Apparatus and methods for providing efficient space-time structures for preambles, pilots and data for multi-input, multi-output (MIMO) communications systems are provided. One such embodiment includes providing a computer program that includes logic configured to provide an initial structure. The computer program further includes logic configured to verify that the rows of the initial structure are linearly independent and logic configured to apply an orthonormalization procedure to the initial structure to obtain a space-time structure. Methods are also provided for providing efficient space-time structures for preambles, pilots and data for MIMO communications systems.
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0. 12. A method of providing a space-time structure for preambles, pilots, and data for a multi-input, multi-output communications system, the method comprising:
verifying that rows of an initial structure are linearly independent;
applying an orthonormalization procedure to the initial structure at an encoder to obtain the space-time structure for a preamble structure, a pilot structure, or a data structure in a time or frequency domain; and
applying, at the encoder, the space-time structure as a space-time preamble structure, a space-time pilot structure, or a space-time data structure in the time or frequency domain and combining the space-time preamble structure, the space-time pilot structure, or the space-time data structure with one or more data symbols for transmission in the multi-input, multi-output communications system.
0. 24. A communication system comprising:
an encoder having a pilot/training symbol inserter, the pilot/training symbol inserter configured to insert pilot symbols or training symbols into data blocks, wherein the inserted pilot symbols or training symbols include a space-time structure formed using an orthonormalization procedure, wherein the encoder is configured to:
verify that rows of an initial structure are linearly independent; and
apply the orthonormalization procedure to the initial structure to form the space-time structure;
at least one modulator coupled to the encoder, each modulator outputting a frame structure comprising a data structure and at least a preamble structure or a pilot structure; and
at least one transmit antenna, each transmit antenna corresponding to a respective one of the at least one modulator, each transmit antenna transmitting the frame structure output from the corresponding modulator.
1. A computer program embodied in a non-transitory computer-readable storage medium having a computer program stored thereon for providing efficient a space-time structures structure for preambles, pilots, and data for a multi-input, multi-output communications systems system, the computer program comprising:
logic configured to provide an initial structure;
logic configured to verify that rows of said the initial structure are linearly independent;
logic configured to apply an orthonormalization procedure to said the initial structure to obtain a space-time structure for a preamble structure, a pilot structure, or pilot a data structure in a time or frequency domain; and
logic configured to insert apply the space-time structure as a space-time preamble structure, a space-time pilot structure, or pilot a space-time data structure in the time or frequency domain and combine the space-time preamble structure, the space-time pilot structure, or the space-time data structure with one or more data symbols for transmission in the multi-input, multi-output (MIMO) communications system.
2. The computer program computer-readable storage medium of
logic configured to choose a symbol alphabet to provide symbols for said the initial structure; and
logic configured to choose an initial configuration of said the initial structure.
3. The computer program computer-readable storage medium of
logic configured to confirm that symbols of said the space-time structure are within a predetermined distance of symbols of said the initial structure;
logic configured to construct a space-time sequence structure from a plurality of said the space-time structures; and
logic configured to verify that a peak-to-average power ratio (PAPR) of said the space-time structure is less than a predetermined value.
4. The computer program computer-readable storage medium of
5. The computer program computer-readable storage medium of
6. The computer program computer-readable storage medium of
0. 7. The computer-readable storage medium of claim 1, wherein said logic configured to verify comprises logic configured to determine if a rank of the initial structure is equal to a number of associated transmit antennas.
0. 8. The computer-readable storage medium of claim 1, wherein the space-time structure has properties of a unitary signal transmission matrix.
0. 9. The computer-readable storage medium of claim 3, further comprising logic configured to store the space-time structure in a memory of the MIMO communications system if the symbols of the space-time structure are within the predetermined distance of the initial structure.
0. 10. The computer-readable storage medium of claim 3, wherein said logic configured to verify that a PAPR of the space-time structure is less than a predetermined value comprises logic configured to convert the space-time structure to a time domain and calculate the PAPR of resultant signal samples.
0. 11. The computer-readable storage medium of claim 3, further comprising logic configured to store the space-time preamble structure, the space-time pilot structure, or the space-time data structure in an encoder.
0. 13. The method of claim 12, further comprising choosing a symbol alphabet to provide symbols for the initial structure and choosing an initial configuration of the initial structure.
0. 14. The method of claim 12, further comprising:
confirming that symbols of the space-time structure are within a predetermined distance of symbols of the initial structure; and
constructing a space-time sequence structure from a plurality of the space-time structures.
