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Description  |
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BACKGROUND OF THE INVENTION
The present invention relates generally to data communications. The
invention more particularly relates to timing recovery techniques used in
data receivers which have automatic and/or adaptive equalizers.
Accurate reception of high-speed data signals transmitted over a
bandlimited channel with unknown transmission characteristics requires the
use of an automatic equalizer. The equalizer, which is resident in the
receiver portion of a data set, or modem, is generally in the form of a
transversal filter. Samples of the received data signal, referred to
herein as line samples, are formed at a predetermined sampling rate. In a
so-called T/2 equalizer, for example, the line samples are formed at twice
the transmitted symbol rate. The line samples are applied to the
transversal filter, in which each is multiplied by a respective one of a
queue of coefficients. The resulting products are added together and, if
necessary, demodulated to generate a baseband signal, referred to herein
as an equalizer output. The value of each equalizer output is used as the
basis for forming a decision as to the value of a respective transmitted
data symbol.
In addition, an error signal is formed equal to the difference between each
equalizer output and a reference signal which represents its respective
data symbol. In the so-called adaptive type of automatic equalizer in
particular, the reference signal is derived from the above-mentioned
decision. The error signal is used to update, or adapt, the transversal
filter coefficient values in such a way as to minimize a measure of the
channel-induced distortion--assumed to be primarily intersymbol
interference--in the equalizer outputs.
An important equalizer operating parameter, in addition to the rate at
which the line samples are formed, is their time occurrence with respect
to the received signal. On the one hand, the coefficient values subsisting
in the equalizer at any given time are determined with the received signal
having been sampled at a particular set of time points on the received
signal. On the other hand, the frequencies of the transmitter and receiver
clocks invariably differ from one another, if only by a very small amount.
Over time, this frequency difference, if not compensated for, would cause
the received signal to be sampled further and further away from the
appropriate time points, this phenomenon being referred to as "timing
drift". As long as the sampling frequency is high enough, the equalizer
does have the ability to compensate for this clock frequency difference
(as long as it is not too large) via the coefficient update process. This
is not an effective long-term solution, however, because the distribution
of coefficient values will eventually become skewed to one end of the
coefficient queue and equalizer performance will degrade sharply.
To deal with this problem, the receiver is conventionally provided with a
so-called timing recovery circuit. The timing recovery circuit determines
whether the line samples are being formed earlier (later) than they should
be and, in response, adjusts the phase of the line sample forming
circuitry such that the line samples are formed a little later (sooner)
then they otherwise would be. This phase adjustment process is referred to
as retarding (advancing) the "receiver timing" or, alternatively, as
retarding (advancing) the sampling phase.
A commonly used timing recovery technique is so-called envelope-derived
timing recovery, disclosed, for example, in the Bell System Technical
Journal, Vol. 54, p. 569 et seq, March 1975. This technique extracts a
symbol-rate tone from the received signal and uses the phase of that tone
to control receiver timing. Envelope-derived timing recovery performs
satisfactorily for many applications. In some situations, however--such as
a narrow rolloff system--the recovered tone may be so weak that accurate
timing recovery is not possible when random data is being received.
An alternative timing recovery technique, referred to herein as
"coefficient tracking", controls receiver timing as a function of
coefficient distribution within the queue. One such technique is disclosed
in U.S. Pat. No. 4,004,226, issued Jan. 18, 1977 to S. U. H. Qureshi et
al. A particular coefficient location--typically at or near the queue
midpoint--is designated as the one at which the coefficient with the
largest magnitude should reside. Periodically, e.g., in each symbol
interval, the coefficient which actually has the largest magnitude is
identified. If it is found to be at the designated location, no timing
adjustment is made. Otherwise, the receiver timing is advanced or
retarded, as appropriate, by a predetermined fixed timing adjustment
increment such that subsequent coefficient adaptation over a number of
symbol intervals causes the largest coefficient to appear at the
designated location.
Another coefficient tracking timing recovery technique is disclosed in U.S.
Pat. No. 4,334,313, issued Jan. 8, 1982 to R. D. Gitlin et al. In
accordance with that technique, the coefficient queue is divided into
front and back portions only. When the largest coefficient is found in the
front portion, receiver timing is retarded. If it is found in the back
portion, receiver timing is advanced.
