Optimal Data Transmission in Backscatter Communication for Passive Sensing Systems

CRFID’s duty cycle is relative to storage capacitor charging and discharging process. After a period of sleep and the dormant process to obtain the RF energy from the reader, the capacitor begins to charge, then the voltage begins to rise^{[ 15 ]}, when it reaches the working voltage, CRFID goes into work mode and starts sending data to the reader. As one version of CRFID, for example, WISP, with the MSP430’s operating voltage of 1.8 V, when the voltage rises to 2.0 V, it can wake up the WISP sensor to send data. The charging voltage of the WISP node is in the following:

(1) $$U_{(t)}=U_{(0)}{\text{e}}^{-t/\tau }+U_{\max }(1-{\text{e}}^{-t/\tau })$$

where $U_{(0)}$ is the voltage at the beginning of the charge, $\tau$ is the time constant of the RC circuit, and $U_{\max }$ is the maximum charge voltage that the capacitor can achieve under the current energy capture conditions. With a change of environmental conditions, the maximum charge voltage that the capacitor can achieve will change. When the energy capture conditions are poor, $U_{\max }$ reduces and $\tau$ increases. This means the WISP tag requires a longer charging time to reach the radio of operating voltage. In order to adapt to different energy capture conditions, this paper proposes a dynamic selection of frame length and charging time for WISP tags. Another advantage of dynamically selecting the frame length is the ability to adapt to the changing channel quality. By combining the optimized frame length, coding redundancy, and charging time, the throughput of the backscatter communication is improved.

At first, we need the expression of the effective throughput for specific energy capture conditions and channel conditions. When the charging time is t, the amount of energy WISP can capture is

(2) $$E(t)=\frac{1}{2}C(U_{(t)}^2-U_{\min }^2)$$

where C represents the size of the charge capacitance in WISP and $U_{\min }$ represents the minimum operating voltage of the WISP. Here we assume that the average energy consumed by each bit of data that WISP tag receives or sends is $E_{\mathrm{bit}}$ . We can then calculate the energy WISP needs to send a frame in each duty cycle,

(3) $$E_{\mathrm{frame}}=E_{\mathrm{bit}}\times (\alpha +l+\omega )$$

where $\alpha$ represents the frame header length and $l$ represents the length of the data frame.

We are here to meet one of the conditions that is limited by the capture of energy,

(4) $$E_{(t_k)}⩾{\chi }_k{E}_{\mathrm{frame}}$$

where $E_{(t_k)}$ represents the energy captured by the tag node when charging time of the k-th cycle is $t$ , ${\chi }_k$ indicates the number of data frames transmitted by WISP during the k-th duty cycle, According to Eq. ( 4 ), we can then conclude that the number of data frames sent by WISP in the $k$ -th duty cycle satisfies the following condition:

(5) $${\chi }_k⩽\min \{\lfloor \frac{E_{(t_k)}}{E_{\text{frame}}}\rfloor ,N+C-\underset{i=1}{\overset{k-1}{\sum }}{\chi }_i\}$$

where $\lfloor \cdot \rfloor$ is rounded down, ${\displaystyle {\sum }_{i=1}^{k-1}}{\chi }_i$ indicates the total number of data frames transmitted before the $k-1$ duty cycles. Substituting Eqs. ( 2 ) and ( 3 ) into Eq. ( 4 ), while letting $U_{(0)}=U_{\min }$ in Eq. ( 2 ), we can get the k-th duty cycle of the charging time, satisfying the following inequality:

(6) $$t_k⩾-\tau \ln \frac{\sqrt{2{\chi }_k{E}_{\text{bit}}(\alpha +l+\omega )|C+U_{\min }^2}-U_{\max }}{U_{\min }-U_{\max }}$$

According to Eq. ( 6 ), we can send each frame of data, the charging time is described in the following formula:

(7) $$\bar{t}=\frac{t_k}{{\chi }_k}$$

The main idea of dynamic frame length adjustment control is to adjust the frame length and coding redundancy according to the throughput measurement of the reader at runtime and to control the charging time, thus improving the throughput of the data. The specific ideas are as follows. The first WISP cycle tracks the storage capacitor voltage, gives access to dynamic $U_{\max }$ and $\tau$ values. When WISP moves to the reader and has data to be sent to the reader, N source data frames of length $l$ are extracted from the cache, encoded as N + M frames, where $M$ is the number of redundant frames, the initial frame length is set to 32 bits. When a round of data transmission is completed, the reader records the time taken and the effective throughput during this time, and compares the current throughput with the previous round.

If the current throughput is higher than that of the previous round, the return ACK informs the tags to increase the transmit frame length; otherwise, the transmit frame length is reduced, the unit of the frame length is increased or decreased by 16 bits each time. In order to avoid frequent changes in the frame length, we introduce an adjustment parameter, if the increasing or decreasing proportion of the throughput does not exceed this threshold, the frame length is not changed^{[ 16 ]}. At the same time, after receiving the reader’s specified frame length, the WISP^{[ 17 ]} tag selects the number of frames transmitted in the current working period according to the current energy capture condition, enters the sleep charging state after sending, waits for the next work cycle to transmit the data. This continues until the current round of data frames are all finished, or until the reader sends a QueryRep frame. After a round of transmission, if the transmission fails to send the number of redundant frames $M$ , the reader checks the current actual number of redundant frames according to Eq. ( 8 ) on the next round,

(8) $$M_{\text{next}}=\lceil (1-\gamma )\times M_{\text{avg}}+\gamma \times M_{\text{cur}}+M_{\min }\rceil ,$$ $$\gamma \in (0,1),M_{\text{next}}\in (M_{\min },M_{\max })$$

where $M_{\text{avg}}$ indicates the average number of redundant frames sent historically, $M_{\text{cur}}$ indicates the number of redundant frames sent in this round, $M_{\min }$ and $M_{\max }$ are the minimum and maximum number of redundant frames, respectively, and the weight coefficient $\lceil \rceil$ indicates rounding up.

The simulation of the CRC check capability is controlled by the program. In the program, the control model runs 500 times. After each simulation model running, it is checked for an error code. We look at whether an error has occurred, using $e$ and $e_1$ to represent the actual number of errors of the final statistics and the number of errors of CRC’ s detection, respectively. In this way, we can count the error rate of the CRC check, which is calulated as follows:

(9) $$\eta =\frac{e_1}{e}\times 100\%$$

If the reader requests a fixed-length response data from WISP, we use different redundant CRC check codes to analyze it. The results obtained by this method are shown in Table 4 .

It can be seen from the results that 100% error detection can be achieved when the number of CRC check bits $r$ and the bit length of the total frames satisfy the relationship $n=2^r-1$ . For example, when $r=8$ , according to the relationship, it can detect 127 bits of data, while the simulation detect 96 bits, thus less than 127 bits, the error rate is 100%. However, if $r$ is less than the required value of the relationship, the CRC check will have a large leak rate. For example, when $r=4$ , it can check a maximum of 15 bits; if the 4-bit parity is added to the 100-bit data behind, this will result in 12% errors missed. The above results show that it is not necessary to select a longer CRC check if the selected CRC bit number $r$ satisfies the actual job requirement. Having the appropriate number of CRC bits ensures no errors are missed, while making the data transmission efficiency as high as possible.

Therefore, in actual RFID applications, CRC-8 check can be selected if the length of a frame of data is less than 256 bits. If the length of the data is greater than 256 bits and less than 64 Kb, the CRC-16 check can be selected. In actual RFID applications, there is essentially no data frame with a length greater than 64 Kb, so there is no need for a 32-bit CRC to be introduced in an RFID application. According to the above problem, our main aim in this paper is to dynamically adjust the WISP data frame length and the corresponding number of CRC checks, to improve the data transmission rate, thereby increasing the throughput and reliability of the entire backscatter link.

