System and method for adapting to a change in constellation density while receiving a signal Number:7,522,679 from the United States Patent and Trademark Office (PTO) owispatent One embodiment comprises a transmitter precoder operating on an input signal to produce an output signal for modulation. The output signal is transmitted using a constellation with a constellation density. The transmitter precoder comprises scaler logic, precoder logic and selection logic. The scaler logic scales the input signal to maintain constant power on the input signal independent of the constellation density. The precoder logic precodes the scaled signal to produce a dither signal limited to a range of values. The range is inversely proportional to the constellation density. The selection logic is operable in two states. In the first state, the selection logic combines the scaled signal with the dither signal to produce the output signal. In the second state, the selection provides the scaled signal as the output signal.<H constellation, receiving, System, and, meth, 7522679.html, Patent, pto, 7522679

Title: System and method for adapting to a change in constellation density while receiving a signal

Abstract: One embodiment comprises a transmitter precoder operating on an input signal to produce an output signal for modulation. The output signal is transmitted using a constellation with a constellation density. The transmitter precoder comprises scaler logic, precoder logic and selection logic. The scaler logic scales the input signal to maintain constant power on the input signal independent of the constellation density. The precoder logic precodes the scaled signal to produce a dither signal limited to a range of values. The range is inversely proportional to the constellation density. The selection logic is operable in two states. In the first state, the selection logic combines the scaled signal with the dither signal to produce the output signal. In the second state, the selection provides the scaled signal as the output signal.

Patent Number: 7,522,679 Issued on 04/21/2009 to Betts

Inventors: Betts; William L. (St. Petersburg, FL)
Assignee: Paradyne Corporation (Largo, FL)

Appl. No.: 10/956,780

Filed: October 1, 2004

Related U.S. Patent Documents


Application Number	Filing Date	Patent Number	Issue Date
60507885	Oct., 2003

Current U.S. Class: 375/341 ; 714/792

Current International Class: H03D 1/00 (20060101); H03M 13/03 (20060101)

Field of Search: 375/341,265,316,295 714/746,792

References Cited [Referenced By]

U.S. Patent Documents


4945549	July 1990	Simon et al.
5394440	February 1995	Goldstein et al.
5400322	March 1995	Hunt et al.
5928377	July 1999	Doran et al.
5966412	October 1999	Ramaswamy
5987637	November 1999	Thomas
6134273	October 2000	Wu et al.
6246694	June 2001	Chen
6310909	October 2001	Jones
2001/0001616	May 2001	Rakib et al.
2002/0114398	August 2002	Lin et al.

Foreign Patent Documents


2366495	Jun., 2002	GB
WO0221699	Mar., 2002	WO

Other References

"Memory efficient pipelined Viterbi decoder with look-ahead trace back" Jung-Gi Baek; Sang-HUn Yoon; Jong-Wha Chong; Electronics, Circuits and Systems, 2001. ICECS 2001. The 8th IEEE International Conference on vol. 2, Sep. 2-5, 2001 pp. 769-772 vol. 2. cited by examiner .
"Multicarrier Modulation for Data Transmission: An Idea Whose Time Has Come" by John A.C. Bingham, Published May 1990 in IEEE Communications Magazine. cited by other.

Primary Examiner: Bocure; Tesfaldet
Assistant Examiner: Hassan; Sarah
Attorney, Agent or Firm: Thomas, Kayden, Horstemeyer & Risley LLP

Parent Case Text

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of U.S. Provisional No. 60/507,885, filed Oct. 1, 2003.

Claims

The invention claimed is:

1. A method for receiving a first series of trellis-encoded symbols transmitted with a first constellation density and a second series of trellis-encoded symbols transmitted with a second constellation density less than the first, wherein a plurality of the second series of symbols comprises a terminating pad of a predetermined length, comprising: receiving and decoding the first series of trellis-encoded symbols in a trellis decoder; determining if one of the decoded symbols corresponds to an end-of-data marker identifying when the terminating pad will be transmitted; receiving the second series of trellis-encoded symbols comprising the terminating pad; and responsive to receiving a last symbol in the terminating pad, decoding all symbols remaining in a Viterbi traceback buffer by selecting a start state based on the last symbol, and repeatedly tracing back through a reduced number of symbols in the traceback buffer using the same selected start state on each trace cycle while reducing the number of symbols by one each trace cycle.

2. The method of claim 1, further comprising: decoding the symbols in the first series by filling the Viterbi traceback buffer; selecting a start state based on the last received symbol; and tracing back through the traceback buffer to flush from the buffer a decoded symbol.

3. The method of claim 1, wherein the pad has a fixed number of symbols.

4. The method of claim 1, further comprising: responsive to receiving an initial symbol of the terminating pad, changing a scale factor used to scale power on received symbols before the symbols are trellis-decoded, then restoring the scale factor before precoder reconstruction.

5. A method for adapting to a change in constellation density of received trellis-encoded symbols, comprising: receiving a trellis-encoded symbol transmitted with a first constellation density in a trellis decoder; determining if the trellis-encoded symbol corresponds to an end-of-data marker, the end-of-data marker identifying when the terminating pad will be transmitted at a second constellation density less than the first; receiving an initial symbol of the terminating pad; receiving a final pad symbol transmitted with the second constellation density; and responsive to receiving the final pad symbol, decoding all buffered symbols based on previously buffered trellis states without computing new error metrics, wherein the computed error metrics associated with the pad symbols produce reliable decoding.

6. The method of claim 5, further comprising: decoding a first previously buffered symbol based on a first computed trellis metric associated with the initial pad symbol and on at least one previously buffered trellis state, wherein the first metric is reliable as a result of the initial pad symbol being transmitted at the second constellation density.

7. The method of claim 6, wherein the step of decoding the first previously buffered symbol is responsive to receiving the initial pad symbol.

8. The method of claim 6, further comprising: decoding a second previously buffered symbol based on a second computed trellis metric associated with the final pad symbol and on at least one previously buffered trellis state, wherein the second metric is reliable as a result of the final pad symbol being transmitted at the second constellation density.