Although coefficient tracking seems to be generally satisfactory, it can
lead to instabilities, at least in theory. In addition, coefficient
tracking is not particularly amenable to quantitative analysis, making it
difficult to "fine-tune" the timing recovery process.
SUMMARY OF THE INVENTION
As in coefficient tracking, the present invention uses the values of the
equalizer coefficients as the basis for timing recovery. However, rather
than search for the largest coefficient, our technique uses a parameter we
refer to as the "center of gravity" of the equalizer coefficients.
Specifically, we have discovered that the center of gravity of the
coefficients is a direct measure of the timing offset and that, therefore,
its time rate of change is a measure of the timing drift. Thus by
computing the center of gravity (or at least a quantity that is reflective
of the value thereof) on a periodic basis, the current timing offset and
drift can be determined and the receiver timing can be adjusted so as to
reduce these toward zero.
This approach has a number of advantages. In particular, it provides a
quantitative measure of timing drift and offset, thereby facilitating
rapid and accurate timing adjustments. Also, being an equalizer-based
timing recovery scheme, timing can be recovered on any channel capable of
equalization. Such is not the case in, for example, an envelope-derived
recovery scheme when there is an absence of signal energy at the band
edges.
BRIEF DESCRIPTION OF THE DRAWING
In the drawing,
FIG. 1 shows an illustrative embodiment of a data signal receiver which
includes timing recovery circuitry embodying the principles of the
invention; and
FIG. 2 depicts a flowchart of the timing recovery technique used within the
receiver of FIG. 1 in accordance with the invention.
DETAILED DESCRIPTION
Receiver 100 shown in FIG. 1 is adapted for use in a voiceband data modem
in a communication system employing quadrature-amplitude modulation (QAM).
Four information bits are communicated once every T=1/2400 sec. The symbol
rate is thus 2400 baud, yielding a binary data transmission rate of 9600
bits per second. It will, of course, be appreciated that the invention is
applicable to transmission systems using other modulation schemes as well
as to baseband systems and can be used in systems using other baud and bit
rates.
The four bits to be transmitted are encoded into two signal levels, each of
which can take on one of the four values +1, -1, +3, -3. The two signal
levels amplitude modulate respective 1800 Hz in-phase and quadrature-phase
carrier waves which, in combination, comprise the transmitted QAM signal.
The QAM signal, representing a succession of data symbols transmitted at a
rate of 1/T symbols per second, is received by receiver 100 on lead 116.
This passband input signal, r(t), passes to analog input circuitry 120,
principally comprised of a bandpass filter. The output of circuitry 120 on
lead 121 is passed to an A/D converter 125.
A master clock 130 generates 4096 master clock pulses every T seconds on
lead 131. These are received by receiver timing generator 135. The latter
counts the pulses on lead 131 and generates timing signals on a number of
output leads to control the sequencing of the various signal processing
functions within the receiver. One of these leads, shown explicitly, is
lead 136. The latter extends pulses to A/D converter 125 at a rate which
causes A/D converter 125 to generate line samples at 4/T samples per
second. A/D converter 125 thus generates four passband line samples
r.sub.1m, r.sub.2m, r.sub.3m and r.sub.4m during the m.sup.th receiver
symbol interval. These are passed to Hilbert filter 128 which, for each
symbol interval, generates on lead 129 digital Hilbert transform pair
r.sub.m /r.sub.m and digital Hilbert transform pair r'.sub.m /r'.sub.m.
That is, a Hilbert transform pair appears on lead 129 once every T/2
seconds.
The aforesaid Hilbert transform pairs are demodulated to baseband in
demodulator 140 to form baseband pairs p.sub.m /p.sub.m ; p'.sub.m
/p'.sub.m on lead 141. One can think of p.sub.m and p.sub.m (p'.sub.m and
p'.sub.m) as the real and imaginary parts of a complex baseband sample
P.sub.m (P'.sub.m). In performing the demodulation, demodulator 140 uses
the values of cos (.omega..sub.c t) and sin (.omega..sub.c t) which it
receives from carrier source 165 on lead 166, .omega..sub.c being the
radian carrier frequency.
