CA1042523A - Phase filter for reducing the effects of the noise components altering discrete phase modulated signals - Google Patents

Abstract

PHASE FILTER FOR REDUCING THE EFFECTS OF THE NOISE
COMPONENTS ALTERING DISCRETE PHASE MODULATED SIGNALS

Abstract of the Disclosure A digital phase filter including two decision filters connected in cascade or parallel for minimizing the noise components in a received phase modulated digital signal.
The first decision filter cancels the residual noise compo-nent representing the phase intercept component and the phase shift component introduced by frequency shift, and the second decision filter minimizes the random noise component representing phase jitter and white noise.

Description

12 Background of The Invention 13 This invention relates to digital data transmission 14 systems and more particulsrly, to a phase filter for reducing the effects of the noise components altering the signals 16 transmitted through a system in which the phase of the trans-17 mitted signals is modulated by discrete values, at discrete 18 times.
19 Phase modulation is a technique widely used in data transmission systems and a detailed description thereof is 21 provided, for example, in "Data Transmission" by W. R. Bennett 22 and J. R. Davey, chapter 10, McGraw-Hill, New York 1965 and 23 "Principles of Data Communications" by R. W. Lucky, J. Salz 24 and E. J. Weldon Jr., chapter 9, McGraw-~lill, New York 1968.
In those digital data transmission systems that utilize phase 26 modulation, the digital data to be transmitted modulate the 27 phase of a carrier at particular instants or sampling times.
28 The direct modulation method, called "Coherent Phase Modula-29 tion", consists in making a predetermined phase absolute value to correspond to a data group or character. For instance, : `` ' . ' ' :
.: : .

104;~523 1 in an elght-phase system, i.e., in a system in which the phase

2 of the transmitted signal can assume eight distinct discrete

3 values, it is possible to make the eight phase absolute values

4 ~t8, 3~/8, ...... , 15~t8, respectively, to correspond to the eight characters 000, 001, 011, 010, 110, 111, 101 and 100 6 as shown in the diagram of Figure la. The modulation method 7 the most currently used, called "Differential Phase Modula- ~;
8 tion", consists in making a phase change rather than a phase g absolute value to correspond to a character. Always in an ~-eight-phase system, it is possible to make phase variations 8, 3~/8, ..., 15~/8 to correspond to the eight characters 12 000, 001, 011, .. ...., lO0. This type of modulation can also be ~-13 illustrated by the diagram of Figure la by taking the phase 14 value of the signal emitted st the preceding sampling time as reference axis OX. The so-modulated carrier frequency is sent, 16 through a transmission medium, to a receiver coupled thereto.
17 At the receiver, the phase value of the signal received at 18 sampllng times is detected, and th~n the value o the trans-19 mitted data is extrscted therefrom. This extraction is generally done by comparing the phase of the signal received 21 at a given sampling time with a reference phase available in 22 the receiver or with the phase of the signal received at the -23 precedlng sampling time according to the use of a coherent or -~
24 differential detection9 as described in the above-referenced books.
26 It is to be noted that phase modulation as briefly 27 described above, is not the only one process for transmitting 28 dlgital data, ln whlch the phase of tbe transmitted signals 29 repre~ents the data. For instance, it is the case of ~he quadrature amplitude modul8tion a description o~ which can be ~R973-011 - 2 -.: . . . .
. ~ .

1~4ZSZ3 1 found in the above-referenced book by R. W. Lucky et al, 2 chapter 7 and more particularly ~n paragraph 7.1.5. Briefly, 3 the quadrature amplitude modulation consists in modulating 4 the amplitude of two csrrlers in quadrature by discrete values, said carriers being emitted in the same time. The 6 following table shows the correspondence between the digital ,.
7 characteræ, the amplitude of each of the carriers in quadra-8 ture A and B and the phase and the amplitude of the signal g resulting from the combi~ation of these carriers, in an eight-10 state system illustrated by the diagram of Figure lb.

I
11 Digital AmplitudeAmplitude Phase of Amplitude 12 characters of Aof B the result- of the 13 - ing signal resulting 14 signal 15` 000 + 3 0 0 + 3 16001 + 1 + 1 ~/4 +~
17011 0 + 3 2~/4 ~ 3 18010 - 1 + 1 3~/4 +~ ~
19110 - 3 0 4~/4 + 3 ~ -20111 - 1 - 1 5~/4 +
21101 0 - 3 6~/4 + 3 22]o~ + 1 - 1 7~/4 +~

.:
23 From thls table, it appears that, in this example, 24 the data can be directly derived from the value of the phase f the emitted resulting signal 26 It would be desirable if the emitted signals should 27 be reeeived without distortion in the receiver. In practice~
28 howe~Jer, the tranamission media introduce disturbances such 29 as intersymbol lnterference and the noise components mainly _ _ , . , ... , .................. ,. . . . , . , .... _ , 1 due to frequency shift, phase intercept, phase jitter and 2 white noise, which disturbances alter the emitted signals 3 when transmitted through the transmission medium.
4 Intersymbol interference i9 due to an interaction between successive emitted signals, which interaction is 6 caused by amplitude and phase distortions introduced by the 7 transmission medium. When intersymbol interference 8 appreciably affects the quality of the received signals, it g is cancelled or reduced by an appropriate device called an "Equalizer". Within the scope of this invention, it is 11 assumed that intersymbol interference is cancelled by an 12 appropriate eq~lali~er, as necessary.
13 Frequency shift is a disturbance affecting the 14 emitted signals when they are transmitted through a trans-mission medium in which they are submitted to an intermediate 16 processing and more particularly, when telephone lines are 17 used aa the transmission medium. Said intermediate process-18 ing mainly includes the transposition of the emitted signals - ~;
19 from one frequency band to another as required by the public network. A frequency shift fs introduces a phase shift 21 ~s ~ 2~fSt, where t i9 the time, which phase shift directly 22 affects the phase of the received signal.
23 Phase intercept is due to the presence of a differ-24 ence between the actual phase of a frequency and the phase corresponding to the ideal channel phase/frequency character-26 istic, at the ends of the frequency bandwidth of the trans-27 mission channel. Thls phase intercept ineroduces an 28 arbitrary con~tant in the received phase value.

FR973-011 - 4 _ : ' ;

1 Phase ~itter results from a random noise frequency 2 modulatlon of the signals when passing through the trans-3 mission medium. Often, it is due to the variation of the 4 power sources of the devices used to carry out the above-indicated intermediate processing.
6 White noise ls due to the additive noise in the 7 transmission medium and to the residual intersymbol interfer-8 ence. It is characterized by a flat frequency spectrum with 9 equal contributions for all the frequencies, but in which the various frequencies exhibit random phases.
11 These noise components have practically no effect 12 in the low speed digital data transmission systems but avoid 13 correct data detection in high speed systems. In those 14 systems using phsse modulation, an increase of the trans-mission speed is generally obtained by increasing the numbe~ ~
16 of the distinct discrete values which can be assumed by the ~ `
17 phase of the emitted signal, which increase appears as a ~ -18 decrease of the gap between two atjoining phase values. For 19 instance, in a four-phase system, this gap is of 90, but it i8 only of 22.5 in a sixteen-phase system. Then it is often 21 not possible to discriminate between two possible phase values 22 in presence of the various above-indicated disturbances and 23 it becomes imperative to provide a device to cancel or reduce 24 the effec~s of these noise components before detecting data.
U. S. patent 3,855,539, filed March 16, 1973 on 26 behalf of Alain Croisier, which patent is assigned to the 27 assignee of the present invention, describes a method and a 28 device for reducing the effects of the noise components alter-29 ing the phase of discrete phase modulsted signals. According , ,'' ~ ~ : ~ ' ,, 1 to this method, a correction value is subtracted from the received signal phase value. The result of this subtraction is multiplied by a first factor proportional to the number of distinct discrete values that the phase of the emitted signal can assume. The integral part of this product represents the data while the fractional part is used to provide said correction value. Said correction value is ob-tained by multiplying said fractional part by a second factor and by integrating the result of this last multiplication. Controlling the value of the second factor enables to minimize the effects of the various above-indicated components selectively. However, this method shows a drawback consisting in the fact that the reduction of the effect of the phase jitter is necessarily accompanied by a decrease of the performance with respect to the white noise. The curves shown in Figure 3 of the above-indicated U.S. patent enables one to appreciate the difficulty encountered to find a good decrease of phase jitter/
increase of white noise compromise.
One of the objects of this invention is to overcome this drawback by providing an optimum phase filter for minimizing, the effects of phase intercept, frequency shift, phase jitter and white noise alter-ing discrete phase modulated signals.
Another object of this invention is to provide an adapti~e phasefilter allowing to reduce the phase jitter effect to a minimum, what-ever the noise modulation frequency causing said jitter, may be.
The objects of this invention are generally obtained by providing a phase filter including two decision filters ~..
''~3 1~4Z5Z3 1 which can be linked through a cascade or a parallel 2 connection. The first decision fi~ter cancels the phase 3 intercept component and the phase shift component introduced 4 by the frequency shift, and the second decision filter mini-mizes the random component representing phase Jitter and 6 white noise. In the first decision filter, a first error 7 signal corresponding to an estimated value of the phase 8 intercept and phase shift components is subtracted from the 9 phase value, of the received signal. The result of this first subtraction is applied to a detector-separator which separates 11 the data and the residual noise component. The residual noise 12 component is applied to a linear filter generating from the 13 previous residual noise components, the estimated value of 14 the phase intercept and phase shift components. In the second decision filter, a second error signal corresponding to an 16 estimated value of the random component is subtracted from 17 the result of the first subtraction. The result of this second 18 s~btraction is applied to a detector-separator which fetches 19 the data and the resldual random component out. Said compo- "
nent ls applied to a linear filter which generates ~rom the 21 previous residual random components, the estimated value of '22 the random component. When the phase Jitter characteristics 23 are unknown or if they are varying in time, the estimated 24 value of the random component can be obtained by an adaptive predictive fllter.
26 These and other obJects, advantages and features of 27 the pre~ent invention will become more readily apparent from 28 the following specification when ta~en in conJunctlon with the ,~
29 drawings.

