# Patent

## Title
Flexible PRBS architecture for a transceiver

## Abstract

An apparatus is provided. The apparatus comprises a polynomial register having a plurality of bits, a first bus, a second bus, and a transceiver that is coupled to the first bus, the second bus, and the polynomial register. The polynomial register is configured to store a user-defined polynomial, and the transceiver includes a pseudorandom bit sequence (PRBS) generator is configured to generate a scrambled signal from the user-defined polynomial and a PRBS checker that is configured to generate a descrambled signal from a second signal using the user-defined polynomial.

## Background
RELATED APPLICATION
This application claims the benefit under 35 U.S.C. §120 of non-provisional application Ser. No. 13/660,841, filed Oct. 25, 2012, issued as U.S. Pat. No. 9,124,462 on Sep. 1, 2015, the entirety of which is incorporated herein by reference.
TECHNICAL FIELD
The invention relates generally to a transceiver and, more particularly, to a physical transceiver (PHY) having a flexible architecture.
BACKGROUND
Turning to FIG. 1, an example of a conventional system 100 can be seen. In this system 100, hosts 102-1 to 102-N (which can be; for example, a computer, router, or switch) are able to communicate with one another over communications medium 112 (which can; for example, be an optical fiber, backplane, or twisted pair) through network interfaces 104-1 to 104-N. In this example, the network interfaces 104-1 to 104-N employ Ethernet over Electrical Backplanes and, more specifically, 10GBase-KR. A description of 10GBase-KR can be found in the Institute of Electrical and Electronics Engineers (IEEE) standard 802.3-2008 (which is dated Dec. 26, 2008 and which is incorporated by reference herein for all purposes). These network interfaces 104-1 to 104-N employ media access control (MAC) circuits 106-1 to 106-N that communicate with PHYs 110-1 to 110-N via, media independent interfaces (MIIs) 108-1 to 108-N (which can typically have half-duplex or full-duplex operation), each of which is described in IEEE standard 802.3-2008.
Of interest here, however, are PHYs 110-1 to 110-N, and, as can be seen in greater detail in FIG. 2, PHYs 110-1 to 110-N (hereinafter PHY 110), PHY 110 employs several sublayers. This PHY 110 can be an independent integrated circuit (IC) or can be integrated with a MAC circuit (i.e., MAC circuit 106-1) and an MII 108. As shown, the PHY 110 is generally comprised of physical medium dependent (PMD) sublayer logic 212; physical medium attachment (PMA) sublayer logic 210, forward error correction (FEC) sublayer logic 204, and physical coding sublayer (PCS) logic 202. These sublayer logic circuits 202, 204, 210, and 212 interact with one another to provide communications between MII 108 and communications medium 112. For transmission, the FEC sublayer logic 204 employs an encoder 206 as described in IEEE standard 802.3-2008, clause 74, and, for reception, the FEC sublayer logic 204 employs a decoder 208 as described in IEEE standard 802.3-2008, clause 74.
As can be seen in FIG. 3, the PCS logic 202 can be a transceiver, having a PCS transmitter 302 and a PCS receiver 304. The transmitter 302, in this example, is able to receive data from MII 108, encode the data with encoder 306, scramble the encoded data with scrambler 308, and convert (so as to be used by FEC sublayer logic 204) with gearbox 310. The receiver 304, in this example, is able to convert data from FEC sublayer logic 204 using gearbox 312, descramble the data with descrambler 314, and decode the data (for use with MII 108) with decoder 316. The details of PCS logic 202 can, for example, be seen in IEEE standard 802.3-2008, clauses 48 and 74.
Of interest here are the scrambler 308 and descrambler 314. In this example, the scrambler 308 and descrambler 314 are able to perform data scrambling/descrambling and error checking. One purpose in scrambling/descrambling data with the PHYs 110-1 to 110-N is to substantially randomize the data to reduce the impact of electromagnetic interference (EMI) and improve signal integrity. This is typically accomplished by the use of a pseudorandom bit sequence (PRBS) generated with a specified polynomial. For example, for 8b/10b encoding, a PRBS-7 (or 1+x6+x7) can be employed, and, for synchronous optical networking or SONET (as specified in ITU O.150), PRBS-23 (or X23+X18+1). Similarly, this PRBS signaling can be employed for error checking
However, as demonstrated above, one polynomial is generally not applicable to all standards (e.g., 802.3-2008 and SONET); each standard usually specifies its own polynomial. Conventionally, this meant that each PHY (e.g., 110-1) would be designed for a particular standard (e.g., PRBS-7 for 802.3-2008) and would lack the flexibility to be used with other standards. A reason for this is that the serial and parallel implementations for the PHYs (e.g., 110-1) would be too costly in terms of area, price, and power consumption to be generally applicable.
Therefore, there is a need for a flexible transceiver architecture.