(It should be noted at this point that, due to processing delays, baseband
pairs p.sub.m /p.sub.m ; p'.sub.m /p'.sub.m are not necessarily generated
during the m.sup.th receiver interval, the latter being defined as the
T-second interval during which line samples r.sub.1m, r.sub.2m, r.sub.3m
and r.sub.4m are generated. The subscript m thus does not identify when
the baseband pairs are generated but, rather, identifies them as being
derived from samples r.sub.1m, r.sub.2m, r.sub.3m and r.sub.4m. Similar
considerations apply to all of the m-subscripted variables herein.)
The baseband pairs on lead 141 are passed to finite-impulse-response
equalizer 150 of conventional design. Since the equalizer receives and
processes more than one input for each symbol interval, it is referred to
as a "fractionally spaced" equalizer and, more specifically, as a T/2 type
of fractionally spaced equalizer since it receives and processes inputs at
a two-per-symbol-interval rate. The outputs of equalizer 150 on lead 151
are generated once per symbol interval and are, respectively, the real and
imaginary components z.sub.m and z.sub.m of a baseband equalizer output
Z.sub.m.
The components of baseband equalizer output Z.sub.m pass to carrier
recovery circuit 153 which corrects for discrepancies in the phase of the
carrier signal generated by carrier source 165 with respect to the
modulating carrier signal in the transmitter. Such discrepancies are due,
for example, to transmitter/receiver frequency and/or carrier phase
differences, channel-induced phase offset, etc. In particular, carrier
recovery circuit 153 responds to the error on lead 162 (generated as
described below) to form, in conventional fashion, a phase correction
estimate .theta.*.sub.m. Circuit 153 then applies that correction to
baseband equalizer output Z.sub.m by multiplying Z.sub.m by
e.sup.-j.theta.*.sbsp.m to generate on lead 156 a phase-corrected signal
Y.sub.m having real and imaginary components y.sub.m and y.sub.m.
Components y.sub.m and y.sub.m are quantized in decision circuit 160. The
resulting outputs, provided on lead 161, are decisions a*.sub.m and
a*.sub.m as to the signal levels which represent components a.sub.m and
a.sub.m of a particular transmitted symbol A.sub.m. Decisions a*.sub.m and
a*.sub.m can be thought of as the real and imaginary components of a
complex decision A*.sub.m.
Decision circuit 160 also provides on lead 162 the real and imaginary
components of a complex baseband error signal E.sub.m having real and
imaginary components e.sub.m and e.sub.m, where e.sub.m =(y.sub.m
-a*.sub.m) and e.sub.m =(y.sub.m -a*.sub.m). These are supplied on lead
162 to carrier recovery circuit 153, as noted above, and are also extended
to equalizer 150 for the purpose of coefficient updating.
In particular, for each symbol interval, equalizer 150 multiplies the
(2M+2) newest, i.e., most-recently formed, baseband samples applied
thereto by respective complex coefficients stored therein and forms the
sum of the resulting products to form equalizer output Z.sub.m in
accordance with
##EQU1##
In Eq. (1), the C.sub.i (m)'s are complex-valued coefficients each having
a particular value associated with the m.sup.th receiver symbol interval.
Each odd-indexed coefficient C.sub.1 (m), C.sub.3 (m), etc. is multiplied
by a respective one of the "unprimed" samples P.sub.m, P.sub.m-1, etc.,
while each even-indexed coefficient C.sub.2 (m), C.sub.4 (m), etc. is
multiplied by a respective one of the "primed" samples P'.sub.m,
P'.sub.m-1, etc. The values of the entire ensemble of coefficients at any
point in time define the transfer function of the equalizer.
Upon having generated Z.sub.m in accordance with Eq. (1), equalizer 150
thereupon updates the coefficient values stored therein to provide
coefficient values associated with the (m+1).sup.st symbol interval. The
updating rules illustratively used are
C.sub.2i-1 (m+1)=C.sub.2i-1 (m)-.beta.E.sub.m-d P.sub.m-i-d+1
-.gamma.[C.sub.2i-1 (m)] (2)
C.sub.2i (m+1)=C.sub.2i (m)-.beta.E.sub.m-d P'.sub.m-i-d+1
-.gamma.[C.sub.2i (m)] (3)
where .beta. and .gamma. are selected constants. The parameter d is a
predetermined number-illustratively equal to 2--whose introduction into
the updating rules takes account of the delay between generation of
baseband samples P.sub.m and P'.sub.m and the generation of error signal
E.sub.m. The updating rules of Eqs. (2) and (3) embody the so-called
stochastic mean-squared error updating algorithm, with the final term in
each of these equations representing so-called "tap leakage".