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' ' 1 Summary of The Inventlon 2 According to the present invention, a phase fllter 3 is disclosed which minimi~es the effects of noise components 4 altering the value of the phase of a digital signal in a digital data transmission system in which the phase of the 6 digital signal can assume Q distinct values representing 7 the data, with the phase filter including two decision filters.
8 The first decision filter cancells residual noise components 9 representing the phase intercept component and the phase shift component introduced by frequency shift, and the 11 second decision filter minimi~es the random noise component 12 representing phase jitter and whi~e noise. The first decision 13 filter includes a first means for providlng a first differ-14 ence signal in response to taking the difference between the received digital signal and a first error signal correspond-16 ing to an estimated value of the residual noise component.
17 Also included is a first detector which is responsive to the 18 first difference signal for separating the data portion from 19 the residual noise portion of said first difference signal.
Further included i8 a fir~t linear filteT which is responsive 21 to the residual noise portion of the first difference signal 22 for generating the first error signal. The second decision 23 filter lncludes a second mean~ for prov~ding a second differ-24 ence signal in response to taking the difference between the first difference ~ignal and a second error signal corres-26 ponding to an estimated value of the random noise component.
27 Also included is a second detector which i8 responsive to the 28 second difference signal for separating the data portion 29 from the random n~ise portlon of the second difference signal, .

.

.... .. ., , .. . . _ . . ....... , ___ . .. , . ____ _ _ .
, , .

11~)4Z5~3 1 with a data representative slgnal being provided at a first 2 output and a random noise representative signal being 3 provided at a second output. Further included is a second 4 linear filter which is responsive to the random noise representative signal for generating the second error signal.
''.' .
6 Brief Description of The Drawin~s .

7 Figures lA and lB show diagrams respectively 8 illustrating an eight-phase modulation and an eight-state 9 quadrature amplitude modulation. -Pigure 2 is a schematic illustration of a phase 11 filter in cascade form according to this invention.
12 Figure 3 is a diagram making the analysis of the 13 first decision filter operation, easier.
14 Figure 4 schematically shows as an example, a digital embodiment of linear filter W(~), 12, of Figure 2.
16 Figure 5 shows the shape of spectrum Rx(e~2~fT) of 17 random component xn in example 1 of second decision filter 8.
18 Figure 6 schematically shows another possible embodi-.
19 ment of second decision filter 8 of Figure 2.
Figure~7 schematically shows as an example, an 21 embodiment of a Wiener adaptive optimal predictive digital 22 filter which can be used in second decision filter 8.
23 Figure 8 shows another possible embodiment of second 24 decision filter 8, including an adaptive predictive fllter.
Figure 9A shows the phase filter of Figure 2 in 26 whlch second declsion filter 8 is as shown on Figure 6.
27 Figure 9B shows the phase filter in parallel form 28 corresponding to the phase filter in cascade form of Figure 9A.

29 Figure lOA shows the phase filter of Figure 2 in .

104~5Z3 1 which second decision filter 8 is as shown in Figure 8.
2 Figure lOB shows the phase filter in parallel form 3 corresponding to the phase filter in cascade form of Figure 4 lOA
Figure 11 schematically shows an ~xample, of a 6 detector-separator used in this invention.
7 Figure 12 is a waveshape relationship diagram of 8 certain signals present in the circuit of Figure 11.

g Description of The Prèferred Embodiment The phase filter of the invention operating on 11 sequences of discrete values, the Z-transform will be used as 12 a mathematical tool. For the description of the theory of 13 the Z-transform and of its applications, it is possible to 14 refer, for example, to the articles entitled: "Z-Transforms and Their Applications in Control Engineering" by Y. Azar, 16 published in "The Radio and Electronic Engineer", July 1965, 17 pages 53 to 57 and "Digital Filter Design Techniques in the 18 Frequency Domainl' by C. M. Rader and B. Gold, "Proceedings 19 of the IEEE", Vol. 55, No. 2, February 1967, pages 149 to 171, and for a detailed study,-it would be possible to refer, for 21 example, to "Theory and Application of the Z-Transform Method"
22 by E. I. Jury, John Wiley, New York, 1967.
23 To relieve the reader of the necessity of refPrring 24 to other documents, a summary of the Z-Transform theory and the results which will be used in this description is given 26 in the following.
27 Suppose a linear sys~em characterized Ln the time-28 domain by its continuous impulse response h(t). In response 1(~4ZSZ3 1 to a continuous input slgnal s(t), this filter provites a 2 continuous output Yignal~g(t~ defined by the following convo-3 lution relation g(t) - ~ h(y) s(t-y) dy (1) If the Laplace transformation is used in the 6 frequency domain~ relation (1) becomes 7 G(p) = H(p) S(p) (2) 8 with p = complex frequency 9 in which G(p), H(p) and S(p) are the Laplace transforms of g(t), h(t) and s(t), respectively. H(p) is called the "transfer 11 function" of the system. The Laplace transform is used very -~
12 much in studying continuous systems since it allows to replace 13 the integratiOQ operation defined in (1) by the simple alge- -14 braic operation defined in (2).
The Z-transform operates as the Laplace transform, la in studying discrete systems. By definition, the Z-transform 17 of the sequence of discrete values s(nT), n-positive integer, 18 and T=appearance periods of the discrete value, is~

19 S(z) = ~ s(nT) z (3) n-0 where z is a complex variable z=eP .
21 Relation (2) becomes ~
22 G(z) = H(z) S(z) (4) ~ -23 where S(z) and G(z) are the Z-transforms of the sequences of 24 discrete values s(nT) and g(nT), and H(z) i9 the Z-~ransform of di~crete impulse respon6e h(nT).
26 Thus, the Z-transform of the output sequence of a 27 discrete filter i9 the product of ~he Z-tran~form of the input 28 sequence and function H(z) characterizirg the filter, which .
. ' ' , ,' . ' l functlon i9 ~imilar to a transer function.
2 A table for convertlng the sequences of values and ; 3 their Z-transforms can be found, for example, in the above-4 indicated book by E. I. Jury.
It should be simply noted here that 6 the Z-tran6form of sequence x~nT) defined by 7 x(nT) ~ 0 for n < 0 8 x(nT) = 1 for n > 0 9 is X(z) = ~ z-n ~ 1 (5) n 0 1-~
ll the Z-transform of a sequence delayed by an 12 elementary delay T i8 equal to the Z-transform of the initial 13 sequence multiplied by z l 1~ The phase filter according to the invention will be described, now, in a cascade form and a parallel form, success-16 ively.
17 Referring to Figure 2 which shows the phase filter 18 in cascade form, it is seen that the discrete value of phase 9 ~ ID fetched out from the received signal and which is supposed to be digitally encoded without limiting the scope of the in-21 vention, is applied to one of the inputs of a binary multiplier 22 2, via line 1. The second input of multiplier 2 is connected 23 to the output of a storage element 4 which can be, for example, 24 a binary register, or a simple prearranged connection assembly, Vi8 llne 3. The output of multiplier 2 is connected via line 26 5 to the input of a first decision filter 6 the output of 27 which is, in turn, connected vla line 7, to the input of a 28 second decision fi~ter 8 which delivers as an output on line 29 9, the ~ignals representing the detected data. The first :

' .

