IEEE P802.11
Wireless LANs
Proposed Draft Text: Mathematical description of signals	
Date:  2020-12-03	
Author(s):	
Name	Affiliation	Address	Phone	email	
Yan Zhang	NXP				
Ruchen Duan	Samgsung				

Abstract
This submission shows 
•	Mathematical description of signal

Revisions:
•	Rev 0: Initial version of the document.
•	Rev 1: Highlighted TBD topics in yellow
•	Rev 2: Fixed tone index for 320/160+160MHz phase rotation Equations
•	Rev 3: Added Hightlighted TBD text in yellow
•	Rev 4: Removed TBD in D0.2 based on latest motions, Motion 137 ( Noncontiguous 80+80 MHz and 160+160 MHz are not defined), Motion 135 (no midamble in R1)
•	Rev 5: Added Note to Table 36-17.
•	Rev 6: Added a note to remind Editor to remove other draft texts related to frequency segments.
•	Rev 7: Remove subcarrier index from  since it is a constant over 20MHz channel
•	Rev 9: Change  to 

36.3.10 Mathematical description of signals

36.3.10.1 Notation

For a description of the conventions used for the mathematical description of the signals, see 17.3.2.5 (Mathematical
conventions in the signal descriptions). In addition, the following notational conventions are used in Clause 36 (Extremely high throughput (EHT) PHY specification):
 indicates the element in row  and column  of matrix, where  and  .
 and  are the number of rows and columns, respectively, of the matrix .
 indicates a matrix consisting of columns  to  of matrix .

36.3.10.2 Subcarrier indices in use

For a description on subcarrier indices over which the signal is transmitted for non-HT, HT and VHT PPDUs, see 21.3.7 (Mathematical description of signals).  For a description on subcarrier indices over which the signal is transmitted for HE PPDUs, see 27.3.10 (Mathematical description of signals).

For a 20 MHz non-OFDMA EHT PPDU transmission, the 20 MHz is divided into 256 subcarriers. The signal
is transmitted on all or a subset of subcarriers -122 to -2 and 2 to 122, with 0 being the center subcarrier.

For a 20 MHz OFDMA EHT PPDU transmission, the 20 MHz is divided into 256 subcarriers. The signal is
transmitted on all or a subset of the subcarriers -122 to -4 and 4 to 122, with 0 being the center subcarrier.

For a 40 MHz non-OFDMA EHT PPDU transmission, the 40 MHz is divided into 512 subcarriers. The signal
is transmitted on subcarriers -244 to -3 and 3 to 244, with 0 being the center subcarrier.

For a 40 MHz OFDMA EHT PPDU transmission, the 40 MHz is divided into 512 subcarriers. The signal is
transmitted on all or a subset of subcarriers  -244 to -3 and 3 to 244, with 0 being the center subcarrier.

For an 80 MHz non-OFDMA EHT PPDU transmission, the 80 MHz is divided into 1024 subcarriers. The signal
is transmitted on subcarriers -500 to -3 and 3 to 500, with 0 being the center subcarrier.

For an 80 MHz OFDMA EHT PPDU transmission, the 80 MHz is divided into 1024 subcarriers. The signal is
transmitted on all or a subset of the subcarriers -500 to -259, -253 to -12, 12 to 253, and 259 to 500, with 0 being the center subcarrier.

For a 160 MHz EHT PPDU transmission, each half 80 MHz bandwidth is divided into 1024 subcarriers, and the subcarriers on which the signal is transmitted in each 80 MHz bandwidth is identical to an 80 MHz EHT PPDU transmission, depending on non-OFDMA or OFDMA transmission within the corresponding 80 MHz.

For a 320 MHz EHT PPDU transmission, each half 160 MHz bandwidth is divided into 2048 subcarriers, and the subcarriers on which the signal is transmitted in each 160 MHz bandwidth is identical to an 160 MHz EHT PPDU transmission, depending on non-OFDMA or OFDMA transmission within the corresponding 160 MHz.