The updated coefficient values, in addition to being used internally within
equalizer 150, are passed on lead 159 to timing recovery circuit 145. The
latter determines, based on the principles of the invention, whether
receiver timing should be advanced or retarded (if either) and provides a
signal indicative of same on add/delete lead 146, which extends to
receiver timing generator 135. The latter, in turn, then appropriately
adjusts the phase of the pulses on lead 136, and, therefore, the phase
with which the samples on lead 126 are formed, by either deleting
(ignoring) one of the clock pulses on lead 131 or adding an extra one,
depending on whether receiving timing is to be advanced or retarded.
The theoreticl underpinings of our invention, which is directed to the
timing recovery technique performed within timing recovery circuit 145,
will now be described. For simplicity, this discussion assumes a one- as
opposed to two-dimensional signal so that signal samples and coefficients
can be represented by real, as opposed to complex, values. The discussion
is equally valid for the two-dimensional case, however.
Assume that a pulse defined by a sequence of samples
p=[p.sub.-K, p.sub.-K+1, . . . p.sub.0, . . . p.sub.K-1, p.sub.K ](4)
is applied to a finite-impulse-response equalizer whose transfer function
is defined by a sequence of coefficients
c=[c.sub.-L, c.sub.-L+1, . . . c.sub.0, . . . c.sub.L-1, c.sub.L ](5)
thereby providing a filtered output sequence given by
z=[z.sub.-(L+K), z.sub.-(L+K)+1, . . . z.sub.0, . . . z.sub.(L+K-1),
z.sub.(L+K) ], (6)
where the elements of z are
##EQU2##
We define the center of gravity of the pulse and the center of gravity of
the coefficients as
##EQU3##
respectively. We similarly define the center of gravity of the filtered
output sequence as
##EQU4##
which, upon substituting (7), yields:
##EQU5##
One of the significant underpinnings of our invention is our realization
that the centers of gravity above are related by the relation
CG.sub.z =CG.sub.p +CG.sub.c, (11)
which can be shown by first writing
##EQU6##
Considering the m.sup.th element of p and the l.sup.th element of c, this
expands to
##EQU7##
Now expanding Eq. (10), we can write
##EQU8##
Finally, substituting k for (l+m) in Eq. (13), we see that Eqs. (13) and
(14) are equivalent and, therefore, CG.sub.p +CG.sub.c =CG.sub.z.
Stated in words, then, the "position" of the output sequence z--defined,
somewhat arbitrarily, as the position of its center of gravity--is given
by the sum of the center of gravity, or "position", of the unfiltered
pulse and the center of gravity of the filter coefficients.
Equation (11) can be rewritten as
CG.sub.c =CG.sub.z -CG.sub.p, (15)
thus showing that the center of gravity of the coefficients is, in fact, a
numerical measure of the change in the position of the pulse p after it
has been filtered. This change, however, is simply the delay introduced by
the equalizer. Thus it is seen that once the desired delay has been
decided upon--this typically being half the "length" of the equalizer,
i.e., half the time it takes for a sample to pass all the way through
it--the current value of c can be used to determine whether that delay is
greater or less than the desired amount and by how much.
Specifically, if the index n is defined as spanning a range of positive and
negative values with n=0 being the desired position of the sampled pulse
within the equalizer, then the coefficient center of gravity CG.sub.c is,
in fact, a measure of the distance, measured in units of half-symbol
periods (since there are two coefficients per symbol), between the desired
and present position of the sampled pulse, that distance being referred to
herein as the "timing offset". In addition, the observed rate of change of
CG.sub.c over time can be used to determine the amount of timing "drift",
i.e., the difference in symbol clock frequency between the transmitter and
receiver. The receiver timing can then be adjusted as appropriate to both
(a) match the receiver symbol clock to that of the transmitter so as to
halt further drift and (b) adjust the equalizer delay to that desired,
i.e., remove the timing offset.
