Access technique of channel hopping communications system

Claim 1

A method for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the method comprising the steps of: activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; during each activation time period, causing the standby unit to monitor a selected channel for receipt of a paging message that includes a unique address associated with the standby unit, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, whereby the paging unit and the standby unit achieve channel hopping synchronization based only on the unique address and knowledge of which one of the subset of the plurality of channels the paging message was received on by the standby unit.

Claim 2

A method for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the method comprising the steps of: activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; during each activation time period, causing the standby unit to monitor a selected channel for receipt of a paging message that includes a unique address associated with the standby unit, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, whereby the paging unit and the standby unit achieve channel hopping synchronization based only on the unique address and knowledge of which one of the subset of the plurality of channels the paging message was received on by the standby unit.

Claim 6

A method for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the method comprising the steps of: activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; during each activation time period, causing the standby unit to monitor a selected channel for receipt of a paging message, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein: the selection of a channel to be monitored by the standby unit is a function of a value of a free-running clock in the standby unit; and the page train is selected from a plurality of page trains, the selection being a function of a value of a free-running clock in the paging unit.

Claim 7

A method for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the method comprising the steps of: activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; during each activation time period, causing the standby unit to monitor a selected channel for receipt of a paging message, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein: the selection of a channel to be monitored by the standby unit is a function of a value of a free-running clock in the standby unit; and the page train is selected from a plurality of page trains, the selection being a function of a value of a free-running clock in the paging unit.

Claim 8

A method for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the method comprising the steps of: activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; during each activation time period, causing the standby unit to monitor a selected channel for receipt of a paging message, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein the duration of each page train is substantially equal to the activation time period, T wake .

Claim 9

A method for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the method comprising the steps of: activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; during each activation time period, causing the standby unit to monitor a selected channel for receipt of a paging message, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein the duration of each page train is substantially equal to the activation time period, T wake .

Claim 11

A method for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the method comprising the steps of: activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; during each activation time period, causing the standby unit to monitor a selected channel for receipt of a paging message, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein the step of repeatedly transmitting the page train from the paging unit to the standby unit comprises: selecting a first page train for use during a first standby time period; repeatedly transmitting the first page train from the paging unit to the standby unit during the first standby time period; and selecting a second page train for use during a subsequent standby time period, wherein the second page train is transmitted on a different subset of channels from that used for transmitting the first page train.

Claim 12

A method for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the method comprising the steps of: activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; during each activation time period, causing the standby unit to monitor a selected channel for receipt of a paging message, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein the step of repeatedly transmitting the page train from the paging unit to the standby unit comprises: selecting a first page train for use during a first standby time period; repeatedly transmitting the first page train from the paging unit to the standby unit during the first standby time period; and selecting a second page train for use during a subsequent standby time period, wherein the second page train is transmitted on a different subset of channels from that used for transmitting the first page train.

Claim 13

A method for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the method comprising the steps of: activating the standby unit for an activation time period, T wake , out of very standby time period , T standby ; during each activation time period, causing the standby unit to monitor a selected channel for receipt of a paging message, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein each page train is transmitted on a subset of channels that are selected from the plurality of channels in accordance with the following equation: train i={hop.sub.modN (k.sub.p +iM), hop.sub.modN (k.sub.p +iM+1), . . . , hop.sub.modN (k.sub.p +iM+(M-1))} where k p is a clock value of the paging unit, the paging unit clock value being updated every T standby period, N is the number of channels in the hopping sequence, T page is the duration of a page message, M=INT(T wake /T page )-1, where INT( ) is a function that leaves only the integer part of a variable, the number of page trains, N T , is given by N T =RNDUP(N/M), where RNDUP( ) is a function that rounds any non-integer up to the nearest integer, i=0, . . . , (N T -1), and hop modN (x)=hop(x mod N).

Claim 14

A method for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the method comprising the steps of: activating the standby unit for an activation time period, T wake , out of very standby time period , T standby ; during each activation time period, causing the standby unit to monitor a selected channel for receipt of a paging message, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein each page train is transmitted on a subset of channels that are selected from the plurality of channels in accordance with the following equation: train i={hop.sub.modN (k.sub.p +iM), hop.sub.modN (k.sub.p +iM+1), . . . , hop.sub.modN (k.sub.p +iM+(M-1))} where k p is a clock value of the paging unit, the paging unit clock value being updated every T standby period, N is the number of channels in the hopping sequence, T page is the duration of a page message, M=INT(T wake /T page )-1, where INT( ) is a function that leaves only the integer part of a variable, the number of page trains, N T , is given by N T =RNDUP(N/M), where RNDUP( ) is a function that rounds any non-integer up to the nearest integer, i=0, . . . , (N T -1), and hop modN (x)=hop(x mod N).