Some examples of conventional systems are: U.S. Pat. Nos. 4,744,104; 5,267,316; 6,820,230; 6,907,062; 7,124,158; 7,414,112; 7,486,725; 7,505,589; U.S. Patent Pre-Grant Publ. Nos. 2003/0014451; U.S. Patent Pre-Grant Publ. No. 2007/008997; and U.S. Patent Pre-Grant Publ. No. 2007/0098160.
SUMMARY
In accordance with an embodiment of the present invention, an apparatus is provided. The apparatus comprises a polynomial register having a plurality of bits, wherein the polynomial register is configured to store a user-defined polynomial; a first bus; a second bus; a transceiver that is coupled to the first bus, the second bus, and the polynomial register, wherein the transceiver includes: a pseudorandom bit sequence (PRBS) generator is configured to generate a scrambled signal from the user-defined polynomial; and a PRBS checker that is configured to generate a descrambled signal from a second signal using the user-defined polynomial.
In accordance with an embodiment of the present invention, the first bus further comprises a first input bus and a second input bus, and wherein the second bus further comprises a first output bus and a second output bus, and wherein the PRBS generator is coupled to the first output bus, and wherein the PRBS checker is coupled to the second input bus.
In accordance with an embodiment of the present invention, the first input bus has a programmable width.
In accordance with an embodiment of the present invention, the PRBS checker further comprises: a first matrix circuit that is configured to include a first matrix corresponding to the user-defined polynomial; a second matrix circuit that is configured to include a first matrix corresponding to the user-defined polynomial; a first multiplier that is coupled to the second matrix circuit and that is coupled to the respective one of the encoder and the second input bus; a data register that is coupled to the second input bus; a second multiplier that is coupled to the first matrix circuit and the data register; an XOR circuit that is coupled to the first and second multipliers; and an error counter that is coupled to the XOR circuit.
In accordance with an embodiment of the present invention, the data register further comprises a first data register, and wherein the PRBS generator further comprises: a third matrix circuit that is configured to include a third matrix corresponding to the user-defined polynomial; a third multiplier that is coupled to the third matrix circuit; a first multiplexer that is coupled to the third multiplier and that is configured to receive a seed; a second data register that is coupled to the first multiplexer and the second data register.
In accordance with an embodiment of the present invention, the transceiver further comprises a detector that is coupled to the PBRS generator and the PRBS checker.
In accordance with an embodiment of the present invention, the polynomial register has 32 bits.
In accordance with an embodiment of the present invention, an apparatus is provided. The apparatus comprises a media access control (MAC) circuit; a interface that is coupled to the MAC circuit; a physical transceiver (PHY) having: a polynomial register having a plurality of bits, wherein the polynomial register is configured to store a user-defined polynomial; a first bus that is coupled to the interface; a second bus; a transceiver that is coupled to the first bus, the second bus, and the polynomial register, wherein the transceiver includes: a pseudorandom bit sequence (PRBS) generator is configured to generate a scrambled signal from the user-defined polynomial; and a PRBS checker that is configured to generate a descrambled signal from a second signal using the user-defined polynomial.
In accordance with an embodiment of the present invention, the PHY further comprises a detector that is coupled to the PBRS generator and the PRBS checker.
In accordance with an embodiment of the present invention, the apparatus further comprise a communications medium that is coupled to the PHY.
In accordance with an embodiment of the present invention, the detector is configure use the PRBS generator and the PRBS checker to characterize the communications medium.
In accordance with an embodiment of the present invention, the apparatus further comprises a host that is coupled to the MAC circuit.
In accordance with an embodiment of the present invention, a method is provided. the method comprises retrieving a user-defined polynomial from a polynomial register having a plurality of bits; generating first, second, and third matrices based at least in part on the user-defined polynomial; generating a first PRBS data set using the first matrix; transmitting the first PRBS data set over a communications medium; receiving a second PRBS data set over the communications medium; and determining a number of bit errors with the second PRBS data set using the second and third matrices.
In accordance with an embodiment of the present invention, the method further comprises: adjusting the first PRBS data set; and repeating the steps of transmitting, receiving, and determining.
In accordance with an embodiment of the present invention, the method further comprises characterizing the communication channel based at least in part on the number of bit errors.