At present, the dynamic data frame length used in WISP tags is not a new topic in academia, but there are some gaps in the use of redundant check bits for different data frame lengths. The resulting of large time interval manifests an excessively long response latency, which will lead directly to the failure of data transmission.

The C1G2 protocol for RFID tags is designed to inventory large tag populations over a number of communication rounds. To realize this protocol, an RFID must traverse a simple state machine and respond appropriately to a set of reader commands.

Following the identification procedure, the RFID reader further requests for more data in the read procedure. The RFID reader establishes a handshake by {ReqRN, Handle}. Similar to {Query, RN16} handshake in the identification procedure, {ReqRN, Handle} is short, serving the collision arbitration purpose. The reader then acknowledges the 16-bit handle and requests a large amount of data. According to the C1G2 standard, the reader is able to collect up to 510 bytes per {Read, Data} exchange. Different from {ACK, EPC}, which is primitive in the identification procedure, the {Read, Data} primitive in the read procedure is tailored for bulk data transfer with variable lengths. But in the previous code test, when more than 8 bytes of data are requested in the read procedure, the success rates suddenly drop to zero even within a small interrogation range.

The data frame format used in this paper is similar to that in the EPC Gen2 protocol. The length of the data load in the EPC Gen2 protocol is defined by the EPC length field in the protocol control field, in the range of 16 - 496 bits, and is an integer multiple of 16 bits, in this case, the number of bits is incremented or decremented by 8 bits each time. Considering that the current device only supports 96 bits, the maximum is 01100, for EPC Gen2, the ACK frame format specified by the protocol consists of 2 bits command and 16 bits random number bits. However, in the algorithm presented in this paper, the frame length and the number of redundant frames need to be adjusted according to the channel condition and the capture condition, which is fed back to the WISP tag by the reader through the ACK frame. Therefore, it involves two additional 5 bits command in the ACK, stating the frame length of the next round and the number of redundant frames in the ACK frame format.

The combination RN16+CRC16 is the most commonly used response of the tag in the uplink, but we have already explained that the CRC-16 verification for a data frame whose length of less than 256 bits involves a lot of energy waste. When the tag returns RN16, CRC-5 is sufficient to complete the inspection task, which saves energy and reduces MCU computing time. For the dynamic EPC data frame, the different checksum bits are determined according to the different frame lengths. Here, assuming that $x$ is the length (bit) of the EPC data to be transmitted, the throughput of the channel and the power of the storage capacitor are determined by the data frame length. The following formula determines the checksum bits for calibrating the frame length of the next round of transmission:

(10) $$\text{CRC-}X=\{\begin{array}{cccccccccccccccccccc}\text{CRC-}5,\hfill & x<32;\hfill & & & & & & & & & & & & & & & & & & \\ \text{CRC-}8,\hfill & 32⩽x<256;\hfill & & & & & & & & & & & & & & & & & & \\ \text{CRC-}16,\hfill & 256⩽x<6400\hfill & & & & & & & & & & & & & & & & & & \end{array}$$

The key to the CRC check is to generate a cyclic redundancy code based on the generator polynomial and the information to be sent. According to the EPC Gen2 protocol, the cyclic redundancy code is suitable for the small memory space (the code takes up less memory space), but the whole process takes a long time and the real-time performance is poor. The reader and tag have two CRC checks: CRC-5 and CRC-16. For CRC-5, this paper presents a byte-by-table method. Since the amount of data exchanged between the reader and the tag is not an integer multiple of the byte, it is not sufficient to find the CRC in bytes. We use a look-up table to solve this problem, with the basic idea that CRC-5 will be translated into CRC-8, then the corresponding generation polynomial is found; according to the generated polynomial, cycle redundancy table is generated, and then through the look-up table, CRC-8 is found and finally converted back to CRC-5.

