Source: https://insight.rpxcorp.com/pat/US7145954B1
Timestamp: 2019-10-14 01:51:05
Document Index: 285057891

Matched Legal Cases: ['art 1', 'art 2', 'art 3', 'art 1', 'art 2', 'art 3', 'art 1530', 'art 1522']

Patent US 7,145,954 B1
US 7,711,028 B2
US 7,321,608 B2
Precision-resolution constrained coding scheme
US 7,443,319 B2
US 20070030883A1
US 20070164881A1
US 20040213325A1
US 6,636,556 B2
US 6,330,233 B1
(b) applying said delta code relative to said at least one non-fixed reference, wherein the delta code is a deterministic delta code, wherein the deterministic delta code is generated using the sequential delta code generation methodology, wherein the deterministic delta code is generated using the Rational Congruential Sequential Delta Code generation methodology, wherein the rational function employed in the Rational Congruential Sequential Delta Code generation methodology is of the form ƒ(x;
a)=axn mod M, where ƒ is a function of variable x, M is an integer modulus, a is a parameter, with possible values of 1, 2, . . . M−1, and n is a nonzero integer.
a)=ax−1 mod M, where ƒ is a function of variable x, M is an integer modulus, a is a parameter, with possible values of 1, 2, . . . M−1.
a)=ax mod M, where ƒ is a function of variable x, M is an integer modulus, a is a parameter, with possible values of 1, 2, . . . M−1.
a)=ax2 mod M, where ƒ is a function of variable x, M is an integer modulus, a is a parameter, with possible values of 1, 2, . . . M−1.
a)=ax3 mod M, where ƒ is a function of variable x, M is an integer modulus, a is a parameter, with possible values of 1, 2, . . . M−1.
(b) applying said delta code relative to said at least one non-fixed reference, wherein the delta code is a deterministic delta code, wherein the deterministic delta code is generated using the iterative delta code generation methodology, wherein the deterministic delta code is generated using the Rational Congruential Iterative Delta Code generation methodology, wherein the rational function employed in the Rational Congruential Iterative Delta Code generation methodology is of the form ƒ(x;
said Ultra Wideband Transmitter and said Ultra Wideband Receiver employ a delta code of a plurality of delta codes, wherein said delta code specifies pulse characteristics relative to at least one non-fixed reference, wherein said pulse characteristics define one of a plurality of communication channels defined by said plurality of delta codes, wherein the delta code is a deterministic delta code, wherein the deterministic delta code is generated using the sequential delta code generation methodology, wherein the deterministic delta code is generated using the Rational Congruential Sequential Delta Code generation methodology, wherein the rational function employed in the Rational Congruential Sequential Delta Code generation methodology is of the form ƒ(x;
said Ultra Wideband Transmitter and said Ultra Wideband Receiver employ a delta code of a plurality of delta codes, wherein said delta code specifies pulse characteristics relative to at least one non-fixed reference, wherein said pulse characteristics define one of a plurality of communication channels defined by said plurality of delta codes, wherein the delta code is a deterministic delta code, wherein the deterministic delta code is generated using the iterative delta code generation methodology, wherein the deterministic delta code is generated using the Rational Congruential Iterative Delta Code generation methodology, wherein the rational function employed in the Rational Congruential Iterative Delta Code generation methodology is of the form ƒ(x;
FIG. 4a illustrates an exemplary pulse time position delta value range layout of four exemplary components subdivided into nine exemplary sub-components;
FIG. 4b illustrates an exemplary pulse amplitude delta value range layout of seven exemplary components;
FIG. 4c illustrates an exemplary pulse width delta value range layout of five exemplary components;
FIG. 6a illustrates non-allowable regions relative to a preceding pulse position within a temporal pulse characteristic delta value range layout;
FIG. 6b illustrates non-allowable regions relative to any preceding pulse position within a temporal pulse characteristic delta value range layout;
FIG. 6c illustrates non-allowable regions relative to a succeeding pulse position within a temporal pulse characteristic delta value range layout;
FIG. 6d illustrates non-allowable regions relative to any succeeding pulse position within a temporal pulse characteristic delta value range layout;
FIG. 8a illustrates an exemplary discrete delta value layout of thirty-seven exemplary evenly distributed delta values including exemplary layout parameters;
FIG. 8b illustrates an exemplary generic discrete delta value layout of six exemplary non-evenly distributed delta values including exemplary layout parameters;
FIG. 9a illustrates an exemplary discrete delta time position value layout;
FIG. 9b illustrates an exemplary discrete delta pulse amplitude value layout;
FIG. 9c illustrates an exemplary discrete delta pulse width value layout;
FIG. 11a illustrates an exemplary single code element per reference temporal delta coding approach.
FIG. 11b illustrates an exemplary single code element per reference non-temporal delta coding approach.
FIG. 12a illustrates an exemplary multiple code elements per reference temporal delta coding approach.
FIG. 12b illustrates an exemplary multiple code elements per reference non-temporal delta coding approach.
FIG. 13a illustrates an exemplary delta code mapping approach, depicting exemplary pulses mapped to exemplary time position delta value range layouts based on integer code element values of a code, where one integer code element exists per reference, layouts are relative to an initial reference position or to the previous pulse position, and pulse positions are further specified using an absolute or relative offset value;
FIG. 13b illustrates an exemplary delta code mapping approach, depicting exemplary pulses mapped to exemplary amplitude delta value range layouts based on integer code element values of a code, where one integer code element exists per reference, layouts are relative to an initial reference amplitude or to the previous pulse amplitude, and pulse amplitude values are further specified using an absolute or relative offset value;
FIG. 14a illustrates an exemplary delta code mapping approach, depicting exemplary pulses mapped to exemplary time position discrete delta value layouts based on integer code element values of a code, where one integer code element exists per reference and layouts are relative to an initial reference position or to the previous pulse;
FIG. 14b illustrates an exemplary delta code mapping approach, depicting exemplary pulses mapped to values within exemplary time position discrete delta value layouts based on integer code element values of a code, where two integer code elements exist per reference and layouts are relative to an initial reference position or to the second pulse position;
FIG. 14c illustrates an exemplary delta code mapping approach, depicting exemplary pulses mapped to exemplary amplitude discrete delta value layouts based on integer code element values of a code, where one integer code element exists per reference and layouts are relative to an initial reference amplitude or to the previous pulse amplitude;
FIG. 15a illustrates exemplary mapping pulses to positions within an exemplary time position delta value range layout based on exemplary floating-point code element values of a code via an exemplary single code element per reference temporal delta coding approach;
FIG. 15b illustrates an exemplary diagram depicting mapping pulses to components within an exemplary temporal delta value range layout using the non-fractional part of a floating point code element value of a code and mapping pulses to exemplary positions within components using the fractional part of the floating-point code element values via an exemplary multiple code element per reference temporal delta coding approach;
FIG. 15c illustrates an exemplary diagram depicting mapping pulse amplitudes to components within exemplary pulse amplitude delta value range layouts using the non-fractional part of a floating point code element value and mapping to exemplary exact amplitude delta values within components using the fractional part of the floating-point code element values via an exemplary single code element per reference non-temporal delta coding approach;
FIG. 16a illustrates a doublet pulse type that can be used in UWB transmitters;
FIG. 16b illustrates a triplet pulse type that can be used in UWB transmitters;
FIG. 17a illustrates a typical UWB pulse train of doublet waveforms;
FIG. 17b illustrates a typical UWB pulse train of triplet waveforms;
FIG. 20a illustrates part 1 of 3 parts of a plot of the power spectral density function for a specific Poisson Code;
FIG. 20b illustrates part 2 of 3 parts of a plot of the power spectral density function for a specific Poisson Code;
FIG. 20c illustrates part 3 of 3 parts of a plot of the power spectral density function for a specific Poisson Code;
FIG. 23a illustrates part 1 of 3 parts of a plot of the power spectral density function for a Uniform Delta Code;
FIG. 23b illustrates part 2 of 3 parts of a plot of the power spectral density function for a Uniform Delta Code;
FIG. 23c illustrates part 3 of 3 parts of a plot of the power spectral density function for a Uniform Delta Code;
FIG. 26a is an exemplary diagram depicting an exemplary linear feedback shift-register pseudorandom number generator;
FIG. 26b is an exemplary diagram of an additive Lagged-Fibonacci shift register pseudorandom number generator;
FIG. 1 depicts a train of pulse time positions 102 and a temporal delta value layout 104 used to map the time position of a pulse 106, t<sub>k</sub>, relative to the time position of the previous pulse 108, t<sub>k−1</sub>. The position of the preceding pulse 108, t<sub>k−1</sub>, is given a zero delta value and all succeeding points in time are some delta time value, ΔT, from that reference point. In a similar manner, FIG. 2 depicts non-temporal characteristic values 202 and a non-temporal delta value layout 204 used to map a non-temporal characteristic value of a pulse 206, v<sub>k</sub>, relative to the non-temporal characteristic value of the preceding pulse 208, v<sub>k−1</sub>. The non-temporal characteristic value of the preceding pulse 208, v<sub>k−1</sub>, is given a zero delta value and all preceding and succeeding non-temporal characteristic values are some delta non-temporal characteristic value, ΔV, from that reference point. A fundamental difference between temporal and non-temporal characteristics is evident when comparing FIG. 1 to FIG. 2, whereas temporal characteristics have an inherent order to them (e.g., t<sub>k+1 </sub>is always greater than t<sub>k</sub>), non-temporal characteristics do not (i.e., v<sub>k+1 </sub>can be less than, greater than, or the same as v<sub>k</sub>).
FIG. 3 illustrates an exemplary delta value range layout which can represent a temporal, non-temporal, or combination of the two, pulse characteristic such as, e.g., timing of a pulse, and amplitude. FIG. 3 includes a delta value range layout 302. The pulse can take on characteristic delta values between a minimum delta value Δv<sub>0 </sub>310 and a maximum delta value Δv<sub>max </sub>312 in layout 302. Layout 302, as shown, can be subdivided into components 304. Components 304 can in turn be divided into sub-components 306. Sub-components 306 can in turn be divided into smaller components 308. Smaller components 308 can then be divided into even smaller components, as shown. The process of subdividing components can be repeated, ad infinitum, so that smaller and smaller components can be obtained.
