BACKGROUND OF THE INVENTION

The present invention relates to an analog-digital coder comprising a charge transfer coded voltage generator.

It is well known in the art to determine by successive approximations the coefficients a_o, a₁ . . . a_i . . . a_n, equal to 0 or 1, making it possible to digitally code an unknown analog voltage V_x by writing it in the form:

|Vx |=ao ·VR +a1 ·VR /2+a2 ·VR /22 + . . . +ai ·VR /2i + . . . +an ·VR /2n 

in which V_R is a reference voltage. For this purpose:

|V_x is firstly compared with V_R --if | Vx is less than V_R, then a_o is equal to 0, otherwise it is equal to 1;

|V_x | is then compared with V_R1 =a_o V_R +V_R /2--if |VV_x | is less than V_R1, then a₁ is equal to 0, otherwise a₁ is equal to 1;

then V_x | is compared to V_R2 =a_o V_R +a₁ ·V_R /2+V_R /4·--if |V_x | is below V_R2, then a₂ equals 0, otherwise a₂ equals 1;

and so on until all the coefficients a_o . . . a_n have been determined.

Thus, for determining by successive approximations the coefficients a_o . . . a_n it is necessary to have the voltages V_R and

VR1 =ao VR +a1 ·VR /2+a2 ·VR /22 + . . . +ai-1 ·VR /2i-1 +VR /2i with i=1 . . . n.

We know a charge transfer coded voltage generator which is shown in FIG. 1. It will be described in greater detail hereinafter and supplies the voltages V_R and V_Ri. This generator for example is described in U.S. Pat. No. 4,350,976, assigned to Thomson-CSF.

A reference charge quantity 2Q_R is injected into the generator at the start of the processing of each sample |V_x |. This generator is constituted by a charge transfer device (CTD), which is divided up into two parallel channels. Half the charge in the CTD before division is collected in each channel.

A charge reading device (CRD) is connected to the two channels--to the storage grid G₁ following the division for the first channel and to the third storage grid G₄ following the division for the second channel.

The CRD then collects the quantity of charges Q_R beneath G₁ and supplies the voltage V_R and then coefficient a₀ can be determined. If a₀ =0 the quantity of charges Q_R stored in the second channel is discharged and if a₀ =1 this charge quantity is stored beneath the second storage grid G₃ following the division of the CTD.

In all cases the charge quantity Q_R stored beneath grid G₁ of the first channel performs a round trip on either side of the zone where the CTD is divided into two channels and thus a charge quantity Q_R /2 is stored beneath grids G₁ and G₂.

Thus, after the transfer of the possible content of G₃, which will be called a₀ Q_R to beneath G₄, the CRD collects a charge quantity equal to a₀ Q_R +Q_R /2 and thus supplies V_R1 making it possible to determine a₁. This is continued until the determination of a_n takes place after the processing of V_Rn.

The following problems occur when it is desired to provide an analog-digital coder with the charge transfer coded voltage generator in question. The coded voltage generator has the defects inherent in charge transfer devices and which essentially consist of the inefficiency of transfer, the dark current and the leakage current resulting from crystal imperfections or interface impurities (and particularly occurring with surface transfer CTD). However, the scale of these defects increases in the case of a coded voltage generator where an initial charge quantity 2Q_R has to perform n round trips on either side of the division of the CTD to finally obtain Q_R /2ⁿ and where the storage time for this charge quantity is therefore relatively long.

With regard to the inefficiency of transfer it is possible to use a CTD with a buried channel, whose transfer coefficient is substantially equal to 1, to within a few 10^-5 units.

However, with regard to the dark and leakage currents it is necessary to limit the number n of successive divisions which can be validly performed, whilst retaining a final charge which is differentiated in a completely satisfactory manner from the thermal background noise.

A further problem is connected with the offset voltages appearing at all the active members adjacent to the generator for forming the coder, particularly when these members are constructed according to MOS technology.

BRIEF SUMMARY OF THE INVENTION

The present invention makes it possible to solve these problems. In the coders according to the present invention the determination by successive approximations of the coefficients a₀ . . . a_n takes place in two stages:

during the first stage the generator processes the voltages V_R and V_Ri with i=1 to k-1, making it possible to determine the coefficients a₀ to a_k-1 ;

during the second stage the generator processes the voltages V_R and V_Ri with i=1 to n-k;

the coder comprising means for ensuring on the one hand the division by 2^k of the processed voltages and on the other their summation with the voltage V_R(k-1) produced at the end of the first stage.--thus, these means supply voltages V_Ri with i=k to n, which makes it possible to determine the coefficients a_k to a_n.