Claim 15

A method for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the method comprising the steps of: activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; during each activation time period, causing the standby unit to monitor a selected channel for receipt of a paging message, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein each page train is transmitted on a subset of channels that are selected from the plurality of channels in accordance with the following equation: train i={hop.sub.modN (k.sub.s '-α+iM), hop.sub.modN (k.sub.s '-α+iM+1), . . . , hop.sub.modN (k.sub.s '-α+iM+(M-1))} where k s ' is an estimate of a clock value of the standby unit, the standby unit's clock value being updated every T standby period, α is a fixed offset value larger than zero, N is the number of channels in the hopping sequence, T page is the duration of a page message, M=INT(T wake /T page )-1, where INT( ) is a function that leaves only the integer part of a variable, the number of page trains, N T , is given by N T =RNDUP(N/M), where RNDUP( ) is a function that rounds any non-integer up to the nearest integer, i=0, . . . , (N T -1), and hop modN (x)=hop(x mod N).

Claim 16

A method for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the method comprising the steps of: activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; during each activation time period, causing the standby unit to monitor a selected channel for receipt of a paging message, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein each page train is transmitted on a subset of channels that are selected from the plurality of channels in accordance with the following equation: train i={hop.sub.modN (k.sub.s '-α+iM), hop.sub.modN (k.sub.s '-α+iM+1), . . . , hop.sub.modN (k.sub.s '-α+iM+(M-1))} where k s ' is an estimate of a clock value of the standby unit, the standby unit's clock value being updated every T standby period, α is a fixed offset value larger than zero, N is the number of channels in the hopping sequence, T page is the duration of a page message, M=INT(T wake /T page )-1, where INT( ) is a function that leaves only the integer part of a variable, the number of page trains, N T , is given by N T =RNDUP(N/M), where RNDUP( ) is a function that rounds any non-integer up to the nearest integer, i=0, . . . , (N T -1), and hop modN (x)=hop(x mod N).

Claim 21

An apparatus for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the apparatus comprising: means for activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; means for causing the standby unit to monitor a selected channel for receipt of a paging message during each activation time period, wherein the paging message includes a unique address associated with the standby unit wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and means for repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, whereby the paging unit and the standby unit achieve channel hopping synchronization based only on the unique address and knowledge of which one of the subset of the plurality of channels the paging message was received on by the standby unit.

Claim 22

An apparatus for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the apparatus comprising: means for activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; means for causing the standby unit to monitor a selected channel for receipt of a paging message during each activation time period, wherein the paging message includes a unique address associated with the standby unit wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and means for repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, whereby the paging unit and the standby unit achieve channel hopping synchronization based only on the unique address and knowledge of which one of the subset of the plurality of channels the paging message was received on by the standby unit.

Claim 26

An apparatus for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the apparatus comprising: means for activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; means for causing the standby unit to monitor a selected channel for receipt of a paging message during each activation time period, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and means for repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein: the selection of a channel to be monitored by the standby unit is a function of a value of a free-running clock in the standby unit; and the page train is selected from a plurality of page trains, the selection being a function of a value of a free-running clock in the paging unit.

Claim 27

An apparatus for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the apparatus comprising: means for activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; means for causing the standby unit to monitor a selected channel for receipt of a paging message during each activation time period, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and means for repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein: the selection of a channel to be monitored by the standby unit is a function of a value of a free-running clock in the standby unit; and the page train is selected from a plurality of page trains, the selection being a function of a value of a free-running clock in the paging unit.

Claim 28

An apparatus for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the apparatus comprising: means for activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; means for causing the standby unit to monitor a selected channel for receipt of a paging message during each activation time period, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and means for repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein the duration of each page train is substantially equal to the activation time period, T wake .

Claim 29

An apparatus for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the apparatus comprising: means for activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; means for causing the standby unit to monitor a selected channel for receipt of a paging message during each activation time period, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and means for repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein the duration of each page train is substantially equal to the activation time period, T wake .

Claim 31

An apparatus for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the apparatus comprising: means for activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; means for causing the standby unit to monitor a selected channel for receipt of a paging message during each activation time period, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and means for repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein the means for repeatedly transmitting the page train from the paging unit to the standby unit comprises: means for selecting a first page train for use during a first standby time period; means for repeatedly transmitting the first page train from the paging unit to the standby unit during the first standby time period; and means for selecting a second page train for use during a subsequent standby time period, wherein the second page train is transmitted on a different subset of channels from that used for transmitting the first page train.