In the beginning, the principle of converting CRC-5 to CRC-8 is shown in Fig. 5 . Assuming that $A(x)$ is a polynomial of the information symbol to be transmitted by the UHF reader^{[ 18 ]} and $G(x)$ is a polynomial of CRC-5 (short form 0x09), in the formula for finding the cyclic redundancy code,

(11) $$\frac{2^{16}A(x)}{G_5(x)}=\frac{2^{16}A(x)x^3}{G_5(x)x^3}$$

let $\bar{A}(x)=A(x)x^3$ and $G_8(x)=G_5(x)x^3=x^8+x^6+x^3$ , then Eq. ( 11 ) can be converted to

(12) $$\frac{2^{16}A(x)}{G_5(x)}=\frac{2^{16}\bar{A}(x)}{G_8(x)}$$

(13) $$\bar{A}(x)=2^{8n}{\bar{A}}_n(x)+2^{8(n-1)}{\bar{A}}_{n-1}(x)+\mathrm{\cdots }+2^0\bar{A}x$$

10.26599/TST.2019.9010037.F5 Fig. 5Principle of converting CRC-5 to CRC-8 flow chart.

According to the principle of the look-up table formula we can deduce Eq. ( 13 ),

(14) $${\displaystyle \frac{2^8{A}^{\prime }(x)}{G_8(x)}}=2^{8n}{Q}_n(x)+2^{8(n-1)}\times$$ $${\displaystyle \frac{2^8[R_n(x)+A{}_{n-1}{}^{\prime }(x)]}{G_8(x)}}+\mathrm{\cdots }+{\displaystyle \frac{2^0{2}^8{A}^{\prime }(x)}{G_8(x)}}$$

where $Q_n(x)$ and $R_n(x)$ are complete and residual functions of CRC-8 for $\bar{A}(x)$ , respectively. From Eq. ( 14 ), the redundant period is the XOR operation of the period of the redundant code and the redundant code, and then the CRC-8 cycle redundancy table is generated, from which we can get the generated polynomial of CRC-8. Generator polynomial should be written to 0x48 with the abbreviated version 0x09 of the CRC-5, which is written as 0x80 of the CRC-8.

CRFID’s duty cycle is relative to storage capacitor charging and discharging process. After a period of sleep and the dormant process to obtain the RF energy from the reader, the capacitor begins to charge, then the voltage begins to rise^{[ 15 ]}, when it reaches the working voltage, CRFID goes into work mode and starts sending data to the reader. As one version of CRFID, for example, WISP, with the MSP430’s operating voltage of 1.8 V, when the voltage rises to 2.0 V, it can wake up the WISP sensor to send data. The charging voltage of the WISP node is in the following:

(1) $$U_{(t)}=U_{(0)}{\text{e}}^{-t/\tau }+U_{\max }(1-{\text{e}}^{-t/\tau })$$

where $U_{(0)}$ is the voltage at the beginning of the charge, $\tau$ is the time constant of the RC circuit, and $U_{\max }$ is the maximum charge voltage that the capacitor can achieve under the current energy capture conditions. With a change of environmental conditions, the maximum charge voltage that the capacitor can achieve will change. When the energy capture conditions are poor, $U_{\max }$ reduces and $\tau$ increases. This means the WISP tag requires a longer charging time to reach the radio of operating voltage. In order to adapt to different energy capture conditions, this paper proposes a dynamic selection of frame length and charging time for WISP tags. Another advantage of dynamically selecting the frame length is the ability to adapt to the changing channel quality. By combining the optimized frame length, coding redundancy, and charging time, the throughput of the backscatter communication is improved.

At first, we need the expression of the effective throughput for specific energy capture conditions and channel conditions. When the charging time is t, the amount of energy WISP can capture is

(2) $$E(t)=\frac{1}{2}C(U_{(t)}^2-U_{\min }^2)$$

where C represents the size of the charge capacitance in WISP and $U_{\min }$ represents the minimum operating voltage of the WISP. Here we assume that the average energy consumed by each bit of data that WISP tag receives or sends is $E_{\mathrm{bit}}$ . We can then calculate the energy WISP needs to send a frame in each duty cycle,

(3) $$E_{\mathrm{frame}}=E_{\mathrm{bit}}\times (\alpha +l+\omega )$$

where $\alpha$ represents the frame header length and $l$ represents the length of the data frame.