Thus, according to the invention, by selecting k=(n+1)/2 the number of round trips of the reference charge quantity 2Q_R initially introduced into the generator is divided by two and the storage time in the CTD of this charge quantity is divided by two. It is pointed out that the coding of each sample |V_x | requires the successive injection into the generator of two reference charge quantities 2Q_R.

The invention makes it possible to obtain a coder with an increased precision or makes it possible to use CTD with lower performance levels with regard to the inefficiency of transfer, as well as the dark and leakage currents.

Finally according to a preferred embodiment the coder according to the invention compensates the offset voltages appearing at the active members.

BRIEF DESCRIPTION OF THE DRAWING

The invention is described in greater detail hereinafter relative to non-limitative embodiments and with reference to the attached drawings, wherein show:

FIG. 1: a plan view of the charge transfer coded voltage generator and the electrical diagram of the charge reading device (CRD), associated therewith, said generator and said CRD being used in the coder according to the invention.

FIG. 2: a diagrammatic view of the analog-digital coder according to the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the drawings the same references designate the same elements, but for reasons of clarity the dimensions and proportions of the different elements have not been respected.

FIG. 1 is a plan view of the charge transfer device constituting the coded voltage generator used in the coder according to the invention, as well as the electrical diagram of the charge reading device, hereinafter called CRD which is associated therewith.

The charge transfer device is preferably of the charge coupled device type (CCD).

There can be either a surface or a volume charge transfer. Finally this device may or may not have a buried channel. The following description relates to the case of a CTD with a channel N created by the overdoping P⁺ of a P-type substrate. In this case timing signals develop between the zero level and a high positive level. It is obviously possible to produce the CTD with a P channel in an N-type substrate.

The active area of the semiconductor substrate in which the transfer and storage of the charges takes place is located within the broken line l. Outside the area delimited by this line substrate overdoping raises the inversion threshold and thus opposes any storage of charges.

At one end of the active area, to the left in the drawing, there is a diode D_e making it possible to introduce a reference charge 2Q_R beneath a storage grid G_e via a transfer grid T_o. Diode D_e and grids RT_o and G_e constitute the device for injecting the reference charge 2Q_R into the generator, designated by the overall reference 1 in the drawing.

Diode D_e is also used for the removal of surplus charges during the operation of the generator.

Grid G_e is followed by a transfer grid T₁ and a storage grid G_o. Beneath G_o an insulating diffusion advances to half the width of the active charge transfer area to carry out a pre-division of the charge stored beneath G_o into two equal charge quantities. This insulating diffusion divides the active area following G_o into two parallel channels which generally have the same width. Thus, two charge quantities equal to half the charge stored beneath G_o are transferred beneath the storage grids G₁ and G₂ positioned on each channel after G_o and separated from the latter by transfer grids T₂ and T₃. This division of a charge quantity by a diffused area is known from the prior art.

Thus, one of the channels is terminated by the storage grid G₁, whilst the other channel has, following the storage grid G₂, two storage grids G₃ and G₄ and three transfer grids T₄ between G₂ and G₃, T₅ between G₃ and G₄ and T₆ between G₄ and a collecting diode D_c. The collecting diode D_c, which terminates the channel, is connected to a d.c. voltage V_DD and makes it possible to remove the charges at the end of processing a sample V_x.

Grids G₁ and G₄ are connected at a point P to the CRD, which performs a non-destructive reading of the charges stored beneath G₁ and G₄.

The CRD can be a current charge reading device known from the prior art, like that described in French Patent Application No. 77 13857, published as No. 2 389.899 in the name of THOMSON-CSF.

The potential at point P is kept constant during the arrival of the charges due to an MOS transistor Q₂ connected between point P and a point A. Transistor Q₂, controlled by a periodic timing signal φ₂, is then polarized in saturation. The current traversing Q₂ during the influx of charges is integrated into a capacitor C_A connected between point A and earth.