0. 15. The method of claim 14, wherein said confirming comprises applying a Euclidean distance metric to determine the distance between the symbols of the space-time structure and the symbols of the initial structure.
0. 16. The method of claim 14, further comprising storing the space-time structure in a memory of the MIMO communications system if the symbols of the space-time structure are within the predetermined distance of the initial structure.
0. 17. The method of claim 12, further comprising verifying that a peak-to-averagc power ratio (PAPR) of the space-time structure is less than a predetermined value.
0. 18. The method of claim 17, wherein said verifying comprises converting the space-time structure to a time domain and calculating the PAPR of resultant signal samples.
0. 19. The method of claim 12, wherein said verifying that the rows of the initial structure are linearly independent comprises determining rank of the initial structure.
0. 20. The method of claim 12, wherein said applying an orthonormalization procedure comprises applying a row-wise Gram-Schmidt procedure to the initial structure to obtain the space-time structure.
0. 21. The method of claim 12, wherein said verifying comprises determining if a rank of the initial structure is equal to a number of associated transmit antennas.
0. 22. The method of claim 12, wherein the space-time structure has properties of a unitary signal transmission matrix.
0. 23. The method of claim 12, further comprising storing the space-time preamble structure, the space-time pilot structure, or the space-time data structure in an encoder.
0. 25. The communication system of claim 24, wherein the encoder is further configured to choose a symbol alphabet to provide symbols for the initial structure and choose an initial configuration of the initial structure.
0. 26. The communication system of claim 24, wherein the encoder is further configured to:
confirm that symbols of the space-time structure are within a predetermined distance of symbols of the initial structure; and
construct a space-time sequence structure from a plurality of the space-time structures.
0. 27. The communication system of claim 26, wherein the encoder is further configured to apply a Euclidean distance metric to determine the distance between the symbols of the space-time structure and the symbols of the initial structure.
0. 28. The communication system of claim 26, further comprising a memory in which the space-time structure is stored if the symbols of the space-time structure are within the predetermined distance of the initial structure.
0. 29. The communication system of claim 24, wherein the encoder is further configured to determine a rank of the initial structure.
0. 30. The communication system of claim 24, wherein the encoder is further configured to apply a row-wise Gram-Schmidt procedure to the initial structure to obtain the space-time structure.
0. 31. The communication system of claim 24, wherein the encoder is further configured to determine if a rank of the initial structure is equal to a number of associated transmit antennas.
0. 32. The communication system of claim 24, wherein the encoder is further configured to verify that a peak-to-average power ratio of the space-time structure is less than a predetermined value.
0. 33. The communication system of claim 32, wherein the encoder is further configured to convert the space-time structure to a time domain and calculate the PAPR of resultant signal samples.
0. 34. The communication system of claim 24, wherein the space-time structure has properties of a unitary signal transmission matrix.
0. 35. The communication system of claim 24, wherein the encoder is further configured to store the space-time preamble structure, the space-time pilot structure, or the space-time data structure.
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With respect to EQ. 1, k represents the sub-carrier or sub-channel of received demodulated signals and T represents a dimension variable that is typically equivalent to Q, although it may have other values. As discussed above, Q and L represent, respectively, the number of modulators 16 and respective transmit antennas 18 and the number of demodulators 22 and respective receive antennas 20 with respect to a typical MIMO communications system 6.
As also depicted in
The length NI of a training block 58 may be shorter than the length N of a data block 59 in a signal structure 52. Typically, the length NI of a training block 58 in the preamble 54 is established as a fraction of the length N of a data block 59 in the data section 56 to provide the relationship of NI being equivalent to N/I, where I is some positive integer. For example, NI may be equivalent to N/4 (i.e., I=4). If the length NI of a training block 58 is not established, the length NI may be assumed to be equivalent to N (i.e., I=1). Typically, the length of a training symbol 53 (i.e., G+NI) is equivalent to the length of a data symbol 55 (i.e., G+N). However, it is feasible for the training symbol 53 to be shorter than the data symbol 55 in the context of the signal structure 52.
A primary purpose of the preamble 54 is to enable the receiver 10 (
A space-time preamble structure, which may also be referred to as a space-time training structure, may be represented by a signal transmission matrix Sk. In accordance with an embodiment of the present invention, the signal transmission matrix Sk of an efficient space-time preamble structure should be a unitary transmission matrix in the frequency domain and have a low PAPR in the time domain. In this regard, efficient space-time preamble structures provide enhanced performance in MIMO communications systems.