Timing recovery circuitry 145 implements the foregoing by maintaining a
variable X.sub.m. The value of this variable is updated, i.e., incremented
or decremented, for each symbol interval as a function of (a) a parameter
.DELTA.X.sub.m, whose value is related to the present timing offset, and
(b) a parameter V.sub.m, whose value is related to the present rate of
change thereof, i.e., timing drift. In particular, X.sub.m is updated in
accordance with the relation
X.sub.m+1 =X.sub.m +.DELTA.X.sub.m +V.sub.m. (16)
That is, the value of this variable associated with the (m+1).sup.st symbol
interval is given by its value associated with m.sup.th symbol interval
updated by the values of .DELTA.X.sub.m and V.sub.m. Once the magnitude of
X.sub.m exceeds a predetermined threshold in a positive (negative)
direction, this is indicative of a situation in which the timing offset
and/or timing drift have been sufficiently positive (negative) over some
preceding number of symbol intervals that an advance (retarding) of
receiver timing is deemed appropriate and such a timing adjustment is
thereupon made. The value of X.sub.m is then reset to zero and the process
continues. In this embodiment, .DELTA.X.sub.m and V.sub.m have small
fractional values and the threshold for X.sub.m is of magnitude unity.
Specifically, the parameter .DELTA.X.sub.m is illustratively given by
.DELTA.X.sub.m =.nu..multidot.SGN(x.sub.m), (17)
where .nu. is a predetermined constant, SGN is a function whose magnitude
is unity and whose sign is the sign of its argument, and x.sub.m is the
present timing offset. The value of x.sub.m is determined, in accordance
with the invention, from the coefficient center of gravity as discussed
hereinbelow. Thus whenever the timing offset is positive (negative), an
increment of fixed magnitude is added to (subtracted from) X.sub.m in each
successive symbol interval.
The other parameter in Eq. (16), viz., the parameter V.sub.m, takes account
of timing drift, which is principally due to the inevitable (albeit
possibly very small) discrepancy between the transmitter and receiver
master clock frequencies. This discrepancy, if not accounted for, would
manifest itself in the form of an ongoing incremental change in the
magnitude of the timing offset. By adding V.sub.m (which may have either a
positive or a negative value) to X.sub.m for each symbol interval, we
effectively cancel the incremental change in timing offset that would
otherwise occur.
Like X.sub.m, the parameter V.sub.m is also arrived at through an iterative
updating process, viz.,
V.sub.m =V.sub.m-1 +.DELTA.V.sub.m. (18)
The parameter .DELTA.V.sub.m is illustratively given by
##EQU9##
where .mu. and .alpha. are predetermined constants and .upsilon..sub.m is
the present rate of change, or "velocity", of the timing offset. The value
of .upsilon..sub.m, like the value of x.sub.m, is determined, in
accordance with the invention, from the coefficient center of gravity, as
discussed hereinbelow.
The parameter .upsilon..sub.m will be non-zero as long as V.sub.m has not
reached a steady-state value. Thus, the value of V.sub.m is subject to a
continual updating until, in fact, a steady-state value is reached, that
value being such as to substantially compensate for what would otherwise
be a continual timing drift principally due, as mentioned above, to
transmitter/receiver clock frequency differences. From that point on, the
value of .upsilon..sub.m, and thus of .DELTA.V.sub.m, will be
substantially zero.
In general, it is desirable to assure that timing changes are made in small
increments over time. It is for this reason that the magnitude of
.DELTA.X.sub.m is not a function of the magnitude of x.sub.m but, rather,
only of its sign. Specifically, if the magnitude of .DELTA.X.sub.m were to
be a function of the magnitude of x.sub.m, .nu. would have to be made
sufficiently small to ensure that .DELTA.X.sub.m would not be unduly large
when x.sub.m took on its maximum value. Such a small value of .nu. would,
however, result in an inadequately long timing adjustment when x.sub.m was
small. Using .DELTA.X.sub.m =.nu..multidot.SGN(x.sub.m) solves this
problem.
On the other hand, .upsilon..sub.m takes on a sufficiently small range of
values--its maximum value being limited by the maximum allowable frequency
deviation in the modem master clock--that, by judicious choice of the
values of .mu. and .alpha., the magnitude of .upsilon..sub.m can be used
in determining .DELTA.V.sub.m without engendering unduly abrupt timing
changes.