9. A method for receiving a first series of trellis symbols transmitted with a first constellation density and a second series of trellis symbols transmitted with a second constellation density less than the first, wherein a plurality of the second series of symbols comprises a terminating pad, comprising: receiving and decoding the first series of trellis symbols in a trellis decoder to determine receipt of an end-of-data marker, the end-of-data marker identifying when the terminating pad will be transmitted at a second constellation density less than the first; receiving a series of vectors, each vector corresponding to one symbol in the second series of trellis symbols; and responsive to determining that one of the vectors in the vector series corresponds to a final pad symbol: computing a trellis metric based on the vector, wherein the trellis metric is reliable as a result of the vector being at the second constellation density; loading a trellis state corresponding to the vector into a Viterbi traceback buffer; selecting a start state of the traceback buffer based on the trellis metric; and repeatedly tracing back through the traceback buffer, starting each trace cycle with the same selected start state while reducing the number of symbols by one each trace cycle, to flush all trellis states from the traceback buffer.

10. The method of claim 9 further comprising: responsive to determining that one of the vectors in the vector series corresponds to a first pad symbol, computing a trellis metric based on the vector, wherein the trellis metric is reliable as a result of the vector being at the second constellation density; loading a trellis state corresponding to the vector into a Viterbi traceback buffer; selecting a start state of the traceback buffer based on the trellis metric; and tracing back through the traceback buffer, starting with the selected start state, to flush one trellis state from the buffer.

11. A system, comprising: means for receiving a first series of trellis-encoded symbols transmitted with a first constellation density; means for decoding the first series of trellis-encoded symbols; means for determining if one of the decoded symbols corresponds to an end-of-data marker identifying when a terminating pad of a predetermined length will be transmitted, where a plurality of a second series of trellis-encoded symbols comprises the terminating pad, the second series of trellis-encoded symbols transmitted with a second constellation density less than the first; means for receiving the second series of trellis-encoded symbols comprising the terminating pad; and means for decoding all symbols remaining in a Viterbi traceback buffer, responsive to receiving a last symbol in the terminating pad, by selecting a start state based on the last symbol, and repeatedly tracing back through a reduced number of symbols in the traceback buffer using the same selected start state on each trace cycle while reducing the number of symbols by one each trace cycle.

12. The system of claim 11, further comprising: means for decoding the symbols in the first series by filling the Viterbi traceback buffer; means for selecting a start state based on the last received symbol; and means for tracing back through the traceback buffer to flush from the buffer a decoded symbol.

13. The system of claim 11, wherein the pad has a fixed number of symbols.

14. The system of claim 11, further comprising: means for changing a scale factor used to scale power on received symbols before the symbols are trellis-decoded responsive to receiving an initial symbol of the terminating pad; and means for restoring the scale factor before precoder reconstruction.

15. A system for adapting to a change in constellation density of received trellis-encoded symbols, comprising: means for receiving a trellis-encoded symbol transmitted with a first constellation density; means for determining if the trellis-encoded symbol corresponds to an end-of-data marker, the end-of-data marker identifying when the terminating pad will be transmitted at a second constellation density less than the first; means for receiving an initial symbol of the terminating pad; means for receiving a final pad symbol transmitted with the second constellation density; and means for responsive to receiving the final pad symbol, decoding all buffered symbols based on previously buffered trellis states without computing new error metrics, wherein the computed error metrics associated with the pad symbols produce reliable decoding.

16. The system of claim 15, further comprising: means for decoding a first previously buffered symbol based on a first computed trellis metric associated with the initial pad symbol and on at least one previously buffered trellis state, wherein the first metric is reliable as a result of the initial pad symbol being transmitted at the second constellation density.

17. The system of claim 16, wherein decoding the first previously buffered symbol is responsive to receiving the initial pad symbol.

18. The system of claim 16, further comprising: means for decoding a second previously buffered symbol based on a second computed trellis metric associated with the final pad symbol and on at least one previously buffered trellis state, wherein the second metric is reliable as a result of the final pad symbol being transmitted at the second constellation density.

19. A system, comprising: means for receiving a first series of trellis symbols transmitted with a first constellation density; means for decoding the first series of trellis symbols to determine receipt of an end-of-data marker, the end-of-data marker identifying when the terminating pad will be transmitted at a second constellation density less than the first; means for receiving a series of vectors, each vector corresponding to one symbol in a second series of trellis symbols transmitted with a second constellation density less than the first, wherein a plurality of the second series of symbols comprises a terminating pad; and means for determining that one of the vectors in the vector series corresponds to a final pad symbol; means for computing a trellis metric based on the vector responsive to determining that the vector corresponds to the final pad symbol, wherein the trellis metric is reliable as a result of the vector being at the second constellation density; means for loading a trellis state corresponding to the vector into a Viterbi traceback buffer responsive to determining that the vector corresponds to the final pad symbol; means for selecting a start state of the traceback buffer based on the trellis metric, responsive to determining that the vector corresponds to the final pad symbol, and means for repeatedly tracing back through the traceback buffer, responsive to determining that the vector corresponds to the final pad symbol, starting each trace cycle with the same selected start state while reducing the number of symbols by one each trace cycle, to flush all trellis states from the traceback buffer.

20. The system of claim 19, further comprising: means for determining that one of the vectors in the vector series corresponds to a first pad symbol; means for computing a trellis metric based on the vector responsive to determining that the vector corresponds to the first pad symbol, wherein the trellis metric is reliable as a result of the vector being at the second constellation density; means for loading a trellis state corresponding to the vector into a Viterbi traceback buffer responsive to determining that the vector corresponds to the first pad symbol; means for selecting a start state of the traceback buffer based on the trellis metric responsive to determining that the vector corresponds to the first pad symbol; and means for tracing back through the traceback buffer, responsive to determining that the vector corresponds to the first pad symbol, starting with the selected start state, to flush one trellis state from the buffer.

Description

FIELD OF THE INVENTION

The present invention relates generally to data communications, and more particularly, to a system and method for adapting to a change in constellation density while receiving a signal.

BACKGROUND

A well-known technique for providing multi-point modulation over a single communications channel is Time Domain Duplexing (TDD). In TDD, a control modem transmits data downstream to one or more tributary modems, then the control modem receives transmissions upstream from any of the tributary modems on a shared channel. If the downstream and upstream channels are isolated (spatially, by frequency division, or by other means) then the downstream transmission can be continuous even though the upstream channel is shared. Each modem must terminate its transmission to allow other modems sharing the same channel to transmit. Examples of TDD systems include Multiple Virtual Line, Digital Subscriber Line (DSL) and ReachDSL.RTM..

Block framed burst communication systems, such as DSL Discrete Multitone Modulation (DMT), also terminate transmission. These systems terminate transmission at the end of each DMT symbol period.

