Application of puncturing of a CRC code for WLANs Monisha Ghosh, Senior Member, IEEE and Frank LaSita, Member, IEEE

{monisha.ghosh, frank.lasita}@interdigital.com InterDigital Communications, LLC

Abstract— In some cases, like the proposed IEEE 802.11ah draft specification [3], there arises a need to have a short Cyclic Redundancy Check (CRC) code to detect errors on the information bits in the various signal fields. While an optimum CRC of the required length can be chosen for this, in order to minimize complexity it may be desirable to “puncture” an existing CRC generator to reduce the number of parity bits transmitted. In this letter we evaluate puncturing of a CRC-8 generator to generate a 4-bit CRC such that the minimum Hamming distance of the resultant code is 2.

Index Terms—CRC, puncturing, 802.11

I. INTRODUCTION YCLIC Redundancy Codes (CRCs) are widely used in many wired and wireless standard specifications as a

means of error detection. The simplest CRC is a 1-bit even-parity as used for example in the IEEE 802.11a specification [1] to protect the information bits in the signal field (SIG bits). The CRC enables the receiver to detect whether the SIG bits were received error-free, or not. As such, it is only used for error detection, and not error correction. The 1-bit parity check ensures that all single bit errors will be detected. Since the SIG bits carry information such as the Modulation Coding Scheme (MCS) used and number of bytes in the following data packet, it is very important that these bits be decoded correctly. The SIG field is usually transmitted at the lowest supported MCS for maximum robustness.

CRCs of various lengths have been extensively studied and computer searches have been performed to determine optimal generator polynomials. However, the minimum Hamming distance of a CRC depends not only on the generator polynomial but also on the length of the information bits being protected by the CRC. It has been shown in [2] that a number of commonly used CRC generators are in fact suboptimal from a minimum Hamming distance point of view for certain data lengths.

Recently, IEEE 802.11ah [3] has begun developing a physical (PHY) and medium access control (MAC) layer specification for the Sub-1 GHz non television white spaces (non-TVWS) frequencies in the 902-928 MHz [3]. The PHY is proposed to be a down clocked version of the IEEE 802.11ac PHY which is the standard for the 5 GHz ISM band [4]. 802.11ac specifies an 8-bit CRC-8 for the SIG field with generator polynomial 𝑥! + 𝑥! + 𝑥 + 1. However, the SIG field in 802.11ah [3] is redefined to include only a

4-bit CRC. In order to keep the implementation complexity low, it was proposed in [5] to just take the 4 least significant bits (LSBs) of the 8-bit parity generated by the CRC-8 polynomial. We show in this paper that this choice leads to a CRC with minimum Hamming distance of 1, i.e. there are certain single bit error patterns that cannot be detected. We also perform a codeword weight analysis and demonstrate that there are other puncturing choices that can deliver a 4-bit CRC with Hamming distance 2 and hence are more suitable than the 4 LSBs.

The paper is organized as follows: Section II provides a brief background of the IEEE 802.11ah specification. Section III develops the codeword weight analysis and Section IV presents analytical and simulation results using the puncture patterns in Section III. Finally, Section V concludes the paper.

II. THE IEEE 802.11AH SPECIFICATION In order to accommodate different spectral allocations

worldwide, IEEE 802.11ah specifies two mandatory modes of operation: a 1 MHz mode and a 2 MHz mode in the 900 MHz frequency band. One of the primary functional requirements of the 802.11ah specification is that it shall support a range of 1 km with at least 100 kbps throughput. This capability will enable applications like remote monitoring and smart metering. In order to obtain this range, the specification proposes a new MCS called “MCS0-Rep2” which is a mode with rate ½ convolutional coding using the standard [133,171] convolutional code, with BPSK modulation, followed by a repetition of the coded symbol. This mode is specified only for the 1 MHz mode of operation. The lowest data rate for the 2 MHz mode is the “MCS0” mode which is simply rate ½ BPSK. Since the SIG field is transmitted with the lowest supported data rate, the 1 MHz mode SIG field uses MCS0-Rep2, while the 2 MHz mode SIG field uses MCS0.