The process of subdividing a delta value range layout into components is now described in detail. In particular, FIG. 3 depicts an exemplary embodiment of pulse characteristic delta value range layout parameters. Specifically, a pulse characteristic delta value range 302 is shown. As depicted in FIG. 3, two layout parameters, Δv<sub>0 </sub>and Δv<sub>max</sub>, can be specified to define a pulse characteristic delta value range 302 bounded by a minimum delta value of Δv<sub>0 </sub>and a maximum delta value of Δv<sub>max</sub>. A second layout parameter, N<sub>components</sub>, can be specified to divide the delta value range 302 into one or more components 304 of the same size, or of different sizes, with each component 304 (indexed by the letter-n) having a minimum delta value, Δv<sub>min</sub>(n), and a maximum delta value, Δv<sub>max</sub>(n), where n=1 to N<sub>components</sub>.
The number and size of components 304 used in a given layout can be selected for various reasons. For example, the number and size of components 304 can be tailored to meet, e.g., specific application requirements, to remain within system implementation limits, to achieve one or more of a variety of system characteristics in areas such as, e.g., performance (i.e., bit rate), reliability (i.e., bit error rate), system-simplicity, ease-of-use, inter alia. When different sized components 304 are employed, minimum and maximum values can be specified for each component 304 indexed by n, wherein the minimum delta value for a given component, Δv<sub>min</sub>(n), equals the maximum delta value of the preceding component, Δv<sub>max</sub>(n−1), or Δv<sub>0</sub>, and the maximum delta value of a given component, Δv<sub>max</sub>(n), equals the minimum delta value for the following component, Δv<sub>min</sub>(n+1), or Δv<sub>max</sub>. When same sized components 304 are employed, the delta value range is evenly divided such that Δv<sub>max</sub>(n)−Δv<sub>min</sub>(n) is equal for each component 304 indexed by n.
An array of layout parameters, N<sub>sub-components</sub>(N<sub>components</sub>), can be specified to subdivide each component 304 into sub-components 306 of the same size, or different sizes, with each sub-component 306 (indexed by m) of the component 304 (indexed by n) having a minimum delta value, Δv<sub>min</sub>(n,m), and a maximum delta value, Δv<sub>max</sub>(n,m), where n=1 to N<sub>components </sub>and m=1 to N<sub>sub-components</sub>(n). As with components 304, the number and size of sub-components 306 for a given component 304 used in a given delta value range layout 302 can also be tailored to meet, e.g., specific application requirements, to remain within system implementation limits, to achieve one or more of a variety of system characteristics in areas such as, e.g., performance (i.e., bit rate), reliability (i.e., bit error rate), system-simplicity, ease-of-use, etc., and/or for many other reasons. When different sized sub-components 306 are employed, minimum and maximum delta values are specified for each sub-component 306 indexed by m of each component 304 indexed by n, wherein the minimum delta value for a given sub-component, Δv<sub>min</sub>(n,m), equals the maximum delta value of the preceding sub-component, Δv<sub>max</sub>(n,m−1), or the minimum delta value of the component in which the sub-component resides, Δv<sub>min</sub>(n), and the maximum delta value of a given sub-component, Δv<sub>max</sub>(n,m), equals the minimum delta value for the following sub-component, Δv<sub>min</sub>(n,m+1), or the maximum delta value of the component in which the sub-component resides, Δv<sub>max</sub>(n). When same sized sub-components 306 are employed, components are evenly divided such that Δv<sub>max</sub>(n,m)−Δv<sub>min</sub>(n,m) is equal for each sub-component 306 indexed by m of a component 304 indexed by n or for all components such that all sub-components 306 of a given component 304 are of the same size, wherein sub-component sizes may vary from component to component or all sub-components of all components are of the same size depending on the sizes of the components and the numbers of sub-components in the components.
In a manner consistent with the subdivision of components into sub-components, additional multi-dimensional arrays of layout parameters can be used to further subdivide sub-components 306 into smaller components 308 (as shown) of the same or different sizes, ad infinitum, until a smallest desirable component resolution is attained, with components at each resolution level having a minimum delta value, Δv<sub>min</sub>(n, m, . . . , a), and a maximum delta value, Δv<sub>max</sub>(n, m, . . . , a), where n=1 to N<sub>components</sub>, m=1 to N<sub>sub-components</sub>(n), . . . , and a=1 to N<sub>smallest components</sub>(n, m, . . . ). Such further subdivision of sub-components into smaller and smaller components enables systems with finer and finer tuning resolution and thus higher and higher fidelity, increases modulation accuracy, and can be useful for other purposes. As with components 304 and sub-components 306, the number and size of these smaller components 308 can also be tailored, e.g., to meet specific application requirements, to remain within system implementation limits, to achieve one or more of a variety of system characteristics in areas such as performance (i.e., bit rate), reliability (i.e., bit error rate), system-simplicity, ease-of-use, etc., and/or for many other reasons. When different sizes of these smaller components 308 are employed, minimum and maximum delta values are specified for each smaller component 308 (indexed by a), wherein the minimum delta value for a component, Δv<sub>min</sub>(n, m, . . . , a), equals the maximum delta value of the preceding component, Δv<sub>max</sub>(n, m, . . . , a−1), or the minimum delta value of the next higher level component in which the component resides, Δv<sub>min</sub>(n, m, . . . ), and the maximum delta value of a given component, Δv<sub>max</sub>(n, m, . . . , a), equals the minimum delta value for the following component, Δv<sub>max</sub>(n, m, . . . , a+1), or the maximum delta value of the next higher level component in which the component resides, Δv<sub>max</sub>(n, m, . . . ). When same sized smaller components 308 are employed, the next higher level components 306 are evenly divided such that Δv<sub>max</sub>(n, m, . . . , a)−Δv<sub>min</sub>(n, m, . . . , a) is equal for each smaller component 308 indexed by a of a given next higher level component or for all next higher level components such that all components of a given next higher level component are of the same size, wherein component sizes may vary from next higher level component to next higher level component or all components of all higher level components are of the same size depending on the sizes of the next higher level components and the numbers of components in the next higher level components.
At the top of FIG. 3, pulse characteristic delta value range 302 is depicted that is bounded by endpoints of Δv<sub>0 </sub>310 and Δv<sub>max </sub>312. Beneath this illustration an equivalent delta value range 302 is shown that has been subdivided into four components 304 by setting the layout parameter N<sub>components </sub>to a value of four (4), and the size of each component has been established by setting the minimum and maximum delta values of each component, Δv<sub>min</sub>(n) and Δv<sub>max</sub>(n), where n=1 to 4. An enlargement of the second component 304 is then shown where the component has been subdivided into twenty sub-components 306 by setting the layout parameter N<sub>sub-components</sub>(2) to a value of twenty (20), and the size of each sub-component 306 has been established by setting the minimum and maximum delta values of the sub-components 308 within component two 304, Δv<sub>min</sub>(n,m) and Δv<sub>max</sub>(n,m), where n=2 and m=1 to 20. As illustrated, there are 20 sub-components 306 in component 304, indexed by n=2, and m=1−20, labeled Δv<sub>min</sub>(2,1) and Δv<sub>max</sub>(2,20).
An enlargement of the eighth sub-component 306 of component two 304 is then shown where the sub-component 306 has been subdivided into ten smaller components 308 by setting the layout parameter N<sub>smaller</sub><sub><sub2>—</sub2></sub><sub>components</sub>(2,8) to a value of ten (10), and the size of each smaller component 308 has been established by setting the minimum and maximum delta values of the smaller components within sub-component eight 306 of component two 304, Δv<sub>min</sub>(n,m,l) and Δv<sub>max</sub>(n,m,l), where n=2, m=8, and l=1 to 10. As illustrated, there are 10 smaller components 308 in sub-component 306, indexed by n=2, m=8, and l=1 to 10, labeled Δv<sub>min</sub>(2,8,1) and Δv<sub>max</sub>(2,8,10).
It is then shown that these smaller components 308 could be subdivided into x even smaller components (whose size is not shown) using another layout parameter [e.g., N<sub>even</sub><sub><sub2>—</sub2></sub><sub>smaller</sub><sub><sub2>—</sub2></sub><sub>components</sub>(2,8,5)=x], which can be further subdivided, ad infinitum. Also not shown in FIG. 3, are enlargements of the other components 304, sub-components 306, and smaller components 308, which in an exemplary embodiment could also contain twenty sub-components 306, ten smaller components 308, and x even smaller components, respectively.
FIGS. 4a through 4c illustrate the use of delta value range layouts to specify ranges of time position delta values, pulse amplitude delta values, and pulse width delta values. It will be apparent to those skilled in the art that delta value range layouts can be used to specify delta value ranges for other pulse characteristics in accord with the present invention.
Beginning with FIG. 4a, the figure depicts in an exemplary embodiment a diagram 400 illustrating a delta value range 401 of time position delta values between to 402 and t<sub>max </sub>404. The time delta value range 401 has been subdivided into four (4) components (the first two components are labeled 406, 408) that have been subdivided into nine (9) sub-components 410, graphed horizontally. Hence, diagram 400 of FIG. 4a illustrates the delta value range 302 of FIG. 3, graphed horizontally, where the pulse characteristic specifies pulse time position. Each component 406 and sub-component 410 has a minimum time delta value 412 and a maximum time delta value 414 specifying a range of delta time as shown with the enlargement of the fifth sub-component 410 of the second component 408 which has a minimum time delta value 412 of Δt<sub>min</sub>(2,5) and a maximum time delta value 414 of Δt<sub>max</sub>(2,5). When a code element value maps to a component 406, sub-component 410, or smaller component within a time delta value range layout 401, a pulse 416 can be positioned at any time delta value within the delta value range specified by the component 406, sub-component 410, or smaller component. An exemplary embodiment of such mapping to a pulse position is shown in FIG. 4a where a pulse 416 is positioned between Δt<sub>min</sub>(2,5) 412 and Δt<sub>max</sub>(2,5) 414, in this example, approximately halfway in between.
