..sub.n. With the 
adjustments 
EQU .tau..rarw..tau.-.gamma..DELTA..tau..sub.n -.DELTA..tau..sub.s,n,(6a) 
EQU .DELTA..tau..sub.s,n+1 =.DELTA..tau..sub.s,n +.zeta..DELTA..tau..sub.n,(6b) 
illustrated in FIG. 2, the function of a second-order phase-locked loop 
(PLL) is achieved. The quantities .gamma.(&gt;0) and .zeta.(&gt;0) are the 
first- and second-order loop gains, and .DELTA..tau..sub.s,n represents 
the estimated timing drift per modulation interval between the timing of 
the received signal and the free-running receiver timing. The timing-phase 
adjustment arrangement shown in FIG. 2 comprises delay element 39, adding 
means 41, and multiplying means 43, constituting the first-order part of 
the PLL; and further comprises delay element 45, adding means 47, and 
multiplying means 49, constituting the second-order part of the PLL. 
The principle purpose of the sequencing control unit is to operate the 
disclosed scheme first in "capture" mode, during which the timing-phase 
vector settles to a valid value, then to store this value as the reference 
timing-phase vector, and finally control the sampling phase such that 
phase represented by the reference vector is maintained. Additional 
functions are described later. 
5. More details on the timing-control operation 
Bandpass Filters 
The bandpass filters BPF-U and BPF-L operate at sampling rate 2/T. Their 
transfer functions are conveniently described by 
EQU S.sup.u (f)=S(f-1/2T), S.sup.L (f)=S(f+1/2T), (7) 
where 
##EQU3## 
denotes the transfer function, with period 2/T, of an equivalent 
time-discrete single-pole lowpass filter. Appropriate values for 
.rho..sub.B are in the range between 7/8 and 15/16. With g.sub.B 
=1-.rho..sub.B, the BPF's achieve unit gain at their center frequencies. 
Consecutive BPF output signals are computed by the recursions 
EQU X.sub.k.sup.U =+j.rho..sub.B X.sub.k-1.sup.U +g.sub.B X.sub.k (.tau.),(9a) 
EQU X.sub.k.sup.L =-j.tau..sub.B X.sub.k-1.sup.L +g.sub.B X.sub.k (.tau.),(9b) 
which are executed first for k=2n, and then for k=2n+1. 
Properties of the correlation products 
For the following derivation, it is assumed that a random sequence {a.sub.i 
}characterized by (4) is transmitted, and that the bandwidth of the 
received signal is less than twice the modulation rate, i.e., H(f) and 
H(f+l/T) exhibit no spectral overlap for .vertline.l.vertline..gtoreq.2, 
where H(f) is the Fourier transform of the signal element h(t). The signal 
elements obtained from h(t) at the output of the bandpass filters BPF-U 
and BPF-L are denoted h.sup.U (t) and h.sup.L (t), with Fourier transforms 
EQU H.sup.U (f)=H(f)S(f-1/2T),H.sup.L (f)=H.sup.L (f)=H(f)S(f+1/2T).(10) 
The expectation of correlation products X.sub.2n+m.sup.U X.sub.2n+m.sup.L, 
for m=0,1, is obtained as follows: 
##EQU4## 
Substitution of Eq. (10) and observing that the integrals are zero for 
.vertline.l.vertline..gtoreq.2 and negligible for l=-1, yields 
##EQU5## 
It can be seen that A' is zero only if the bandpass filters provide 
complete spectral separation, i.e., S(f-1/2t)S(f+1/2T)=0, and that the 
magnitude of C' depends strongly on the signal power in the bandedge 
regions around .+-.1/2T. 
Taking the difference of two T/2-spaced correlation products leads to the 
expectation 
EQU E{X.sub.2n.sup.U. X.sub.2n.sup.L -X.sub.2n+1.sup.U.X.sub.2n+1.sup.L 
}=Ce.sup.j2.pi..tau./T, C=2C', (13) 
in which the undesired bias quantity A' is eliminated. 
Timing-Phase Vector 
The timing-phase vector is obtained by the recursive lowpass filter 
operation 
EQU tpv.sub.n (.tau.)=.rho..sub.LPF tpv.sub.n-1 (.tau.)+g.sub.LPF 
[X.sub.2n.sup.U X.sub.2n.sup.L -X.sub.2n +1.sup.U X.sub.2n+1.sup.L ].(14) 
An appropriate value for .rho..sub.LPF is 127/128. 