BRIEF DESCRIPTION OF THE DRAWINGS
For a more complete understanding of the present invention, and the advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, in which:
FIG. 1 is a diagram of an example of a conventional system;
FIG. 2 is a diagram of an example of a PHY of FIG. 1;
FIG. 3 is a diagram of a PCS logic of FIG. 2;
FIGS. 4 and 5 are diagrams of an example of a PCS logic in accordance with embodiments of the present invention;
FIG. 6 is a diagram of an example of the programmable transmission and reception circuits of FIGS. 4 and 5;
FIG. 7 is a diagram of an example of the PRBS generator of FIG. 6; and
FIG. 8 is a diagram of an example of the PRBS checker of FIG. 6.



DETAILED DESCRIPTION
Refer now to the drawings wherein depicted elements are, for the sake of clarity, not necessarily shown to scale and wherein like or similar elements are designated by the same reference numeral through the several views.
Turning to FIGS. 4 and 5, example of the transceivers 400-A and 400-B can be seen. As shown with the example shown in FIG. 4, transceiver 400-A can be used as part of PCS logic 202 of FIG. 3, and, as shown in the example of FIG. 5, transceiver 400-B can be used to communicate with a serializer/deserializer (SERDES) device. Other implementations can be used with the transceivers 400-A and 400-B, including implementations that omit encoder 306 and decoder 316. In each case, the transceivers 400-A and 400-B employ programmable transmission and reception circuits 406-A/406-B and 408-A/408-B that can perform scrambling/descrambling and error checking based on a user-specified or user-defined polynomial.
Turning to FIG. 6, the programmable transmission and reception circuits 406-A/406-B and 408-A/408-B (which are referred to hereinafter as 406 and 408) can be seen in greater detail. Collectively, circuits 406 and 408 can be considered to be a transceiver. As shown in this example, circuit 406 generally comprises a PRBS generator 504 and a scrambler 502, while circuit 408 generally comprises a descrambler 506 and PRBS checker 508. As shown, there can also be a detector 512 that is in communication with the PRBS generator 504 and the PRBS checker 508. This detector 512 can cause the PRBS generator 504 to transmit PRBS data sets over a communications medium (e.g., 112) and receive the bit errors from the PRBS checker 508. Based on this information, the detector 512 can search for optimal settings by transmitting repeated PRBS data sets (after each adjustment iteration) and receiving the bit errors, or it can characterize the communication channel (e.g., 112), allowing the detector 512 to detect the communication medium type (e.g., twist pair, optical, and so forth). Additionally, the busses that communicate with the scrambler 502 and descrambler 506 can have a programmable width (e.g., a maximum width of 32 bits but adjustable down to 1 bit).
Also, as can be seen in the example of FIG. 6, there is a polynomial register 510 shown. This polynomial register 510 typically has a predetermined width or number of bits (e.g., 32 bit) that is accessible to a user. The user is able to write to this register 510 so as to store a user-defined polynomial. As an example, if a user chooses to use PRBS-7 (which has a polynomial of 1+x6+x7) for scrambler 502, the user can write the following to a 32-bit register (e.g., register 510):