34.3.10.3 Channel frequencies
Let
                (36-1)
                                          (36-2)
                         (36-3)

where 
     and  are defined in Table 36-15 (Fields to specify EHT channels).

                                                          Table 36-15—Fields to specify EHT channels
Field	Meaning	
	Channel width. Possible values represent 20 MHz, 40 MHz, 80 MHz, 160 MHz, and 320 MHz channels.	
	For a 20 MHz, 40 MHz, 80 MHz, 160 MHz, or 320 MHz channel, denotes the channel center frequency. Valid range is 1 to 200 for 5GHz band, and 1 to 253 for 6GHz band.	
	Denotes the location of the primary 20 MHz channel. 
Valid range is 1 to 200 for 5GHz band, and 1 to 253 for 6GHz band.	

When  is 20 MHz, 40 MHz, 80 MHz, or160 MHz, the relationship between  and  is specified in Equation (21-5) in 21.3.7.3  (Channel frequencies).

When  is 320 MHz,  and  shall have the relationship specified in Equation (36-4).

                                   (36-4)       

where
    .     is an integer with possible range .

When  is 40 MHz, 80 MHz, 160 MHz, or 320 MHz, the relationship between  and  is specified in Equation (21-6), and the relationship between  and  is specified Equation (21-7) in 21.3.7.3  (Channel frequencies).

When  is 80 MHz, 160 MHz, or 320 MHz, the relationship between  and  is specified in Equation (21-8), and the relationship between  and  are specified in Equation (21-9) in 21.3.7.3  (Channel frequencies).

When  is 160 MHz, or 320 MHz, the relationship between  and  are specified in Equation (21-10) in 21.3.7.3  (Channel frequencies).

When  is 320 MHz, 
    —   The primary 160 MHz channel is the channel with 160 MHz bandwidth centered at     
         MHz, where  is given in Equation (36-5)
    —   The secondary 160 MHz channel is the channel with 160 MHz bandwidth centered at     
         MHz, where  is given in Equation (36-6)

                                   (36-5)       

                                        (36-6)

where .

36.3.10.4 Transmitted signal 

Editor Note: Remove (i_Seg) at 240.53 and 240.62 in D0.2. 

The transmitted signal is described in complex baseband signal notation. The actual transmitted signal on
transmit chain  is related to the complex baseband signal by the relation
shown in Equation (36-7).

   (36-7)

where

       represents the complex baseband signal of transmit chain .

      represents the center frequency of the transmitted PPDU. Table 36-16 (Center frequency of the transmitted PPDU) shows as a function of the channel starting frequency, dot11CurrentChannelWidth  and CH_BANDWIDTH, where , , ,  ,  , and  are described in 36.3.10.3 (Channel frequencies). 

Table 36-16—Center frequency of the transmitted PPDU 

	CH_BANDWIDTH		
				
20 MHz	CBW20			
40 MHz	CBW20			
	CBW40			
80 MHz	CBW20			
	CBW40			
	CBW80			
160 MHz	CBW20			
	CBW40			
	CBW80			
	CBW160			
320 MHz	CBW20			
	CBW40			
	CBW80			
	CBW160			
	CBW320			

The transmitted RF signal is derived by upconverting the complex baseband signal, which consists of several fields. The timing boundaries for the various fields are shown in Figure 36-32 (Timing boundaries for EHT PPDU fields), where  is the number of EHT-LTF symbols and is defined in Table 36-14 (Frequently used parameters),  is the number of OFDM symbols in the EHT-SIG field present in an EHT MU PPDU, and  is the number of data OFDM symbols.

Figure 36-32 – Timing Boundaries for EHT PPDU fields 
.

The time offset, , determines the starting time of the corresponding field relative to the start of L-STF ().

The signal transmitted on transmit chain  shall be as shown in Equation (36-8). 

     (36-8)

where
 is only applicable to an EHT MU PPDU

In an EHT MU PPDU, for each field excluding the PE field,  is defined as the summation of one or more subfields. Each subfield, , is defined to be an inverse Fourier transform in Equation (36-9).