A unitary transmission matrix contains rows and columns that are orthogonal to each other, and the energy of the signals represented by each row or column is unity. In mathematical terms, a unitary transmission matrix has the properties represented by the following equations:
where Si,j represents the constituent symbols of the unitary transmission matrix.
Providing a space-time preamble structure that is a unitary signal transmission matrix Sk reduces or eliminates noise enhancement during channel estimation of the received signals. Moreover, providing a space-time preamble structure that possesses a low PAPR reduces or eliminates signal non-linearities and spurious, out-of-band signal transmissions. As will be discussed below, data structures formed by space-time processing (i.e., space-time data structures) to be a unitary transmission matrix also provide enhanced performance in MIMO communications systems.
The following descriptions present several examples of data structures that, in accordance with the present invention, can be applied and/or modified to provide space-time preamble structures that are unitary transmission matrices. As a first example, a diagonal data structure can be applied and/or modified to provide a space-time preamble structure in accordance with the present invention. In this regard, the resulting diagonal space-time preamble structure is a unitary transmission matrix. The following diagonal structure SD1 is an example of this unitary transmission matrix that can be applied as a space-time preamble in a MIMO communications system with Q antennas:
The foregoing diagonal space-time preamble structure SD1 can be simplified so that the same training symbol (e.g., S1) can be transmitted from each antenna, instead of Q different training symbols (i.e., S1, S2, etc.), as shown by the following simplified diagonal structure SDS that can be applied, in accordance with the present invention, as a space-time preamble structure in a MIMO communications system with Q antennas:
When the foregoing diagonal structures SD, SDS are applied as space-time preamble structures in a MIMO communications system, the training symbols are transmitted sequentially in time from each corresponding transmit antenna, and the parameters of the received symbols are estimated by the receivers connected to each receive antenna. Due to their unitary characteristic, the diagonal structures SD, SDS, provide simplified signal acquisition (i.e., synchronization) and parameter estimation when applied as a space-time preamble structure in a MIMO communications system. These diagonal structures SD, SDS, are preferably applied as space-time preamble structures, in MIMO communications systems that use two transmit antennas. As the number (Q) of transmit antennas in the MIMO system is increased, the power output from each transmit antenna typically has to be reduced by a factor of Q due to the nature of MIMO systems. As a result, the efficiency of the diagonal space-time preamble structures SD, SDS may decrease in MIMO systems with more than two transmit antennas, since the diagonal structures SD, SDS only include symbols on the main diagonal (i.e., spanning from the top-left to the bottom).
A data structure that was introduced by S. Alamouti is another example of a data structure that can be applied and/or modified, in accordance with the present invention, to provide a space-time preamble structure SA. This data structure is a unitary transmission matrix, and it can be applied as a space-time preamble structure SA, in MIMO communications systems that employ two transmit antennas. The space-time preamble structure SA has the following form:
In the above space-time preamble structure SA, the “*” symbol indicates a complex conjugate operation. The foregoing space-time preamble structure SA can also be simplified, in accordance with the present invention, so that the same training symbol is transmitted from each of the two antennas of the MIMO system, as shown by the following simplified space-time preamble structure SAS:
Several orthogonal structures that were introduced by V. Tarokh, et al. are examples of data structures that can be applied and/or modified, in accordance with the present invention, to provide space-time preamble structures that are unitary transmission matrices. These data structures can be applied as space-time preamble structures, in accordance with the present invention, in MIMO communications systems that employ four or eight transmit antennas. For a four-antenna MIMO system, the following space-time preamble structure ST4 can be applied, in accordance with present invention, when the constituent symbols have real number values:
The foregoing space-time preamble structure ST4 can be simplified, in accordance with the present invention, so that the same training symbol is transmitted from each of the four antennas of the MIMO system, as shown by the following simplified space-time preamble structure ST4S. The symbols of this structure ST4S may have complex values (e.g., W+jX):
The foregoing simplified structures SAS, ST4S (i.e., EQ. 6 and EQ. 8) typically form unitary transmission matrices when applied as space-time preamble structures, without further modification. Furthermore, the PAPR of the simplified space-time preamble structures SAS, ST4S are typically unity when the symbols consist of chirp-type sequences, such as:
Therefore, these simplified structures SAS, ST4S are typically efficient (i.e., they provide time and frequency synchronization, estimation of noise variance and channel parameters, and low PAPR) when applied, in accordance with the present invention, as space-time preamble structures.