TDD systems typically used uncoded modulation rather than Trellis Coded Modulation (TCM) even though TCM provides 3-6 dB of performance gain, because Trellis decoding significantly increases line turn-around time (the time it takes a particular tributary modem to stop receiving and start transmitting). Trellis decoding typically requires a lengthy delay through a Viterbi decoder. During this time, the channel is out of service because the receiver must complete Viterbi decoding of a frame before transmission.

This same Viterbi decoder delay makes Trellis coding incompatible with use of a Decision Feedback Equalizer (DFE) in the receiver, since decoder delay prevents timely generation of reference vectors needed by the DFE. Therefore, in systems using TCM, intersymbol interference is typically handled with a precoder in the transmitter rather than a DFE in the receiver. However, conventional precoder designs are incompatible with systems that switch constellation densities or that switch between coded/uncoded transmissions (e.g. ReachDSL V2.RTM.). With a conventional precoder, power is scaled at the precoder output, and the power of the signal within the precoder's Finite Impulse Response (FIR) varies directly in proportion to constellation density. A change in constellation density, as would occur for a change in data transmission rate, therefore causes a power discontinuity within the precoder, which results in errors.

DESCRIPTION OF THE DRAWINGS

The accompanying drawings illustrate several aspects of the present invention, and together with the description serve to explain the principles of the invention.

FIG. 1 is a block diagram of a communications system using an exemplary embodiment.

FIG. 2 is a block diagram of a frame structure used by the communications system of FIG. 1.

FIG. 3 is a block diagram of the transmitter used by the communications system of FIG. 1.

FIG. 4 is a high level block diagram of the receiver used by the communications system of FIG. 1.

FIG. 5 is a block diagram of the receiver used by the communications system of FIG. 1 with DFE details.

FIG. 6 is a block diagram of the receiver used by the communications system of FIG. 1 with precoder details.

FIG. 7 is a block diagram of the receiver used by the communications system of FIG. 1 with additional details.

FIG. 8 is a trace buffer diagram of the receiver used by the communications system of FIG. 1.

FIG. 9 is a timing diagram of the receiver used by the communications system of FIG. 1.

FIG. 10 illustrates the sequence produced by one embodiment of a transmitter with rate-adaptive symbol interleaving.

FIG. 11 is a block diagram of a transmitter with multi-dimensional Trellis coding and rate-adaptive symbol interleaving.

FIG. 12 is a block diagram of a receiver with multi-dimensional Trellis coding and rate-adaptive symbol interleaving.

DETAILED DESCRIPTION

One embodiment uses truncated Trellis decoding that reduces delay through the Viterbi decoder. A standard Trellis coding/decoding system transmits data with a constellation density of L, where L is a function of the channel signal-to-noise ratio (SNR) and data transmission rate. With Truncated Trellis coding/decoding, the transmitter appends a terminating pad to the end of each frame, and the pad is transmitted with a reduced constellation density. The pad could contain user data, cyclic redundancy check (CRC) symbols, etc., or it could be discarded.

In this embodiment, this reduced constellation density may be used such that in the receiver the Trellis path metrics of the best path are improved by more than the gain of the Viterbi decoder. Viterbi decoding of symbols received before the pad can then be truncated since the increased margin makes the pad symbols and their associated path metrics very reliable.

Another embodiment employs a rate-adaptive precoder that seamlessly adapts to changing constellation densities and coding modes. In this embodiment, input to the precoder may be scaled to maintain a constant power level within the precoder, independent of constellation density. A power discontinuity within the precoder FIR is thus avoided when the constellation density changes. Although the precoder is not active during uncoded transmissions (e.g., frame headers), the precoder's FIR is filled with reference vectors for the header. Accordingly, intersymbol interference (ISI) from the uncoded header is accommodated by the precoder.

Yet another embodiment employs a rate-adaptive symbol interleaver for multi-dimensional Trellis codes. When symbol interleaving is used, the early symbol component and the late symbol component of a particular multi-dimensional Trellis symbol are not transmitted consecutively, but are interleaved with symbol components from other multi-dimensional Trellis symbols. In yet another embodiment, constellation-encoded portions of a particular Trellis symbol component are transmitted sequentially and are not aligned with their corresponding multi-dimensional Trellis symbol. At the receiver, the equalizer output is scaled to a uniform grid for slicing and decoding. The uniform grid ensures consistent error metric calculations, because error vectors are compressed on constellations with lower densities. The constellation density and corresponding scale factor can change on any fractional frame boundary if the embodiment uses fractional encoding. These embodiments may be used alone or in various combinations for optimal performance. A non-limiting list of combinations includes: truncated Trellis decoding; truncated Trellis decoding and rate-adaptive preceding; truncated Trellis decoding and rate-adaptive Trellis symbol interleaving; rate-adaptive preceding and rate-adaptive Trellis symbol interleaving.

FIG. 1 is a block diagram of a communications system incorporating an exemplary embodiment. Communications devices 101 and 102 communicate over channel 103. Channel 103 can use a wired medium (e.g., twisted pair, coax, hybrid-fiber coax, or other suitable wire-based medium), or can be wireless (e.g., wireless Local Area Network, satellite, mobile phone, near field communication device, or other suitable wireless medium). In this simplified diagram, device 101 is shown as the transmitter, and device 102 is shown as the receiver, as may be the case in broadcast applications. However, it will be understood that each device contains both transmit and receive functionality, as may be the case in duplex applications. In one embodiment, channel 103 is a subscriber loop, and devices 102 and 101 are Digital Subscriber Loop (DSL) modems.

FIG. 2 is a block diagram of a Physical Media Dependent (PMD) layer frame structure used by the communications system of FIG. 1. Frame 201 consists of header 202, data 203, and terminating pad 204. Terminating pad 204 is transmitted with a lower constellation density than data 203, so that receiver 102 must adapt to this change in constellation density at the end of data 203, before terminating pad 204 is received.

The method used by receiver 102 to determine receipt of terminating pad 204 depends on details of the PMD-layer frame structure. With this exemplary embodiment, terminating pad 204 is a fixed size, so determining the last symbol of terminating pad 204 is trivial once the initial symbol of terminating pad 204 is known. Determining the receipt of the initial symbol of terminating pad 204 depends on determining the end of data 203.