A. 1 MHz SIG Field The 1 MHz mode uses a 32-point FFT with a carrier

spacing of 31.25 kHz. Of the 32 available frequency carriers, 24 are used for data, 2 are in-band pilots, one is the DC which is not used and the rest are band-edge carriers. The SIG field is defined to have 26 bits of information, followed by a 4-bit CRC and 6 tail bits (to terminate the convolution code) for a total of 36 bits. Using MCS0-Rep2, the SIG field thus requires 6 OFDM symbols. In [5] it was proposed that the 4-bit CRC be derived from the CRC-8

C

polynomial by using only the 4 LSBs.

B. 2 MHz SIG Field The 2 MHz mode uses a 64-point FFT with a carrier

spacing of 31.25 kHz. Of the 64 available frequency carriers, 52 are used for data, 4 are in-band pilots, one is the DC which is not used and the rest are band-edge carriers. Depending on the transmission mode (multi-user or single user), there are two SIG fields specified: SIGA and SIGB. The SIGA field is defined to have 38 bits of information, followed by a 4-bit CRC and 6 tail bits (to terminate the convolution code) for a total of 48 bits. Using MCS0, the SIGA field thus requires 2 OFDM symbols. In [5] it was proposed that the 4-bit CRC in SIGA be derived from the 8-bit CRC generated by the CRC-8 polynomial by using only the 4 LSBs. The SIGB field carries less information bits and hence has room for the full 8-bit CRC derived from CRC-8.

III. CODEWORD WEIGHT ANALYSIS Let us assume that there are 𝑁 information bits. Then, we

can form the generator matrix, G, for the polynomial 𝑥! + 𝑥! + 𝑥 + 1 as 𝐺 =   𝐼! 𝑃 where 𝐼! is an 𝑁  ×𝑁 identity matrix and 𝑃 is a 𝑁  ×8 parity matrix. The corresponding parity check matrix, H, can then be written as 𝐻 =   𝑃! 𝐼! , where 𝐼! is an 8  ×8 identity matrix.

In order to reduce the number of parity bits from 8 to 4, 4 rows are removed from matrix 𝑃! to get a 4  ×𝑁 matrix (or 4 columns may be removed from 𝑃 ) to obtain a matrix denoted as 𝑃!!. The new parity check matrix describing this punctured code is then given by 𝐻! =   𝑃!! 𝐼! , where 𝐼! is an 4  ×4 identity matrix.

In order to avoid a minimum Hamming distance of 1 for a code created as described above, a requirement is that no column of the parity check matrix be composed of all zeros [6]. Therefore, when 4 rows are removed in the example above, it is desirable for the matrix 𝐻! not to have an all-zero column. The complete Hamming Weight (HW) distribution of a code may be derived from the parity check matrix as follows: the number of codewords with HW = i is the number of combinations of i columns of the parity check matrix such that a linear combination is equal to a zero vector [6].

For the CRC-8 code, there are 70 combinations of 4-bit puncturing patterns, i.e., there are 70 ways to puncture 8 parity bits to obtain 4 parity bits. Let 𝑐!𝑐!𝑐!𝑐!𝑐!𝑐!𝑐!𝑐!  denote the 8 parity bits generated from the CRC-8 polynomial where 𝑐! is the least significant bit (LSB) and 𝑐! is the most significant bit (MSB). Table 1 shows the HW distribution for 𝑁 = 26 information bits when a 4-bit CRC is derived by puncturing the CRC-8 code. Only those combinations that have minimum HW = 2 are shown, other than the MSBs and LSBs which have minimum HW of 1. As comparison, the HW distribution of the optimal 4-bit CRC generator using the CRC-4 polynomial 𝑥! + 𝑥 + 1  is also shown. Table 2, on the other hand, shows a HW distribution for 𝑁 = 38 information bits

when a 4-bit CRC is derived by puncturing the CRC-8 code.