FIG. 4b depicts a diagram 420 illustrating an exemplary embodiment of a range of amplitude delta values from Δa<sub>0 </sub>422 to Δa<sub>max </sub>424 that has been subdivided into seven (7) components (the first two components are labeled 426, 428), vertically shown on the left side of diagram 420. Hence, the delta value range 302 of FIG. 3 is shown in diagram 420 of FIG. 4b as a vertical axis, corresponding to an amplitude pulse characteristic. Seven (7) exemplary pulses 430a, 430b, 430c, 430d, 430e, 430f and 430g, are shown in diagram 420 which correspond to representative pulses 430a–430g that could be specified using the layout relative to a some reference pulse amplitude. In an exemplary embodiment, the exact amplitude delta value of each pulse 430a–430g is determined by arbitrarily selecting the approximate middle delta value of the range component to which a code element value (i.e., 1 through 7) would map. It will be apparent to those skilled in the art that alternative delta values to the middle delta value of the delta value range can be similarly used. The exact delta values selected within the components 426, 428 could be any other common offset value or a different offset value for each pulse 430a–430g. The layout could also be further subdivided into smaller components (i.e., not shown) to provide greater value resolution.
FIG. 4c illustrates an exemplary embodiment of diagram 440 depicting a pulse width delta value range between Δw<sub>0 </sub>444 and Δw<sub>max </sub>446 that has been subdivided into five (5) components (the first two components are labeled 448, 450) and shows five (5) representative pulses 452 that could be specified using the layout relative to some reference pulse width. Hence, diagram 440 of FIG. 4c illustrates the delta value range 302 of FIG. 3, graphed horizontally, corresponding to a pulse width characteristic. As with the pulse amplitude delta values of FIG. 4b, the pulse width delta values used for the five (5) representative pulses 452a, 452b, 452c, 452d and 452e were arbitrarily selected as the middle delta value within each component 448, 450. As will be apparent to those skilled in the relevant art, alternative values to the middle value of the range can be similarly used.
Referring to FIGS. 4b and 4c, for non-temporal pulse characteristics, delta value range layouts may include positive and negative delta range values. Accordingly, the reference characteristic value to which a non-temporal delta value range layout is relative may reside at any point within the layout.
FIGS. 6a through 6d depict non-allowable regions within temporal pulse characteristic delta value range layouts.
In FIG. 6a, non-allowable regions relative to a preceding pulse time position are shown. Specifically, a layout relative to a preceding pulse 602 at time position, t<sub>k−1</sub>, is shown, where t<sub>a</sub>, t<sub>b</sub>, t<sub>c</sub>, and t<sub>d </sub>are time values relative to the preceding pulse. Because time has an inherent order (i.e., by definition the time position value of the pulse in question succeeds the time position value of the preceding pulse), only those values after the preceding pulse are included in the non-fixed layout.
Four delta values, Δt<sub>a</sub>, Δt<sub>b</sub>, Δt<sub>c</sub>, and Δt<sub>d</sub>, illustrate how the time values can be translated into delta values in which case the time values t<sub>k−1</sub>, t<sub>a</sub>, t<sub>b</sub>, t<sub>c</sub>, and t<sub>d </sub>become 0, Δt<sub>a</sub>, Δt<sub>b</sub>, Δt<sub>c</sub>, and Δt<sub>d</sub>.
Three non-allowable regions 606–610, shown shaded with diagonal lines, are defined. The first region 606 enforces a minimum proximity limit, Δt<sub>min</sub>, where Δt<sub>k </sub>cannot reside between 0 and Δt<sub>a </sub>and therefore t<sub>k </sub>cannot reside between t<sub>k−1 </sub>and t<sub>a</sub>. The second region 608 is bounded by minimum and maximum delta values, such that Δt<sub>k </sub>cannot reside between Δt<sub>b </sub>and Δt<sub>c </sub>and therefore t<sub>k </sub>cannot reside between t<sub>b </sub>and t<sub>c</sub>. The third region enforces a maximum proximity limit, Δt<sub>max</sub>, where Δt<sub>k </sub>cannot be greater than or equal to Δt<sub>d </sub>and therefore t<sub>k </sub>cannot be greater than or equal to t<sub>d</sub>.
In FIG. 6b, non-allowable regions relative to any preceding pulse position are shown. Specifically, FIG. 6b illustrates that the non-allowable regions can be defined the same way relative to any preceding pulse, so that the only difference between FIGS. 6a and 6b is that t<sub>k−1 </sub>becomes t<sub>k−n</sub>.
FIGS. 6c and 6d illustrate that similar non-allowable regions can be defined relative to the succeeding pulse 612 position, t<sub>k+1</sub>, and to any succeeding pulse 612 position, t<sub>k+n</sub>, respectively. In the two figures, the first region 618 enforces a minimum proximity limit, Δt<sub>min</sub>, where Δt<sub>k </sub>cannot reside between −Δt<sub>h </sub>and 0, and therefore t<sub>k </sub>cannot reside between t<sub>h </sub>and t<sub>k+n</sub>. The second region 616 is bounded by minimum and maximum delta values, such that Δt<sub>k </sub>cannot reside between −Δt<sub>f </sub>and −Δt<sub>g </sub>and therefore t<sub>k </sub>cannot reside between t<sub>f </sub>and t<sub>g</sub>. The third region 614 enforces a maximum proximity limit, Δt<sub>max</sub>, where Δt<sub>k </sub>cannot be less than or equal to −Δt<sub>e </sub>and therefore t<sub>k </sub>cannot be less than or equal to t<sub>e</sub>.
FIG. 7 illustrates non-allowable regions within a non-fixed non-temporal characteristic value layout relative to any other pulse having some characteristic value, v<sub>k±n</sub>. This figure is consistent with (and similar to a combination of) FIGS. 6b and 6d. Here, six non-allowable regions 702–712 are shown with the three leftmost regions 702–706 mirroring the three rightmost regions 708–712. Note the figure is two-sided as opposed to one-sided since a non-temporal characteristic value of a pulse can be less than, greater than, or the same as that of any other pulse.
Another exemplary embodiment of the present invention defines a pulse characteristic delta value layout by specifying a layout of discrete delta values to which individual code elements can map. As depicted in FIGS. 8a and 8b, a layout parameter, N<sub>discrete values </sub>can be specified to identify some number of discrete delta values within a layout 802 having a delta value, Δv(n), with an index n, where n=1 to N<sub>discrete values</sub>. Discrete delta values may, e.g., be evenly distributed, or not, as depicted in FIGS. 8a and 8b, respectively.
Beginning with FIG. 8a, a diagram 800 illustrates an exemplary embodiment in which thirty-seven (37) evenly-distributed (the first two width values are labeled 808, 810) discrete delta values, Δv(1) 804 through Δv(37) 806, are shown. In the exemplary embodiment, the number of discrete values 804, 806 within layout 802 is thirty-seven (37) and is referred to as N<sub>discrete values</sub>=37.
FIG. 8b depicts, in an exemplary embodiment, a diagram 812 illustrating six (6) non-evenly-distributed (the first two width values are labeled 818 and 820) discrete delta values, Δv(1) 814 through Δv(6) 816. In the exemplary embodiment, the number of discrete delta values 814, 816 within layout 822 is six (6) and is referred to as N<sub>discrete values</sub>=6.
Referring to FIGS. 8a and 8b, for temporal characteristics, Δv(1) must be greater than zero since inherently a succeeding pulse occurs at some time after a preceding pulse. However, for non-temporal pulse characteristics, delta values may be positive or negative. Accordingly, the absolute characteristic value to which a non-temporal discrete delta value layout is relative may reside at any discrete value within the layout, e.g., Δv(3), in which case Δv(3)=0.
FIGS. 9a through 9c illustratively depict exemplary embodiments of discrete delta value layouts for exemplary temporal and non-temporal pulse characteristics, including, e.g., time position, amplitude, and width pulse characteristics, respectively.
FIG. 9a depicts a diagram 902, illustrating an exemplary embodiment of the invention having thirty-seven (37) discrete time values from Δt(1) 904 through Δt(37) 906. Diagram 902 includes five (5) representative pulses 908a, 908b, 908c, 908d and 908e, representing placement at five of the 37 time positions from Δt(1) 904 through Δt(37) 906 relative to a reference position equal to a Δt(1) of zero.
FIG. 9b depicts a diagram 910, illustrating an exemplary embodiment of the invention having seven (7) discrete delta values 914a, 914b, 914c, 914d, 914e, 914f and 914g, corresponding to exemplary voltage amplitude delta values, Δ1.2v, Δ2.34v, Δ3.35v, Δ5.05v, Δ6.2v, Δ7v and Δ9.35v, respectively. Diagram 910 includes representative pulses 912a, 912b, 912c, 912d, 912e, 912f and 912g, depicted having amplitudes 1.2v–9.35v greater than some reference amplitude 916 (e.g., 0.1v), that correspond to the seven discrete delta values 914a–914g, respectively. Unlike the representative pulses depicted in FIG. 4b, which can have any amplitude delta value within a given delta value range component, the amplitude delta values Δ1.2v–Δ9.35v of discrete pulse amplitude delta values 914a–914g of the seven representative pulses 912a–912g of the exemplary embodiment of FIG. 9b must equal a discrete amplitude delta value.
Similarly to FIG. 9b, FIG. 9c depicts a diagram 924, illustrating an exemplary embodiment of the invention having a layout of five (5) discrete pulse width delta values 920a, 920b, 920c, 920d and 920e, and five (5) representative pulses 918a, 918b, 918c, 918d and 918e. Each pulse 918a–918e has a width that corresponds to the discrete pulse width delta values 920a–920e plus some reference pulse width 922.
In one exemplary embodiment of the present invention, the discrete delta value layout embodiments illustrated in FIGS. 8a and 8b, described above, can be combined with an embodiment of a delta value range layout such as, e.g., the layout 401 of FIG. 4a, enabling code element values to specify, e.g., a component 406, 408 within the delta value range layout 401 and a discrete delta value layout within the component 406, 408 (not shown). The use of a combination of the discrete layout and value range layout approaches is shown in FIG. 10.