1~4;~523 1 decision fllter 6 include~ a blnary subtr~ctor 10, a 2 detector-separator 11 and a digltal linear filter 12. The 3 output of multiplier 2 i9 connected via line 5 to the (+) 4 input of subtractor 10 the output of which i8 conneeted via line 13 to the input of detector-separator 11 which will be 6 described with more details with reference to Figure 11.
7 Output Er of detector-separator 11 i9 connected vla line 8 ~14 to the input of filter 12 the output of which is connected 9 via line 15 to the (-) input of subtractor 10. Second ::
decision filter 8 includes two binary subtractors 16 and 17, 11 a detector-separator 18, a digital predictive filter 19 12 and two delay elements 20 and 21, each introducing an 13 elementary delay T equal to the sampling time period.~ The 14 output of subtractor 10 of decision filter 6 is connected via line 7 to the (+) input of subtractor 16 the output of 16 which is connected to the input of detector-separator 18 17 whose output is connected to output line 9. The output of 18 detector-separator 18, i8 further connected via line 22, .
19 to the input of delay element 20 the output of which is connected to the (-) input of subtractor 17 via line 23. The .
21 (+) input of subtractor 17 is connected to the output of delay `
22 ele~ent 21 whose input i8 connected to line 7 via line 24. ~
23 The output of subtractor 17 is connected via line 25, to the ~:
24 input of predictive filter 19 the output of which is connected to the (-) input of subtractor 18 via line 26.
26 The operation of the device schematically shown on 27 Figure 2 will now be described.
28 Within the scope of this invention, assume the case 29 of a digital data transmission system ln which the phase value :
FR973-011 - 13 - ~

.. . . . . .
.
' ': '' ` ' . ~ ' , ;: ' ~ , l of the signsl emitted at sampling times, represents the data.
2 The receiver of the system should process the recelved slgnal 3 to allow correct detection of the phase of the signal received 4 at sampling times to fetch the data out. Processing the received signal which is not within the scope of the invention, ; 6 generally includes, automatic control of the received signal 7 amplitude to allow further processing of constant mean power 8 signals, sampling of the signal received at times t=nT and 9 the conversion of the sampled signal in digital form. In addition, it includes, if necessary, the equalization of the , 11 received signal to reduce the effects of intersymbol inter-.; ~
12 ference. Detecting the value of the received signal phase, 13 also sampled and equalized if necessary, at sampling times, 14 is not within the scope of the invention either, and can, for instance, be ensuret by the phase detector described in U. S.
16 Patent 3,825,737, filed December 11, 1972 on behalf of Alain 17 Crosier, which patent is assigned to the assignee of the 18 present invention.
19 These phase values form the input signals of the 20~ device of this invention.
21 ~ The signal applied to the input of the phase filter 22 of the invention, shown on Figure 2, appears, therefore, as 23 a sequence of discrete phase values (~'n)~ where ~'n is the 24 phase value of the signal received at sampling time t=nT.
Phase ~'n can be expressed as follows 26 ~ n ~n + ~n t6) , .
27 where 28 ~ is the discrete val~e of the phase of the signal 29 emitted st tlme t=nT, and represents the data, , :
` FR973-011 - 14 -:~ , ' , 1 and 104ZSZ3 2 n represent~ all disturbances or noises introduced 3 during the slgnal transmission.
4 In a Q phase system, Q being a positive integer, i.e., in a system in which the emitted signal phase can 6 assume Q distinct discrete values at each sampling time, ~n 7 can take the following values ;

~ 0 2~ 4~ ~Q~ (7) ~ -9 Noise en can be expressed as follows ~n Q (aO + aln + xn) (8) 11 where 12 aO is ~ constant representing phase intercept 13 al ls a constant representing the frequency shift ~
14 between two successive sampling times, and Xn is a random noise component representing 16 phase ~itter and white noise.
17 Phase ~' is applied via line 1, to an input of 18 multiplier 2 which multiplies it by factor Q/2~ stored in 19 storage element 4. Multiplier 2 provides at its output, on line 5, signal Yn-21 Yn Q ~ n 22 Substltuting ~' by lts value obtained for expression (6), 23 into expresslon (9) 24 Yn Q (~n + ~n) (10) which can be expressed as follows 26 Yn ~ ln + bn ' ' ~

....... . . .
.

~04~S23 1 with 2~ I n and bn ' Q ~n 3 From (7) and (8), one has, respectively ln = - 1, 2, ... , (Q-l) (12) bn = aO + aln + xn (13) 6 It is to be noted that ln is an integer representing 7 the data and that bn represents all the noise components 8 introduced during the transmission. It is to be noted that 9 the very particular purpose of the multiplication of value ~'n by factor Q/2~ is to apply to the input of first decision 11 filter 6, a signal Yn equal to an integer in the absence of 12 noise, which appears in equations (11) and (13) by making 13 bn30.
14 The ob~ect of the system of this invention is to retrieve ln and to minimize the effects of bn. According to 16 this invention, the reduction of the effects of bn to a 17 minimum is obtained by cancelling phase intercept a and 18 phase shift aln introduced by frequency shift al, by using 19 first decision filter 6 and by reducing random component xn by using second decision filter 8.
~21 The operation of first decision filter 6 will now 22 . be analyzed.
23 Signal Yn available on line 5 is applied to the 24 (+) input of subtractor 10 which receives at its (-) input an error signal un supplied by linear Eilter 12 the transfer .`
26 function of which is referenced W(z). Subtractor 10 supplies Z7 at its output, difference yn-un which is applied to the input 28 of detector-~eparator 11.
'.

FR973-011 - 16 - . ~

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.
' . : . .
:
~ ': , , ' -~ `

1 The analysls of the system according to this in~
vention will be carried out a6suming that no detection 3 errors appear, said errors being supposed to be very rare.
4 From (11), we have ~ -; 5 y -u - 1 + b - u n n n n n Yn Un = ln + f with fn ' bn ~ Un (14) ~

7 Since ln is a positive integer, saying that there ~ -8 is no detection errors means that f is a fractional number 9 the absolute value of which is comprised between 0 and 1.
The function of detector-separator 11 to be described in 11 detail with reference to Figure 11, consists in fetching 1 l~ ~;
12 and fn out of difference yn-un applied to it. In detector-13 separator 11, only output Er providing residual noise component i 14 fn~ is used ant connected to line 14. Signal fn is applied to ~ 15 linear filter 12 with transfer function W(z), the function of ! 16 which consists in generating error signal un from the sequence 17 of residual noise components {fn}
18 It should be noted that when only first decision ¦ 19 filter 6 is necessary, i.e., when the effects of the random component can be neglected, output Da of detector-separator Zl 11 providing ln as an output of the phase ~filter, will be l 22 used.
1~ 23 Now, trsnsfer function W(z) will be determined so !
24 that first decision filter 6 cancels phase intercept aO and phase shift aln introduced by the frequency shift. For that, 26 it i8 always assumed that there is no detection error~ and ~7 the response of decision filter 6 will be analyzad with 28 respect to nolse bn only. In this case, decision filter 6 ' ~9 can be as shown on Figure 3. It should be noted that in the ;
FR973~ 17 -.

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, l()~ZSZ3 1 filter shown on the flgure, detector-separator 11 i8 cancelled 2 since it has no effect on residual noise component fn when 3 there is no detection errors. On the other hand, random 4 component xn in noise bn will be neglected, which will be verified later.
6 Therefore, it is assumed that:
7 bn = aO ~ aln n - 0, 1, 2, .................. (15) 8 Determining W(z) so that first decision filter 6 cancels aO
9 and aln, i.e., cancels bn as defined by (15~, means determin-ing W(z) so that fn= when bn=aO+aln.
11 Assume that B(z) and F(z) are the Z~transforms of 12 b and f , respectively, from (3) 13 B(z) = ~ bk z k (16) k=0 14 F(z) = + ~ fk Z (17) k=0 From Figure 3, we have 16 F(z) = B(z) - W(z) F(z) 17 or 18 F(z) = 1 B(æ) 1 + W(z) 19 or F(z) = G(z) B(z) (18) 21 with 22 G(Z~ ~ 1 +lW( ) (19) 23 It should be noted that G(z) represents the trsnsfer 24 function of first de~ision filter 6.
.

.. . .. . _ . .
.. ..
:: . ~
.. , "
.' ' ' ~

:~ :
1 Fro~ (16) and (15~ 4 2 5 Z 3 2 B(z~ ~ + ~ bk ~ k _ + ~ (a +alk) z k Y! k=0 3 = + ~ aOz k + + ~ alk z k (20) k=0 k-0 4 From (5) + ~ a z k a _1 (21) k=o 1 z 6 Furthermore i 7 ~ alk z k = al [z 1 + 2z 2 + 3z 3 + .--] (22) ~ k=0 j 8 Relation (5) can be expressed as follows i~ :
~: 1 + z-l + z-2 + = 1 (23) l- z 1~10 The derivative from (23) with respect to z can be expressed :~
'Y 11 as follows: ; -. .