The voltage at point A is read by an MOS transistor or TMOS Q₅ in the form of a follow-up system, whose grid is connected to point A, whereof one of the electrodes is connected to a voltage V_DD and whose other electrode supplies a reading voltage to the terminals of a resistor R_S or to the drain of an MOS transistor Q₆. For reading the voltage at point A TMOS Q₅ can be replaced by an operational amplifier of gain 1 and with a high input impedence.

A TMOS Q₃, whose grid is connected to B is also connected between point A and point B. A capacitor C_B is connected between point B and the timing signal φ₂. Finally a TMOS Q₄ is connected between point B and the voltage V_DD, the grid of Q₄ also being connected to V_DD.

Finally a TMOS Q₁ is connected between grids G₁ and G₄ and earth. The TMOS Q₁ is controlled by a periodic timing signal φ₁ and, when conductive, brings about the zeroing of the grids to which it is connected.

We will now study the operation of the generator shown in FIG. 1.

It is possible to differentiate five operating sequences.

These consist of sequence t_o when a charge 2Q_R is introduced between grid G_e in the form of the wellknown "fill and spill" method. For this purpose diode D_e is successively raised to a low level permitting the passage of charges from D_e beneath T_o and G_e, then to a high level permitting the storage of a given charge quantity 2Q_R beneath G_e. Grids T_o and G_e receive the same voltage V_GE. The reference charge quantity 2Q_R is stable to the first order, i.e. is independent of the voltage variations V_GE applied to T_o and G_e, if the surface potential curves as a function of the voltage of this pair of grids are parallel. To this end the threshold voltages beneath T_o and G_e are adjusted by implantation beneath T_o.

The other transfer electrodes of generator T₁ . . . T₆ can be realized on an extra thickness of oxide. However, like T_o they can also be produced by implantation or by any other known process.

This is followed by sequence t₁ when charge 2Q_R is transferred beneath G_o. The timing signal φ₂ is at high level V.sub.φ in such a way that the point B, initially precharged by Q₄ to V_DD -V_T passes to V_B =V_DD -V_T +V.sub.φ and point A is raised by Q₃ to V_B -V_T in which V_T is the threshold voltage common to the TMOS Q₁, Q₂, Q₃. Point P and grids G₁ and G₄ are pre-charged to V.sub.φ -V_T. The TMOS Q₂ is polarized in saturation because the potential at A V_AO =V_B -V_T is greater than the potential at P V.sub.φ -V_T.

In sequence T₂ the charge 2Q_R present beneath G_o is transferred to G₁ and G₂ by the zeroing of G_o, because grids T₂ and T₃ are at fixed intermediate potential. The charges are therefore transferred above the potential barriers induced beneath T₂ and T₃, which eliminates interference which could bring about the zeroing of T₂ and T₃ on reading grids G₁ and G₂.

Thus, charge Q_R is beneath G₁ and charge Q_R beneath G₂ due to the division of the charges performed by insulating diffusion.

At the same time T₄ and G₃ are zeroed and T₅ is brought to a fixed intermediate potential corresponding to the transfer of an optional charge from G₃ to G₄.

During the first sequence t₂ there are no charges to be transferred from G₃ to G₄ and it is the arrival of Q_R beneath G₁ which brings about the passage of a current through Q₂, which maintains the potential at point P at V.sub.φ -V_T by removing the charge from point P to point A, so that the potential at A passes from V_AO to V_A =V_AO -Q_R /C_A =V_AO -V_R.

Thus, by means of the TMOS Q₅ in the form of a follow-up system, there is a voltage V_S proportional to V_R =Q_R /C_A. Voltage V_R makes it possible for the coder according to the invention to supply the value of a_o :

ao =0 if |VX |<VR 

ao =1 if |VX |>VR 

This is followed by sequence t₃ and depending on the value of a_o the charge transfers differ. If a_o =0 charge Q_R is transferred from G₂ beneath G_o and then below G_e and is finally removed by diode D_e. If a_o =1 charge Q_R is transferred from G₂ to beneath G₃ and the charge a_o Q_R is then stored beneath G₃.

The final sequence is t₄ with the passage of φ₁ to high level and φ₂ to low level making conductive the TMOS Q₁ and bringing about the zeroing of G₁. Thus, Q_R is transferred from G₁ to G_o, G₄ is zeroed and a optional charge is transferred from G₄ to G₃.