Specifically, FIG. 10 illustratively depicts a diagram 1000 showing an exemplary embodiment of the invention using a combination of a discrete delta value layout similar to the one depicted in FIG. 8a, and a delta value range layout the same as shown in FIG. 4a. Referring now to FIG. 10, a delta value range layout 401 (as shown in FIG. 4a) is subdivided into four (4) components (the first two are labeled 1002, 1004) that are subdivided into nine (9) sub-components (the first two are labeled 1006, 1008). Component 21004 is shown with a sub-component 1006 which has been further expanded to illustrate discrete delta values from Δv<sub>min</sub>(2,5) 1010 to Δv<sub>max</sub>(2,5) 1012. Thus, as illustrated, each sub-component 1006 contains 27 discrete delta values. The layout 1001 of the exemplary embodiment of diagram 1000 could, e.g., be specified by setting N<sub>components</sub>=4, N<sub>sub-components</sub>(1–4)=9, and N<sub>discrete values</sub>(1–4, 1–9)=27.
According to the present invention, delta codes may be mapped relative to one or more reference values, a reference value may change with each code element, and reference values may change after some number of code elements (e.g., every third code element). FIGS. 11a and 11b depict examples of temporal and non-temporal delta coding where a single code element exists per reference value. In FIG. 1a, diagram 1100 illustrates six temporal delta value layouts 1102a–1102f and six pulse time positions 1104a–1104f specified by six delta values ΔT1–ΔT6 (1106a–1106f), where the first delta value 1106a is mapped to a delta value range layout 1102a that is relative to an initial reference 1108 and the other five delta values 1106b–1106f are mapped to delta value range layouts 1102b–1102f that are relative to the time positions of the preceding pulses. Specifically, ΔT11106a indicates a difference in time to the first pulse 1104a on a layout 1102a that is relative to an initial reference time 1108. ΔT21106b indicates a difference in time to the second pulse 1104b on a layout 1102b that it is relative to the time position of the first pulse 1104a. ΔT31106c indicates a difference in time to the third pulse 1104c on a layout 1102c that is relative to the time position of the second pulse 1104b. ΔT41106d indicates a difference in time to the fourth pulse 1104d on a layout 1102d that is relative to the time position of the third pulse 1104c. ΔT51106e indicates a difference in time to the fifth pulse 1104e on a layout 1102e that is relative to the time position of the fourth pulse 1104d. ΔT61106f indicates a difference in time to the sixth pulse 104f on a layout 1102f that is relative to the time position of the fifth pulse 1104e.
FIG. 11b provides a diagram 1120 that depicts six non-temporal delta value layouts 1122a–1122f and the characteristic values of six pulses 1124a–1124f specified by mapping six delta values ΔV1–ΔV6 (1126a–1126f), where the first delta value 1126a is mapped to a delta value range layout 1122a that is relative to an initial reference value 1128 and the other five delta values 1126b–1126f are mapped to delta value layouts 1122b–1122f that are relative to the characteristic values of the preceding pulses. Similar to FIG. 11a, ΔV1126a indicates a characteristic value difference to the characteristic value of the first pulse 1124a on a layout 1122a that is relative to an initial reference value 1128. ΔV21126b indicates a characteristic value difference to the characteristic value of the second pulse 1124b on a layout 1122b that is relative to the characteristic value of the first pulse 1124a. ΔV31126c indicates a characteristic value difference to the characteristic value of the third pulse 1124c on a layout 1122c that is relative to the characteristic value of the second pulse 1124b. ΔV41126d indicates a characteristic value difference to the characteristic value of the fourth pulse 1124d on a layout 1122d that is relative to the characteristic value of the third pulse 1124c. ΔV51126e indicates a characteristic value difference to the characteristic value of the fifth pulse 1124e on a layout 1122e that is relative to the characteristic value of the fourth pulse 1124d. ΔV61126f indicates a characteristic value difference to the characteristic value of the sixth pulse 1124f on a layout 1122f that is relative to the characteristic value of the fifth pulse 1124e. Negative delta values were included in FIG. 11b to illustrate that, as described previously, non-temporal characteristic values do not have an inherent order, as do temporal characteristic values.
FIGS. 12a and 12b depict examples of temporal and non-temporal delta coding where multiple code elements exist per reference value. In FIG. 12a, diagram 1200 illustrates two temporal delta value layouts 1202a and 1202b, and six pulse time positions 1204a–1204f specified by six delta values ΔT1−ΔT6 (1206a–1206f), where the first three delta values 1206a–1206c are mapped to a delta value layout 1202a that is relative to an initial time reference 1208 and the last three delta values 1206d–1206f are mapped to a delta value layout 1202b that is relative to the time position of the third pulse 1204c. Specifically, ΔT11206a indicates a difference in time to the first pulse 1204a on a layout 1202a that is relative to an initial reference time 1208. ΔT21206b indicates a difference in time to the second pulse 1204b on a layout 1202a that is relative to an initial reference time 1208. ΔT31206c indicates a difference in time to the third pulse 1204c on a layout 1202a that is relative to an initial reference time 1208. ΔT41206d indicates a difference in time to the fourth pulse 1204d on a layout 1202b that is relative to the time of the third pulse 1204c. ΔT51206e indicates a difference in time to the fifth pulse 1204e on a layout 1202b that is relative to the time of the third pulse 1204c. ΔT61206f indicates a difference in time to the sixth pulse 1204f on a layout 1202b that is relative to the time of the third pulse 1204c.
FIG. 12b provides a diagram 1220 that depicts three non-temporal delta value layouts 1222a–1222c and the characteristic values of six pulses 1224a–1224f specified by six delta values ΔV1−ΔV6 (1226a–1226f), where the first two delta values 1226a–1226b are mapped to a delta value layout 1224a that is relative to an initial reference value 1228, the third and fourth delta values 1226c–1226d are mapped to a delta value layout 1224b that is relative to the characteristic value of the second pulse 1224b, and the fifth and sixth delta values 1226e–1226f are mapped to a delta value layout 1224c that is relative to the characteristic value of the fourth pulse 1224d. Similar to FIG. 12a, ΔV11226a indicates a characteristic value difference to the characteristic value of the first pulse 1224a on a layout 1222a that is relative to the initial value reference 1228. ΔV21226b indicates a characteristic value difference to the characteristic value of the second pulse 1224b on a layout 1222a that is relative to the initial value reference 1228. ΔV31226c indicates a characteristic value difference to the characteristic value of the third pulse 1224c on a layout 1222b that is relative to the characteristic value of the second pulse 1224b. ΔV41226d indicates a characteristic value difference to the characteristic value of the fourth pulse 1224d on a layout 1222b that is relative to the characteristic value of the second pulse 1224b. ΔV51226e indicates a characteristic value difference to the characteristic value of the fifth pulse 1224e on a layout 1222c that is relative to the characteristic value of the fourth pulse 1224d. ΔV61226f indicates a characteristic value difference to the characteristic value of the sixth pulse 1224f on a layout 1222c that is relative to the characteristic value of the fourth pulse 1224d. A negative delta value was also included in FIG. 12b to again illustrate that, as described previously, non-temporal characteristic values do not have inherent ordering, as do temporal characteristic values.
Unlike integer codes, where an absolute or relative offset value can be used to specify exact characteristic delta values in a common manner, the fractional portions of floating-point values can vary per code element. The variance of fractional portions of floating-point values per code element, allows each pulse characteristic value to be established independent of other pulse characteristic values. FIG. 13a through FIG. 13c below, illustrate an exemplary embodiment of the component numbering and code mapping embodiments involving integer code element values mapping to delta value range layouts.
FIG. 13a depicts a diagram 1300 of an exemplary embodiment illustrating pulses 1310a–1310d mapped to positions within (in the exemplary embodiment) components 1304 numbered one through nine (1–9) within delta value range layouts 1302a–1302d shown relative to an initial reference 1312 and to preceding pulse position values 1310a–1310c. Diagram 1300 includes a delta code 1314 of integer code elements 1316a–1316d that map to the four layouts 1302a–1302d. Each integer code element 1316a–1316d is, for example, mapped to a component 1304 within its corresponding layout. Specifically, the first code element value 1316a maps to the third component 1304 of the first delta value range layout 1302a that is relative to the initial reference 1312. The second code element value 1316b maps to the sixth component 1304 of the second delta value range layout 1302b that is relative to the first pulse position 1310a. The third code element value 1316c maps to the fourth component 1304 of the third delta value range layout 1302c that is relative to the second pulse position 1310b. The fourth code element value 1316d maps to the eighth component 1304 of the fourth delta value range layout 1302d that is relative to the third pulse position 1310c. The exact positions of the pulses 1310a–1310d within the components 1304 specified by integer codes 1316a–1316d can be determined using, e.g., an absolute, or a relative position offset value. An offset equal to approximately half of a delta value range component was used in the figure. Note that FIG. 13a provides an example of employing a delta value range layout with the single code element per reference temporal delta coding approach described above and illustrated in FIG. 11a.
FIG. 13b depicts a diagram 1320 of an exemplary embodiment illustrating pulses 1330a–1330d mapped to amplitude values within (in the exemplary embodiment) components 1324 numbered −4, −3, −2, −1, 1, 2, 3, 4 within delta value range layouts 1322a–1322d shown relative to an initial value reference 1332 and to preceding pulse amplitude values 1330a–1330c. Diagram 1320 includes a delta code 1334 of integer code elements 1336a–1336d that map to the four layouts 1322a–1322d. Each integer code element 1336a–1336d is, for example, mapped to a component 1324 within its corresponding layout. Specifically, the first code element value 1336a maps to the eighth component 1324 of the first delta value range layout 1322a that is relative to the initial reference 1332. The second code element value 1336b maps to the sixth component 1324 of the second delta value range layout 1322b that is relative to the first pulse amplitude value 1330a. The third code element value 1336c maps to the second component 1324 of the third delta value range layout 1322c that is relative to the second pulse amplitude value 1330b. The fourth code element value 1336d maps to the sixth component 1324 of the fourth delta value range layout 1322d that is relative to the third pulse amplitude value 1330c. The exact delta amplitude values of the pulses 1330a–1330d within the components 1324 specified by integer codes 1336a–1336d can be determined using, e.g., an absolute, or a relative position offset value. An offset equal to approximately half of a delta value range component was used in the figure. Three non-allowable regions 1338a, 1338b, 1338d, shown shaded with diagonal lines, are defined due to a requirement that pulse amplitude be a positive value. The non-allowable region 1338a within the first delta value range layout 1322a encompasses the first four components 1324 that would map to amplitude values less than zero. The non-allowable region 1338b within the second delta value range layout 1322b encompasses a portion of the first component 1324 that would map to amplitude values less that zero. The non-allowable region 1338d within the fourth delta value range layout 1322d encompasses the first component 1324 and a portion of the second component 1324 that would map to amplitude values less than zero. Note that FIG. 13b provides an example of employing a delta value range layout with the single code element per reference non-temporal delta coding approach described above and illustrated in FIG. 11b.
