Modulation with fundamental group

Embodiments of a system and method for providing fundamental group modulation are generally described herein. In some embodiments, a trajectory mapper is arranged to receive a modulation symbol sequence. A signal trajectory sample memory is arranged to store a representation of signal trajectories for a topological space having a set of predetermined removed regions therein. The trajectory mapper accesses the signal trajectory sample memory to select a signal trajectory relative to the set of predetermined removed regions in the topological space based on the received modulation symbol sequence and produces a sequence of in-phase (I) and quadrature (Q) sample values at a specified sample rate in response to the selected signal trajectory, the I and Q sample values serving as a basis for an amplified radio frequency signal.

BACKGROUND

Telecommunication systems are widely deployed to provide various telecommunication services. A communications system may be characterized as a collection of transmitters, receivers, and communications channels that send messages to one another. The transmission of a raw electrical signal, i.e., baseband signal, has several limitations, including bandwidth limitations, distance limitations, etc. To address these issues, many different modulation techniques have been developed. Modulation involves encoding a baseband source signal Sm(t) onto a carrier signal. The carrier waveform is then varied in a manner directly related to the baseband signal.

The move to digital modulation provides more information capacity, compatibility with digital data services, higher data security, better quality communications, and quicker system availability. Digital modulation converts information-bearing discrete-time symbols into a continuous-time waveform. The choice of digital modulation scheme will significantly affect the characteristics, performance and resulting physical realization of a communication system. Traditional linear modulation systems map collections of data bits to a point in a two-dimensional space,2, e.g., binary phase shift keying (BPSK), quadrature phase shift keying (QPSK), 16-state quadrature amplitude modulation (16-QAM). For example, relatively simple modulation such as QPSK offers excellent bit error rate (BER) performance at relatively low signal strengths. QPSK however uses a large bandwidth. 16-QAM is more bandwidth efficient, but requires on stronger signal strength than QAM to achieve a low BER. This is particularly so for the more dense bandwidth schemes such as 64-state QAM (64-QAM).

There are three characteristics of a signal that are typically changed over time: amplitude, phase, or frequency. Note that phase and frequency are related. Amplitude and phase can be modulated simultaneously and independently. Such signals can also be represented by in-phase (I) and Quadrature (Q) components, which are the rectangular representation of the polar signals. It is common for digital modulations to map the data to a number of discrete points on the I/Q (complex) plane. These are known as constellation points.

Between symbol clock transitions, the carrier is modulated by an amplitude and phase or, equivalently, an I and Q value that maps to a constellation point in the complex plane. A constellation point encodes a specific data sequence, which includes one or more data bits. A constellation diagram shows the valid locations (i.e., the magnitude and phase relative to the carrier) for permitted symbols. To demodulate the incoming data, the exact magnitude and phase of the received signal is estimated. The layout of the constellation diagram and its ideal symbol locations is determined generically by the chosen modulation format.

Modulation symbols are used to represent information, wherein a symbol may represent one bit or a number of bits. In a symbol period, a digital modulator maps bits to a transmitted waveform from a pre-defined set of possible waveforms, wherein a waveform corresponds to an information symbol. However, existing modulation systems operate in the complex plane or R2. These systems do not rely on the topological of alternative spaces or path relative to such a space to directly carry information.

DETAILED DESCRIPTION

A fundamental group is an algebraic group on equivalence classes associated with a topological space wherein any two paths that start and end at a fixed base point and, that can be continuously deformed into each other are considered members of the same equivalence class. A fundamental group records information about the basic shape, or holes, in a topological space. Fundamental group modulation may be applied to wireless or wired communications. Fundamental group modulation is also applicable to a user (links) or networks. Fundamental group modulations involves modulation of a data sequence on a carrier by transmitting paths relative to a set of removed regions or “holes” {dn} in a topological space. A topological space is defined by the sets the space contains and a specific set of relations between the sets. In topological terms, Fundamental Group Modulation assigns some subset of the fundamental group of a topology to carries information. Fundamental group modulation is not limited to 2-dimensional space (2), but is extensible ton. Fundamental group modulation may be used to also apply information assignment to the direction (orientation) that a path is traversed about a given dn.