Hereinafter the different sequences will be described starting from sequence t₁, sequence t_o not taking place because charge Q_R is beneath G_o. Thus, we have in succession:

sequence t₁ : pre-charge at points A and P;

sequence t₂ :

transfer of Q_R /2 beneath G₁ and of Q_R /2 beneath G₂ ;

transfer of a_o Q_R from G₃ to G₄ ;

reading of a_o Q_R +Q_R /2;

comparison by the coder of

VR1 =(ao ·QR +QR /2)/CA with |VX |

and determination of a₁.

sequence t₃ :

if a₁ =0 removal of Q_R /2 from G₂ to G_o, G_e and D_e ;

if a₁ =1 transfer of Q_R /2 from G₂ beneath G₃ ;

sequence t₄ :

transfer of Q_R /2 from G₁ to G_o ;

transfer of a_o Q_R from G₄ to G₃, giving the charge a_o Q_R +a₁ Q_R /2 beneath G₃.

The different sequences are thus described starting from t₁. For the determination of each coefficient a_i it is therefore necessary to have a pre-charge sequence t₁ and three successive transfer sequences t₂, t₃, t₄, whereof some comprise a plurality of simultaneous transfers.

When all the coefficients have been determined charges are removed from G₁ to G_o, G_e and D_e and the charges stored beneath G₄ are transferred to D_c.

FIG. 2 relates to a diagrammatic representation of the analog-digital coder according to the invention. The charge transfer coded voltage generator and the CRD are connected in series and are symbolized by rectangles.

In accordance with the selected embodiment of the CTD with a channel N, the charges collected at point A of the CRD are electrons and therefore correspond to the processing of negative voltages -V_R and -V_Ri.

The CRD output is connected to two sample and hold means which are symbolically represented by a switch S₃ and S₄, followed by a capacitor connected to earth C₃ and C₄.

It is obvious that in actual fact the switches of the sample and hold means are constituted in per se known manner by one or more switching MOS transistors.

Each sample and hold means is connected to one of the two inputs of a two-channel, capacitive voltage divider. One of the channels of this divider, in FIG. 1 the upper channel following switch S₃ and capacitor C₃, comprises a capacitor of value 2^k C, whilst the other channel comprises a capacitor of value C.

The output of the divider is called R, i.e. the common point for the two capacitors C and 2^k C.

It is known to construct such a capacitive divider in integrated form and with a high precision. Thus, the article which appeared in the IEEE Journal of Solid-State Circuits, volume SC 10, no. 6, December 1975, pp. 371 to 379 describes systems of integrated capacitors with weighted values for use in charge redistribution capacitive coders.

In the coder according to the invention the determination of the coefficients a_o to a_n making it possible to code each sample V_x is carried out in two stages.

During the first stage the switch S₃ is closed, whilst switch S₄ is open. The reference charge quantity 2Q_R is introduced into the generator and the CRD successively transmits to the upper channel of the voltage divider, i.e. to capacitor 2^k C, the voltages -V_R and -V_Ri with i=1 to k-1.

On coding voltage V_x by using 12 coefficients a_o to a_n ; k is generally taken as equal to 6.

The voltages -V_R and -V_R1 to V_R5 are thus successively transmitted to the capacitor 2⁶.C.

The voltage V at the output, i.e. at point R of the capacitive voltage divider is therefore successively equal to: ##EQU1## with i=1 to 5, in which C_p represents the total stray capacitance at point R, which is represented by dotted lines in FIG. 2.

During the second stage switch S₃ is open, whilst switch S₄ is closed.

It is pointed out that the capacitor C₃ of the sample and hold means of the upper channel remains charged to the final voltage -V_R5 transmitted b the CRD during the first stage.

A new reference charge quantity 2Q_R is introduced into the generator and the CRD then successively transmits to the lower channel of the voltage divider, i.e. to capacitor C, the voltages -V_R and -V_Ri with i=1 to n-k.

With the numerical example referred to hereinbefore where n equals 11 (because there are 12 coefficients a_o to a_n) and in which k is equal to 6, it is the voltages -V_R and -V_Ri with i=1 to 5, which are successively transmitted to capacitor C.

The voltage V at the output of the divider is therefore successively equal to: ##EQU2## with i=1 to 5.

It is apparent that the second term of these two equations is constituted by voltages processed by the generator during the second stage, which are multiplied by 1/X. The voltages processed by the generator during the second stage are therefore divided by 2⁶ compared with those processed by the generator during the first stage, which appeared at point R multiplied by the coefficient 2⁶ /K.