Magnitude Control of the Timing-Phase Vector 
The magnitude of tpv.sub.n (.tau.) is continuously monitored. If it 
deviates from the unit radius by more than a specified amount, e.g., 
.+-.5%, the LPF gain g.sub.LPF and tpv.sub.n (.tau.) are scaled instantly, 
provided g.sub.LPF does not exceed a maximally allowed value g.sub.LPFmax. 
The following algorithm performs this function: 
EQU .DELTA.=.vertline.tpv.sub.n (.tau.).vertline..sup.2 -1 (15a) 
EQU .alpha.=1-.DELTA./4 (15b) 
If .vertline..DELTA..vertline.&gt;0.1 and .alpha.g.sub.LPF 
.ltoreq.g.sub.LPFmax then: 
EQU g.sub.LPF .rarw..alpha.g.sub.LPF, tpv.sub.n (.tau.) .rarw..alpha.tpv.sub.n 
(.tau.). (15c) 
6. Summary of problems solved by the invention 
Essential features of the invention disclosed above are as follows 
(equations are simplified and denoted by Roman numerals): 
First feature: Bias-free correlation term 
The output signals of the upper and lower bandpass filters are sampled 
twice per modulation interval. The samples are denoted 
EQU X.sub.2n.sup.U =X.sup.U (nT+.tau.), X.sub.2n +1.sup.U =X.sup.U 
(nT+T/2+.tau.) (1a) 
EQU and 
EQU X.sub.2n.sup.L =X.sup.L (nT+.tau.), X.sub.2n +1.sup.L =X.sup.L 
(nT+T/2+.tau.), (1b) 
for the upper and lower bandpass filters, respectively. A correlation term 
consisting of the difference of two correlation products is used: 
EQU .DELTA.tpv.sub.n (.tau.)=X.sub.2n.sup.U (.tau.)X.sub.2n.sup.L 
(.tau.)-X.sub.2n+1.sup.U (.tau.)X.sub.2n+1.sup.L (.tau.). (11) 
The expectation of this new correlation term, given by Eq. (13) above, 
takes on the form 
EQU E{.DELTA.tpv.sub.n (.tau.)}=Ce.sup.j2.pi..tau./T. (111) 
The exact circular dependence on the sampling phase is obtained without 
requiring bandpass filters with strong frequency separation. Hence, the 
use of first-order complex bandpass filters with a single imaginary-valued 
pole is sufficient. 
Second feature: Lowpass filtering and power control 
The variance of the correlation term given by Eq.(11) is significantly 
reduced by lowpass filtering. The resulting complex signal is called 
"timing-phase vector" and obtained by 
EQU tpv.sub.n (.tau.)=.rho..sub.LPF tpv.sub.n-1 (.tau.)+g.sub.LPF 
.DELTA.tpv.sub.n (.tau.), (IV) 
where .rho..sub.LPF represents a real-valued pole close to, but smaller 
than unity. 
The gain g.sub.LPF and the magnitude of the timing-phase vector are almost 
instantly adjusted by a mechanism described above, such that the 
timing-phase vector remains close to a value on the unit circle. In this 
this way, the timing-phase vector is essentially made independent of the 
signal power in the bandedge regions of the received signal. Its angle 
represents the currently estimated sampling phase. 
Third feature: Capturing the initial sampling phase 
At the beginning of receiver training, the timing control scheme operates 
in capture mode. The timing-phase vector is updated according to Eq.(IV), 
but its value is not yet used for sampling-phase control. When the time 
interval specified for the capture mode elapses, the current value of the 
timing-phase vector is stored as a "reference timing-phase vector", 
tpv.sub.REF. Its angle represents the sampling phase to be maintained 
during subsequent receiver operations. 
During the capture period, which extends typically over N =100 . . . 300 
modulation intervals, the sampling phase of the receiver can drift 
relative to the phase of the received signal. However, with a maximum rate 
uncertainty of 10-.sup.4, as specified by CCITT, the effect of this drift 
is negligible. 
Fourth feature: Sampling-phase tracking 
During the subsequent tracking mode, updating of the timing-phase vector by 
Eq.(IV) continues. From the current timing-phase vector and the stored 
reference timing-phase vector, the phase error is computed: 
EQU .DELTA..tau..sub.n =lm{tpv.sub.n (.tau.)tpv.sub.REF }. (V) 
Eq.(V) yields a good approximation of the actual phase difference, because 
the magnitudes of the timing-phase vectors are controlled and phase 
differences are normally small. 