In summary, then, the values of .mu., .alpha. and .nu. are chosen in such a
way as to ensure that the timing recovery technique can, in fact, account
for worst-case timing drift and correct for same in a reasonable time
period without, on the other hand, making timing adjustments too abruptly.
Illustratively, .mu.=2.sup.-8, .alpha.=2.sup.-8, and .nu.=2.sup.-12.
In order to determine x.sub.m and .upsilon..sub.m from the coefficient
center of gravity pursuant to the invention, timing recovery circuit 145
needs, first of all, to determine that center of gravity. Since the
equalizer coefficients are complex numbers, their center of gravity is
likewise a complex number. It turns out, however, that all timing
information is contained within the real part of that complex number.
Timing recovery circuit 145 therefore works strictly with the real part of
the center of gravity.
In particular, if the n.sup.th (complex) coefficient, C.sub.n (m), of the
overall coefficient queue is represented as the complex number c.sub.n
(m)+jd.sub.n (m), then the coefficient center of gravity associated with
the m.sup.th symbol interval can be written as
##EQU10##
Thus the real part of the coefficient center of gravity associated with
the m.sup.th symbol interval, denoted CG.sub.c.sup.r (m), is given by
##EQU11##
Computing CG.sub.c.sup.r (m) from Eq. (21) obviously involves performing a
division. However, timing recovery circuit 145 is illustratively
implemented as a programmed digital signal processing (DSP) chip on which
division is relatively slow and awkward. To get around this, circuit 145
approximates CG.sub.c.sup.r (m) using the following adaptive division:
CG.sub.c.sup.r (m)=CG.sub.c.sup.r
(m-1)+k.multidot.SGN[Num-Den.multidot.CG.sub.c.sup.r (m-1)](22)
where Num and Den are, respectively, the numerator and denominator from Eq.
(21). The parameter k is chosen taking into account the fact that the
maximum difference in freqwuency between the transmitter and receiver
clocks is 1 part in 5000 so that, in particular, 1/5000<k<<1. Moreover, by
choosing k sufficiently small, the adaptive division represented by Eq.
(22) serves as a smoothing filter on the measured center of gravity.
Illustratively, k=2.sup.-8. It will, of course, be appreciated that
center-of-gravity signal CG.sub.c.sup.r (m) is not precisely equal to the
coefficient center of gravity, due to the fact that CG.sub.c.sup.r (m) is
computed via an approximation process. It is, however, a signal which is
indicative of the center of gravity and, as such, is wholly adequate to
carry out the present timing recovery technique.
FIG. 2 is a flowchart of the processing performed within timing recovery
circuit 145. As shown therein, timing recovery circuit 145, after first
initializing CG.sub.c.sup.r (0) and two parameters F(0) and G(0)
(discussed below) all to zero at step 201, proceeds to calculate Den and
Num from Eq. (22) at steps 204 and 206. CG.sub.c.sup.r (m) is then
determined at step 208. As indicated at step 211, CG.sub.c.sup.r (m) is
then filtered by passing it through a first stage of linear filtering to
generate a filtered signal F(m) in accordance with
F(m)=F(m-1)+.alpha.[CG.sub.c.sup.r (m)-F(m-1)] (23)
In Eq. (23), .alpha. is the same .alpha. used in Eq. (19). F(m) is then
passed through a second stage of linear filtering to generate a filtered
signal G(m) in accordance with
G(m)=G(m-1)+.alpha.[F(m)-G(m-1)] (24)
As shown in step 215, the variables x.sub.m and .upsilon..sub.m --actually
a scaled version thereof .upsilon..sub.m /.alpha.--are then determined
from F(m) and G(m) in accordance with
x.sub.m =2F(m-1)-G(m-2) (25)
##EQU12##
Equations (25) and (26) are arrived at by simply rearranging terms in the
expression
.alpha.[x.sub.m -F(m-1)]=.alpha.[F(m)-G(m-1)]=.upsilon..sub.m, (27)
the validity of which can be verified by equating x.sub.m with
CG.sub.c.sup.r (m) in Eq. (23)--since, as discussed above, the timing
offset x.sub.m is, in fact, represented by the coefficient center of
gravity CG.sub.c.sup.r (m)--and then substituting from Eqs. (23) and (24)
into Eq. (27).