~12 ~ -2. 2z-3 - 3Z-4' '' ' = ~ (24) ~13 : : Relation (24) can be written 14 . ~ . z-l + 2z-2 + 3z~3 + ~ ' Z -1 2 (25) J~15 Substituting (25) into (22) 1)2 l26) ~ 17 Sub~tltuting (21) and (26) into (20) ?~ . B(z) = aO + (al-aO) z 1 (27) ~ 18 (1 z-1)2 'iy 'I~ ' ' ~
~ FR973-011 - 19 -.

f . ~ _ __ __ _ ~': ' ' 1~4~S23 1 Now, it will be shown that a filter the transfer function of 2 which is as follows G(z) = (l-z ) with 0 < a < 1 (28) (1-CL~ ) 4 cancel a noise B(z) as defined by (27).
In effect, by substituting the values of B(z) and 6 G(z) given in expresslons (27) and (28), respectively, into 7 (18), we have the following expression of F(z) F(z) = aO + (al-aO) z 1 (1 -1)2 9 corresponding sequence {fn} of which will now be determined.
Expression (29) can be expressed as follows -~
- 1 . -11 F(z) = -1 2 1 o _ (30) (l-~z ) (1 ~z~ )2 12 From (25) aO [1+2 -l + 3~2z-2 + ... , + (k+l)kk Z +-~-]
13 (1 ~z-1)2 14 = + ~ aO(k+l) akz (31) . k=0 and (al-aO) z = (a -a ) z 1 [1+2az + 3~ z + 4~ z +.. ]
16 - , ~ 1 o 17 ( ) [ -1 + 2~z-2 + 3~2z-3 +...... + k~ Z

18 ~ (al-ao) k~k lz k (32) ~ , .

19 Substituting (31) and (32) into (30) F(z) = ~ [aO (k+l)a ~ (al-aO)k~ 1] z k k=0 ..

.. . . ; -`

~042S23 1 From the ~-ersnsform definition recalled in (3), we have 2 fn ~ aO (n~ n + (al-aO)nan l 3 or a -a (1-~) 4 fn = ~ [aO + 1 o~ n~ (34) Since 0 < ~ < 1, it is seen from expression (34), 6 that fn' when n ~ ~. In other words, decision filter 6 with 7 transfer function G(æ) as defined by (28), cancels phase 8 intercept aO and phase shift aln when n + ~, i.e., in steady 9 condition.
It is to be noted, from expression (28), that 11 G(z)tl when ~)1, i.e., that the effect of the first decision 12 filter on the components other than aO and aln, and in 13 particular on random component x , can be neglected whe~
14 ~1, which verifies the original assumption according to which xn was neglected in the analysis of first decision filter 6.
16 However, when a)l, the time response of the filter 17 becomes infinite. Therefore, it is necessary to choose a so 18 that decision filter 6 cancels components aO and aln and does 19 not have appreciable effect on component xn while having a correct response time. A good compromise is obtained, for 21 example 9 with ~=0.9.
~22 Transfer function G(z) of first decision filter 6 23 being determined by expr~ssion (28), transfer function W(z) 24 of llnear filter 12 is obtained by calculating W(z) in function of G(z) from (19) and by replacing G(z) by its value derived 26 from (28). We have 27 W(z) = C(z) 28 and 29 W(z) = 2(1-~)z-l + (~2_l~z~2 1-2z 1 + z-2 1 Figure 4 schematically shows an example of a digital 2 embodiment of a digital linear filter 12 in canonical form with 3 transfer function W(z) as deflned by relatlon (35). The embodi-4 ment of a digital filter from its transfer function, is a technique known in the art and a description of which can be 6 found, for example, in the article entit~ed: "Digital Filter 7 Design Techniques in the Frequency Domain" by C. M. Rader and 8 B. Gold, published in the "Proceedings of the IEEE", Vol. 55, 9 No. 2, February 1967, pages 149 to 171. In Figure 4, lines 14 and 15 have the same references as in Figure 2. The filter 11 shown on the figure conventionally includes two binary 12 summing devices 110 and 112, two delay elements each shown as a blocl~ T, each one introducing an elementary delay Tl and 14 four binary multiplying devices multiplying the signals delivered from the outputs of the delay elements by factors 16 +2, -1, 2(1-a), and (~ -1) respectively, said factors appear-17 ing in expression (35) of transfer function W(z). In the 18 embodiment shown on the figure, the output of summing device 19 112 is connected to the input of a module Q logic unit 40 the output of which is connected to line 15.
~!~ 21 The use of a modulo Q logic unit is made necessary i 22 by the type of the signals processed in the system of the 23 invention. In effect, input signal ~'n applied to line 1 24 representG a phase value lylng within the range 0 -2~ radians.
Then, this phase value is multiplied by factor Q/2~ by 26 multiplylng device 2, therefore, result Yn of this multiplica-~i 27 tion is a value lying within the range 0 - Q. As seen above, ~; 28 y is applied to the (+) input of subtractor 10 the (-) input 29 of which receives error signal u . To avoid saturation of .. . .

1 the system, it is ~herefore necessary for signal un to be also 2 comprised between 0 and Q, i.e., for the output signal of 3 filter 12 to be included between 0 and Q, which involves the 4 use of a modulo Q logic unit at the output of this filter.
By referring again to Figure 4, it can be seen that 6 the modulo Q logic unit is, in fact, very simple when opera-7 ting on binary numbers. In effect, it is known that the binary 8 representation of a number by m bits is a modulo 2m represen-9 tation of this number and that the representations of x and x+p2m (p is an integer) are identical for the m low-order bits.
11 The representation of the negative numbers lS obtained by 12 using the 2's complement, i.e., by adding the negative value 13 of x to 2 . It i8 to be noted that, as in the case of the 14 binary representation with a sign bit, the highest order bit is 0 for x > 0 and 1 for x < 0.
16 In the embodiment shown on Figure 4, Q is coded 17 with 8 bits and then, the modulo Q logic unit is a binary 18 modulo 2 logic unit. Inside the filter itself, the use of 19 16 bit binary numbers has been chosen to have a better accuracy in the calculations, thus the outputs of summing devices 110 21 and 112 are l6 bit numbers. The operation consisting of 22 taking only the modulo 28 value of the filter output signal, 23 i.e., of the summing device 112 output signal, is simply 24 ensured by taking only the 8 low-order bits of the signal pro-vided by summing device 112. Logic unit 40 is limited to a 26 shift register 41 of 16 bit~ the input of which is connected 27 to the output of summing device 112 and to a two input AND
28 gate 42. The output of register 41 i9 connected to an input 29 of AND gate 42 which is rendered contuctive only for the 8 .

,,_, _ . . _ ., _ .__ , ,., _ . . . ---- -- -- . -- ~, .
: . . . ,.. :
,. . ..
. .

l low-order bits, i.e., the first 8 bits of the number stored 2 in register 41. AND gate 42 is conditioned by a gate control 3 signal Tr applied to the second input of AND gate 42. AND
4 gate 42 delivers at its output, modulo 28 error signal un which is applied to the (-) input of subtractor 11 through 6 line 15 (Figure 2).
7 The operation of second decision filter 8 will be '-8 analysed in the next paragraph.
g It is assumed that transfer function W(z) of filter 12 is chosen as determined by expression (35) so that the 11 noise component at the input of second decision filter 8 has 12 essentially the same characteristics as the ones of noise 13 bn applied to the input of first decision filter 6, except 14 that components aO and aln are supposed to be removed by deci~ion filter 6. With reference to expression (13), it is 16 seen that the noise at the input of second decision filter ~
17 includes, then, only random component xn. The signal applied 18 to the input of decision, filter 8 is, therefore, the sum of ,' 19 signal ln representing the data,,the value of which is not -~
affected by the first decision filter (when there is no 21 detection error) and of random noise component xn. Signal 22 ln+xn is applied via line 7 to the t~) input of subtractor 16 23 which receives, at its the (-) output, an error signal which 24 is, in fact, an estimated value of xn, referenced xn. The , output signal of subtractor 16 can be written ln+en, where 26 en represents the difference between xn and xn, i.e., the 27 residual random component. Signal ln+en is applied to the 28 input of detector-separator 18 identical to detector-separator 29 12, except that only the output delivering signal ln repre-sentin8 the detected data, is used. Signal 1 detected by .- ' .

. _,,, . _ .__ _. . __ . ,, , . __ _.. _._ _ . . . . . . .. .. ... .
.. . .
~: ' ,. , : , . .