  (36-9)

Editor’s Note: Per the authors of 20/1337r3,  in Equation (36-9) is TBD.
In an EHT TB PPDU, transmitted by user u in the r-th occupied RU or MRU, each subfield,  , is defined in Equation (36-10).

                             (36-10)

In the remainder of this subclause, pre-EHT modulated fields refer to the L-STF, L-LTF, L-SIG, RL-SIG, U-SIG and EHT-SIG fields, while EHT modulated fields refer to EHT-STF, EHT-LTF, Data and PE fields, as shown in Figure 36-32 (Timing boundaries for EHT PPDU fields).

For an EHT MU PPDU, the total power of the time domain EHT modulated field signals summed over all transmit chains should not exceed the total power of the time domain pre-EHT modulated field signals summer over all transmit chains.

For an EHT TB PPDU, the total power of the time domain EHT modulated field signals summed over all transmit chains may exceed the total power of the time domain pre-EHT modulated field signals summer over all transmit chains by up to 3dB.

For notational simplicity, the parameter BW is omitted from some bandwidth dependent terms.

In Equation (36-9) and Equation (36-10) the following notations are used:
     is a windowing function. An example function, , is given in 17.3.2.5 (Mathematical conventions in the signal descriptions).
        is defined in Table 36-14 (Frequently used parameters)
      For pre-EHT modulated fields, . For EHT modulated fields,  for an EHT MU PPDU, and  for an EHT TB PPDU, where  and  are given in Table 36-14 (Frequently used parameters).
          is the power boost factor in the range  of the r-th occupied RU or MRU in an EHT MU PPDU. For an EHT MU PPDU, an AP shall limit the ratio between the maximum value of  and the minimum value of  to 2 unless the Power Boost Factor Support subfield of the EHT PHY Capabilities Information field in the EHT Capabilities element from all recipient STAs is 1, in which case the AP can use a ratio of up to 4. For an EHT MU PPDU transmitted to single user,  is always set to 1 (TBD).

          For pre-EHT modulated fields,  is the set of subcarriers indices in the allocated 20 MHz channels. For EHT modulated fields in a non-punctured non-OFDMA EHT PPDU,  is the set of subcarriers indices from  to  as defined in Table 36-10 (Subcarrier allocation related constants for the EHT modulated fields in a full bandwidth non-OFDMA EHT PPDU) excluding DC subcarriers. For EHT modulated fields in a punctured non-OFDMA EHT PPDU and an OFDMA EHT PPDU,  is the set of subcarriers indices for the tones in the r-th RU or MRU. Data and pilot subcarrier indices for RUs are defined in Table 27-7 (Data and pilot subcarrier indices for RUs in a 20 MHz HE PPDU and in a non-OFDMA 20 MHz HE PPDU), Table 27-8 (Data and pilot subcarrier indices for RUs in a 40 MHz HE PPDU and in a non-OFDMA 40 MHz HE PPDU), Table 36-5 (Data and pilot subcarrier indices for RUs in an 80 MHz EHT PPDU), Table 36-6 (Data and pilot subcarrier indices for RUs in a 160 MHz EHT), and Table 36-7 (Data and pilot subcarrier indices for RUs in a 320 MHz EHT PPDU). Data and pilot subcarrier indices for MRUs consist of the data and pilot subcarrier indices of all component RUs.

     is the power normalization factor and is defined in Equation (36-11).

Editor’s Note: Per the authors of 20/1337r2,  in Equation (36-11) is TBD.

                             (36-11)

    Table 36-17 (Number of modulated subcarriers and guard interval duration values for EHT PPDU fields) summarizes the various values of  as a function of bandwidth.