In one embodiment, header 202 and data 203 are also fixed size, so end of data 203 is at a fixed symbol position relative to the start of header 202. In another embodiment, header 202 contains a frame length field, which can be used to calculate end of data 203 relative to start of header 202. In yet another embodiment, data 203 contains fixed-size cells, and each cell contains a flag field which indicates whether or not that cell is the last cell in a frame. This flag can be used to calculate end of data 203 relative to the last cell. These embodiments can be generalized as "determining an end-of-data identifier." Other details about the size and contents of header 202 and data 203 vary according to implementation.

In one embodiment, receiver 102 adapts to the receipt of terminating pad 204 in other ways as well. In this embodiment, header 202 is transmitted without channel encoding (e.g., Trellis coding) and without preceding, while data 203 and terminating pad 204 are transmitted using channel encoding and preceding. Therefore, receipt of the last symbol of terminating pad 204 triggers changes in the operation of the precoder and channel encoder of receiver 102, as will be discussed in further detail hereinafter. Other combinations of coding and preceding are possible, for example, channel encoding header 202 without preceding, etc.

FIG. 3 is a block diagram of one embodiment of the transmitter 101 of FIG. 1, employing rate-adaptive preceding and Truncated Trellis decoding. Transmit data (TXD) is collected in a buffer 301 to facilitate extraction of data at a variable data rate. A conventional constellation encoder 302 encodes a variable number of data bits per symbol b using a constellation with density D. The constellation encoder may include a fractional encoder, such as a modulus converter, to encode a fractional number of bits to a constellation size D that is not a power of 2. If truncated Trellis Decoding is used in receiver 102, the controller 303 varies constellation density D so that a smaller density is used when transmitting terminating pad 204.

The encoded symbols are then mapped to signal points in a constellation by a conventional mapper 304. In this example embodiment, the mapping is rotationally invariant to avoid phase ambiguity in the channel. However, other mappings such as Gray coding can be used in other embodiments.

Switch 305 determines whether or not Trellis encoder 306 injects redundant bits into the input data path for the mapper 304. Controller 303 opens switch 305 when header 202 is being transmitted, and closes it whenever data 203 or terminating pad 204 is being transmitted. Although switch 305 and other switches are shown in these diagrams, the switches may be a logical construct in other embodiments, and a particular implementation may or may not use physical switches.

Scaler 307 at the output of mapper 304 scales the power on the constellation signal points by factor S.sub.b, which is provided by controller 303. This scale factor S.sub.b is inversely proportional to the constellation density D, and is typically half the distance between points in the constellation. Table 1 lists exemplary scale factors used by one embodiment for various combinations of constellation density D and bits-per-symbol b.

TABLE-US-00001 TABLE 1 F/s.sub.b Constellation Bits/Symbol s.sub.b (F = 2.sup.23) Power Density 2 (Header) 32760 256 2.000 4 1.5 28368 296 2.000 4 2 21448 391 4.000 8 2.5 17944 467 6.000 8 3 16736 501 7.000 12 3.5 14192 591 10.000 16 4 11952 702 14.375 24 4.5 10192 823 20.000 32 5 8544 982 28.750 48 5.5 7176 1169 41.000 64 6 6080 1380 57.391 92 6.5 5104 1644 81.750 128 7 4304 1949 115.082 184 7.5 3624 2315 162.750 256 8 3048 2752 230.634 364 8.5 2560 3277 326.188 512 9 2160 3884 460.794 728 9.5 1816 4619 650.817 1024 10 1528 5490 921.722 1452 10.5 1280 6554 1304.864 2048 11 1080 7767 1845.191 2900 11.5 904 9279 2610.819 4096

In this exemplary embodiment, all values of S.sub.b are integer multiples of 8 and the largest value of S.sub.b is a 15-bit integer. These integers improve resolution and enable the use of conventional 16-bit multipliers. Other values of S.sub.b can be used in alternative embodiments, as long as the local transmitter 101 and remote receiver 102 use the same S.sub.b values for a given combination of constellation density D and bits-per-symbol b. In Table 1, S.sub.b is the controlled value, and the other values vary appropriately. S.sub.b is controlled because it will be used by the precoder modulator, and so must be identical in the local transmitter and in the remote receiver. S.sub.b controls the relative transmit power with respect to constellation density and has been scaled to accommodate the dither signal power of the circular precoder. F/sb is a compression factor that can have reduced precision with little impact on the accuracy of final decoding.

Switch 308, operated by controller 303, determines whether precoding is applied to the signal. If switch 308 is closed, the output of precoder 309 is subtracted from the scaled mapper output to produce precoded signal x (on connection 310). If switch 308 is open, preceding is not applied and subtractor 311 has no effect. The output of subtractor 311 is passed to modulator 312, where the signal is modulated by conventional means. After modulation, other conventional means (not shown) convert the modulated signal to an analog waveform, amplify the waveform and couple it to channel 103.

In addition to being supplied to modulator 312, the output of subtractor 311 is also fed back into precoder 309. Precoded signal x (on connection 310) is supplied to precoder 309 as input 313, and, after a unit delay operation 314, to FIR filter 315. The output of FIR 315 (x.sub.p) is then supplied on connection 316 to modulo operator 317. The coefficients of FIR 315 are typically derived from the remote receiver 102. Unit delay operation 314 signifies that the sampled signal being currently computed is not to change the state of FIR 315 until the next sample interval.

The precoder 309 operates as follows. The output of FIR 315 (x.sub.p) on connection 316 is reduced by the modulo operator 317 to produce dither signal d (on connection 318). It is dither signal d which is selected by switch 308 for subtraction from the scaled mapper output to produce the precoded signal x (on connection 310). Input 313 is added to the FIR output x.sub.p on connection 316 then scaled by factor F/S.sub.b. The product of S.sub.b*(F/S.sub.b) is F, which in one embodiment is 2.sup.23. While specific values of F are implementation-dependent, it is advantageous for F to be a uniform power of two.

After scaling by factor F/S.sub.b, the signal is sliced. The action of slicer 319 is to determine the precoder Trellis state. Because precoder 309 modifies the transmitted signal, it also modifies the Trellis state of the signal. The precoder Trellis state, along with the modulo count from modulo operator 317, are used by Trellis encoder 306 to compute its next state.