FIG. 14a through FIG. 14c below, illustrate an exemplary embodiment of the component numbering and code mapping embodiments involving integer code element values mapping to discrete delta value layouts.
FIG. 14a depicts a diagram 1400 of an exemplary embodiment illustrating pulses 1410a–1410d mapped to positions within (in the exemplary embodiment) discrete delta values 1404 numbered one through six (1–6) within discrete delta value layouts 1402a–1402d shown relative to an initial time reference 1412 and to preceding pulse values 1410a–1410c. Diagram 1400 includes a delta code 1414 of integer code elements 1416a–1416d that map to the four layouts 1402a–1402d. Each integer code element 1416a–1416d is, for example, mapped to a discrete delta value 1404 within its corresponding layout. Specifically, the first code element value 1416a maps to the second discrete value 1404 of the first discrete delta value layout 1402a that is relative to the initial reference 1412. The second code element value 1416b maps to the third discrete value 1404 of the second discrete delta value layout 1402b that is relative to the first pulse position 1410a. The third code element value 1416c maps to the fifth discrete value 1404 of the third discrete delta value layout 1402c that is relative to the second pulse position 1410b. The fourth code element value 1416d maps to the fourth delta value 1404 of the fourth discrete delta value layout 1402d that is relative to the third pulse position 1410b. Note that FIG. 14a provides an example of employing a discrete delta value layout with the single code element per reference temporal delta coding approach described above and illustrated in FIG. 11a.
FIG. 14b depicts diagram 1420 including an exemplary embodiment of the invention illustrating pulses 1424a–1424d mapped to discrete time position values (1432) within two temporal discrete delta value layouts 1422a and 1422b, each numbered 1–21. Layout 1422a is relative to an initial time reference 1430 and layout 1422b is relative to the second pulse position 1424b. With this embodiment, integer code element values 1428a–1428d of a delta code 1426 specify discrete values where the first and second code elements are mapped to the first layout 1422a that is relative to an initial reference 1430, and the third and fourth code elements are mapped to the second layout 1422b that is relative to the position of the second pulse 1424b. Specifically, the first code element 1428a indicates the number of discrete values 1432 from the initial reference 1430 at which to position the first pulse 1424a. Similarly, the second code element 1428b indicates the number of discrete values 1432 from the initial reference 1430 at which to position the second pulse 1424b. After positioning the second pulse the reference changes to the position of the second pulse 1424b. The third code element 1428c indicates the number of discrete values 1432 from the second pulse position 1424b at which to position the third pulse 1424c. Similarly, the fourth code element 1428d indicates the number of discrete values 1432 from the second pulse position 1424b at which to position the fourth pulse 1424d. Note that FIG. 14b provides an example of employing a discrete delta value layout with the multiple code elements per reference temporal delta coding approach described above and illustrated in FIG. 12a.
FIG. 14c depicts a diagram 1460 including an exemplary embodiment of the invention illustrating code elements 1470a–1470d of a code 1468 mapped to discrete amplitude delta values 1466a–1466g within discrete value layouts 1462a–1462d. With this embodiment, integer code element values 1470a–1470d specify discrete values 1466a–1466g within layouts 1462a–1462d, where the first layout 1462a is relative to an initial reference amplitude 1472, e.g., zero volts, and the remaining layouts 1462b–1462d are relative to the amplitude of the preceding pulses 1464a–1464c. In the exemplary embodiment, an integer code element 1470a–1470d exists per pulse 1464a–1464d. Specifically, the first code element 1470a indicates the discrete delta value within the first layout 1462a specifying a delta amplitude value 1466g of 5.35v from an initial reference 1472 of zero volts and thereby specifying the amplitude of the first pulse as 5.35v. The second code element 1470b indicates the discrete delta value within the second layout 1462b specifying a delta amplitude 1466e of 2.2v relative to the amplitude of the first pulse 1464a and thereby specifying the amplitude of the second pulse 1464b as 7.55v (i.e., 5.35v+2.2v). The third code element 1470c indicates the discrete delta value within the third layout 1462c specifying a delta amplitude 1466b of −3.95v relative to the amplitude of the second pulse 1464b and thereby specifying the amplitude of the third pulse 1464c as 3.6v (i.e., 7.55v−3.95v). The fourth code element 1470d indicates the discrete delta value within the fourth layout 1462d specifying a delta amplitude 1466g of 5.35v relative to the amplitude of the third pulse 1464c and thereby specifying the amplitude of the fourth pulse 1464d as 8.95v (i.e., 3.6v+5.35v). Two non-allowable regions 1474a and 1474d, shown shaded with diagonal lines, are defined due to a requirement that pulse amplitude be a positive value. The non-allowable region 1474a within the first discrete delta value layout 1462a encompasses the first four discrete values 1466a–1466d that would map to amplitude values less than or equal to zero. The non-allowable region 1474d within the fourth discrete delta value layout 1462d encompasses the first two discrete values 1466a and 1466b that would map to amplitude values less than zero. Note that FIG. 14c provides an example of employing a discrete delta value layout with the single code element per reference non-temporal delta coding approach described above and illustrated in FIG. 11b.
FIGS. 15a through 15c depict three exemplary embodiments of the invention illustrating methods mapping floating-point code element values to delta value range layouts. FIG. 15a depicts a diagram 1502 including an exemplary embodiment illustrating each floating-point code element value 1506a–1506d (collectively 1508) mapping a pulse 1510a–1510d to a position in time (as shown). With this embodiment, floating-point code element values 1506a–1506d of a code 1508 specify differences in time between consecutive pulses where the first code element 1506a is relative to an initial reference 1512, and the remaining code elements 1506b–1506d are relative to the preceding pulse position. Specifically, assuming nanosecond time units, the first pulse 1510a is positioned 52.5 nanoseconds after the initial time reference 1512, the second pulse 1510b is positioned 71.2 nanoseconds after the first pulse 1510a, the third pulse 1510c is positioned 161.1 nanoseconds after the second pulse 1510b, and the fourth pulse 1510d is positioned 45.73 nanoseconds after the third pulse 1510c.
FIG. 15b depicts a diagram 1520 including an exemplary embodiment of the present invention illustrating the non-fractional part 1530a–1530h and fractional part 1522a–1522h of each floating-point code element value of the code 1524 mapping a pulse 1538a–1538h, respectively, to a position in time per one of two delta value range layouts. With this embodiment, the first four code elements map to a delta value range layout 1526a relative to an initial reference 1528 of zero and the last four code elements map to a delta value range layout 1526b relative to the position in of time of the fourth pulse 1530d. In an exemplary embodiment of the invention, the non-fractional part of each code element value 1530 indicates the component 1532 within the given delta value range layout 1526 to which the code element is mapped. The fractional part of each code element value 1522 indicates a fraction 1534 of the distance 1536 in time between the minimum and time values of the component 1532 specified by the non-fractional part of the code element value 1530 at which to place a pulse 1538. Note that FIG. 15b provides an example of employing a delta value range layout with the multiple code elements per reference temporal delta coding approach described above and illustrated in FIG. 12a.
FIG. 15c depicts diagram 1540 illustrating non-fractional portions 1542a–1542d and fractional portions 1544a–1544d of each floating-point code element value of the code 1546 mapping a pulse 1558a–1558d to an amplitude value per one of four delta value range layouts 1552a–1552d. With this embodiment, the non-fractional portion of the first floating-point code element value 1542a maps to a delta value range layout 1552a that is relative to an initial value reference 1554 of zero, and the non-fractional portion of the last three floating-point code element values 1542b–1542d map to delta value layouts 1552b–1552d that are relative to the amplitude value of the preceding pulse. Diagram 1540 also includes fractional portions 1544a–1544d of each floating-point code element specifying the fractional difference between the minimum and maximum amplitude values of each mapped component 1548 used to determine the exact amplitude value for a pulse 1558a–1558d. Specifically, the non-fractional part of the first code element 1542a indicates the fourth component 1548 from the initial reference value 1554 and thereby specifies the eighth component in the layout 1552a. The fractional part of the first code 1544a specifies that the first pulse amplitude be at a value that is 51 percent of the eighth component. The non-fractional part of the second code element 1542b indicates the second component 1548 from the amplitude value of the first pulse 1558a and thereby specifies the sixth component in the layout 1552b. The fractional part of the second code 1544b specifies that the second pulse amplitude be at a value that is 65 percent of the sixth component 1548. The non-fractional part of the third code element 1542c indicates a delta of minus three components 1548 relative to the amplitude of the second pulse 1554b and thus specifies the second component 1548 in the layout 1552c. The fractional part of the third code 1544c specifies that the third pulse amplitude be at a value that is 26 percent of the second component 1548. The non-fractional part of the fourth code element 1542d indicates a delta of three components 1548 from the amplitude of the third pulse 1554c and thus specifies the seventh component 1548 in the layout 1552d. The fractional part of the fourth code 1544d specifies that the fourth pulse amplitude be at a value that is 68 percent of the seventh component 1548. Three non-allowable regions 1556a, 1556b, and 1556d, shown shaded with diagonal lines, are defined because of a requirement that pulse amplitude be a positive value. The non-allowable region 1556a within the first delta value range layout 1552a encompasses the first four components 1548 that would map to amplitude values less than or equal to zero. The non-allowable region 1556b within the second delta value range layout 1552b encompasses a portion of the first component 1548 that would map to amplitude values less than or equal to zero. The non-allowable region 1556d within the fourth discrete delta value layout 1552d encompasses the first component 1548 and the portion of the second component 1550 that would map to amplitude values less than or equal to zero. Note that FIG. 15c provides an example of employing a delta value range layout with the single code element per reference non-temporal delta coding approach described above and illustrated in FIG. 11b.