CRC bits HW =

1

HW =

2

HW =

3

HW =

4

HW =

5

c7c6c5c4

(LSBs)

3 28 254 1663 8839

c3c2c1c0

(MSBs)

2 24 247 1687 8969

c3c2c1c0

(optimal

4-bit

CRC)

0 15 280 1785 8736

c5c2c1c0 0 31 247 1658 9029

c5c3c2c0 0 34 247 1635 9029

c5c4c1c0 0 27 265 1670 8879

c5c4c2c0 0 31 252 1657 8976

c5c4c3c1 0 29 264 1658 8890

𝑁 c6c3c1c0 0 35 246 1648 8992

c6c3c2c1 0 29 260 1662 8930

c6c4c3c1 0 31 263 1646 8901

c6c5c2c0 0 33 259 1640 8909

c6c5c3c2 0 32 259 1645 8925

c7c4c2c1 0 33 248 1646 9018

c7c4c3c0 0 29 264 1658 8890

c7c4c3c1 0 29 260 1662 8930

c7c5c2c0 0 34 255 1635 8949

c7c5c4c1 0 32 266 1629 8874

c7c6c3c2 0 29 262 1674 8896

Table 1: HW distribution when a 4-bit CRC is derived from the CRC-8 code for 26 information bits.

CRC

Bits

HW =

1

HW =

2

HW =

3

HW =

4

HW = 5

c7c6c5c4

(LSBs)

3 52 707 6932 53267

c3c2c1c0

(MSBs)

3 51 697 6947 53377

c3c2c1c0

(optimal

4-bit

CRC)

0 39 765 7101 52761

c5c3c2c0 0 65 692 6867 53564

c5c4c2c0 0 58 711 6942 53325

c6c4c3c1 0 56 720 6938 53292

c6c5c2c0 0 62 704 6894 53426

c6c5c3c2 0 58 717 6906 53339

c7c4c2c1 0 65 699 6865 53501

c7c5c4c1 0 60 721 6876 53323

Table 2: HW distribution when a 4-bit CRC is derived from the CRC-8 code for 38 information bits.

As shown in Tables 1 and 2, if the 4 CRC bits are chosen to be 𝑐!𝑐!𝑐!𝑐! (i.e., the four LSBs) or bits 𝑐!𝑐!𝑐!𝑐! (i.e., the four MSBs), then the minimum Hamming distance is 1, i.e. there are 2 to 3 single-bit error patterns that will not be detected by the resulting code. Hence these are not good choices for a CRC. Instead, there are other choices shown in Tables 1 and 2 that deliver a minimum Hamming distance of 2 and hence are better choices for a CRC.

We also observe that as the number of data bits, 𝑁 , increases, the number of combinations that result in a code with Hamming distance 2 decreases, e.g. from 16 to 7 when 𝑁 increases from 26 to 38. Figure 1 shows that if the number of information bits exceeds 56, then no puncturing combination will yield a 4-bit CRC with minimum Hamming distance 2 that is derived from CRC-8. Instead one would have to puncture a longer CRC. One such possibility is the CRC-32, with polynomial 𝑥!" + 𝑥!" +𝑥!" + 𝑥!! + 𝑥!" + 𝑥!" + 𝑥!! + 𝑥!" + 𝑥! + 𝑥! + 𝑥! +𝑥! + 𝑥! + 𝑥 + 1 that is used in 802.11ah as the frame-check sequence. Figure 1 shows that it is possible to obtain a 4-bit CRC by puncturing the CRC-32 code for a larger number of information bits, up to 148.

Figure 1 Generating a 4-bit CRC by puncturing CRC-8 and CRC-32

IV. RESULTS The probability of false positives (which is the same as

the probability of undetected error) is the probability that the CRC indicates that the SIG bits have been received correctly when they have not, and should be as low as possible. A false positive detection on the CRC leads to unsuccessful attempts at trying to decode the data packet that follows the SIG, with erroneous information for example on MCS and length from the incorrectly decoded SIG bits. This leads to wasted power consumption.

It is possible to use the closed-form expression for probability of undetected error, 𝑃!", in (1) below for the additive white Gaussian noise (AWGN) case where it is assumed that the information bits are uncoded, 𝑃! is the probability of bit error in AWGN as a function of signal-to-noise-ratio (SNR) given by 𝑃! = 0.5erfc 0.5SNR , erfc is the complementary error function, HW(𝑖) is the Hamming weight distribution of the code and 𝑁!  is the total number of coded bits (information + parity) [6].

𝑃!" = HW(𝑖)𝑃!! 1 − 𝑃! !!!!!