1 detector-separator 18 ls applied via line 22, to the input of 2 elementary delay element 20 with transfer function z snd 3 providing signal ln 1 at its output. Signal ln 1 is applled 4 to the (-) input of subtractor 17 the (+) input of which receives signal ln+xn delayed by elementary delay element 21, 6 i.e., signal ln_l~xn_l. The output signal of subtractor 17 7 representing random component xn 1 when there is no detection 8 error, is applied to the input of predictive filter 19 with 9 transfer function P(æ), via line 25. The function of predic-tive filter 19 the output of which is connected to the (-) 11 input of subtractor 16, consists in predicting the estimated 12 value of x , xn, from the prior values of the random component, 13 to minimize residual random component en at the input of 14 detector-separator 18.
Predictive filter lq with transfer function P(z) 16 will be now determined so that second decision filter 8 17 minimizes random component xn. For this purpose, it will be 18 successively assumed that the spectrum of random component 19 xn is known, then that it is undetermined or time-varying.
First assumption: The shape of the spectrum of 21 xn is known.
22 It is assumed that sequence {xn} has a known 23 rational spectrum. The power spectral density Rx(æ) of xn 24 ~ can be expressed as follows:

R (z) ~ N(z) NtZ ) (36) x ( ) ( -1) 26 where N(z) and D(z) are z-polynomi~als all of whose zeroes lie 27 o~tslde the unit circle, and N(z 1~ and D(z 1) are the conju-28 gated polynomials of N(z) and D(z) respectively, readily 29 obtained by replacing z by æ 1. Ie can be shown that any ' ' ,: .

1 finite energy rational noise spectrum can be transferred in~o 2 this form by applying the "spectral factorization" method. It 3 can be shown that optlmal predictive filter P(z) of xn, in the 4 mean square sense, i.e., predictive filter P(z) which minimizes en2, is 6 P(z) z [N(z ) - aD(z )] (37) N( -1) 7 where 8 a ' D(o)' '~
g Now, it is possible to determine the whole transfer ~ ~ -function of second decision filter 8. For this purpose, it i5 11 always assumed that there i8 no error detection and only the 12 noise components are considered. By referring to Figure 2 13 and using the Z-transform. -14 E(z) = (l-z P(z) ~ X(z) (38) where 16 E(z) and X(z) are the Z-transforms of en and xn, 17 respectiyely.
1~ Expres~ion (38) can be expressed as follows 19 E(z) = H(z) X(z) (89) with 21 H( ) 1 z-l P(z) (40) 22 where 23 H(z) is the transfer function of second decision 24 filter 8.
By substituting the value of P(z) derived from ex-26 press~on (37) into expre~sion (40), one obtains transfer 27 function H(z) of second decision filter 8 which reduces random 28 component xn in the assumption under consideration. We have 29 H(z) , D(~ ) (41) N(z ' ' ~-~04;~523 Re(Z) = IHtZ) I R (z) 2 = ~H(Z) H(Z 1) RX(Z) (41') 3 From (36) and (41) 4 Re(z) = a (42) This expression shows that the optimum filter is the 6 one which produces a white noise spectrum at the input of 7 detector-separator 18. On the other hand, a2 represents total 8 noise power e 2 .
.
g . EXAMPLES.
Let the power spectral density Rx(z) of sequence 11 {x } be of the form 12 R(z) = k +k +k {kl [ ( ) ( ~1)~+ k2 ~ ( ) ( lJ k3}

(43) 13 where 14 P1 = 2 (2rrfiT)2 (1-~i2 ) lS gi(Z) = -1-2~i cos (2~fiT)z + ~i 16 T = sa=pling period -~17 x 2 ~ 1 18 The two terms in brackets in expression (43) represent the noise components concentrated in frequency bandwidths proportional to (l-~i) a~d centered at fiHz, and term k3 I 21 rePreSents white noise.
22 Example 1 1 23 Let kl=4, k2~k3=1 ' -104'~5Z3 1 T = 1800H~, fl-55Hz, f2-llOHz and ~ 2~0.85 2 Spectrum~R (e~2~fT) corresponding to thls example 3 is shown on Figure 5.
4 , Combining terms in expression (43), we obtain a rational expression that factors lnto the form (36) where 6 N(z) = 0.63(1-1.34z+0.56z2) (1-1.13z+0.4z2) D(z) = (1-1.67z+0.7225z ) (1-1.58~+0.7225z 8 a2 = 0.39 9 We obtain transfer functlon P(z) P(z) = 0.81 - 1.61z-1 + 1.18z 2 _ 0.278~ -N(z Example 2 12 Repeatlng the above example with k3=1/4 gives ~ -13 ', N(z) = 0.4(1-1.46z+0.61z2) (1-0.78z+0.25z2) 1 14 D(z) same as example 1 15' a2 = 0.163 ' ',, 16 It should be noted that in example 2, a2 is much 17 smaller than in example 1, lndicating that the filter is more 18 ,effective when the white noise component is small compared ' ,~' ~,~ 19 with the other components. ' ~ `
'I 20 The embod-iment of digital predictive filter 19 will 21 not be described here since it can be conventionally obtained 22 from the expression of P(z), as described, for example, in the 1 23 above-mentioned article by C. M. Rader and B. Gold.
24 It should be noted that the configurati.on of decision 1 25 filter 8 as shown in Figure 2, can be modified to obtain a con-26 figuration identical to the one of the first decision filter 27 shown in Figure 2. Figure 6 shows a possible configuration of ~04Z523 1 second decision filter 8, designated 8'. In this configuration, 2 line 7 is connected to the (+) input of a binary subtractor 30 3 the output of which i9 connected to the input of a detector-4 separator 31 identical to detector-separators 11 and 18. A
first output of the detector-separator is connected to data 6 output line 9 while a second output is connected via linP 32 7 to the input of a digital linear filter 33 the transfer 8 function of which is referenced L(z). The output of filter 9 33 is connected to the (-) input of subtractor 30 via line 34. In operation, signal ln+xn is applied via line 7, to the 11 (+) input of subtractor 30 the (-) input of which receives an 12 error signal vn provided by filter 33. The subtractor provides 13 at its output, signal ln+en where en=xn-vn is the residual 14 random component at the input of detector-separator 31. As ~
indicated above, the function of the detector consists in :
16 detecting ln and en. When there is no detection error, 1 ::
17 is provided on line 9 while residual random component en is ~ :
18 supplied by the second output of detector-separator 31 connec-19 ted to line 32. The function of filter 33 consists in generating error signal vn from the prior values of residual 21 random component en.
22 It will now be shown that the configurations of :
23 second decision filters 8 and 8' as they appear in Figures 2 24 and 6, respectively, are equivalent and L(z) will be determined so that the conflguration of Figure 6 reduces random component 26 xn to a mlnimum. It will be always assumed that there i9 no 27 detection error and only the noise components will be con-28 sidered. By referring to Figure 6 and using the Z-transform, 29 we have: E~z) - X(z) - L(z) E(z) (44) or ~ ~ , l E(z) = l+L(z) X(z) (45) 2 where E(z) and X(z) sre the Z-transforms of e and x , n n 3 respectively.
4 Expression (45) can be expressed as follows E(z) = H'(z) X(z) (46) 6 with 7 H'(z) = l+L(z) (47) 8 where ~'(z) is the transfer function of the configuration of 9 the filter shown on Figure 6.
In order to have the two configurations of second 11 decision filters 8 and 8' as they appear on Figures 2 and 6, 12 respectively, equivalent, it is necessary for said two configu- ~
13 rations to have the same transfer function, i.e., ~ -14 H'(z~ = H(z) (48) or from (40) and (47) 16 ) l+L(Z) (49) 17 From which, it can be deduced that we must have 18 P(z) = z L(Z? (50) ' 19 or !
~ 20 ( ) -l P(Z) (51) l-z P(z) 21 The value of L(z) so that the second configuration of second j 22 decision filter 8 shown in Figure 8, eancels random component 23 xn, i9 resdlly obtalned by replaeing P(Z? in (54) by its value ¦ 24 derived from (37). It yields:

L(z) 5 N(z-l) D( ~
aD(z FR973-011 - 3n .
' - -~ .

1 Reconsidering example 1 2 L(z) = 1.24z - 2.55z + 1.87z - 0.4z 4 (52') - 1 ~

3 So far, one has determined the two possible configura-4 tions of second decision filter 8, when the shape of the spectrum of random component x is known. The second assumption will now -6 be studied.
7 Second Assumption: The xn spectrum shape is un-8 determined or time-varying.
9 It was seen above that random component xn is mainly due to phase jitter and white noise. Now, we will consider 11 - the case when the phase jitter effect is preponderant over the 12 white noise effect and random component xn will be considered -13 as phase ~itter.
14 Since phase jitter is generally provided with a spectrum limited between 0 and 300 Hz while the sampling -16 frequency, i.e., the character transmission rate in a digital 17 data transmission system is very often at least equal to 1200 18 Hz, it is possible to ob`tain a good estlmated value of x by 19 usi~g an adaptive predictive filter, for instance a Wiener adaptive predictive filter. It will be briefly recalled below 21 what is a Wiener adaptive predictive filter of order p.
22 Assume that sn 1 i8 the (n-l~th sample of 8 low 23 frequency signal, and that s^n i9 an estimated value of the next 24 sample. Estimated value s^n is obtained by carrying out an e~trapolation on the last p samples. This can be expressed 26 as follows . ~

. . .
.. .. . ..