FIG. 1shows a comparison100between fundamental group modulation according to an embodiment and traditional modulations techniques. As shown inFIG. 1, traditional techniques110attempt to mathematical invert120the channel effects prior to detection130, e.g., equalization. However, fundamental group modulation150directly detects160information in the image space, i.e., a map of the original modulation space mapped through the channel operator. Through direct detection of information in the image space, fundamental group modulation150may improve communications through channel distortion180, improve stability for mobile wireless networked communications, increase communications responsiveness due to higher data efficiencies over longer ranges and less re-routing through the network connections and increase information density per packet.

FIG. 2illustrates an example of traditional modulation technique200. As shown inFIG. 2, a 4QAM (QPSK) system transmits a path in a subset of C×R, i.e., the Cartesian product of the complex plane with the real line, by sending a continuous complex signal trajectory210over time220.FIG. 2shows the modulation as a path in C×R projected into a constellation/trajectory in C230. Thus, existing digital modulation techniques map information to a point on a complex signal trajectory or constellation230. Channel impairments are addressed through mathematical correction, i.e., equalization, which balances modulation symbol distortion for corruption by noise or interference sources.

The channel may be viewed as an operator, F(•), where y(t)=F(x(t)) with x(t) the transmit signal and y(t) the signal at the RF front end. F(•) is typically characterized as convolution with a time-varying function. Additive Gaussian noise imparted by the RF front end corrupts the signal that is then processed for detection. The signal is sampled and processed to acquire unknown frequency and timing offsets so that the detector can recover an estimate of the modulated stream. Accordingly, receiver noise corrupts the transmitted signal's trajectory. In the example cases, the demodulator's n detectors determine whether the path encircles the puncture point once or not (non-oriented) and in which direction (oriented).

FIGS. 3a-billustrates graphs300,350showing fundamental group modulation and detection according to an embodiment and traditional modulation and detection.FIG. 3aillustrates a traditional system300receiving data310that treats the channel as a distortion function320and attempts to invert the distortion330. Traditional systems, whether single carrier, multi-carrier OFDM, or continuous phase modulation (CPM), apply an equalizer330that attempts to mathematically invert the channel effects. Single carrier and CPM systems essentially approximate deconvolution while OFDM systems leverage a cyclic waveform property. Traditional systems address interference in two ways. Some systems estimate the interference and then perform linear cancellation. This is useful in the sense that it removes the source of degradation. Equalization330reduces the interference by minimizing an error metric, e.g., mean-square error, and this balances the distortion caused by the channel and interference to attempt data recovery.FIG. 3billustrates fundamental group modulation performing detection of the signal directly in the image space of the channel function. Input data360is directly mapped370to detect the data380.

FIG. 4shows a fundamental group modulator400according to an embodiment. InFIG. 1, a data bit sequence b(k)410enters a bits-to-symbol map412, wherein the data bit sequence b(k)410, binary information, is mapped to symbols to produce a modulation symbol sequence m(k)414. For the case where the fundamental group is applied for some {dn}, there are 2npossible distinct equivalences determined by inclusion of the holes. The 2npossible distinct equivalences are the same as the cardinality (set size) of the power set of {dn}. If fundamental groups with orientation are applied, a rotation of −1, 0, +1 may be achieved so the natural information map is ternary. Therefore, there are 3npossible distinct equivalence classes of paths with orientation.

The modulation symbol sequence m(k)414is provided to signal trajectory mapper418, wherein the trajectory mapper418accesses a signal trajectory sample memory420to select a signal trajectory. The signal trajectory sample memory420includes a stored representation of the possible signal trajectories. In another embodiment, an algorithm may be used to form either digital or analog trajectories. The trajectory mapper418using the signal trajectory memory420produces a sequence of in-phase (I) and quadrature (Q) sample values Ik(n)422and Qk(n)424at a specified sample rate in response to the modulation symbol sequence m(k)414. The in-phase and quadrature sample values, Ik(n)422and Qk(n)424, are provided to digital-to-analog (D/A) converters430,432. The D/A converters430,432accept a discrete sample value (e.g., base2number) in and produces a baseband analog signal representation at the output. The in-phase and quadrature sample values, Ik(n)422and Qk(n)424, are provided at a selected sample rate.

The signal out of the D/A converters430,432are provided to analog reconstruction filters440,442. The signal from the D/A converters430,432includes harmonic content, which is attenuated by the analog reconstruction filters440,442. The in-phase I(t)444and quadrature Q(t)446waveforms provided at the output of the analog reconstruction filters440,442are provided to an I/Q mixer450where the in-phase I(t)444and quadrature Q(t)446waveforms are mixed with a radio frequency (RF) signal462established by a local oscillator (LO)460. For example, the LO signal is given by:c(t)=cos(ωct) withωc=2πfcandfc=100 Mhz.