In a conventional precoder, such as the one used in ITU-T Recommendation V.34, the modulo operator operates on intervals that are a power of two. This limitation of the modulo operator simplifies interoperability within standardized products, but also results in a constellation that expands with constellation density. This expansion of the constellation increases the power of the conventionally precoded signal within FIR 315. In the conventional precoder, power is scaled at the output of the precoder, and the power of the precoded signal within the FIR varies directly in proportion to constellation density. Thus, a change in constellation density when using a conventional precoder causes a precoder power discontinuity within the FIR 315, which leads to transmission errors.

In contrast, various embodiments may employ a different modulo technique to compute dither signal d. Accordingly, the transmitter embodiment of FIG. 3 scales the mapper output by S.sub.b before providing input to precoder 309. Because S.sub.b varies according to constellation density, the precoded signal x, including the contents of the precoder FIR 315, maintains a constant power level, independent of constellation density. The modulo value used by modulo operator 317 is also the constellation-dependent scalar S.sub.b. Modulo operator 317 subtracts 2S.sub.b from any positive signal greater than S.sub.b, until it is less than or equal to .+-.S.sub.b. Modulo operator 317 adds 2S.sub.b to any negative signal less than -S.sub.b, until it is greater than or equal to -S.sub.b. The result is a dither signal d with a magnitude less than or equal to .+-.S.sub.b. The modulo count is a base 2 count of the number of additions or subtractions performed.

Note that uncoded header 202 fills FIR 315 even though the precoder is not active (switch 308 is open). Thus, the transmitter embodiment of FIG. 3 continuously fills the memory of transmitter precoder FIR 315, thus maintaining postcursors from uncoded header 202 in FIR 315. Similarly, the receiver embodiment of FIG. 4 maintains postcursors from the header in the filter of the receiver precoder. By maintaining postcursors from uncoded header 202, precursor symbols will be accurately replicated by FIR 315 when switch 308 closes and the first precoded symbol (in data 203) is transmitted. In this way, a discontinuity is avoided when the precoder is enabled, and switching of coding mode and constellation density is seamless.

FIGS. 4-7 are block diagrams illustrating various embodiments of the receiver 102 of FIG. 1. FIG. 4 is a high-level block diagram, and FIGS. 5-7 illustrate various embodiments in further detail. The embodiment of FIG. 5 illustrates rate-adaptive noise-whitening and rate-adaptive decision-feedback equalization. The embodiment of FIG. 6 illustrates rate-adaptive precoding. The embodiment of FIG. 7 illustrates truncated Trellis decoding. The features of these embodiments may also combined in other ways not illustrated.

In FIG. 4, the received signal is extracted by a conventional hybrid circuit, filtered, converted from analog to digital, and digitally filtered (not shown), then supplied as input to a conventional adaptive equalizer 401. The output of adaptive equalizer 401 is supplied to noise-whitening block 402.

The noise-whitened signal on output 403 is supplied as input to a decision feedback equalizer (DFE) block 404. After equalization by DFE block 404, the equalized signal is supplied as input to uncoded slicer 405, Trellis decoder 406, and Truncated Trellis decoding logic 407. Truncated Trellis decoding logic 407 operates in conjunction with Trellis decoder 406 to perform Truncated Trellis Decoding.

The output of Trellis decoder 406 is supplied to precoder reconstruction unit 408. Precoder reconstruction unit 408 also receives input from the uncoded slicer 405 through a path which bypasses the Trellis decoder 406 (since headers are not Trellis coded), and another input (409) from Truncated Trellis decoding logic 407. Precoder reconstruction unit 408 reconstructs the symbol which was precoded by transmitter 101. After reconstruction, the constellation decoder 410 outputs received data (RXDATA) consisting of b data bits.

A. Rate-Adaptive Noise-Whitening and Decision Feedback Equalization in a Receiver

FIG. 5 is a block diagram showing more details of the noise-whitening block 402 and DFE block 404. Input from adaptive equalizer 401 is fed (after a unit delay) into a noise-whitening FIR 501 with coefficients identical to those in the precoder FIR of the remote transmitter 101. The output of noise-whitening FIR 501 is then added to the output of adaptive equalizer 401. Switch 502 selects either the output of the adaptive equalizer 401 (scaled by factor K1) when operating without precoding, or selects the combined output of the noise-whitening FIR 501 and the adaptive equalizer 401 (scaled by factor K2) when operating with precoding. The scale factors K1 and K2 provide flexibility in the scale of the equalized signal. In this embodiment, a precoded signal has a larger peak signal power, which is accommodated by a smaller scale factor K2.

DFE block 404 receives as input the (scaled) reference vector output of uncoded slicer 405, which is subtracted from the noise-whitened signal on output 403. After a unit delay, this signal is provided to DFE FIR 503. The output of DFE FIR 503 is gated by switch 504 and subtracted from noise-whitened signal on output 403, then scaled by factor F/S.sub.b for input to uncoded slicer 405 and to Trellis decoder 406. The scale factor F/S.sub.b reduces the size of the constellation to a uniform grid suitable for slicing or decoding. (Values for F and F/S.sub.b were discussed above in connection with Table 1.) On a uniform grid, constellation points are all equal distance resulting in power proportional to constellation density (or data rate). Advantageously, at lower densities the slicer signal is compressed by a smaller value of factor F/S.sub.b. In the process, noise or distortion on the received signal is directly compressed also, resulting in increased margin as the density is decreased.

Both the noise-whitening block 402 and DFE block 404 can operate simultaneously. The inputs are always active to fill the FIR memories in these two blocks for seamless mode transitions. In this embodiment, switch 502 and switch 504 operate in concert to activate either the noise-whitening block 402 for coded modulation or the DFE block 404 for uncoded modulation. In this embodiment, DFE block 404 implements a noise-whitening DFE, so ideal reference vectors (x.sub.i) output by the uncoded slicer 405 are subtracted (after scaling by factor S.sub.b to normalize power) from the noise-whitened signal at the input to the DFE block 404. This noise-whitening operation of DFE block 404 is valuable for adapting and seamlessly updating precoder coefficients. An alternative embodiment uses an Inter-Symbol Interference (ISI) DFE instead. In that embodiment, the reference vectors are input to the DFE block 404 without being combined with the noisy equalizer signal.