Thus, for an example register (e.g., 601) having a width of 32 bits, the user can specify any of approximately 2×109 polynomials. This user-defined polynomial (which can be retrieved from register 510) can be used by the scrambler 502, PRBS generator 504, descrambler 506, and PRBS checker 508 accordingly. Alternatively, there can be multiple polynomial registers (e.g. 510), and each of the scrambler 502, PRBS generator 504, descrambler 506, and PRBS checker 508 may have a separate polynomial register (e.g., 510).

Turning to FIGS. 7 and 8, an example of the PRBS generator 504 and PRBS checker 508 can be seen in greater detail. Each of the example PRBS generator 504 and PRBS checker 508 use a user-defined polynomial that can be retrieved from register 510. One of the purposes in having a PRBS system (e.g., PRBS generator 504 and PRBS checker 508) is to allow for bit error testing of high speed serial links, and there are many communications standards or protocols that define or call out specific polynomials (e.g., PRBS-7). The PRBS generator 504 and PRBS checker 508 shown in this example are generally unaware of the communication protocol or standard and can be used for nearly all known standards.
The PRBS system (e.g., PRBS generator 504 and PRBS checker 508) in this example is based in part on the generation of polynomial state and data matrices (which can respectively be referred to as the P-matrix and D-matrix). In operation, the signal POLY (which generally corresponds to the user-defined polynomial stored in register 510) can be used to generate matrices, which can be referred to. The P- and D-matrices P and D are typically square binary matrices that are a function of or based at least in part on the user-defined polynomial. The basis for forming the P- and D-matrices P and D are identity matrices IP and ID (respectively), which typically have uniquely assigned vectors for each column of the first row of the P- and D-matrices P and D (i.e., P0,j and D0,j). An example of identity matrix IP can be seen below:




   
  
    [
    
      
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
      
    
    ]
  




The identity matrix ID is generally comprised of the matrix IP that is shifted or adjusted based on the desired input bus width. For example, the identity matrix ID (which is derived from the matrix IP shown above) can be as follows for a 20-bit bus width:





   
  
    [
    
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
      
    
    ]
  




An adjustment vector {right arrow over (A)} is also determined. Typically, when the signal POLY1 is transmitted, the lowest bit is truncated, and a ‘0’ is appended to signal POLY1 to form adjustment vector {right arrow over (A)}. For example, with the PRBS-7 polynomial used above, the adjustment vector {right arrow over (A)} would be:





The P- and D-matrices P and D can then be determined.

Looking first to the P-matrix P it can be determined on a row-by-row basis through the use of a set of matrices (e.g., 32−32×32 matrices), which can be referred to as working matrices WP[r], where r denote the P-matrix P row. These working matrices WP[r], in this example, are based at least in part on the identity matrix IP and can be determined using the following formula:




  
    
      
        
          
            WP
            ⁡
            
              [
              r
              ]
            
          
          
            _
            _
          
        
        =
        
          {
          
            
              
                
                  
                    
                      WP
                      ⁡
                      
                        [
                        0
                        ]
                      
                    
                    
                      _
                      _
                    
                  
                  =
                  
                    
                      I
                      P
                    
                    
                      _
                      _
                    
                  
                
              
            
            
              
                
                  
                    
                      
                        WP
                        
                          i
                          ,
                          j
                        
                      
                      ⁡
                      
                        [
                        r
                        ]
                      