The exemplary code mapping embodiments shown in FIGS. 15a through 15c can be applied at any component, sub-component, smaller component, even smaller component, ad infinitum, level defined within a delta value range layout.
FIGS. 16a and 16b depict typical pulse shapes used in UWB transmission systems. Specifically, a first pulse shape 1602 and a second pulse shape 1604 are illustrated.
FIGS. 17a and 17b depict two time-coded pulse trains, which are sequences of such pulses transmitted according to a time-hopping code C=(T<sub>k</sub>, 0≦k≦N−1). The first pulse train 1702 comprises pulses 1602, while the second pulse train 1704 comprises pulses 1604. The length of the time-hopping code for both pulse trains is N, and the values T<sub>k </sub>specify the transmission times of the individual pulses in the time-coded pulse train.
For the purposes of this discussion, the length N of the time-hopping code C can either be finite or infinite. If N is finite, the code is presumed to repeat after a finite time T, where T>T<sub>N−1</sub>. Infinite-length codes do not occur in practice, of course, but serve as useful models for repeating codes with very long repeat periods, as well as for long codes that are generated dynamically during transmission.
The difference sequence ΔC=(ΔT<sub>k</sub>=T<sub>k</sub>−T<sub>k−1</sub>, 1≦k≦N−1) represents the inter-pulse times, i.e. the time delays between the transmissions of successive pulses in the time-coded pulse train. Note that since T<sub>k</sub>=T<sub>k−1</sub>+ΔT<sub>k</sub>, the time-hopping code C is completely specified by an initial time T<sub>0</sub>, a difference sequence ΔC, and in the case of repeating codes, a repeat time T:
<maths id="MATH-US-00001" num="00001"><math overflow="scroll"><mrow><mrow><msub><mi>T</mi><mi>k</mi></msub><mo>=</mo><mrow><msub><mi>T</mi><mn>0</mn></msub><mo>+</mo><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo>⁢</mo><mrow><mi>Δ</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo>⁢</mo><msub><mi>T</mi><mi>j</mi></msub></mrow></mrow></mrow></mrow><mo>,</mo><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></mrow></mrow></math></maths>Thus the difference sequence effectively specifies the time positioning of the kth pulse relative to the previous pulse, through the relation T<sub>k</sub>=T<sub>k−1</sub>+ΔT<sub>k</sub>. In one embodiment, the method of the present invention is to design time-hopping codes by constructing difference sequences in certain specific ways. Time-hopping codes produced in these ways will be referred to as one type of delta codes.
The second-order difference sequence, Δ<sup>2</sup>C=(Δ<sup>2</sup>T<sub>k</sub>=T<sub>k</sub>−T<sub>k−2</sub>, 2≦k≦N−1), specifies the distance between non-adjacent pulse pairs which have one intervening pulse between them. Similarly, the r<sup>th </sup>order difference sequence<FORM>Δ<sup>r</sup>C=(Δ<sup>r</sup>T<sub>k</sub>=T<sub>k</sub>−T<sub>k−r</sub>,r≦k≦N−1)</FORM>specifies the distance between non-adjacent pulse pairs which have r−1 intervening pulses between them. These higher order difference sequences can be used to position a pulse relative to not only the preceding pulse, but relative to some or all of the preceding pulses. These sequences can also be used to position pulses with respect to succeeding, as well as preceding pulses, although such “temporal look-ahead” methods obviously must be done off-line.
<maths id="MATH-US-00002" num="00002"><math overflow="scroll"><mrow><mrow><mi>S</mi><mo>⁡</mo><mrow><mo>(</mo><mi>ω</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mi>S</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>ω</mi><mo>,</mo><mrow><mi>Δ</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo>⁢</mo><mi>C</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi>N</mi><mo>+</mo><mrow><mn>2</mn><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></mrow><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></munderover><mo>⁢</mo><mrow><mrow><mi>cos</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>ω</mi><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></mrow><mi>k</mi></munderover><mo>⁢</mo><mrow><mi>Δ</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo>⁢</mo><msub><mi>T</mi><mi>l</mi></msub></mrow></mrow></mrow><mo>)</mo></mrow></mrow><mo>.</mo></mrow></mrow></mrow></mrow></mrow></mrow></mrow></math></maths>
The auto-correlation function of the code can be found through an inverse Fourier transform of the above power-spectral density function. Control over the inter-pulse times can be achieved in many ways. For instance, a constraint of the form<FORM>t<sub>low</sub>≦ΔT<sub>k</sub>≦t<sub>high </sub></FORM>is easily implemented at code design-time, and ensures that no two pulses are ever closer than t<sub>low </sub>in time, nor further apart than t<sub>high</sub>. Such a constraint also guarantees a PRF that is between PRF<sub>low</sub>=t<sub>high</sub><sup>−1 </sup>and PRF<sub>high</sub>=t<sub>low</sub><sup>−1</sup>. There are additional ways of controlling the PRF through the difference sequence, such as by controlling the average value of the ΔT<sub>k</sub>'s that form the difference sequence. The current invention addresses all these parameters through the construction of the difference sequences used to generate the time-hopping code.
Delta codes constructed through difference sequences that have been generated via some random process can possess statistical properties that make them highly desirable for implementing robust impulse communication systems. In this case we have<FORM>ΔT<sub>k</sub>=η<sub>k</sub>,0≦k≦N−1</FORM>where the η<sub>k</sub>'s are samples drawn from some probability distribution P. Standard random-number generation techniques can be used to produce the sequence of η<sub>k </sub>values used to generate the delta-code. There are standard techniques for relating the expected power-spectral density and auto-correlation functions for these codes to the probability distribution P. Additionally, the value-range and discrete layouts described above, as well as combinations of these, can be implemented through the selection of an appropriate probability distribution function P. Layout specifications of non-allowable regions or values simply translate into requirements that the probability density or mass functions for P be zero for certain values.
<maths id="MATH-US-00003" num="00003"><math overflow="scroll"><mrow><mrow><mi>p</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>⁢</mo><mn>1</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow><mo>)</mo></mrow><mo>⁢</mo><msup><mi>ⅇ</mi><mrow><mrow><mo>-</mo><mi>x</mi></mrow><mo>/</mo><mi>T</mi></mrow></msup></mrow></mrow></math></maths>of this distribution, for several different values of T. (Here, the notation 1(P), where P is a proposition, refers to the function that is 1 when P is true, and 0 otherwise. Thus 1(x≧0) is the function that is 0 when x is negative, and 1 otherwise.) Specifically, T=25 for the first graph 1802, T=50 for the second graph 1804, and T=100 for the third graph 1806. Because the inter-pulse times are exponentially distributed random variables, the pulse transmission times generated by this method form realizations of a Poisson process, a fundamental process in the theory of probability. For this reason, we will refer to delta-codes generated by this method as Poisson codes.
<maths id="MATH-US-00004" num="00004"><math overflow="scroll"><mrow><mrow><mrow><mi>S</mi><mo>⁡</mo><mrow><mo>(</mo><mi>ω</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mi>N</mi><mo>+</mo><mrow><mn>2</mn><mo>⁢</mo><mrow><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo>⁢</mo><mrow><mrow><mo>(</mo><mrow><mi>N</mi><mo>-</mo><mi>k</mi></mrow><mo>)</mo></mrow><mo>⁢</mo><mrow><msup><mi>cos</mi><mi>k</mi></msup><mo>⁡</mo><mrow><mo>(</mo><mi>θ</mi><mo>)</mo></mrow></mrow><mo>⁢</mo><mrow><mi>cos</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>k</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo>⁢</mo><mi>θ</mi></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></maths>where θ=arctan(ωT). FIG. 19 shows a graph 1902 of this expected power-spectral density function (the values N=16 and T=100 were used in producing the graph). The frequency units in FIG. 19 are gigahertz; the expected power-spectral density is flat for frequencies above 4 megahertz. The low-frequency artifacts are a function of the length of the code, and become smaller as the code length increases. In the N→∞ limit (and after a suitable resealing), the average power-spectral density function converges to
<maths id="MATH-US-00005" num="00005"><math overflow="scroll"><mrow><mrow><msub><mi>S</mi><mi>∞</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mi>ω</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>⁢</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mfrac><mrow><mn>2</mn><mo>⁢</mo><mi>π</mi></mrow><mi>T</mi></mfrac><mo>⁢</mo><mrow><mi>δ</mi><mo>⁡</mo><mrow><mo>(</mo><mi>ω</mi><mo>)</mo></mrow></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math></maths>(here δ(w) is the familiar Dirac impulse function). As is readily apparent, this function is completely flat, except for a spike at ω=0. The corresponding auto-correlation function is given by
<maths id="MATH-US-00006" num="00006"><math overflow="scroll"><mrow><mrow><msub><mi>R</mi><mi>∞</mi></msub><mo>⁡</mo><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>⁢</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>+</mo><mrow><mi>δ</mi><mo>⁡</mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow></math></maths>which again is close to ideal. Of course, the spectral and auto-correlation functions for any specific Poisson code will oscillate randomly about these expected values. FIGS. 20a, 20b, and 20c together depict a graph 2002 of the power spectral density for a Poisson code of length N=16.
<maths id="MATH-US-00007" num="00007"><math overflow="scroll"><mrow><mover><mi>PRF</mi><mi>_</mi></mover><mo>=</mo><mfrac><mn>1</mn><mi>T</mi></mfrac></mrow></math></maths>It should be noted that pure Poisson codes have no constraints on the inter-pulse times ΔT<sub>k</sub>. As a consequence, long Poisson codes will probably contain pulse pairs that are extremely closely spaced, as well as other successive pulse pairs that are very widely separated. Such occurrences may well violate hardware constraints, and thus there needs to be a mechanism for correcting them. Constraints on the inter-pulse times, such as t<sub>low</sub>≦ΔT<sub>k</sub>≦t<sub>high</sub>, can achieve this. These constraints can be implemented by changing the distribution P that controls the generation of the inter-pulse times. Two embodiments that implement this idea are constrained Poisson codes and uniform delta codes.