B. Rate-Adaptive Precoder Reconstruction in a Receiver

FIG. 6 is a block diagram showing more details of the precoder reconstruction unit 408. Precoder reconstruction unit 408 has an enable input 409 supplied by Truncated Trellis decoding logic 407. This enable input 409 runs the precoder reconstruction unit 408 while symbols in header 202 are being received, and also while data is exiting the Viterbi traceback buffer 702. Precoder reconstruction unit 408 is not running while Viterbi traceback buffer 702 is initially filling. The precoder reconstruction unit 408 begins operation when the first Trellis-decoded symbol exits the Viterbi traceback buffer 702.

Precoder reconstruction unit 408 has two reference vector inputs: one from uncoded slicer 405 (not coded); and one from Trellis decoder 406 (originally coded by the remote transmitter). Note that internal to the Trellis decoder 406, the delayed equalizer output was scaled down for slicing then scaled back up to generate a recovered ideal reference vector. Reconstruction FIR 601 receives one of these reference vector inputs, based on the state of switch 602. In uncoded mode, switch 602 initializes reconstruction FIR 601 with reference vectors (x.sub.i) from uncoded slicer 405. In coded mode, switch 602 selects the input to reconstruction FIR 601 to be the signal on output 603. This signal on output 603 is itself produced by subtracting the output of reconstruction FIR 601 from the ideal reference vector. This switching operation is an important step allowing seamless transition between Trellis precoded or uncoded modes of communication.

The output of reconstruction FIR 601 is also operated on by modulo operator 604 to produce the reconstructed dither signal (d'). The reconstructed dither signal (d') is added to the signal on output 603 to generate the final reconstructed ideal reference signal (x.sub.i'). The reconstructed signal (x.sub.i') is scaled by factor (F/S.sub.b') then sliced and decoded to generate b bits of received data (RXD).

Now that both precoder reconstruction unit 408 and DFE block 404 have been discussed in detail, the interaction between them will now be described. When operating in precoded mode, switch 504 is open and the DFE block 404 is thus disabled. Conversely, when operating without preceding, switch 504 is closed and the DFE block 404 is enabled. The DFE is a powerful equalizer, which adapts in real time to changing line conditions. It avoids noise enhancement that may occur in a conventional feed forward equalizer. However, Trellis coding does not provide sufficient margin to compute reliable reference vectors (x.sub.i) for immediate input to the DFE.

Therefore, the exemplary embodiments of FIGS. 3-7 utilize precoder 405 and DFE block 404 as follows. During initialization, the receiver 102 runs without preceding and without Trellis coding, utilizing the DFE instead. After initialization, the receiver 102 switches to precoded mode. The DFE block 404 is disabled and the coefficients of DFE FIR 503 are loaded into noise-whitening FIR 501 and into precoder reconstruction unit 408. The coefficients are also sent to the transmitter 101 to be loaded in the transmitter's precoder FIR.

Several methods are used by exemplary embodiments to compute the coefficients. One of the simplest embodiments uses the coefficients of a noise whitening DFE directly. With this embodiment, the DFE coefficients are initially set by a conventional equalizer training sequence. Then during normal operating mode, the DFE coefficients track changing line conditions while receiving the uncoded header. Other methods include Levinson-Durbin calculations based on the autocorrelation of the equalizer error vectors. With either embodiment, the coefficients can be sent to the transmitter 101, without disrupting data traffic, by modifying the header transmitted by the local modem to include a packet of coefficients.

C. Truncated Trellis Decoding in a Receiver

FIG. 7 is a block diagram showing more details of the Trellis decoder 406 and Truncated Trellis decoding logic 407. Trellis decoder 406 works in conjunction with Truncated Trellis decoding logic 407 to perform truncated Trellis decoding. Like any conventional Trellis decoder, the series of symbols produced by the Trellis encoder in the transmitter depends not only on the bits input to the Trellis encoder, but on the preceding symbols as well. Thus, the Trellis encoder uses state information to encode symbols, and the Trellis decoder likewise keeps track of state information when decoding symbols. A conventional Trellis decoder uses state information as input in order to output a symbol.

In the embodiment of FIG. 7, Trellis decoder 406 processes a number of symbols conventionally using new state information, and then is able, using input from Truncated Trellis decoding logic 407, to output the remaining symbols of the frame without the need for further state information. The Trellis decoding is thus "truncated" at this point, and the total delay for decoding an entire frame is reduced. Details of the truncation process will be explained with reference to FIG. 8.

In one embodiment, Trellis decoder 406 consists of Trellis metric logic 701, Viterbi traceback buffer 702, demapper 703, and equalizer delay buffer 704. The equalized and noise-whitened received signal at the input to Trellis decoder 406 consists of a series of expanded noisy vectors (v.sub.i), where the constellation signal point transmitted by remote transmitter 101 is a point somewhere along that expanded vector. The operation of the remote precoder and local noise whitening filter results in an expanded signal constellation that maintains the valid Trellis state.

Trellis metric logic 701 computes an error metric representing the error between the received vector v.sub.i and possible Trellis states. (Although multiple error metrics may be used, for ease of explanation, the metrics associated with a particular state will be referred to as an error metric. The metric may also be referred to as a Trellis metric). Trellis metric logic 701 feeds the error metric e.sub.i and pointers to the respective previous states to the Viterbi traceback buffer 702. The Viterbi traceback buffer 702 uses the error metric e.sub.i and the pointers to trace back through a state machine and determine the Trellis state T.sub.i corresponding to that vector v.sub.i. Viterbi traceback buffer 702 accumulates V states before the first Trellis state T.sub.i can be output to demapper 703. This number V is known as the depth of the Viterbi traceback buffer.

Note that the vector input to demapper 703 is not v.sub.i, because v.sub.i corresponds to states entering, not exiting, the Viterbi traceback buffer 702. Instead, equalizer delay buffer 704 is used to delay the expanded noisy vectors v.sub.i (by Viterbi depth V) as states accumulate in Viterbi traceback buffer 702. Then, as states exit the Viterbi traceback buffer 702, equalizer delay buffer 704 provides to demapper 703 the vectors v.sub.i corresponding to those accumulated states. The expanded noisy vectors v.sub.i are scaled by F/S.sub.b' to a uniform grid for slicing in the demapper 703. For some implementations the equalizer delay buffer 704 may store vectors v.sub.i that have been previously scaled by F/S.sub.b' to reduce complexity.

Once a Trellis state T.sub.i is output from Viterbi traceback buffer 702, demapper 703 uses Trellis state T.sub.i to slice the vector v.sub.i',producing the noise-free expanded ideal reference vector x.sub.i. Scalar 605 scales the output of demapper 703 by factor S.sub.b' to the original power of the expanded vectors v.sub.i. The ideal reference vector x.sub.i will represent a constellation that has been expanded by the operation of the precoder and noise-whitening filter.