The foregoing structures SA, ST4 (i.e., EQ. 5 and EQ. 7) are also typically efficient when applied as space-time preamble structures, in accordance with the present invention. The structure ST4 is typically not efficient when applied as space-time data structures in a MIMO communications system. However, both structures SA, ST4 can be modified and then applied as efficient space-time data structures, in accordance with the present invention. Since the structures SA, ST4 will include symbols with complex values when they are applied as space-time data structures, the resultant data structures will typically not be unitary transmission matrices. Therefore, the structures SA, ST4 can be modified, in accordance with the present invention, to form unitary transmission matrices and, thus, provide efficient space-time data structures. Methods, in accordance with the present invention, to transform these structures SA, ST4 and other structures into efficient space-time data structures will be described below.
The following space-time preamble structure ST8 is based on another data structure by Tarokh, et al., and the structure ST8 can be applied in eight-antenna MIMO communications systems, in accordance with present invention, when the constituent symbols have real number values:
The foregoing space-time preamble structure ST8 can be simplified, in accordance with the present invention, so that the same training symbol is transmitted from each of the eight antennas of the MIMO system, as shown by the following simplified space-time preamble structure ST8S
The foregoing structures ST8, ST8S (i.e., EQ. 10 and EQ. 11) are typically efficient when applied as space-time preamble structures, in accordance with the present invention. However, these structures ST8, ST8S are typically not efficient when applied as space-time data structures in a MIMO communications system. The structure ST8 preferably can be modified and then applied as efficient space-time data structures, in accordance with he present invention. Since the structure ST8 will include symbols with complex values when it is applied as a space-time data structure, the resultant data structure will typically not be a unitary transmission matrix. Therefore, the structure ST8 can be modified, in accordance with the present invention, to form a unitary transmission matrix and, thus, provide an efficient space-time data structure. Methods, in accordance with the present invention, to transform this structure ST8 and other structures into efficient space-time data structures will be described below.
Orthogonal structures, such as those introduced by Tarokh, et al., typically only have applications to MIMO communications systems that employ two, four, or eight transmit antennas. As described above, some of the orthogonal structures can be applied, in accordance with the present invention, in two-antenna MIMO systems as space-time data structures, with complex symbols, that are unitary transmission matrices. However, the application of existing orthogonal structures using complex symbols (e.g., for space-time data structures) in MIMO systems having more than two transmit antennas typically results in a loss of the system diversity gain and/or system bandwidth. For example, the following orthogonal structure ST3 was introduced by Tarokh, et al. for use as a data structure with complex symbols in a three-antenna MIMO system:
When the foregoing structure ST3 is applied in a three-antenna MIMO system, it does not provide the full diversity performance of the system, which is the capability to transmit three symbols over three symbol periods. Instead, the structure ST3 only provides for the transmission of three symbols over four symbol periods, which is apparent since the structure has a four rows instead of three. This lack of full diversity may result in a loss of as much as 25% of system throughput. However, methods, in accordance with the present invention, will be discussed below to transform such inefficient structures into efficient space-time structures (for preambles or data) that provide full diversity performance in MIMO communications systems.
The foregoing space-time preamble structures, in accordance with the present invention, can be applied in a Q-antenna MIMO communications system, such as the system 6 depicted in
In general the transmission matrix for Q transmit antennas over Q symbol intervals can be represented by the following matrix SQ2:
This general transmission matrix SQ2 can be composed using Q2 different symbols (or sequences in the case of OFDM modulation). However, in general, only Q sequences are used to form a structure. As discussed above, the transmission performance of Q symbols over Q symbol periods indicates full diversity performance of the MIMO system and also indicates the utilization of the full bandwidth of the system. Thus, such performance indicates the optimal use of the system resources.
In order to utilize the structure of the foregoing general transmission matrix SQ2 to construct efficient space-time sequence structures for preambles, pilots and data to be applied in MIMO communications systems, the matrix SQ2 is pre-processed and/or pre-conditioned in accordance with the present invention.
Following step 122, the method 120 proceeds to step 124 in which the rows of the initial structure Sin are verified to be linearly independent. The check for linear independence of the rows of the initial structure Sin may be performed by various methods and techniques, which may be known in the art. For example, the rows of the initial structure Sin can be tested for linear independence by determining the rank of the initial structure Sin. If the rank of the initial structure Sin is determined to be Q, the rows of the initial structure Sin are linearly independent. If the rows of the initial structure Sin are determined to be linearly independent, the method 120 proceeds to the next step 126. However, if the rows of the initial structure Sin are determined not to be linearly independent, the method 120 returns to step 122, in which one or more different initial structures Sin are provided and the method 120 proceeds again to step 124.