                    
                    =
                    
                      
                        WP
                        
                          
                            i
                            -
                            1
                          
                          ,
                          
                            j
                            -
                            1
                          
                        
                      
                      ⁡
                      
                        [
                        r
                        ]
                      
                    
                  
                  ,
                  
                    1
                    ≤
                    r
                  
                  ,
                  
                    
                      i
                      ≤
                      n
                    
                    ;
                    
                      2
                      ≤
                      j
                      ≤
                      n
                    
                  
                
              
            
            
              
                
                  
                    
                      
                        WP
                        
                          i
                          ,
                          0
                        
                      
                      ⁡
                      
                        [
                        r
                        ]
                      
                    
                    =
                    
                      FNP
                      ⁡
                      
                        [
                        r
                        ]
                      
                    
                  
                  ,
                  
                    1
                    ≤
                    r
                    ≤
                    n
                  
                
              
            
          
        
      
    
    
      
        (
        1
        )
      
    
  
  
    
      where
    
    
      
        
      
    
  
  
    
      
        
          FNP
          ⁡
          
            [
            r
            ]
          
        
        =
        
          
            (
            
              
                
                  
                    WP
                    j
                  
                  ⁡
                  
                    [
                    
                      r
                      -
                      1
                    
                    ]
                  
                
                →
              
              ⊕
              
                A
                →
              
            
            )
          
          ·
          ...
          ·
          
            (
            
              
                
                  
                    WP
                    0
                  
                  ⁡
                  
                    [
                    
                      r
                      -
                      1
                    
                    ]
                  
                
                →
              
              ⊕
              
                A
                →
              
            
            )
          
        
      
    
    
      
        (
        2
        )
      
    
  




The P-matrix P can then be extracted from working matrices WP[r] by application of the following equation:





  
    
      
        
          P
          
            i
            ,
            j
          
        
        =
        
          {
          
            
              
                
                  0
                  ,
                  
                    i
                    >
                    BW
                  
                
              
            
            
              
                
                  
                    
                      WP
                      
                        i
                        ,
                        j
                      
                    
                    ⁡
                    
                      [
                      
                        BW
                        -
                        1
                      
                      ]
                    
                  
                  ,
                  otherwise
                
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




where BW is the bus width. For example, with the PRBS-7 polynomial used above and a 20-bit bus width BW, the P-matrix P should be:





   
  
    [
    
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
        
          1
        
        
          1
        
        
          0
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
        
          1
        
        
          1
        
        
          0
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
        
          1
        
        
          1
        
        
          0
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
        
          1
        
        
          1
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          1
        
        
          0
        
        
          1
        
        
          1
        
        
          1
        
      
      
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          0
        
        
          1
        
        
          0
        
        
          1
        
        
          0
        
        
          1
        
        
          1
        
      
    
    ]
  




Similarly, with D-matrix D, it can be determined on a row-by-row basis through the use of a set of matrices (e.g., 32−32×32 matrices) or working matrices WD[r]. These working matrices WD[r], in this example, are based at least in part on the identity matrix ID and can be determined using the following formula:




  
    
      
        
          
            WD
            ⁡
            
              [
              r
              ]
            
          
          
            _
            _
          
        
        =
        
          {
          
            
              
                
                  
                    
                      WD
                      ⁡
                      
                        [
                        0
                        ]
                      
                    
                    
                      _
                      _
                    
                  
                  =
                  
                    
                      I
                      D
                    
                    
                      _
                      _
                    
                  
                
              
            
            
              
                
                  
                    
                      
                        WD
                        
                          i
                          ,
                          j
                        
                      
                      ⁡
                      
                        [
                        r
                        ]
                      
                    
                    =
                    
                      
                        WD
                        
                          
                            i
                            -
                            1
                          
                          ,
                          
                            j
                            -
                            1
                          
                        
                      
                      ⁡
                      
                        [
                        r
                        ]
                      
                    
                  