The phase error .DELTA..tau..sub.n is used to control the sampling phase 
.tau. according to the principles of a phase-locked loop. In the 
implementation of the disclosed timing control scheme, a second-order loop 
should be realized to cancel rate-offset between the rate of the received 
signal and the free-running receiver timing. The dynamics of the 
phase-locked loop can precisely be determined, because the employed phase 
errors do not depend on the signal power in the bandedge regions. 
7. Additional feature: Performing functions at reduced rate 
The bandwidth of all signals after the bandpass filters is small compared 
to the modulation rate. Hence, operations can be executed at a lower 
sampling rate without significant loss in performance. This permits the 
achievement of significant savings in the processing power required to 
realize the timing control scheme with a digital signal processor. 
A sequencing of operations is suggested using a sequencing control unit (37 
in FIG. 1; and FIG. 5) comprising a counter (TIMCNTL) which is incremented 
after each modulation interval. When the beginning of signal reception is 
detected, the counter is initialized to a negative value -N, where N is 
the capture period. When non-negative values are reached, the counter is 
limited to count modulo 4. Thus, TIMCNTL assumes values -N, -N+1, . . .-1, 
0, 1, 2, 3, 0, 1, 2, 3, 0, 1, . . . . 
After initialization, the following functions are performed: 
(1) TIMCNTL =even (TC-E): two consective output values for each of the two 
bandpass filters are computed. To compute these values without performing 
recursive filter operations for TIMCNTL =odd, the recursions 
##EQU6## 
are used. 
(2) TIMCNTL =odd (TC-O): the timing-phase vector is updated. Notice that 
this corresponds to replacing in Eq.(14) tpv.sub.n-1 (.tau.) by 
tpv.sub.n-2 (.tau.). Additional functions depend on a closer inspection of 
TIMCNTL. 
If TIMCNTL mod =1(TC-G), the magnitude of the current timing-phase vector 
is controlled and the gain g.sub.LPF is adjusted accordingly. 
If TIMCNTL mod 4=3 and TIMCNTL &lt;-1 no further functions are performed. If 
TIMCNTL =-1, tpv.sub.n (.tau.) is copied into tpv.sub.REF. If TIMCNTL 
&gt;0(TC-P), the phase error is computed from tpv.sub.n (.tau.) and 
tpv.sub.REF and the phase-locked loop functions are performed. This 
results in adjusting the phase of a receiver-timing oscillator, or 
equivalent timing means (e.g., programmable timer interrupts or signal 
interpolation), at rate 1/4. 
The sequencing control (37 in FIG. 1; and FIG. 5) comprising counter 
TIMCNTL distributes the processing load required for the above functions 
evenly among consecutive modulation intervals. 
An implementation of the sequencing control means 37 is shown in FIG. 5. It 
comprises the counter TIMCNTL (51) which is incremented by the modulation 
interval clock (53). Output lines B0, B1, and BS represent the least 
significant bit (B0), the next-to-least significant bit (B1), and the most 
significant sign bit (BS). Using two's complement number representation, 
negative values of TIMCNTL are represented by BS =1. If TIMCNTL reaches 
non-negative values, BS changes to 0 and forces modulo-4 counting (. . . 
0, 1, 2, 3, 0, 1, . . . ) by preventing carries from B1 to the 
next-significant bit. 
A number -N, determining the length of the capture period, is stored in a 
register (55) and loaded into the counter when a START signal (57) becomes 
active, indicating the beginning of receiver training. 
Sequencing control signals TC-E, TC-O, TC-G, TC-P, and TC-REF are obtained 
as follows. B0 is connected to an inverter 61 to produce TC-E, 
representing even counter contents. B0 is gives directly TC-O, indicating 
odd counter contents. B0 and the complement of B1 are combined in AND gate 
63 to form TC-G, which becomes active whenever B1, B0, are 0, 1, (TIMCNTL 
=1 mod 4). B0, B1, and the complement of BS are combined in AND gate 65 to 
form TC-P, which becomes active whenever BS, B1, B0 are 0, 1, 1(TIMCNTL =3 
mod 4, TIMCNTL &gt;0). All counter bits are combined in AND gate 67 to form 
TC-REF, which becomes active when all counter bits are 1's (TIMCNTL =-1).