In the step 126, an orthonormalization (i.e., orthogonalization and normalization) procedure is applied to the initial structure Sin. The orthonormalization procedure may be any procedure that transforms the initial structure Sin to a space-time structure Sout that has the properties of a unitary signal transmission matrix. As discussed above, a unitary transmission matrix has the following mathematical properties:
where Si,j represents the constituent symbols of the unitary transmission matrix. One example of an orthonormalization procedure that may be applied to the initial structure Sin to obtain a space-time structure Sout that is a unitary signal transmission matrix is known as a row-wise Gram-Schmidt procedure. An example application of a row-wise Gram-Schmidt procedure will be presented below.
The resultant space-time structure Sout that is obtained by the step 126 may be applied as an efficient space-time preamble structure or an efficient space-time data structure, depending on the characteristics of the constituent symbols of the structure. For example, as discussed above, an efficient space-time preamble structure includes symbols that provide time and frequency synchronization and estimation of noise variance and channel parameters. In contrast, an efficient space-time data structure typically includes symbols that have complex values, as also discussed above. Further, if OFDM modulation is employed in the communications system, the constituent symbols will be symbol sequences, as also discussed above.
The resultant space-time structure Sout may be applied accordingly as a space-time preamble or data structure in a Q-antenna MIMO communications system, such as the system 6 depicted in
The symbols or symbols sequences may also be derived from polyphase sequences, such as Chirp sequences; Milewski sequences; Frank-Zadoff sequences; Chu sequences; Suehiro polyphase sequences; and Ng et al. sequences, among others known in the art.
Following the step 142, the method 140 concludes with a step 144 in which the initial configuration of the initial structure Sin is chosen. The determination of the initial configuration may add certain specific characteristics to the structure. For example, the initial configuration typically reduces the number of possible symbol combinations from Q2 to Q. The initial configuration may be chosen from any structure configuration. The following are several examples of a possible initial configuration of the initial structure Sin:
Based on the determination of the symbol alphabet and the initial structure configuration in the step 142 and the step 144, respectively, an initial structure Sin can be determined. This initial structure Sin can be used in the method 120, depicted in
If the rows of the initial structure Sin are determined to be linearly independent, the method 160 proceeds from the step 164 to a step 166 in which an orthonormalization procedure is applied to the initial structure Sin to transform the initial structure Sin to a space-time structure Sout that has the properties of a unitary signal transmission matrix. This step 166 is at least substantially similar to the step 126 discussed above with respect to
Following the step 166, the method 160 proceeds to a step 168 in which the alphabet points of the constituent symbols of the resultant space-time structure Sout are checked to be within a tolerable distance of the alphabet points of the constituent symbols of the initial structure Sin. The amplitude of the alphabet points may be modified during the orthonormalization procedure in the step 166. The tolerable distance is typically dependent on the operating capability of components of the MIMO communications system 6, such as digital-to-analog (D/A) converters. The constituent symbols of the space-time structure Sout may be checked to be within a tolerable distance of the original alphabet points by various methods and techniques, which are known in the art. For example, the constituent symbols of the space-time structure Sout may be checked to be within a tolerable distance by application of a Euclidean distance metric represented, for example, by the following equation:
dt,l=∥St−Sl∥2 EQ. 16
If the constituent symbols of the space-time structure Sout are found to be within a tolerable distance from the original alphabet points of the initial structure Sin, the space-time structure Sout is stored in a memory or other device for application in a MIMO communications system. However, if the constituent symbols of the space-time structure Sout are not determined to be within a tolerable distance from the original alphabet points, the method 160 returns to step 162, in which one or more different initial structures Sin are provided and the method 160 proceeds again as described above.
In the case of a MIMO communications system that employs OFDM modulation, the steps 162 through 168 may be repeated until a sufficient number of space-time structures Sout that are unitary signal transmission matrices are obtained and stored, as discussed above.