., : , ., :

-Sp+~ 1 + ~2S2 + '- ~p P

2 ' - ~292 + - + ~psp+l 3 I (53) 4 Sn = ~lSn-p + ~29Pn p+l P n-l J
with n > 2p.
6 Factors ~i~ i=1,2, ... , p are ad~usted so that error 7 En 1~ in the mean square sense, between the actual value of 8 sq and its estimated value s^q, q=p, (p~l), ;.. , (n-l), is at 9 a minimum:
n-l (s - s^ )2 (54) 11 . In this assumption, it is possible to replace predictive filter 12 19 of second decision filter 8 shown in Figure 2, by a Wiener 13 adaptive predictive filter. If a Wiener filter of order p=5, 14. for instance, is uset, estimated value xn of xn is obtained from the following iteration relation 16 . n ~lXn-l + ~2Xn-2 + ~3xn_3 + ~4xn_4 + ~sxn_5 (55) 17 The values of ~i~ 1=1, .... 5, are calculated so that 18 error En 1 in the mean square sense, betwean xq and xq, q=6, 19 ... , (n-l) is at a minimum. We have n-l En-l = ~6 (xq~xq) (56) 21 Figure 7 schematically shows an example of a digital 22 embodiment of such a Wiener adaptive predictive filter 19'.

.

~04Z5Z3 1 Signal xn 1 provided from the output of subtractor 2 17 (Figure 2) is applied to the input of a 5T long delay line 3 50, T being the sampling period via line 25. In a digital 4 embodi~ent, this delay line consists in a shift register including five stages, each one introducing a delay T. This 6 delay line includes six taps 51-l, 51-2, ... , 51-6 positioned 7 with respect to T so that when value xn 1 is applied to the 8 input of delay line 50, and therefore, is available at tap 51-1 value9 x 2~ Xn 3~ xn_4~ xn_5 and xn_6, .
are available at taps 51-2, ... , 51-6.
11 Taps 51-1, ... , 51-5 are respectively connected to 12 a first input of five binary multipliers 52-1, ... , 52-5.
` 13 The oùtput of these m~ltipliers are connected to the inputs 14 of a summing device 53 the output of which supplies estimated `~ 15 value xn on line 26. The output of summing device 53 is ~! 16 further connected via line 54, to the input of a delay element f 17 55 introducing a delay T, the output of said delay element :`
l 18 thus supplying value xn 1 The output of delay element 55 is ~' 19 connected to the (-) input of a binary subtractor 56, the (+) -input of which is connected to line 25 via line 57. The output l` 21 of subtractor 56 providing error signal ~ l=xn l-xn l~ is ~:
22 connected via line 58, to a first input of five binary ~ .
1 23 multipliers 59-1, 59-2, .. ..., 59~5, the second inputs of which .~ 24 are respectively connected to taps 51-2, ... , 51-6. The outputs , 25 of multipllers 59-1, ..... , 59-5 are respectively connected to :, 26 the input of five digital integrators 60-1, 60-2, .~., 60-5 27 which, in the embodiment shown on the figure, consist in revers-28 ible counters. Each of integrators 60-1, .... , 60-5 is provided , 29 with a rese~ting input, respectively connected to lines 61-1, , .
:, ' 1~4ZS23 1 ... , 61-S. The outputs of integrators 60-1, .... , 60-5 are 2 respectively connqcted to the input of five factor ad~usting 3 logic units 62-1, 62-2, ... , 62-5, the outputs of which are 4 respectively connected to the second input of each of multi-pliers 52-1, ... , 52-5. The outputs of logic units 62-1, 6 ... , 62-5, respectively supply the values of factors ~ 2 7 .-. ~5. To simplify the drawing, only logic unit 62-1 is 8 shown in detail, other logic units 62-2, ... , 62-5 being 9 identical to the first one. Logic unit 62-1 includes a two input AND gate 63-1 with two inputs, one of which is connect-11 ed to the output of integrator 60-1 and the second one is 12 connected to a first control signal source (not shown), via 13 line 64-1. The output of A~D gate 63-1 is connected to a 14 first input of a binary multiplier 65-1 the second input of ~
which is connected to a storage element 67-1 via line 66-1. ~ -16 The output of multiplier 65-1 is connected to the (-) input -- ~ -17 of a binary subtractor 68-1 the (+) input of which is connect-18 ed to a second control signal source (not shown) via line 19 69-1. The output of subtractor 68-1 is connected to the -second input of multiplier 52-1.
21 The device shown on Figure 7 will now be described.
22 ~ The device shown on the figure should provide 23 estimated value xn obtained from expression (55), factors 24 i' i=l, ... , 5 being obtained by min,imizing error En 1 defined by expre~sion (56) recalled below.
n-l 26 En_l ~6 (xq xq (57) FR97~-011 - 34 -. ,~ .

, .

~04Z5Z3 1 or n-l 2 En~ eq (58) i:
; 3 with 4 ~q = xq - xq (59) Error En 1 appearing as a square sum Eq , q=6, (n-l), minimi-6 zing En 1 is the same as minlmizing all ~q, q=6, ... , (n-l) 7 succassively and summing the minimum values of Eq obtained in 8 this way. One will consider the instant where signal xn 1 is 9 applied to the input of the Wiener predictive filter and analyse -~
the minimi~ation of ~n 1~ while showing how the device shown on 11 the figure takes the minimum ~alues of the prior s2 into 12 account.
13 ~ Since the only adJustable elements which can be 14 modified to minimize ~n 1 are the values of coefficients ai, i=l, 2, ... , 5, elementary error En 1 will be minimum if the 16 derlvative of en 1 with respect to the various coefficients, is 17 cancelled, i.e., if 18 a~n-l = O for i=l, .... , 5 (60) a,~i 19 we have Z ~n 1 = 2~n 1 a~n-l (61) 21 Substituting ~n 1 by its value derived from (59~
22 9~2_1 = 2(Xn-l-xn-l) n-I n-l (62 i 23 Signal xn 1 being independent from the values of co-24 efficien~s ai, expression (61) becomes ~R973-011 - 35 -.

, -~ a~n-l = ~2(Xn-l-xn-l) a~ (63) ; 2 but, from (55) 3 2 = ~ x + ~ x + ~ x + a x + a x (64) n-l l n-2 2 n-3 3 n-4 4 n-5 5 n-6 4 or Xn-l = ~ i n-l-i (65) 6 Derivating expression (65) with respect to ai a~
7 a n-l = Xn-l-i for i=l~ 5 (66) Xi .: .
8 Then, expression (63) becomes a 2 g ~n-l = -2 (Xn-l-xn-l) Xn-l-i (67) ' 10 Therefore, En 1 is minimi~ed by making f 11 (Xn l~Xn-l) Xn-l-i for i=l, .... , 5 (68) 12 With reference to Figure 7, it will be shown that i~ 13 ~he device of said figure, applies relation (68).
14 For clarity, only the ad~ustment of the value of ~I~ 15 coefficient ~1' i.e., the application of expression (68) for 1 16 i=l, wlll be described, the ad~ustment of the other co-,~ 17 efficients being similar. Signal xn 2 available on tap 51-2 18 i9 applie`d to the second input of multiplier 59-l the first 19 input of which receives 9ignal En l~Xn l~Xn 1 via line 58-20 ` Sign~l En 1 is provided by the output of subtractor 56 which 21 receives, on,its (+~i,nput~,slgnal xn_l via-line 57, and on 22 its (-) input, slgnal xn 1 available at the output of delay 23 element 55. Therefore, the output of multiplier 59-1 pro-24 ( n-l xn_l) Xn-2 which is applied to the input FR973-011- 36 - ;~ ~

.
.
:

-1 Of integrator 60-1 which accumulates the value of this 2 product with the prior values of said product. A resetting 3 line 61-1 is provided since, in practice, said products are 4 accumulated only during a determined period of time, equal, in general, to the length of delay line 50. The result of 6 this accumulation, which will be referenced Qal, is used to 7 ad~ust the value of al until this result is null. The result 8 of the accumulation is applied to the input of the logic unit 9 62-1 provided for ad~usting coefficients and supplying the new value of al at its output. In practice, this adjustment 11 is not performed on each sampling time but, in general, after 12 a determined period of time equal in general, to the length 13 of delay line 50, and it is the reason why it has been pro-14 vided, at the input of logic unit 62-1, an AND gate 63-1 which is rendered conductive only after this determined period 16 of time has elapsed. Signal a~l is then applied to the first 17 input of multiplier 65-1 which receives on its second input, 18 a val~e ~ stored in element 67-1 and representing the incre-19 ment step of coefficients ai. The output of multiplier 65-1 providing product ~aal is connected to the (-) input of sub-21 tractor 68-1 to the (~) input of which is applied the prior 22 value of 1 23 al new al old ~a~l 24 which is applied to the second lnput of multiplier 52-1.
Multipller 52-1 multiplies the new value of al with signal 26 xn 1 available from tap 52-1. Then, multiplier 52-1 carries 27 product alxn 1 out, said product being applied to one of the 28 inputs of summing device 53. Summing device 53 receives, on 29 ~ts other inputs, in the same way, products a2xn 2~ a3xn 3 ~.. - - ~_ 1 ~4xn 4, ~5xn 5, and supplies as an output, estimated value 2 Xn defined by expr~ession (55). Estimated value xn i9 applied 3 through line 54 to the input of delay element 55 which will 4 apply it at the next sampling time, to the (-) input of subtractor 56. Then, value xn will be used to carry on the 6 coefficient adjusting procedure. This value xn is further 7 applied to the (-) input of subtractor 16 (Figure 2) via line 8 26, for minimizing random component x through second decision g filter 8 (Figure 2).
It should be noted that the adaptative predictor 11 operates exactly as a conventional transversal equalizer, - 12 except that the error criterion is different. The above-13 described predictor converges exactly as an equalizer the 14 taps of which are spaced by the sampling period. It should i 15 also be noted that, since the predictor operates as a low 16 pass filter, the white noise has little effect on the estimate 17 of the phase jitter.
18 Figure 8 shows another possible configuration of 19 decision filter 8 designated as 8", including an adaptive - 20 predictive filter.
21 Signal ln+xn available on line 7 (Figure 2) is 22 applied to the (+~ input of a binary subtractor 70 the (-) ! 23 input of which receives an estimated value of xn, xn. Signal 24 ln+xn-xn supplied from the output of subtractor 70 is applied to the lnput of a detector-fleparator 71 which provides signal 26 1 representing the data on line 9 and residual random com- .
27 ponent x ~xn on line 72. Signal xn-xn i9 applied to the 28 input of an elementary delay element 73 introducing an element-29 ary delay T so that signal xn l-xn 1 is available at its output, FR973-011 ~ 38 -`

104~5;~3 l said signal is applied via line 74, to a first (+) input of 2 a binary adder 75 t~he second (~) input of which receiv~s 3 signal xn 1 via line 76. Thus, adder 75 provides at its 4 output, signal x 1 which is applied to the input of an adaptive predictive f~lter 77 providing signal xn at its 6 output. Signal xn is applied to the (-) input of subtractor 7 70, via line 78, and to the input of an elementary delay 8 element 80 introducing an elementary telay T via line 79.
9 Signal xn 1 available from the output of delay element 80, is applied to the second (+) input of adder 75 via line 76.
11 It should be noted that the function and the opera-12 ting conditions of adaptive predictive filter 77 are identical 13 to the ones of predlctive filter 19 provided in the configura-14 tion shown on Figure 2 and that it is possible to use the Wiener predictive filter described in Figure 7 as an adaptive 16 predictive filter.
17 So far, the system has been described as a phase 18 filter in which both decision filters 6 and 8 are connected in 19 cascade. Now, the system will be described as a phase filter in parallel form, in the cases when the predictive filter of 21 the second decision filter is provided with fixed coefficients, -22 and then when said filter is adaptive.
23 Case of a predictive filter with fixed coefficients 24 For clarity, the phase filter, shown in Figure 9a, i8 in cascade form similar to Figure 2, with the second 26 decision filter 8 having the configuration as shown in Figure 27 6. In Figure 9b, the corresponding parallel form is shown.
28 It should be noted that the same reference numbers are used 29 for the same elements in the structure of Figure 9a as shown FR973-011 ~ 39 -1~4ZSZ3 1 in Figures 2 and 6 since the operation of the respective 2 elements has been ~described before.
3 In Figure 9b, signal Yn available on line 5 is 4 applied to the (+) input of a binary subtractor 90 the output of which is connected to the input of a detector-separator 6 91. The first output of detector-separator 91 connected to 7 line 9, provides the signal representing data ln, while the 8 second output of detector-separator appearing on line 32 is 9 connected in parallel to filters 92 and 93 with transfer functions W'(z) and L'(z), respectively. The outputs of 11 filters 92 and 93 are added in binary adder 94 the output 12 of which is connected to the (-) input of subtractor 90.
i3 Now, it will be shown that the structures of 14 Figures 9a and 9b are equivalent by considering their trans-fer functions.
16 As seen above, transfer functions G(z) and H'(z) 17 of decision filters 6 and 8 of Figure 9a are given by relations 18 (19) and (47), respectively, 19 1 + W(z) ~ H'(z) =

Transfer functions Sc(z) of the cascade configuration ~ -21 shown on Figure 9a can be expressed as follows 22 Sc(z) = G(z) H'(z) 23 or 24 c 1 + W(z) 1 + L(z) (69) Transfer function S (z) of the paralle:l configuration -:
FR973-011 ~ 40 - ~

.. . .. ... . .... .. . .

1 shown on Figure 9b can be directly established from the 2 diagram of Figure 9b p 1 + W'(z) + L'(z) 4 In order to make the cascade and parallel forms equivalent, we must have 6 S (z) = S (z) 7 or 8 (l+W(z))) (l+L(z)) = l+W'(z) + L'(z) (71) 9 For instance, it i9 possible to select W'(z)=W(z) and to derive the value of L'(z) from relation (71). It yields ~ 11 L'(z) = L(z) (l+W(z)) (72) '~ 12 Substituting W(z) and L(z) by their values derived ~' 13 - from (35) and (52') respectively, into (72), we obtain the 14 value of L'(z) i~ the case of above example 1.
Case of the adaPtive predictive filter -~ 16 For clarity, the phase filter shown on Figure lOa, .l~ 17 is in cascade form similar to Figure 2, with the second 3 18 . decision fllter 8 having the adaptive configuration as shown .:
19 . in Figure 8. In Figure lOb the corresponding parallel form is shown. The same reference numbers are used for the same 21 elements in the structure of Figure lOa as shown on Figures 22 2 and 6, since the operation of the elements has been des-I .23 cribed before.
~! 24 In Flgure lOb J signal y available on line 5 i9 applied to the (~) input of a binary subtractor 95 the output 26 of which is connected to the input of a detector-separator 96.

. FR973-011 - 41 -., .

1~)4'~5Z3 1 The first output of detector-separator 96, connected to line 2 9, provides the si~nal representing data ln while the second 3 output of detector-separator 96 is connected in parallel to 4 the input of filter 97 with transfer function W(z), which is identical to filter 12 of Figure 2, and to the input of an 6 elementary delay element 98. The output of filter 97 is 7 connected to the first (+) input of a binary adder 99 the 8 output of which is connected to the (-) input of subtractor 9 95. The output of delay element 98 is connected to the first (+) input of a binary adder 100 the output of which is 11 connected to the input of an adaptive predictive filter 101.
12 The output of filter 101 is connected to the second (~) input 13 of adder 99 and to the input of a delay element 102 the output -14 of which is connected to the second (+) input of adder 100.
It should be noted that the first loop including 16 subtractor 95, detector-separator 96 and filter 97 is identical 17 to first decision filter 6 shown in Figure 2. The operating 18 conditions of the first loop being identical to the ones of 19 fir~t decision filter 6, in the cascade form.
20 ^ The second loop including subtractor 95, detector-21 separator 96, delay elements 98 and 102, adder 100 and adaptive ~22 filter 101, is identical to second decision fllter 8" as des-23 cribed with reference to Figure 8. In the parallel form shown 24 on Figure lOb, the second output of detector-separator 96 supplies both residual component fn and residual random compo-26 nent xn-xn. However, as seen above in the description of the 27 operation of first decision filter 8", residual component 28 fn' in steady condition. In steady conditions, the signal 29 applied to the illpUt of delay element 98 is only comprised FR973-OlI - 42 - -. . ~ -`'.