Precoder reconstruction unit 408 operates on expanded ideal reference vector x.sub.i to reconstruct the bounded constellation vector which was originally produced by the remote transmitter 101. This reconstructed ideal reference vector is then sliced by decoder 410 to recover the data bits originally produced by the remote transmitter 101.

In a conventional Trellis decoder, the output of equalizer delay buffer 704 is delayed by a fixed number of symbol times D, equal to the depth of the Viterbi traceback buffer (V). In the embodiment of FIG. 7, D is variable, and its value is determined by controller 705. Controller 705 sets D=V at the start of a frame, and when the terminating pad 204 is received, controller 705 begins the truncation process by reducing D to zero. (This process will be explained in more detail in connection with FIGS. 8 and 9.) On receipt of the reduced-density terminating pad, controller 705 also changes the values of S.sub.b and F /S.sub.b used by DFE block 404 for scaling. After delay D, these same values are output as S.sub.b' and (F/S.sub.b)' by delayed control block 706. The delayed values are used by demapper 703 and precoder reconstruction unit 408 for scaling.

FIG. 8 is a diagram showing more details of truncated Trellis decoding performed by exemplary embodiments of the Trellis decoder 406 and Truncated Trellis decoding logic 407. The contents of Viterbi traceback buffer 702 and equalizer delay buffer 704 are shown at various points in time. In this example embodiment, Viterbi traceback buffer 702 has depth V=5, delay D is initialized to V, and the number of symbols in the terminating pad is 2. At symbol time t1, incoming vector v.sub.1 (801) enters Trellis metric logic 701, which produces error metric e.sub.1 (802). The error metric e.sub.1 is input to Viterbi traceback buffer 702, which computes a Trellis state T.sub.1 (803) from the error metric. Trellis state T.sub.1 is loaded in the newest position in Viterbi traceback buffer 702. At the same time, incoming vector v.sub.1 enters equalizer delay buffer 704. Since D=V, equalizer delay buffer 704 produces no output yet.

At symbol time t2, incoming vector v.sub.2 enters Trellis metric logic 701, and error metric e.sub.2 is produced. Error metric e.sub.2 is loaded into Viterbi traceback buffer 702 Trellis state T.sub.2 is computed from the error metric. Trellis state T.sub.2 is loaded in the newest position in Viterbi traceback buffer 702, with Trellis state T.sub.1 shifting to the next position. At the same time, incoming vector v.sub.2 enters equalizer delay buffer 704 and vector v.sub.2 is shifted one position. Equalizer delay buffer 704 produces no output.

Processing for symbol times t3 and t4 are analogous, and are not shown. As one skilled in the art understands, trellis state T is actually a set of pointers from all current trellis states to the respective preceding Trellis states.

At symbol time t5, incoming vector v.sub.5 enters Trellis metric logic 701, and error metric e.sub.5 and Trellis state T.sub.5 are produced. At this point, there are states corresponding to 5 symbols in Viterbi traceback buffer 702, equal to the depth V of Viterbi traceback buffer 702. Viterbi traceback buffer 702 has enough state information to begin the traceback through the state machine and produce an output symbol. Thus, Viterbi traceback buffer 702 traces back through the preceding trellis state pointers, starting with state T.sub.5 associated with incoming vector v.sub.5g to recover the oldest Trellis state T.sub.1. This oldest Trellis state T.sub.1 is used by demapper 703 to slice vector v.sub.1 from equalizer delay buffer 704 and produce ideal reference vector x.sub.1 (804) as output. (Decoding will be finished by decoding 407 after precoder reconstruction unit 408.)

At symbol time t6, incoming vector v.sub.6, corresponding to the first symbol in terminating pad 204, enters Trellis metric logic 701. (First pad symbol is determined by receiving an end-of-data marker, as described in connection with FIG. 2). Because the terminating pad 204 was transmitted with a lower constellation density, error metric e.sub.6 is error-free, and consequently, Trellis state T.sub.6 is very reliable. Viterbi traceback buffer 702 traces back starting with state T.sub.6, to recover oldest Trellis state T.sub.2. Viterbi traceback buffer 702 uses Trellis state T.sub.2 in combination with vector v.sub.2 from equalizer delay buffer 704 to produce ideal reference vector x.sub.2 as output.

At symbol time t7, incoming vector v.sub.7, corresponding to the last symbol in terminating pad 204, enters Trellis metric logic 701. (As described in connection with FIG. 2, receipt of last pad symbol is determined by receipt of first pad symbol). Because the terminating pad 204 was transmitted with a lower constellation density, error metric e.sub.8 is error-free, and consequently, Trellis state T.sub.8 is very reliable. Viterbi traceback buffer 702 traces back starting with state T.sub.7, to recover Trellis state T.sub.3. Viterbi traceback buffer 702 uses Trellis state T.sub.3 in combination with vector v.sub.3 from equalizer delay buffer 704 to produce ideal reference vector x.sub.3 as output.

At symbol time t8, the last symbol in the terminating pad 204 has entered Viterbi traceback buffer 702, and the truncated decoding process begins. The presence of reliable Trellis states T.sub.7 and T.sub.8 in Viterbi traceback buffer 702 allows Viterbi traceback buffer 702 to output subsequent symbols without receiving incoming vectors or computing error metric(s) for those vectors. Therefore, the traceback of the Trellis decoder state machine can progress without waiting for new incoming vectors, using only information already stored in Viterbi traceback buffer 702 and in equalizer delay buffer 704.

Controller 705 first reduces D=5 to D=4 (805) to extract data within the Viterbi and equalizer delay buffers without introducing new incoming vectors. Equalizer delay buffer 704, using this reduced delay, produces vector v.sub.4. Viterbi traceback buffer 702, using the reduced delay, uses the oldest Trellis state T.sub.4 in combination with vector v.sub.4 to produce ideal reference vector x.sub.4 as output. Controller 705 then reduces D=4 to D=3, to recover then combine oldest Trellis state T.sub.5 with vector v.sub.5 to produce ideal reference vector x.sub.5 as output. Ideal reference vectors x.sub.6 and x.sub.7 are output in a similar fashion.