Claim 32

An apparatus for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the apparatus comprising: means for activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; means for causing the standby unit to monitor a selected channel for receipt of a paging message during each activation time period, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and means for repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein the means for repeatedly transmitting the page train from the paging unit to the standby unit comprises: means for selecting a first page train for use during a first standby time period; means for repeatedly transmitting the first page train from the paging unit to the standby unit during the first standby time period; and means for selecting a second page train for use during a subsequent standby time period, wherein the second page train is transmitted on a different subset of channels from that used for transmitting the first page train.

Claim 33

An apparatus for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the apparatus comprising: means for activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; means for causing the standby unit to monitor a selected channel for receipt of a paging message during each activation time period, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and means for repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein each page train is transmitted on a subset of channels that are selected from the plurality of channels in accordance with the following equation: train i={hop.sub.modN (k.sub.p +iM), hop.sub.modN (k.sub.p +iM+1), . . . , hop.sub.modN (k.sub.p +iM+(M-1))} where k p is a clock value of the paging unit, the paging unit clock value being updated every T standby period, N is the number of channels in the hopping sequence, T page is the duration of a page message, M=INT(T wake /T page )-1, where INT( ) is a function that leaves only the integer part of a variable, the number of page trains, N T , is given by N T =RNDUP(N/M), where RNDUP( ) is a function that rounds any non-integer up to the nearest integer, i=0, . . . , (N T -1), and hop modN (x)=hop(x mod N).

Claim 34

An apparatus for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the apparatus comprising: means for activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; means for causing the standby unit to monitor a selected channel for receipt of a paging message during each activation time period, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and means for repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein each page train is transmitted on a subset of channels that are selected from the plurality of channels in accordance with the following equation: train i={hop.sub.modN (k.sub.p +iM), hop.sub.modN (k.sub.p +iM+1), . . . , hop.sub.modN (k.sub.p +iM+(M-1))} where k p is a clock value of the paging unit, the paging unit clock value being updated every T standby period, N is the number of channels in the hopping sequence, T page is the duration of a page message, M=INT(T wake /T page )-1, where INT( ) is a function that leaves only the integer part of a variable, the number of page trains, N T , is given by N T =RNDUP(N/M), where RNDUP( ) is a function that rounds any non-integer up to the nearest integer, i=0, . . . , (N T -1), and hop modN (x)=hop(x mod N).

Claim 35

An apparatus for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the apparatus comprising: means for activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; means for causing the standby unit to monitor a selected channel for receipt of a paging message during each activation time period, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and means for repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein each page train is transmitted on a subset of channels that are selected from the plurality of channels in accordance with the following equation: train i={hop.sub.modN (k.sub.s '-α+iM), hop.sub.modN (k.sub.s '-α+iM+1), . . . , hop.sub.modN (k.sub.s '-α+iM+(M-1))} where k s ' is an estimate of a clock value of the standby unit, the standby unit's clock value being updated every T standby period, α is a fixed offset value larger than zero, N is the number of channels in the hopping sequence, T page is the duration of a page message, M=INT(T wake /T page )-1, where INT( ) is a function that leaves only the integer part of a variable, the number of page trains, N T , is given by N T =RNDUP(N/M), where RNDUP( ) is a function that rounds any non-integer up to the nearest integer, i=0, . . . , (N T -1), and hop modN (x)=hop (x mod N).

Claim 36

An apparatus for establishing a connection between a paging unit and a standby unit in a channel hopping communications system, the apparatus comprising: means for activating the standby unit for an activation time period, T wake , out of every standby time period, T standby ; means for causing the standby unit to monitor a selected channel for receipt of a paging message during each activation time period, wherein the selected channel is selected from a plurality of channels, and wherein, for each subsequent activation time period, the selected channel is a subsequent one of the plurality of channels as specified by a hopping sequence; and means for repeatedly transmitting a page train from the paging unit to the standby unit, until a response is received from the standby unit, wherein each page train comprises a plurality of paging messages, each paging message being transmitted on a different one of a subset of the plurality of channels, wherein each page train is transmitted on a subset of channels that are selected from the plurality of channels in accordance with the following equation: train i={hop.sub.modN (k.sub.s '-α+iM), hop.sub.modN (k.sub.s '-α+iM+1), . . . , hop.sub.modN (k.sub.s '-α+iM+(M-1))} where k s ' is an estimate of a clock value of the standby unit, the standby unit's clock value being updated every T standby period, α is a fixed offset value larger than zero, N is the number of channels in the hopping sequence, T page is the duration of a page message, M=INT(T wake /T page )-1, where INT( ) is a function that leaves only the integer part of a variable, the number of page trains, N T , is given by N T =RNDUP(N/M), where RNDUP( ) is a function that rounds any non-integer up to the nearest integer, i=0, . . . , (N T -1), and hop modN (x)=hop (x mod N).