                  ,
                  
                    1
                    ≤
                    r
                  
                  ,
                  
                    
                      i
                      ≤
                      n
                    
                    ;
                    
                      2
                      ≤
                      j
                      ≤
                      n
                    
                  
                
              
            
            
              
                
                  
                    
                      
                        WD
                        
                          i
                          ,
                          0
                        
                      
                      ⁡
                      
                        [
                        r
                        ]
                      
                    
                    =
                    
                      FND
                      ⁡
                      
                        [
                        r
                        ]
                      
                    
                  
                  ,
                  
                    1
                    ≤
                    r
                    ≤
                    n
                  
                
              
            
          
        
      
    
    
      
        (
        4
        )
      
    
  
  
    
      where
    
    
      
        
      
    
  
  
    
      
        
          FND
          ⁡
          
            [
            r
            ]
          
        
        =
        
          
            (
            
              
                
                  
                    WD
                    j
                  
                  ⁡
                  
                    [
                    
                      r
                      -
                      1
                    
                    ]
                  
                
                →
              
              ⊕
              
                A
                →
              
            
            )
          
          ·
          ...
          ·
          
            (
            
              
                
                  
                    WD
                    0
                  
                  ⁡
                  
                    [
                    
                      r
                      -
                      1
                    
                    ]
                  
                
                →
              
              ⊕
              
                A
                →
              
            
            )
          
        
      
    
    
      
        (
        5
        )
      
    
  




The D-matrix D can then be extracted from working matrices WD[r] by application of the following equation:





  
    
      
        
          D
          
            i
            ,
            j
          
        
        =
        
          {
          
            
              
                
                  0
                  ,
                  
                    i
                    >
                    BW
                  
                
              
            
            
              
                
                  
                    
                      WD
                      
                        i
                        ,
                        j
                      
                    
                    ⁡
                    
                      [
                      
                        BW
                        -
                        1
                      
                      ]
                    
                  
                  ,
                  otherwise
                
              
            
          
        
      
    
    
      
        (
        6
        )
      
    
  




Looking first to the PRBS generator 504, it does not rely on a D-matrix. As shown in FIG. 7, the matrix circuit 602 generates the P-matrix P from signal POLYIN as described above. Initially, when the PRBS system is activated, the multiplexer 614 can be set to allow a seed value or vector SEED to be written to register 604. This seed value can, for example, be a pseudorandom number generated from a clock. Once seeded, the multiplexer 614 is set to form a feedback path from the multiplier 606 to registers 604. For each iteration, the multiplier 606 multiplies the P with the value or vector stored in registers 604. The output from the multiplier 606 can form the output vector DATAOUT. Alternatively, an inverse of the output of multiplier 606 can be used as the output vector DATAOUT by using the inversion circuit 610. In this case the AND gate 612 (which receives a system enable signal EN and inverse enable signal INV), controls the multiplexer 608 such that it outputs the inverse of the output of the inversion circuit 610 as the output vector DATAOUT.
The PRBS checker 508, on the other hand, does use both the P- and D-matrices P and D and has a function that is similar to the descrambler 506. In operation, the P- and D-matrices P and D are generated by matrix circuits 702 and 704, respectively, the input data DATAIN for checker 508. This input data DATAIN can also be written to register 710. The multiplier 706 can multiply the input data DATAIN (which can for example be 20-bits wide data vector) by the D-matrix D to generate a D-vector. The P-matrix P can be multiplied by the information (e.g., vector) stored in register 710 with multiplier 712 to generate a P-vector. The outputs of multipliers 706 and 712 (i.e., the D-vector and the P-vector) can then be XORed with an error detector circuit 908 and output to error counter 718 that generates an error count value ECNT and an error flag ERRORFLG. Alternatively, an inverse of the input data DATAIN when the multiplexer is selected to pass the output of inversion circuit 714; this is typically employed when the generator 504 is selected to output an inverted data vectors (e.g., DATAOUT).
One advantage of having such a flexible PRBS system is that the communication channel can be characterized or optimized. For example, a detector 512 can be included that can control the generator 504 and checker 508. This detector 512 can allow for iterative or repeated PRBS transmission over the communication channel, and, based on the error, adjustments can be made so as to substantially optimize transmission over the communication channel. Alternatively, this detector 512 can be used to determine the type of communication channel (e.g., optical, twisted pair, and so forth) using similar repeated transmissions.
Having thus described the present invention by reference to certain of its preferred embodiments, it is noted that the embodiments disclosed are illustrative rather than limiting in nature and that a wide range of variations, modifications, changes, and substitutions are contemplated in the foregoing disclosure and, in some instances, some features of the present invention may be employed without a corresponding use of the other features. Accordingly, it is appropriate that the appended claims be construed broadly and in a manner consistent with the scope of the invention.