Processing of symbols up to the time the last symbol of the terminating pad 204 enters Viterbi traceback buffer 702 occurs using symbol timing (because output depends on the next received symbol). Once the last terminating pad enters, processing is no longer tied to symbol timing. Rather, Trellis decoding is truncated and can proceed as fast as implementation allows.

FIG. 9 is a timing diagram showing how the various components of an exemplary embodiment interact to perform truncated trellis decoding. Equalizer output 901 is the received signal on output 403 (see FIG. 4) after clean-up and possible expansion by noise-whitening block 402. (It is this signal on output 403 which is input to Trellis decoder 406. The DFE is normally disabled during Trellis decoding.) Because this line depicts constellation density, it may consist of multiple lines: a single line in the middle means no symbols received; two lines means low-density symbols; four lines means high-density symbols. Delayed equalizer output 902 is the same signal but delayed by the depth of the Viterbi traceback buffer 702. Switch state 903 is a combined signal indicating the states of switch 305, switch 308, switch 502, and switch 504, which together determine whether precoding reconstruction is performed. The next line, b (904) is the number of bits per symbol. Switch state 905 is the state of switch 602 selecting input to the precoder reconstruction unit 408. The next line, S.sub.b' (906), is the value of scalar S.sub.b, used at the input to Trellis decoder 406. The following line, S.sub.b' (907), is a delayed version of S.sub.b, used at the output of Trellis decoder 406. Enable line (908) is a control signal that determines when the precoder reconstruction runs. The next line, D (909), is a signal that determines how long equalizer delay buffer 704 holds the received signal on output 403. The last line, RxData (910), is decoded data that is output by decoder 410.

Time t1 marks the receipt of the first symbol of header 202. This header symbol was transmitted with a relatively low constellation density. Several signal lines are affected by constellation density. Received data on equalizer output 901 transitions from zero (no symbols received) to two levels, and b (904) transitions from no bits-per-symbol to an intermediate number of bits-per-symbol. As described with reference to the transmitter in FIG. 3, scale factor S.sub.b is inversely proportional to the constellation density. Line S.sub.b (906), which corresponds to the value of scale factor S.sub.b, therefore transitions from zero to a high level at t1.

The header symbol was transmitted without Trellis coding and without precoding, which affects several signal lines. Switch state 903 is low, indicating that switch 305, switch 308, switch 502, and switch 504 are set for uncoded operation. Switch state 905 transitions to high, indicating that switch 602 has selected the reference vectors (x.sub.i) at the output of the uncoded slicer 405 to initialize the reconstruction FIR 601. Enable line (908) is high indicating precoder reconstruction is initializing. Switch state 903 remains low, indicating that noise-whitening block 402 and precoder reconstruction unit 408 are bypassed, and DFE block 404 is used instead. Because the header symbol was also transmitted without Trellis coding, RxData (910) transitions on receipt of this symbol, showing the decoded header is available immediately.

Time t2 marks the receipt of the last symbol of header 202 and the first symbol of data 203. The data symbols were transmitted with a higher constellation density. Thus, received data on line equalizer output 901 transitions from two (low density) to four (high density) levels, and b (904) transitions from an intermediate bits-per-symbol to a high number of bits-per-symbol. Because the constellation density has changed, S.sub.b 906, which is inversely proportional to constellation density, transitions to a reduced (but non-zero) level at t2.

The data symbols were also transmitted with Trellis coding and with preceding. Thus, switch state 905 transitions back to low, indicating that switch 602 has selected the reference vectors (x.sub.i) at the output of Trellis decoder 406 to fill the reconstruction FIR 601. Enable line (908) transitions to low indicating precoder reconstruction is not active but the initialized values are retained in the reconstruction FIR 601. RxData (910) transitions to zero on receipt of the last header symbol, because the data symbols following were transmitted with Trellis coding and will not be decoded until after a delay in the Viterbi traceback buffer 702.

Time t3 occurs after V symbols have entered Viterbi traceback buffer 702. With the traceback buffer full, Trellis decoder 406 now accepts an input vector from equalizer delay buffer 704. This is shown by delayed equalizer output 902 transitioning from zero to multilevel (high-density) symbols. Using this input vector and Trellis states already in the traceback buffer, Trellis decoder 406 runs through the state machine to produce an output symbol, the first one to exit Viterbi traceback buffer 702.

This first output symbol affects several signal lines. Now that the Trellis decoder 406 is outputting data, enable line (908) also transitions to run the precoder reconstruction unit 408. The first output symbol is a high-density data symbol rather than a low-density header symbol. Therefore, received data on RxData (910) transitions from zero to a high level, signifying the change in constellation density. Line S.sub.b' (907), which should match constellation density, also transitions at t3. This scale factor is used by both demapper 703 and precoder reconstruction unit 408.

Time t4 marks the first symbol of terminating pad 204 being received and also entering Viterbi traceback buffer 702. This symbol was transmitted with a low constellation density, and this change in density affects several signal lines. Received data on equalizer output 901 transitions from four levels to two. Line S.sub.b (906) (corresponding to an input to Trellis decoder 406) transitions from an intermediate to a high level (since the scaling is inversely proportional to density), and b (904) transitions from a high number of bits-per-symbol to an intermediate number.

Time t5 marks the last symbol of terminating pad 204 being received and also entering Viterbi traceback buffer 702. Since this pad marks the end of the current frame 201, the receiver sets up to receive the header of the next frame. This is indicated by switch state 903 transitioning back to low, bypassing noise-whitening block 402 and using DFE block 404 instead. As described with reference to FIGS. 7 and 8, once this last pad symbol enters Viterbi traceback buffer 702, the Trellis decoding process can be truncated. This is indicated by the downward step transitions in delay signal D (909). As D steps down, equalizer delay buffer 704 supplies the next vector and Viterbi traceback buffer 702 runs its state machine, without accepting new Trellis metrics, to produce the next reference vector vi.

Conventional Trellis decoding would not finish decoding until time t7. Conventional Trellis decoding requires decoding to be tied to an incoming symbol clock until the entire frame is decoded, since error metrics based on newly received data must be input to the Trellis decoder to produce each new output.

Using truncated Trellis decoding, the last symbol in terminating pad 204 is decoded at time t6, when delay signal D (909) goes to zero. With truncated Trellis decoding, symbols in terminating pad 204 are transmitted with excess margin, and this excess margin allows the Viterbi traceback buffer to be flushed once the last terminating pad symbol enters the traceback buffer, without computing new error metrics.

D