If the symbols are within a tolerable distance, in step 170, the stored space-time structure Sout used to construct space-time sequence structures Sout,k, where k represents a sub-carrier or sub-channel index of the OFDM setup. The space-time sequence structures Sout,k may be constructed by an encoder, as described above with respect to
In the final step 172 of the method 160, the peak-to-average power ratio (PAPR) of the space-time sequence structures Sout,k are tested to determine if the PAPR of the structures is low enough to provide efficient signal transmission and reception in a MIMO OFDM communications system. The PAPR of the training sequences may be tested by various methods and techniques, which may be known in the art. For example, the PAPR of the space-time sequence structures Sout,k may be tested by converting the structures to the time domain (e.g., by inverse Fourier transform or “IFT”) and calculating the PAPR of the resultant signal samples. If the PAPR of the space-time sequence structures Sout,k is found to be acceptable (e.g., at or approaching unity), the structures have been determined to be efficient, in accordance with the present invention, and may be used for preambles or data in a MIMO communications system 6 employing OFDM modulation. However, if the PAPR of the space-time sequence structures Sout,k are found to be unacceptably high, the method 160 returns to step 162, in which one or more different initial structures Sin are provided and the method 160 proceeds again as described above.
In the case of some orthogonal polyphase sequences, complex coefficients bi that are used to modulate the sequences may be useful to form efficient space-time sequence structures Sout,k. In this regard, modulation of the orthogonal polyphase sequences by the complex coefficients bi may make the rows of the corresponding space-time structures Sout linearly independent. Furthermore, the modulation by the complex coefficients bi may also reduce the PAPR of the resulting space-time sequence structures Sout,k that are formed from the space-time structures Sout.
In the step 126 of the method 120 and the step 166 of the method 160, described above with respect to
It is noted that embodiments of the present invention, such as those described above, may be implemented in hardware, software, firmware, or a combination thereof. For example, in some embodiments, the present invention may be implemented as a computer program or application in software or firmware that is stored in a memory and that is executed by a suitable instruction execution system. In other embodiments the present invention may be implemented, for example, with one or a combination of the following technologies, which may be known in the art: one or more discrete logic circuit(s) having logic gates for implementing logic functions upon data signals, an application specific integrated circuit (ASIC) having appropriate combinational logic gates, a programmable gate array(s) (PGA), a field programmable gate array (FPGA), etc.
It is further noted that any process descriptions or blocks in flow charts described above may represent modules, segments, and/or portions of a computer program or application code that includes one or more executable instructions for implementing specific logical functions or steps in the process. Alternate implementations are included within the scope of the present invention in which functions may be executed out of order from that shown in the figures and/or discussed above, including substantially concurrently or in reverse order, depending at least in part on the functionality involved, as will be understood by those skilled in the art.
With regard to any block diagrams described above, although the flow of data or other elements may be depicted as unidirectional or bi-directional, such depictions are merely exemplary and not limiting. Variations of the flows depicted in the block diagrams are included within the scope of the present invention. Furthermore, the functionality of some of the blocks may be implemented by a single combined block within the scope of the present invention.
Moreover, embodiments of the present invention, such as those described above, may comprise an ordered listing of executable instructions for implementing logical functions which can be embodied in any computer-readable medium for use by or in connection with an instruction execution system, apparatus, or device, such as a computer-based system, processor-containing system, or other system that can fetch the instructions from the instruction execution system, apparatus, or device and execute the instructions. In the context of this disclosure, a “computer-readable medium” may be any means that can contain, store, communicate, propagate, or transport the program for use by or in connection with the instruction execution system, apparatus, or device. The computer readable medium may be, for example but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, device, or propagation medium. More specific examples (i.e., a non-exhaustive list) of the computer-readable medium include the following: an electrical connection (electronic) having one or more wires, a portable computer diskette (magnetic), a random access memory (RAM) (electronic), a read-only memory (ROM) (electronic), an erasable programmable read-only memory (EPROM or Flash memory) (electronic), an optical fiber (optical), and a portable compact disc read-only memory (CDROM) (optical). It is noted that the computer-readable medium may even be paper or another suitable medium upon which the program is printed, as the program can be electronically captured, via for instance optical scanning of the paper or other medium, then compiled, interpreted or otherwise processed in a suitable manner if necessary, and then stored in a computer memory.
Finally, it should be emphasized that the above-described embodiments of the present invention are merely possible examples of implementations set forth for a clear understanding of the principles of the invention. Many variations and modifications may be made to the above-described embodiment(s) of the invention without departing substantially from the spirit and principles of the invention. All such modifications and variations are intended to be included herein within the scope of this disclosure and the invention, and protected by the following claims.
Mody, Apurva N., Stuber, Gordon L.
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