We are here to meet one of the conditions that is limited by the capture of energy,

(4) $$E_{(t_k)}⩾{\chi }_k{E}_{\mathrm{frame}}$$

where $E_{(t_k)}$ represents the energy captured by the tag node when charging time of the k-th cycle is $t$ , ${\chi }_k$ indicates the number of data frames transmitted by WISP during the k-th duty cycle, According to Eq. ( 4 ), we can then conclude that the number of data frames sent by WISP in the $k$ -th duty cycle satisfies the following condition:

(5) $${\chi }_k⩽\min \{\lfloor \frac{E_{(t_k)}}{E_{\text{frame}}}\rfloor ,N+C-\underset{i=1}{\overset{k-1}{\sum }}{\chi }_i\}$$

where $\lfloor \cdot \rfloor$ is rounded down, ${\displaystyle {\sum }_{i=1}^{k-1}}{\chi }_i$ indicates the total number of data frames transmitted before the $k-1$ duty cycles. Substituting Eqs. ( 2 ) and ( 3 ) into Eq. ( 4 ), while letting $U_{(0)}=U_{\min }$ in Eq. ( 2 ), we can get the k-th duty cycle of the charging time, satisfying the following inequality:

(6) $$t_k⩾-\tau \ln \frac{\sqrt{2{\chi }_k{E}_{\text{bit}}(\alpha +l+\omega )|C+U_{\min }^2}-U_{\max }}{U_{\min }-U_{\max }}$$

According to Eq. ( 6 ), we can send each frame of data, the charging time is described in the following formula:

(7) $$\bar{t}=\frac{t_k}{{\chi }_k}$$

The main idea of dynamic frame length adjustment control is to adjust the frame length and coding redundancy according to the throughput measurement of the reader at runtime and to control the charging time, thus improving the throughput of the data. The specific ideas are as follows. The first WISP cycle tracks the storage capacitor voltage, gives access to dynamic $U_{\max }$ and $\tau$ values. When WISP moves to the reader and has data to be sent to the reader, N source data frames of length $l$ are extracted from the cache, encoded as N + M frames, where $M$ is the number of redundant frames, the initial frame length is set to 32 bits. When a round of data transmission is completed, the reader records the time taken and the effective throughput during this time, and compares the current throughput with the previous round.

If the current throughput is higher than that of the previous round, the return ACK informs the tags to increase the transmit frame length; otherwise, the transmit frame length is reduced, the unit of the frame length is increased or decreased by 16 bits each time. In order to avoid frequent changes in the frame length, we introduce an adjustment parameter, if the increasing or decreasing proportion of the throughput does not exceed this threshold, the frame length is not changed^{[ 16 ]}. At the same time, after receiving the reader’s specified frame length, the WISP^{[ 17 ]} tag selects the number of frames transmitted in the current working period according to the current energy capture condition, enters the sleep charging state after sending, waits for the next work cycle to transmit the data. This continues until the current round of data frames are all finished, or until the reader sends a QueryRep frame. After a round of transmission, if the transmission fails to send the number of redundant frames $M$ , the reader checks the current actual number of redundant frames according to Eq. ( 8 ) on the next round,

(8) $$M_{\text{next}}=\lceil (1-\gamma )\times M_{\text{avg}}+\gamma \times M_{\text{cur}}+M_{\min }\rceil ,$$ $$\gamma \in (0,1),M_{\text{next}}\in (M_{\min },M_{\max })$$

where $M_{\text{avg}}$ indicates the average number of redundant frames sent historically, $M_{\text{cur}}$ indicates the number of redundant frames sent in this round, $M_{\min }$ and $M_{\max }$ are the minimum and maximum number of redundant frames, respectively, and the weight coefficient $\lceil \rceil$ indicates rounding up.