The voltage V at the output of the divider can therefore be written: ##EQU3##

It is therefore apparent that during the first stage the voltages:

V=-VR ·26 /K and V=-VRi ·26 /K

are obtained at the output of the divider at point R, with i=1 to 5 and during the second stage the voltages:

V=-VRi ·26 /K

are obtained at the output of the divider at point P with i=6 to 11.

Thus, to within the coefficient 2⁶ /K the voltages V_R and V_Ri are obtained with i=1 to 11 and these voltages make it possible to determine the coefficients a_o to a_n.

The output of the capacitive voltage divider, i.e. point R is connected at a point Q to the output of a polarity reversal device, constituted by an amplifier able to assume two different gains +1 and -1. This device receives the voltage V_x to be coded. Initially the coder according to the invention determines (it would be apparent in which way this is carried out hereinafter) the sign bit a_s of each sample V_x to be coded:

as =1 if Vx >0

a=0 if Vx <0

In the case where a_s =0, i.e. V_x is negative, the reversal device receives the order to reverse the polarity of V_x and assumes the gain -1. At the output of the reversal device when the sign bit a_s has been determined we obtain |V_x |.

Thus, at point Q voltage

|Vx |-VR ·26 /K

|Vx |-VRi ·26 /K

with i=1 to 11, is obtained.

Thus, it is easy to apply to V_x the same coefficients 2⁶ /K as at V_R and V_Ri, which can be carried out by a variable gain amplifier.

A follow-up stage, constituted in per se known manner by a differential amplifier A is connected to point Q. This stage limits the attenuation due to the stray capacitance C_p at point Q and protects the latter which is a high impedance floating point against switching interference introduced by switches S₅ and S₆.

To the output of the follow-up stage are connected two sample and hold means represented in the same symbolic manner as the two previously used by a switch S₅, S₆, followed by a capacitor C₅, C₆ connected to earth. Each sample and hold means is connected to one of the two inputs e₁ and e₂ of a comparator. In FIG. 2 the sample and hold means constituted by switch S₅ and capacitor C₅ is connected to comparator input e₁.

This comparator supplies a 1 if the voltage applied to its input e₁ is greater than that applied to its input e₂ and a 0 in the opposite case.

This comparator is realized in per se known manner by a system of MOS transistors and capacitors.

Finally the comparator output is connected to a memory constituted by a type D flip-flop or a shift register in which are stored the values of the sign bit a_s, then coefficients a_o to a_n.

Hereinafter the manner in which the sign bit a_s is determined will be considered.

The polarity reversing device of V_x comprises two switches S₁ and S₂. Switch S₁ is in series with the input of the reversing device and switch S₂ is connected between the input thereof and earth.

For the determination of a_s switch S₁ is open and switch S₂ closed. The input of the reversing device is consequently short-circuited. The reversing device is at gain +1 and transmits its offset voltage to point Q.

Moreover, switches S₃, S₄ and S₆ are closed, whilst switch S₅ is open. The timing signals are applied to the voltage generator and to the CRD, but the injection of charges is inhibited. Thus, capacitor C₆ is charged with the offset voltages of the different circuits of the coder, with the exception of the comparator and the memory. These circuits are generally constructed according to MOS technology and it is particularly important to take account of the offset voltages.

Moreover, at the output of the CRD there is a mean voltage V_PO corresponding to the average polarization level of the CRD prior to the introduction of the reference charge quantity 2Q_R. This voltage is applied to one of the terminals of capacitors 2^k C and C.

Once capacitor C₆ has been charged switches S₂ and S₆ are opened and switches S₁ and S₅ closed. The mean voltage V_PO continues to be applied to capacitors 2^k C and C. Voltage V_x -V_PO added to the offset voltages charges capacitor C₅ and is applied to the input e₁ of the comparator which deducts therefrom the sign bit a_s.

In the case where a_s =0 and where it is therefore necessary to pass the gain of the reversing device from 30 1 to -1, capacitor C₆ is discharged and is recharged with the offset voltages of the different circuits of the coder, with on this occasion the gain of the reversing device at -1.

When capacitor C₆ is charged with these offset voltages it is optionally possible to check that the sign bit a_s is equal to 0.

Once the sign bit a_s has been determined a reference charge quantity 2Q_R is introduced into the coded voltage generator.