With constrained Poisson codes, inter-pulse times are sampled from a modified exponential distribution on the interval [t<sub>low</sub>,t<sub>high</sub>]. In this way, the inter-pulse times are still random, but satisfy the constraint t<sub>low</sub>≦ΔT<sub>k</sub>≦t<sub>high</sub>, thus in effect implementing a delta value range layout with two non-allowable regions, as described previously and illustrated in FIG. 5. The probability density function of the modified exponential distribution is given by
<maths id="MATH-US-00008" num="00008"><math overflow="scroll"><mrow><mrow><mi>p</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mo>{</mo><mtable><mtr><mtd><mrow><mrow><mi>λ</mi><mo>·</mo><mn>1</mn></mrow><mo>⁢</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>∈</mo><mrow><mo>[</mo><mrow><msub><mi>t</mi><mi>low</mi></msub><mo>,</mo><msub><mi>t</mi><mi>high</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>⁢</mo><mfrac><msup><mi>ⅇ</mi><mrow><mo>-</mo><mrow><mi>λ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>-</mo><msub><mi>t</mi><mi>low</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></msup><mrow><mn>1</mn><mo>-</mo><msup><mi>ⅇ</mi><mrow><mo>-</mo><mrow><mi>λ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><msub><mi>t</mi><mi>high</mi></msub><mo>-</mo><msub><mi>t</mi><mi>low</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></msup></mrow></mfrac></mrow></mtd><mtd><mrow><mrow><mi>if</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo>⁢</mo><mi>λ</mi></mrow><mo>≠</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mn>1</mn><mrow><msub><mi>t</mi><mi>high</mi></msub><mo>-</mo><msub><mi>t</mi><mi>low</mi></msub></mrow></mfrac><mo>⁢</mo><mn>1</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>∈</mo><mrow><mo>[</mo><mrow><msub><mi>t</mi><mi>low</mi></msub><mo>,</mo><msub><mi>t</mi><mi>high</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mtd><mtd><mrow><mrow><mi>if</mi><mo>⁢</mo><mstyle><mspace width="0.8em" height="0.8ex"/></mstyle><mo>⁢</mo><mi>λ</mi></mrow><mo>=</mo><mn>0.</mn></mrow></mtd></mtr></mtable></mrow></mrow></math></maths>Here λ is a constant controlling the mean value of the distribution. Graphs of this density function are shown in FIG. 21 for several values of λ. Specifically, λ=1.0 for graph 2102, λ=0.3 for graph 2104, λ=0 for graph 2106, λ=−0.3 for graph 2108, and λ=−1.0 for graph 2110. Notice that when λ=0 we have the uniform distribution on [t<sub>low</sub>,t<sub>high</sub>]; in this case, the resulting codes are called uniform delta codes. The mean inter-pulse time is given by
<maths id="MATH-US-00009" num="00009"><math overflow="scroll"><mrow><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo>⁢</mo><mrow><mrow><mi>T</mi><mo>=</mo><mrow><msub><mi>t</mi><mi>low</mi></msub><mo>+</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mrow><msup><mi>ⅇ</mi><mrow><mo>-</mo><mrow><mi>λ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><msub><mi>t</mi><mi>high</mi></msub><mo>-</mo><msub><mi>t</mi><mi>low</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></msup><mo>⁡</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mrow><mi>λ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><msub><mi>t</mi><mi>high</mi></msub><mo>-</mo><msub><mi>t</mi><mi>low</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>λ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>-</mo><msup><mi>ⅇ</mi><mrow><mo>-</mo><mrow><mi>λ</mi><mo>⁡</mo><mrow><mo>(</mo><mrow><msub><mi>t</mi><mi>high</mi></msub><mo>-</mo><msub><mi>t</mi><mi>low</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow></msup></mrow><mo>)</mo></mrow></mrow></mfrac></mrow></mrow><mo>,</mo></mrow></mrow></math></maths>and the average PRF is again
<maths id="MATH-US-00010" num="00010"><math overflow="scroll"><mrow><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo>⁢</mo><mrow><mover><mi>PRF</mi><mi>_</mi></mover><mo>=</mo><mrow><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>.</mo></mrow></mrow></mrow></math></maths>
The parameter λ, which can be any real number, controls the mean inter-pulse time; it can be shown that as λ runs from −∞ through 0 and on to ∞, the mean inter-pulse time runs from t<sub>high </sub>through (t<sub>low</sub>+t<sub>high</sub>)/2 and on to t<sub>low</sub>. FIG. 22 depicts a graph 2202 of the mean inter-pulse time as a function of the parameter λ.
Naturally, constraining Poisson codes in this way will have an effect on the spectrum and auto-correlation functions of the resulting code. However, such effects can be predicted with theory, and controlled. FIGS. 23a, 23b, and 23c together depict a graph 2302 of the power-spectral density function for a constrained Poisson code of length 16. The value λ=0 was used in generating this particular code, so it is actually a uniform delta code.
The processing steps for an algorithm to generate Poisson codes are given below. Note that for simplicity, T<sub>0 </sub>is set to 0. This is not essential to the algorithm; T<sub>0 </sub>could be set to any arbitrary initial value. FIG. 24 presents a flow diagram 2400 for Poisson code generation algorithm.
Step 1 (2402). Set k=1, and T<sub>0</sub>=0.
Step 3 (2408). Set ΔT<sub>k</sub>=η<sub>k</sub>, where □<sub>k </sub>is the output of an exponential random number generator with mean T.
Step 4 (2410). Set T<sub>k</sub>=T<sub>k−1</sub>+□T<sub>k</sub>, and store T<sub>k</sub>.
The processing steps for an algorithm to generate constrained Poisson codes are given below. Setting the parameter □=0 results in the generation of uniform delta codes. Note that for simplicity, T<sub>0 </sub>is set to 0. This is not essential to the algorithm; T<sub>0 </sub>could be set to any arbitrary initial value. FIG. 25 provides a flow diagram 2500 for a constrained Poisson code generation algorithm.
Step 1 (2502). Set k=1, and T<sub>0</sub>=0.
Step 3 (2508). Set ΔT<sub>k</sub>=η<sub>k</sub>, where □<sub>k </sub>is the output of a modified exponential random number generator with control parameter □, and limits t<sub>low</sub>, t<sub>high</sub>.
Step 4 (2510). Set T<sub>k</sub>=T<sub>k−1</sub>+□T<sub>k</sub>, and store T<sub>k</sub>.
Computer random number generator functions can in one embodiment employ a linear congruential generation (LCG) method, which generates an n-th random number, x<sub>n</sub>, from a previous random number, x<sub>n−1</sub>, using an equation of the general form as follows:<FORM>x<sub>n</sub>=Ax<sub>n−1</sub>+c(mod m)</FORM>where n identifies a given random number in the generated random number sequence, and the generated sequence is characterized by the multiplier A, the additive constant c, the modulus m, and an initial seed x<sub>0</sub>. These random number generator functions can be referred to as LCG(a,c,m,x<sub>0</sub>), which determines the sequence generated.
Another exemplary embodiment of another method that can be used as a computer random number generator is known as a Additive Lagged-Fibonacci Generator (ALFG) method. The approach can be described by an equation of the form:<FORM>x<sub>n</sub>=x<sub>n−j</sub>+x<sub>n−k</sub>(mod 2<sup>m</sup>),j<k </FORM>where n identifies a given random number in the generated random number sequence, and j and k represent offsets to previously generated random numbers. The period of these generators is (2<sup>k</sup>−1)2<sup>m−1 </sup>and they are referred to as ALFG(l,k,m,x<sub>0</sub>), which determines the sequence generated.
Binary shift-register pseudorandom number generators can be implemented in many different ways. In an exemplary embodiment, a linear feedback shift register as illustrated in FIG. 26a, can be used. FIG. 26a illustratively depicts a block diagram 2602 including an exemplary embodiment of an LCG linear feedback shift register, including exclusive OR logic gate 2604 having two inputs 2606, 2608 and one output 2610. Eight bit shift register 2614 (labeled bit 02614a through bit 72614h includes shift out output 2612 coupled to input 2606 of exclusive OR logic gate 2604. Bit 32614d is coupled to input 2608 of exclusive OR logic gate 2604, and output 2610 of exclusive OR logic gate 2604 is coupled as shown to bit 72614h.
<maths id="MATH-US-00011" num="00011"><math overflow="scroll"><mrow><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo>⁢</mo><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msub><mo>=</mo><mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>k</mi><mo>-</mo><mn>1</mn></mrow></munderover><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo>⁢</mo><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>⁢</mo><mrow><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo>⁡</mo><mrow><mo>(</mo><mrow><mi>mod</mi><mo>⁢</mo><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo>⁢</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow></mrow></mrow></mrow></mrow></math></maths>where n identifies a given random number in the generated random number sequence, k is the number of bits in the shift register, a<sub>i </sub>is the value of the i-th bit in the shift register. The sequence of bits that is generated depends on the initial shift-register state and which shift-register bit value 2614d, a<sub>i</sub>, is fed back into the exclusive-OR device 2604 along with the shifted output 2612.
The ALFG method can also be implemented using a shift register and a modulo adder device 2618, as shown in FIG. 26b. FIG. 26b depicts diagram 2616 including an eight-bit shift register having bit 02628a through bit 72628h. Diagram 2616 also includes addition modulo 2 device 2618 having two inputs 2620 (coupled to bit 72628h) and 2622 (coupled to bit 52628f), and an output 2624 which can be outputted and can be fed back into input 2626 of the ALFG shift register at bit 02628a.
The ALFG shift register can be described by an equation of the form:<FORM>x<sub>n</sub>=x<sub>n−j</sub>+x<sub>n−k</sub>(mod 2),j<k </FORM>where n identifies a given code in the generated code sequence, and j and k represent the shift-register bits 2628h, 2628f fed into the modulo adder device 2618.