The simulation of the CRC check capability is controlled by the program. In the program, the control model runs 500 times. After each simulation model running, it is checked for an error code. We look at whether an error has occurred, using $e$ and $e_1$ to represent the actual number of errors of the final statistics and the number of errors of CRC’ s detection, respectively. In this way, we can count the error rate of the CRC check, which is calulated as follows:

(9) $$\eta =\frac{e_1}{e}\times 100\%$$

If the reader requests a fixed-length response data from WISP, we use different redundant CRC check codes to analyze it. The results obtained by this method are shown in Table 4 .

It can be seen from the results that 100% error detection can be achieved when the number of CRC check bits $r$ and the bit length of the total frames satisfy the relationship $n=2^r-1$ . For example, when $r=8$ , according to the relationship, it can detect 127 bits of data, while the simulation detect 96 bits, thus less than 127 bits, the error rate is 100%. However, if $r$ is less than the required value of the relationship, the CRC check will have a large leak rate. For example, when $r=4$ , it can check a maximum of 15 bits; if the 4-bit parity is added to the 100-bit data behind, this will result in 12% errors missed. The above results show that it is not necessary to select a longer CRC check if the selected CRC bit number $r$ satisfies the actual job requirement. Having the appropriate number of CRC bits ensures no errors are missed, while making the data transmission efficiency as high as possible.

Therefore, in actual RFID applications, CRC-8 check can be selected if the length of a frame of data is less than 256 bits. If the length of the data is greater than 256 bits and less than 64 Kb, the CRC-16 check can be selected. In actual RFID applications, there is essentially no data frame with a length greater than 64 Kb, so there is no need for a 32-bit CRC to be introduced in an RFID application. According to the above problem, our main aim in this paper is to dynamically adjust the WISP data frame length and the corresponding number of CRC checks, to improve the data transmission rate, thereby increasing the throughput and reliability of the entire backscatter link.

At present, the dynamic data frame length used in WISP tags is not a new topic in academia, but there are some gaps in the use of redundant check bits for different data frame lengths. The resulting of large time interval manifests an excessively long response latency, which will lead directly to the failure of data transmission.

The C1G2 protocol for RFID tags is designed to inventory large tag populations over a number of communication rounds. To realize this protocol, an RFID must traverse a simple state machine and respond appropriately to a set of reader commands.

Following the identification procedure, the RFID reader further requests for more data in the read procedure. The RFID reader establishes a handshake by {ReqRN, Handle}. Similar to {Query, RN16} handshake in the identification procedure, {ReqRN, Handle} is short, serving the collision arbitration purpose. The reader then acknowledges the 16-bit handle and requests a large amount of data. According to the C1G2 standard, the reader is able to collect up to 510 bytes per {Read, Data} exchange. Different from {ACK, EPC}, which is primitive in the identification procedure, the {Read, Data} primitive in the read procedure is tailored for bulk data transfer with variable lengths. But in the previous code test, when more than 8 bytes of data are requested in the read procedure, the success rates suddenly drop to zero even within a small interrogation range.

The data frame format used in this paper is similar to that in the EPC Gen2 protocol. The length of the data load in the EPC Gen2 protocol is defined by the EPC length field in the protocol control field, in the range of 16 - 496 bits, and is an integer multiple of 16 bits, in this case, the number of bits is incremented or decremented by 8 bits each time. Considering that the current device only supports 96 bits, the maximum is 01100, for EPC Gen2, the ACK frame format specified by the protocol consists of 2 bits command and 16 bits random number bits. However, in the algorithm presented in this paper, the frame length and the number of redundant frames need to be adjusted according to the channel condition and the capture condition, which is fed back to the WISP tag by the reader through the ACK frame. Therefore, it involves two additional 5 bits command in the ACK, stating the frame length of the next round and the number of redundant frames in the ACK frame format.