Given the values of x.sub.m and .upsilon..sub.m /.alpha. from step 215,
timing recovery circuit 145 can then update, in accordance with Eqs. (16)
and (18), the registers that it uses to hold V.sub.m and X.sub.m, as shown
at step 217. It is then determined at step 221 whether X.sub.m has
overflowed, i.e., exceeded +1. If it has, then at step 222, a pulse is
added to lead 131, as previously noted, and X.sub.m is cleared to zero.
Alternatively, if it is determined at step 224 that X.sub.m has
underflowed, i.e., become more negative than -1, then at step 227, a pulse
is deleted from lead 131 and, again, X.sub.m is cleared to zero.
At this point, F(m) and G(m)--whose values will be used at steps 211 and
215 in the next symbol interval (where they will appear as F(m-1) and
G(m-1), respectively)--must be updated. The reason for this is that having
changed X.sub.m (at step 217), we have effectively changed the value of
x.sub.m. The values of F(m) and G(m) must therefore be adjusted so as to
be consistent with the new implicit value of x.sub.m carried forward into
the next symbol interval.
Specifically, in order to determine the amount by which F(m) and G(m)
should be adjusted, we iteratively substitute in Eq. (23) the expression
for F(m-1) derived from Eq. (23) itself, i.e.,
F(m-1)=F(m-2)+.alpha.[CG.sub.c.sup.r (m)-F(m-2),
and so forth for a large value of m, and making the assumption that
.upsilon..sub.m is a constant equal to [CG.sub.c.sup.r (m)-CG.sub.c.sup.r
(m-1)] for all m--an assumption that, indeed, is justifiable because of
the corrections that are in fact made to F(m) and G(m) for each symbol
interval--then we arrive at
##EQU13##
By assuming that (1-.alpha.) is approximately equal to unity, we can write
##EQU14##
Following the same approach, it can be shown that
##EQU15##
From Eqs. (27) and (28), then, we see that when X.sub.m and V.sub.m are
changed by .DELTA.X.sub.m and .DELTA.V.sub.m, respectively, then the
corresponding changes in F(m) and G(m) are
[.DELTA.X.sub.m -.DELTA.V.sub.m /.alpha.]/2.sup.11 (29)
for F(m) and
[.DELTA.X.sub.m -2.DELTA.V.sub.m /.alpha.]/2.sup.11 (30)
for G(m). The factor of 2.sup.11 comes about by virtue of the fact that
F(m) and G(m) are in units of half-symbol periods while .DELTA.X.sub.m and
.DELTA.V.sub.m are in units of 2.sup.-11 half-symbol periods because each
unit change in X.sub.m causes a receiver timing change of 1/4096, i.e.,
2.sup.-12 symbol periods=2.sup.-11 half-symbol periods.
The adjustments set forth in Eqs. (29) and (30) are subtracted from F(m)
and G(m) at step 231.
One particular point that needs to be addressed relates to the fact that in
the present embodiment, fixed carrier demodulation is performed prior to
equalization; in typical applications, all demodulation is performed after
equalization. The reason this is desirable is that CG(m) can be shown to
be equal to the equalizer zero-frequency group delay. However, for a
passband equalizer, i.e., one which operates on passband, non-demodulated
samples, there is substantially no zero-frequency energy, at least in
typical voiceband applications. Performing the demodulation after
equalization, then, would not yield a meaningful coefficient center of
gravity computation. The zero-frequency group delay for a baseband
equalizer, by contrast, is relatively smooth, flat and "well-behaved".
Hence the use of demodulation before the equalizer.
The foregoing merely illustrates the principles of the invention. Thus, for
example, although the receiver of FIG. 1 is shown as being comprised of
individual hardware elements, the equivalent signal processing may be
carried out using one or more programmed processors operating under the
control of microcode, firmware, software, etc. And, of course, the various
frequencies and other numerical parameters set forth herein are merely
illustrative and will differ in varying applications.
It is thus the case that those skilled in the art will be able to devise
numerous arrangements which, although not explicitly shown herein, embody
the principles of the invention.
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