The sequential generation methodology works by directly specifying a sequence of delay values for the delta code. To be specific, let S be some set of states, and let N be a positive integer. Let ƒ be a mapping from {1, 2, . . . , N−1} into S, and let g:S→[0, ∞) be a mapping from S into the non-negative reals. The function ƒ describes how a given sequence is generated, while the function g establishes a correspondence between the sequence values and the delay values to be used in constructing the delta code. Specifically, the difference sequence ΔC=(ΔT<sub>k</sub>=T<sub>k</sub>−T<sub>k−1</sub>, 1≦k≦N−1) is generated according to the relation<FORM>ΔT<sub>k</sub>=g(ƒ(k)),1≦k≦N−1</FORM>FIG. 27 depicts a simplified top-level flow diagram 2702 for sequential generation of delta codes. Specifically, to determine the delta time value to the next pulse, function ƒ 2706 receives as input pulse index k (2704) and produces an output s<sub>k </sub>2708 that is input to function g 2710, which produces an output ΔT<sub>k </sub>2712. Typically, though not always, ƒ will be chosen to provide useful correlation or spectral properties, while g will be chosen to satisfy PRF and other timing constraints.
Rational Congruential delta codes are a specific embodiment of this methodology in the current invention. These codes are generated through functions ƒ of the form<FORM>ƒ(x)=r(x)mod M, </FORM>where N is a positive integer, and r is a rational function defined on Z<sub>M</sub>, the set of integers modulo M. In particular, the following special cases are identified:
Hyperbolic Congruential delta codes: ƒ(x;a)=ax<sup>−1 </sup>mod M
Quadratic Congruential delta codes: ƒ(x;a)=ax<sup>2 </sup>mod M
Cubic Congruential delta codes: ƒ(x;a)=ax<sup>3 </sup>mod M
The processing steps for an algorithm to generate rational congruential sequential delta codes are given below. Note that for simplicity, T<sub>0 </sub>is set to 0. This is not essential to the algorithm; T<sub>0 </sub>could be set to any arbitrary initial value. FIG. 28 provides a flow diagram 2800 for a rational congruential sequential delta code generation algorithm.
Step 1 (2802). Set k=1, and T<sub>0</sub>=0.
Step 3 (2808). Set ΔT<sub>k</sub>=g(r(k) mod M), where r is a rational function defined on Z<sub>M</sub>, the set of integers modulo M.
Step 4 (2810). Set T<sub>k</sub>=T<sub>k−1</sub>+□T<sub>k</sub>, and store T<sub>k</sub>.
ƒ(x)=ax<sup>2 </sup>mod M
The iterated methodology utilizes iterated function systems as the mechanism for generating delta codes deterministically. The basic methodology is as follows. Let S be some set of states, and let ƒ:S→S be a mapping from S into itself. The function ƒ describes how transitions are made from one state to another. Let g:S→[0, ∞) be a mapping from S into the non-negative real numbers. The function g establishes a correspondence between the states in S and the time delays used to generate the delta code. Given an initial state s<sub>1</sub>εS, a difference sequence ΔC=(ΔT<sub>k</sub>=T<sub>k</sub>−T<sub>k−1</sub>, 1≦k≦N−1) can now be generated according to the following recursion:<FORM>ΔT<sub>1</sub>=g(s<sub>1</sub>)</FORM><FORM>s<sub>k+1</sub>=ƒ(s<sub>k</sub>)</FORM><FORM>ΔT<sub>k+1</sub>=g(s<sub>k+1</sub>)</FORM><FORM>i=1,2, . . . ,N−2</FORM>FIG. 29 depicts a simplified top-level flow diagram 2902 for this process. Specifically, given an initial state 2904 as input, function ƒ 2906 outputs a new state 2908 which is input back into function ƒ 2906 and is also input into function g 2910, which outputs ΔT<sub>k+1 </sub>2912. Typically, though not always, ƒ will be chosen to provide useful correlation or spectral properties, while g will be chosen to satisfy PRF and other timing constraints.
<maths id="MATH-US-00012" num="00012"><math overflow="scroll"><mrow><mstyle><mspace width="0.3em" height="0.3ex"/></mstyle><mo>⁢</mo><mrow><mrow><mrow><mi>L</mi><mo>⁡</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mfrac><mi>x</mi><mrow><mn>1</mn><mo>-</mo><mi>α</mi></mrow></mfrac><mo>⁢</mo><mn>1</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>∈</mo><mrow><mo>[</mo><mrow><mn>0</mn><mo>,</mo><mrow><mn>1</mn><mo>-</mo><mi>α</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mrow><mn>2</mn><mo>⁢</mo><mi>α</mi></mrow><mo>-</mo><mn>1</mn></mrow><mi>α</mi></mfrac><mo>⁢</mo><mn>1</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>∈</mo><mrow><mo>[</mo><mrow><mrow><mn>1</mn><mo>-</mo><mi>α</mi></mrow><mo>,</mo><mn>1</mn></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mfrac><mrow><mi>x</mi><mo>-</mo><mi>β</mi></mrow><mi>β</mi></mfrac><mo>⁢</mo><mn>1</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>∈</mo><mrow><mo>[</mo><mrow><mn>1</mn><mo>,</mo><mrow><mn>1</mn><mo>+</mo><mi>β</mi></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>+</mo><mrow><mfrac><mrow><mi>x</mi><mo>-</mo><mrow><mn>2</mn><mo>⁢</mo><mi>β</mi></mrow></mrow><mrow><mn>1</mn><mo>-</mo><mi>β</mi></mrow></mfrac><mo>⁢</mo><mn>1</mn><mo>⁢</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>∈</mo><mrow><mo>[</mo><mrow><mrow><mn>1</mn><mo>+</mo><mi>β</mi></mrow><mo>,</mo><mn>2</mn></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></mrow></math></maths>where □ and □ are arbitrary constants satisfying 0<α, β<1. Specifically, line 3002 plots L(x) for 0≦x≦α, line 3004 plots L(x) for α≦x≦1, line 3006 plots L(x) for 1≦x≦1+β, and line 3008 plots L(x) for 1+β≦x≦2. For xε[0, 2) define<FORM>Q(x)=1(xε[1, 2))</FORM>Note that Q(x)=0 if xε[0, 1), while Q(x)=1 if xε[1, 2). Now define<FORM>η(x)=min {n≧1:Q(L<sup>(n)</sup>(x))≠Q(x)}</FORM>Here L<sup>(n)</sup>(x) denotes the n-fold composition of L with itself, applied at x. If x is such that the set on the right is empty, □ is undefined at that value of x. It can in fact be shown that □ is defined for almost every (in the sense of Lebesgue measure) x in [0, 2). For these values of x, define<FORM>ƒ(x)=L<sup>(η(x))</sup>(x).</FORM>This function now serves as the ƒ in the original recursion that defines the iterated methodology. A number of choices for g are possible in this case, the simplest of which is probably<FORM>g(x)=η(x).</FORM>It is worth noting that, for this choice of ƒ, there exist choices of g that result in delta codes that are good statistical approximations of the pseudo-random Poisson codes introduced above. It is also possible to modify the definition of the function L so that constraints of the form t<sub>low</sub>≦ΔT<sub>k</sub>≦t<sub>high </sub>can be implemented in this method.
The processing steps for an algorithm to generate rational congruential iterative delta codes are given below. Note that for simplicity, T<sub>0 </sub>is set to 0. This is not essential to the algorithm; T<sub>0 </sub>could be set to any arbitrary initial value. FIG. 31 depicts a flow diagram 3100 for a rational congruential iterative delta code generation algorithm.
Step 1 (3102). Set k=1, T<sub>0</sub>=0, s=s<sub>0</sub>, and □T<sub>1</sub>=g(s<sub>0</sub>).
Step 3 (3108). Set s=r(s) mod M, where r is a rational function defined on Z<sub>M</sub>, the set of integers modulo M.
Step 4 (3110). Set □T<sub>k</sub>=g(s).
Step 5 (3112). Set T<sub>k</sub>=T<sub>k−1</sub>+□T<sub>k</sub>, and store T<sub>k</sub>.
The processing steps for an algorithm to generate piece-wise linear iterative delta codes are given below. Note that for simplicity, T<sub>0 </sub>is set to 0. This is not essential to the algorithm; T<sub>0 </sub>could be set to any arbitrary initial value. FIG. 32 depicts a flow diagram 3200 for a rational congruential iterative delta code generation algorithm.
Step 1 (3202). Set k=1, T<sub>0</sub>=0, x=x<sub>0</sub>, and □T<sub>1</sub>=g(x<sub>0</sub>).
Step 3 (3208). Set x=L<sup>(η(x))</sup>(x), where L and □ are as described above.
Step 4 (3210). Set □T<sub>k</sub>=g(x).
Step 5 (3212). Set T<sub>k</sub>=T<sub>k−1</sub>+□T<sub>k</sub>, and store T<sub>k</sub>.
Pulse characteristics may be specified using a pseudorandom delta code such as a Poisson code or a constrained Poisson code, which includes the special case uniform delta code. Alternatively, pulse characteristics may be specified using deterministic delta codes including those generated using a sequential generation methodology, e.g., a rational congruential sequential delta code. In four separate exemplary embodiments of the invention, rational congruential sequential delta codes may employ a hyberbolic congruential, linear congruential, quadratic congruential function, or cubic congruential function. In another exemplary embodiment rational congruential sequential delta codes may employ a function of the form ƒ(x;a)=ax<sup>n </sup>mod M.
Deterministic delta codes may also be generated using an iterative generation methodology, e.g., a rational congruential iterative delta code. In four separate exemplary embodiments of the invention rational congruential iterative delta codes may employ a hyberbolic congruential, linear congruential, quadratic congruential function, or cubic congruential function. In another exemplary embodiment rational congruential iterative delta codes may employ a function of the form ƒ(x;a)=ax<sup>n </sup>mod M. Furthermore, deterministic iterative delta codes may be generated using the piecewise linear iterative delta code generation methodology.
375/138, 375/239, 375/242, 375/247
Method And Apparatus For Mapping Pulses To A Non Fixed Layout