The combination RN16+CRC16 is the most commonly used response of the tag in the uplink, but we have already explained that the CRC-16 verification for a data frame whose length of less than 256 bits involves a lot of energy waste. When the tag returns RN16, CRC-5 is sufficient to complete the inspection task, which saves energy and reduces MCU computing time. For the dynamic EPC data frame, the different checksum bits are determined according to the different frame lengths. Here, assuming that $x$ is the length (bit) of the EPC data to be transmitted, the throughput of the channel and the power of the storage capacitor are determined by the data frame length. The following formula determines the checksum bits for calibrating the frame length of the next round of transmission:

(10) $$\text{CRC-}X=\{\begin{array}{cccccccccccccccccccc}\text{CRC-}5,\hfill & x<32;\hfill & & & & & & & & & & & & & & & & & & \\ \text{CRC-}8,\hfill & 32⩽x<256;\hfill & & & & & & & & & & & & & & & & & & \\ \text{CRC-}16,\hfill & 256⩽x<6400\hfill & & & & & & & & & & & & & & & & & & \end{array}$$

The key to the CRC check is to generate a cyclic redundancy code based on the generator polynomial and the information to be sent. According to the EPC Gen2 protocol, the cyclic redundancy code is suitable for the small memory space (the code takes up less memory space), but the whole process takes a long time and the real-time performance is poor. The reader and tag have two CRC checks: CRC-5 and CRC-16. For CRC-5, this paper presents a byte-by-table method. Since the amount of data exchanged between the reader and the tag is not an integer multiple of the byte, it is not sufficient to find the CRC in bytes. We use a look-up table to solve this problem, with the basic idea that CRC-5 will be translated into CRC-8, then the corresponding generation polynomial is found; according to the generated polynomial, cycle redundancy table is generated, and then through the look-up table, CRC-8 is found and finally converted back to CRC-5.

In the beginning, the principle of converting CRC-5 to CRC-8 is shown in Fig. 5 . Assuming that $A(x)$ is a polynomial of the information symbol to be transmitted by the UHF reader^{[ 18 ]} and $G(x)$ is a polynomial of CRC-5 (short form 0x09), in the formula for finding the cyclic redundancy code,

(11) $$\frac{2^{16}A(x)}{G_5(x)}=\frac{2^{16}A(x)x^3}{G_5(x)x^3}$$

let $\bar{A}(x)=A(x)x^3$ and $G_8(x)=G_5(x)x^3=x^8+x^6+x^3$ , then Eq. ( 11 ) can be converted to

(12) $$\frac{2^{16}A(x)}{G_5(x)}=\frac{2^{16}\bar{A}(x)}{G_8(x)}$$

(13) $$\bar{A}(x)=2^{8n}{\bar{A}}_n(x)+2^{8(n-1)}{\bar{A}}_{n-1}(x)+\mathrm{\cdots }+2^0\bar{A}x$$

10.26599/TST.2019.9010037.F5 Fig. 5Principle of converting CRC-5 to CRC-8 flow chart.

According to the principle of the look-up table formula we can deduce Eq. ( 13 ),

(14) $${\displaystyle \frac{2^8{A}^{\prime }(x)}{G_8(x)}}=2^{8n}{Q}_n(x)+2^{8(n-1)}\times$$ $${\displaystyle \frac{2^8[R_n(x)+A{}_{n-1}{}^{\prime }(x)]}{G_8(x)}}+\mathrm{\cdots }+{\displaystyle \frac{2^0{2}^8{A}^{\prime }(x)}{G_8(x)}}$$

where $Q_n(x)$ and $R_n(x)$ are complete and residual functions of CRC-8 for $\bar{A}(x)$ , respectively. From Eq. ( 14 ), the redundant period is the XOR operation of the period of the redundant code and the redundant code, and then the CRC-8 cycle redundancy table is generated, from which we can get the generated polynomial of CRC-8. Generator polynomial should be written to 0x48 with the abbreviated version 0x09 of the CRC-5, which is written as 0x80 of the CRC-8.