With the precautions taken it is only the voltage variations at the output of the CRD compared with the initial polarization V_PO which are transferred at Q. Thus, the value V_PO is not involved in the determination of the coefficients, provided that it remains constant.

The way in which coefficients a_s . . . a_n are determined will now be examined.

It has been seen hereinbefore that at point Q the voltages |V_x |-V_R ·2⁶ /K and |V_x |-V_Ri ·2⁶ /K with i=1 to 11, are obtained.

Switches S₂ and S₆ are open, whilst switches S₁ and S₅ are closed. Furthermore, switch S₃ or switch S₄ is closed, depending on whether it is the first or second stage of the processing of voltages V_R and V_Ri.

The voltages |V_x |-V_R ·2⁶ /K and |V_x |-V_Ri ·2⁶ /K therefore charge capacitor C₅, whilst capacitor C₆ is still charged with the offset voltages of the coder.

As capacitor C₅ is charged with the voltage |V_x |-V_Ri ·2⁶ /K switch S₅ is opened. The comparator then determines coefficient a_i, which is stored in the memory. The information on the value of the coefficients is transmitted to the generator. For a_i =0 the quantity of charges stored beneath G₂ is discharged to D_e, whilst it is stored beneath G₃ to contribute to the processing of V_Ri+1 if a_i =1. Switch S₅ is then again closed, so that C₅ is charged with |V_x |-V_Ri+1 ·2⁶ /K and a_i +1 is determined.

All the circuits of the coder, i.e. the reversing device, the sample and hold means, the capacitive voltage divider, the follow-up stage, the comparator and the memory are preferably constructed in accordance with N MOS or C MOS technology compatible with the construction of the charge transfer coded voltage generator. Thus, these circuits are essentially constituted by elements which can best be constructed by MOS technology, i.e. MOS capacitors and transistors. In particular all the switches S₁ to S₆ are constituted by one or more MOS transistors.

The coder according to the invention carries out the digital coding of an analog signal by using a linear law. By associating therewith a digital transcoding making it possible to pass from a linear law to a logarithmic law it is possible, for example, to construct a logarithmic coder in accordance with law A or law μ, which are standardized for the coding of voice signals.

It is obviously possible to use the generator of FIG. 1 for obtaining an analog-digital decoder. In this case the coefficients a_o . . . a_n are known, there is no V_x and the decoder makes it possible to process the analog voltage:

Vx =ao ·VR +a1 ·VR /24 . . . +an ·VR /2n 

In this case the TMOS Q₁ is only connected to grid G₁ and the remainder of the CRD is only connected to grid G₄.

A single reading of the charges stared beneath G₄ is then necessary for the processing of the series of coefficients a_o . . . a_n. This reading takes place by the transfer of charges from G₃ to G₄, whilst the quantity of charges

ao ·QR +a1 ·QR /2+ . . . +an ·QR /2n 

has been stored beneath G₃.

It is obvious to carry out a two-stage decoding in the manner described hereinbefore for the coding operation.

During the first stage the values equal to 0 or 1 of the coefficients a_o to a_k- are introduced into the generator. The quantity of charges stored beneath G₄ :

ao . . . QR +a1 ·QR /2+ . . . +ak-1 ·QR /2k-1 

is read. This reading voltage V₁ charges capacitor C₃ of the sample and hold means connected to capacitor 2^k C. The charges are then removed from the generator and a new reference charge quantity is introduced into the generator.

During the second stage the values of coefficients a_k to a_n are introduced into the generator. This is followed by the reading of the quantity of charges stored beneath G₄ : a_k ·Q_R +a_k+1 ·Q_R /2+ . . . . This reading voltage V₂ then charges the capacitor C₄ of the sample and hold means connected to capacitor C.

At the output of the voltage divider at point P the voltage V equals, to within the sign (depending on whether the CRD gives a negative reading or not):

V=V1 ·2k /K+V2 ·1/K

Thus, the voltage V₂ is divided by 2^k compared with V₁ and the voltage V obtained corresponds to the reading, to within the coefficient 2^k /K:

ao ·QR +a1 ·QR +a1 ·QR /2+ . . . +an ·QR /2n 

A polarity reversing device positioned at the output and controlled by the sign bit a_s of the coded word makes it possible to restore the magnitude and sign of the analog voltage V_x.

Reference has been made hereinbefore to the performance of the two-stage coding or decoding, but it is obviously possible to use three, four or in general p stages.