!!! (1) With 𝑁! = 42, we evaluate equation (1) above and plot

the results in Figure 2 for the optimal CRC-4 code, the CRC-8 code with the 4 LSB puncture pattern 𝑐!𝑐!𝑐!𝑐! and the CRC-8 code with puncture pattern 𝑐!𝑐!𝑐!𝑐!. It is clear that the 4 LSB puncture pattern performs significantly worse than the pattern 𝑐!𝑐!𝑐!𝑐! which is very close in performance to the optimal CRC-4. However, it may be argued that bit interleaved coded modulation (BICM) is used on the SIG bits in 802.11ah and hence the result for uncoded bits in Figure 2 does not apply there. Hence, in order to evaluate the various puncture patterns obtained from Section III using BICM, simulations were performed to obtain the probability of false positives.

Figure 2 Theoretical 𝑃𝑢𝑑 performance.

From Tables 1 and 2 above, we see that the 7 combinations of parity bits in Table 2 are a subset of those in Table 1. Since it would be desirable to have the same puncture pattern for both 1 MHz and 2 MHz modes, these 7, along with the 4 MSBs, 4 LSBS and the optimum CRC-4 were simulated in order to choose a common puncture pattern.

Figure 3 shows the false positive rate for the 1 MHz mode in AWGN. The SIG field was generated according to the description in Section II, and ideal channel estimation and detection were assumed. The SIG error rate does not depend on the CRC chosen, but we see that there is almost a 1 dB difference in the false positive rate performance between the 4 LSBs, as proposed in [5], and the best pattern which is 𝑐!𝑐!𝑐!𝑐! . This pattern also gives almost the same performance as the optimal CRC-4 generator.

Figure 4 shows the results obtained with the 2 MHz mode in AWGN with the same 7 patterns, with the SIG field being generated according to the description in Section II. The spread in performance is less than in Figure 3, but nevertheless, the pattern 𝑐!𝑐!𝑐!𝑐! is almost as good as the optimum CRC-4 generator.

Figure 3 False Positive Performance in the 1 MHz mode

SIG Error Rate

False Positive Rate

Figure 4 False Positive Performance in the 2 MHz mode

V. CONCLUSIONS We have shown analytically that puncturing the 8-bit

CRC generated by the polynomial 𝑥! + 𝑥! + 𝑥 + 1 by keeping only the 4 LSBs of the 8 parity bits leads to a code with Hamming distance 1. This is a poor choice for any error detecting CRC and should not be used. Instead, we have shown, analytically and with simulations, that better puncturing choices are available that will result in a punctured code with Hamming distance 2. This allows the use of a shorter CRC without requiring a different CRC generator polynomial. We also show that for lengths beyond 56 bits it is not possible to obtain a Hamming distance 2 code by puncturing CRC-8 and instead puncturing the CRC-32 could be considered.

REFERENCES [1] Local and metropolitan area networks—Specific requirements; Part

11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications; High-speed Physical Layer in the 5 GHz Band, Available: http://easy.intranet.gr/IEEE80211a.pdf

[2] P. Koopman and T. Chakravarty, “Cyclic Redundancy Code (CRC) polynomial selection for embedded networks,” Available: http://www.ece.cmu.edu/~koopman/roses/dsn04/koopman04_crc_poly_embedded.pdf .

[3] “Proposed specification framework for Tgah,” https://mentor.ieee.org/802.11/dcn/11/11-11-1137-11-00ah-specification-framework-for-tgah.docx .

[4] Local and metropolitan area networks—Specific requirements; Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) specifications; Amendment 4: Enhancements for Very High Throughput for Operation in Bands below 6 GHz, IEEE P802.11ac/D3.1, August 2012. http://www.ieee802.org/11/private/Draft_Standards/11ac/DraftP802.11ac_D3.1.pdf.

[5] “SIG Field 4-bit CRC,” https://mentor.ieee.org/802.11/dcn/12/11-12-0596-01-00ah-sig-field-4-bit-crc.pptx

[6] S. Lin and D. J. Costello, Error Control Coding (2nd Edition), Prentice Hall, 2004.

SIG Error Rate

False Positive Rate