-1 of xn-xn and thus, the operating conditions of the second 2 loop are identlcal,to the ones of decision fllter 8" ~hown 3 in Figure 8. Thus, the second loop operates exactly as 4 decision filter 8" of Figure 8 and adaptive filter 101 can be as the Wiener adaptive predictive filter shown in Figure 6 7.
7 With reference to Figure 11, an example of a 8 digital embodiment of the detector-separators used in the 9 invention, will now be described.
It was seen above that the function of the detector- ~:
11 separator used in the inventi.on consists in isolatin~, from 12 the signal applied thereto, term ln representing the data, 13 and the residual components marked f in the case of the 14 detector-separator of first decision filter 6, and en=xn-xn in the case of the detector-separator of second decision 1-6 filter 8.
17 Furthermore, it was seen that ln is a positive 18 integer (relation (12~).
19 ln = - 1, 2, -- (Q-l) and that, when there is no detection error, residual noise ~21 component e , for example, is a fractional number the absolute 22 value of which is comprised between 0 and 1.
: 23 It is possible to isolate ln and en in several 24 different ways. If, for instance, the signsl applied to the input of the detector-separator i8 equal to 2.6, it can be 26 arbitrarily said that:
~27 ln=3 and en~-.04 i.e., that -0.5 < en < 0'5 28 or 29 ln-2 and en~+0.6 i.e., that 0 < en ~ 1 FR973~011 - 43 --1 Figure 11 describes an example of a digital 2 detector-separator using the second possible solution~
3 It is assumed that the case of a eight-phase 4 digital data transmission system is considered and that the signal applied to the input of the detector-separator 6 shown on the figure appears as a eight-bit binary word.
7 The device shown on the figure includes an eight-bit shift 8 register 103, a latch 104, three AND gates 105, 106 and ~ -9 107 and an OR gate 108. The lnput binary word is applied to-the input of shift register 103. The input binary word ~ represents both an integer ln representing the data, and a r 12 fractional number representing the noise. In the case of I3 an eight-phase system, ln can assume one of the eight values 14 ln=0, 1, 2, ... , 7, and can be coded by three bits (23~8 phases) which are the three high-order bits in the input ! 16 binary word. In the device shown on Figure 11, ln is ' 17 detected by connecting the outpue of shift register 103 to a first input of AND gate 105 and by conditioning said AND
19 gate 105 only when the three high-order bits are available at the output of shift register 103. AND gate 105 is con-21 trolled by applying an appropriate transfer control signal 22 Tr 2 on the second input of AND gate 105 via line 109~ The output of AND gate 105 is connected to line 9 (Figure 2) from ~ 24 whlch lln is available. The fractional part of the input 1 25 binary word representing the noise is also obtained by connect-26 ing the output of shift register 103 to a first input of AND
27 gate 106 and by enabling the gate when the five low-order ~' 28 bits are available from the output of shift register 103.
29 AND gate 106 is controlled by applying an appropriate transfer .. : ' ~-.. . . . .
~ .'. ".'' ' ' ' ~ '",' ' ' ~' , ~ : , , . ~ , . . .

1~4~SZ3 l control signal Tr 1 on the second input of AND gate 106 via 2 line 110. The five low-order bits transmitted by AND gate 3 106 are applied to line 14 through OR gate 108. However, 4 these five bits bein8 the five low-order bits of an eight-bit binary word, it is necessary to convert these five bits into -.
6 an eight-bit binary word. In the device shown on the figure, 7 the five bits to eight bits conversion is readily carried 8 out by repeating the last of the five bits, three times. A
9 tap is provided for this purpose at the fifth bit position of shift register 103 and this tap is connected to the input 11 of latch 104 which reads the bit which is in this bit position 1~ when the input binary word is in shift register 103. This 13 reading is controlled by timing signal Tr 3 applied to.this 14 latch via line lll. The binary state assumed by latch 104 : -is transmitted through AND gate 107 and OR gate 108 for three 16 bit times following the first five bits fetched out of shift 17 register 103. AND gate I07 is controlled by applying an 18 . appropriate transfer control signal Tr 2 on the second input 19. of AND gate 107 via line 112. The timing sequence for Trl, Tr2 and Tr3 is illustrated in Figure 12.
21 While the invention has been particularly shown and 22 described with reference to the preferred embodiment thereof, 23 it will be understood by those skilled in the art that various 24 changes in form and details may be made therein without depart-ing from the s.plrit and scope of the invention.
26 What is claimed is:

JA/~mh FR973-011 - 45 _ .. . .
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Claims (8)

The embodiments of the invention on which an exclusive property or privilege is claimed are defined as follows:

1. In a phase filter for minimizing the effects of noise components altering the value of the phase of a digital signal in a digital data transmission system in which the phase of said digital signal can assume Q distinct values representing the data, with said phase filter in-cluding two decision filters, with the first decision filter cancelling residual noise components representing the phase intercept component and the phase shift component introduced by frequency shift, and the second decision filter minimizing the random noise component representing phase jitter and white noise, the combination comprising:
input means for receiving said digital signal;
said first decision filter including;
first means for providing a first difference signal in response to taking the difference between the received digital signal from said input means and a first error signal corresponding to an estimated value of said residual noise component, a first detector responsive to said first difference signal for separating the data portion from the residual noise portion of said first difference signal;
a first linear filter responsive to said residual noise portion of said first difference signal for generating said first error signal.
1. Claim 1 continued:
   Said second decision filter including;
   second means for providing a second difference signal in response to taking the difference between said first difference signal and a second error signal corres-ponding to an estimated value of said random noise compo-nent, second detector responsive to said second difference signal for separating the data portion from the random noise portion of said second difference signal, with a data representative signal being provided at an output, and a second linear filter responsive to said data representative signal for generating said second error signal.
   
   2. In apparatus for minimizing the effects of noise components altering the value of the phase of a digital signal in a digital data transmission system in which the phase of said digital signal can assume Q distinct values representing the data, the combination comprising:
   input means for receiving said digital signal;
   a first comparison means having first and second inputs and an output, with the first input being connected to said input means for receiving said digital signal and with the second input receiving a first error signal;
   a first detection means having an input connected to the output of said first comparison means for detecting the integral and fractional parts of the output signal provided by said first comparison means, with the integral part representing the data and the fractional part represen-ting the noise residual component, with said noise residual signal being provided at an output of said first detection means;
   a first linear filter having an input connected to the output of said first detection means for receiving said noise residual component and providing said first error signal at an output, with said first error signal being applied to the second input of said first comparison means for minimizing the residual noise component appearing at the output of said first comparison means;
2. Claim 2 continued a second comparison means having first and second inputs and an output with the first input receiving the output signal from said first comparison means and the second input receiving a second error signal;
   a second detection means having an input connected to the output of said second comparison means and having an output, with said second detection means detecting the integral and fractional parts of the output signal provided by said second comparison means, with the integral part representing the data and the fractional part representing the random noise component, with said integral part being provided at the output of said second detection means;
   first and second delay means, with said first delay means delaying the signal provided at the output of said first comparison means and said second delay means delaying the signal appearing at the output of said second detection means;
   a third comparison means, for comparing the delayed signals provided at the respective outputs of said first and second delay means; and a second linear filter having an input for receiv-ing the output signal from said third comparison means and providing said second error signal at an output, with said second error signal being applied to the second input of said second comparison means for minimizing the random noise component appearing at the output of said second comparison means.
3. 3. The combination claimed in Claim 2, wherein said input means includes means for multiplying said digital data signal by a coefficient having the value Q/2.pi..
4. 4. The combination claimed in Claim 3, wherein said first, second and third comparison means each comprises a subtractor.
5. 5. The combination claimed in Claim 4, wherein said first linear filter has a transfer function, where, z is the z-transform notation; and .alpha. is a coefficient lying within the range 0 < .alpha. < 1
6. 6. The combination claimed in Claim 5, wherein said second linear filter has a transfer function, where, z is the z transform notation;
   N(z) and D(z) are Z-polynomials all of whose zeroes lie outside the unit circle;
   
   N(z-1) and D(z-1) are the conjugated polynomials of N(z) and D(z) respectively; and
7. 7. The combination claimed in Claim 5, wherein said second linear filter has a transfer function, where, z is the z-transform notation;
   N(z) and D(z) are z-polynomials all of whose zeroes lie outside the unit circle;
   N(z-1) and D(z-1) are the conjugated polynomials of N(z) and D(z) respectively; and
8. 8. In a phase filter for reducing the effects of the noise components altering the phase value of a received signal, in a digital data transmission system in which the phase of the emitted signals can assume Q discrete distinct values representing the data, at the sampling times, the combination comprising:
   input means adapted to receive the phase value of the signal received at the sampling times;
   a subtractor a first input of which is connected to said input means and a second input of which receives an error signal;
   a detector-separator connected to the output of said subtractor, to separate the integral and fractional parts of the signal supplied by said subtractor, said integral and fractional parts respectively corresponding to a signal representing the data and to a noise residual component which includes a residual random component and being re-spectively provided to first and second outputs of said detector-separator;
   a first linear filter with transfer function where, z is the z-transform notation; and .alpha. being a coefficient lying in the range 0 < .alpha. < 1;
   connected to said second output of said detector-separator;
   
   a first delay element introducing an elementary delay T equal to the sampling time period, connected to said second output of said detector-separator;
   a first adder with two inputs, a first input of which is con-nected to the output of said first delay element;
   an adaptive predictive filter the input of which is connected to the output of said first adder to generate said residual random component from the signals supplied by said first adder;
   a second delay element identical to said first delay element to input of which is connected to the output of said adaptive pre-dictive filter and the output of which is connected to the second input of said first adder; and a second adder the inputs of which are respectively connected to the outputs of said first linear filter and said adaptive pre-dictive filter, and the output of which is connected to said second input of said subtractor.