Table 36-17— Number of modulated subcarriers and guard interval duration values for EHT PPDU fields 	
Field	 as a function of bandwidth	Guard interval duration	
	20 MHz	40 MHz	80 MHz	160 MHz	320 MHz		
L-STF	12	24	48	96	192	-	
L-LTF	52	104	208	416	832	TGI,L-LTF	
L-SIG in an EHT PPDU	56	112	224	448	896	TGI,Pre-EHT	
L-SIG in a non-HT duplicate PPDU	-	104	208	416	832		
RL-SIG	56	112	224	448	896	TGI,Pre-EHT	
U-SIG	56	112	224	448	896	TGI,Pre-EHT	
EHT-SIG	56	112	224	448	896	TGI,Pre-EHT	
NON_HT_DUP_OFDM-Data (see NOTE)	-	104	208	416	832	TGI,Pre-EHT	
NOTE-For notational convenience, NON_HT_DUP_OFDM-Data is used as a label for the Data field of a NON_HT PPDU with format type NON_HT_DUP_OFDM.	

  (TBD)

   is a set of 20 MHz channels where pre-EHT modulated fields are located. The set of 20 MHz channels contains one or more values in the range 0 to  for an EHT MU PPDU with preamble puncturing, or an EHT TB PPDU, and it contains all values in the range 0 to  for an EHT MU PPDU without preamble puncturing.

 is the cardinality of the set of 20MHz channels .

 

 is the cardinality of the set of subcarriers 

 is the cardinality of the set of modulated subcarriers within  for the EHT-STF and Data fields. For EHT-LTF field,

  is the power scale factor of a given field within an OFDM symbol for an EHT TB PPDU. For the pre-EHT modulated fields,  is in the range of . The same  value applies to all pre-EHT modulated fields. For EHT modulated fields, . 

    is the spatial mapping matrix for subcarrier k. For EHT modulated fields,    is a matrix with  rows and  columns. For pre-EHT modulated fields  is a column vector with  elements, with element  being , where  represents the cyclic shift for the transmitter chain whose value is defined in 36.3.11.2.1 (Cyclic shift for pre-EHT modulated fields).

   is the spatial mapping matrix for user u on subcarrier k.  For EHT modulated fields,  is a matrix with  rows and  columns. For pre-EHT modulated fields  is a column vector with  elements, with element  being , where  represents the cyclic shift for the transmitter chain whose value is defined in 36.3.11.2.1 (Cyclic shift for pre-EHT modulated fields).

   is the subcarrier frequency spacing. For pre-EHT modulated fields,  given in Table 34-9 (Timing-related constants). For EHT modulated fields,  given in Table 34-9 (Timing-related constants). 

       is given in Table 34-14 (Frequently used parameters).

 is the frequency-domain symbol assigned for subcarrier k of user u in the r-th RU or MRU for the m-th spatial stream. Some of the  within  have a value of zero. Examples of such cases include the DC tones, guard tones on each side of the transmit spectrum, the null subcarriers in an EHT OFDMA PPDU, as well as the unmodulated tones of L-STF, EHT-STF, and EHT-LTF fields.

  is the guard interval duration used for each OFDM symbol in the field. The value for each field is defined in Table 34-9 (Timing-related constants).

 For pre-EHT modulated fields, . For EHT modulated fields,  represents the cyclic shift per spatial stream, whose value is defined in 36.3.11.2.2 (Cyclic shift for EHT modulated fields).

      is used to represent a phase rotation applied to the k-th subcarrier for a given bandwidth , which is  determined by the TXVECTOR parameter CH_BANDWIDTH as defined in Table 36-18 (CH_BANDWIDTH and  for pre-EHT modulated fields). For EHT modulated fields,  for all subcarriers. For pre-EHT modulated fields,  is defined as in 21.3.7.5 (Definition of tone rotation) for 20 MHz, 40 MHz, 80 MHz, and 160 MHz PPDU transmission, and in Equation (36-12) for 320 MHz PPDU transmission.       

              For a 320 MHz PPDU transmission, 
                                                              (36-12)

Table 36-18— CH_BANDWIDTH and  for pre-EHT modulated fields 	
CH_BANDWIDTH		
CBW20	¡k,20	
CBW40	¡k,40	
CBW80	¡k,80	
CBW160	¡k,160	
CBW320	¡k,320	
EHT-CBW-PUNC80 (TBD)	¡k,80	
EHT-CBW-PUNC160 (TBD)	¡k,160	
EHT-CBW-PUNC320 (TBD)	¡k,320