## Claims

1. A transmitter for use in a physical coding sublayer (PCS) of a network transceiver, the transmitter comprising:
a polynomial register configured to store a user-defined polynomial; and
a pseudorandom bit sequence (PRBS) generator including:
a matrix circuit coupled with the polynomial register to receive the user-defined polynomial, the matrix circuit configured to generate a P-matrix based on the user-defined polynomial;
a data register configured to receive a seed vector; and
a multiplier coupled with the matrix circuit and the data register, the multiplier configured to generate a PRBS signal based on the P-matrix and based on data stored in the data register.

2. The transmitter of claim 1, wherein the PRBS generator includes a multiplexer having:
a first input configured to receive the seed vector;
a second input coupled with the multiplier to receive the PRBS signal; and
an output coupled with the data register, the output configured to deliver the seed vector to the data register before the multiplier generates the PRBS signal, and the output configured to deliver the PRBS signal to the data register after the multiplier generates the PRBS signal.

3. The transmitter of claim 1, wherein the PRBS generator includes:
an inversion circuit coupled with the multiplier to receive the PRBS signal, the inversion circuit configured to generate an inverted PRBS signal based on the PRBS signal; and
an output multiplexer coupled with the multiplier and the inversion circuit, the output multiplexer configured to deliver either one of the PRBS signal or the inverted PRBS signal as a PRBS output signal.

4. The transmitter of claim 1, wherein the multiplier is configured to recursively multiply the P-matrix with the data stored in the data register to generate the PRBS signal.

5. The transmitter of claim 1, wherein the matrix circuit is configured to:
receive the user-defined polynomial;
generate an adjustment vector based on the user-defined polynomial;
generate a working P-matrix based on the adjustment vector and based on an identity matrix; and
generate the P-matrix by masking the working P-matrix based on a bus width of the user-defined polynomial.

6. The transmitter of claim 1, wherein the P-matrix is a 32-by-32 matrix having zero value coefficients along row numbers beyond a bus width of the user-defined polynomial.

7. A receiver for use in a physical coding sublayer (PCS) of a network transceiver, the receiver comprising:
a polynomial register configured to store a user-defined polynomial;
a pseudorandom bit sequence (PRBS) checker including:
a data bus configured to deliver reception data received by the network transceiver;
a first matrix circuit coupled with the polynomial register to receive the user-defined polynomial, the first matrix circuit configured to generate a D-matrix based on the user-defined polynomial;
a first multiplier coupled with the first matrix circuit and the data bus, the first multiplier configured to generate a D-vector by multiplying the D-matrix with the reception data;
an error detector circuit coupled with the first multiplier to receive the D-vector, the error detector circuit configured to generate an error signal based on the D-vector.

8. The receiver of claim 7, wherein the PRBS checker includes:
a second matrix circuit coupled with the polynomial register to receive the user-defined polynomial, the second matrix circuit configured to generate a P-matrix based on the user-defined polynomial; and
a second multiplier coupled with the second matrix circuit and the data bus, the second multiplier configured to generate a P-vector by multiplying the P-matrix with the reception data.

9. The receiver of claim 8, wherein the error detector circuit includes an exclusive-or (XOR) gate coupled with the first multiplier and the second multiplier, and the XOR gate configured to generate the error signal based on an exclusive-or operation on the D-vector and P-vector.

10. The receiver of claim 8, wherein the PRBS checker includes:
an inversion circuit coupled with an input bus to receive data from the transceiver, the inversion circuit configured to invert the received data; and
a multiplexer coupled with the input bus and the inversion circuit, the multiplexer configured to deliver either one of the received data or the inverted received data to the data bus.

11. The receiver of claim 8, wherein the second matrix circuit is configured to:
receive the user-defined polynomial;
generate an adjustment vector based on the user-defined polynomial;
generate a working P-matrix based on the adjustment vector and based on a P-identity matrix independent of a bus width of the user-defined polynomial; and
generate the P-matrix by masking the working P-matrix based on the bus width of the user-defined polynomial.

12. The receiver of claim 8, wherein the P-matrix is a 32-by-32 matrix having zero value coefficients along row numbers beyond a bus width of the user-defined polynomial.

13. The receiver of claim 8, wherein the PRBS checker includes a data register coupled with the data bus to register the reception data, the data register coupled with the second multiplier to provide the reception data.

14. The receiver of claim 7, wherein the first matrix circuit is configured to:
receive the user-defined polynomial;
generate an adjustment vector based on the user-defined polynomial;
generate a working D-matrix based on the adjustment vector and based on a D-identity matrix dependent on a bus width of the user-defined polynomial; and
generate the D-matrix by masking the working D-matrix based on the bus width of the user-defined polynomial.

15. The receiver of claim 7, wherein the D-matrix is a 32-by-32 matrix having zero value coefficients along row numbers beyond a bus width of the user-defined polynomial.

16. The receiver of claim 7, wherein the PRBS checker includes an error counter coupled with the error detector circuit to receive the error signal, the error counter configured to update an error count value based on the error signal.

17. A network transceiver comprising:
a polynomial register configured to store a user-defined polynomial;
a pseudorandom bit sequence (PRBS) generator including:
a first matrix circuit coupled with the polynomial register to receive the user-defined polynomial, the first matrix circuit configured to generate a P-matrix based on the user-defined polynomial; and
a first multiplier coupled with the matrix circuit, the first multiplier configured to generate a PRBS signal based on the P-matrix and based on a seed vector; and

a PRBS checker including:
a second matrix circuit coupled with the polynomial register to receive the user-defined polynomial, the second matrix circuit configured to generate a D-matrix based on the user-defined polynomial; and
a second multiplier coupled with the second matrix circuit, the second multiplier configured to generate a D-vector by multiplying the D-matrix with input data received by the network transceiver; and
a error detector circuit coupled with the second multiplier to receive the D-vector, the error detector circuit configured to generate an error signal based on the D-vector.

18. The network transceiver of claim 17, further comprising:
a detector circuit coupled with the PRBS generator and the PRBS checker, the detector circuit configured to determine a condition of a communication channel accessed by the network transceiver based on the PRBS signal and the error signal.

19. The network transceiver of claim 17, further comprising:
a detector circuit coupled with the PRBS generator and the PRBS checker, the detector circuit configured to adjust a transmission over a communication channel based on the PRBS signal and the error signal.

20. The receiver of claim 17, wherein:
the PRBS checker includes:
a third matrix circuit coupled with the polynomial register to receive the user-defined polynomial, the third matrix circuit configured to generate a second P-matrix based on the user-defined polynomial; and
a third multiplier coupled with the third matrix circuit, the third multiplier configured to generate a second P-vector by multiplying the P-matrix with the input data; and

the error detector circuit includes an exclusive-or (XOR) gate coupled with the second multiplier and the third multiplier, and configured to generate the error signal based on an exclusive-or operation on the D-vector and P-vector.