Note that s(t)=r(t)cos(ωct+θ(t)).

An RF circuit470may be applied to convert the output of the I/Q mixer450, s(t)452, to an amplified final RF frequency signal472. In a wireless application, an antenna480is used to radiate the final amplified RF signal472in the form of electromagnetic energy.

Accordingly, the bits-to-symbols mapper412is used to map binary values to base2, base3, etc. modulation symbols. However, modulation symbols may be directly supplied to the signal trajectory sample memory420. The signal trajectory sample memory420is used to define the signal trajectories that will be transmitted. The components of a communication system according to an embodiment share an understanding of the underlying topological space and, specifically, the “holes” {dn} in the space that the paths encircle. Further, the fundamental group modulator400operates in the original complex space mapped through the channel function when the channel map is a local homeomorphism so that the image space over the constraint set has the same topology as the original space. Therefore, fundamental groups are preserved, i.e., if a loop went about a point in the original space then it also does so in the image space. However, embodiment may include admissible channel operators besides homeomorphisms.

The bits-to-symbols mapper412, the trajectory mapper418and/or the signal trajectory sample memory420may be realized in a microprocessor, a field programmable array (FPGA), an application specific integrated circuit (ASIC), etc. The trajectory mapper418and/or signal trajectory sample memory420may be realized as an analog circuit that produces trajectories at an RF frequency directly.

A topology is defined as a collection of points and sets, i.e., a “space” that has certain properties. Two spaces are considered topologically identical if they can be related through a homeomorphism, i.e., a continuous map that has a continuous inverse. Algebraic topology introduces the notion of a fundamental group by observing when two loops in a space that have a common, fixed starting and ending base point can be continuously deformed into each other. Intuitively, the fundamental group records information about the holes of a topological space. Any closed paths relative to a base point that can be continuously deformed into each other are considered equivalent, i.e., classes. The classes are the elements of the group. Equivalent (homeomorphic) topological spaces have the same fundamental groups. The basis of the proposed fundamental group modulator400is that a topology is an invariant under a homeomorphic (channel) transformation and so is the fundamental group. Therefore, the distinction between classes of loops in the modulation space is preserved at the detector—even when the channel evolves. The elements of the group are invariant between the domain and image space of the channel operator. In contrast, systems that communicate via constellation points selected from the complex plane may exhibit points that substantially vary as the channel changes.

Herein, simple topologies are determined by puncturing holes in a region of the complex plane and extending the concept to higher dimensional spaces. The notation C\{a, b, . . . z} is used to refer to the complex plane that is missing the points in the set {a, b, . . . z}, i.e., the points are punctured from the space. Elements of a group that loop around points by rotation clock-wise (−1), no rotation (0), and counter-clockwise (1), i.e., negative, zero, and positive oriented loops respectively, are used.

The modulation trajectories are generated with a traditional quadrature structure that establishes a trajectory in the complex plane. The demodulator measures the rotation about each of the k punctures in the plane. The k detectors compute a winding number by approximating the Cauchy integral of the detected path, g(t), about the punctured point z0as:

Therefore, the demodulator samples the received path sufficiently to approximate this integral over a path that is known to be closed at the transmitter. Acquisition recovers and corrects for frequency error as in traditional systems. Timing acquisition acquires the modulation base point so that integration is accomplished relative to the original closed path as viewed in the channel's image space.

FIGS. 5a-bshow a comparison of non-oriented500and oriented550one-ternary symbols according to an embodiment.FIGS. 5a-bshow spaces C\{a} that have a single punctured point510,560.FIG. 5aillustrates that on the complex plane, C\{a}520, the system sends one bit non-oriented per symbol.FIG. 5billustrates that on the complex plane, C\{a}570, the system sends one ternary symbol (1.6 bits) oriented per symbol. In general, a non-oriented system with k punctures can carry k bits because the set of trajectories has the same cardinality as the power set of the puncture points, i.e., (2k). An oriented system carriers log2(3k) bits and the mapping is more readily represented in terms of ternary symbols rather than binary digits (bits).

InFIGS. 5a-b, the space520contains a single point hole in2at (0,0). So d0=(0,0), i.e., the point hole, wherein d0510,560is marked in each. InFIG. 5a, if the path530encircles d0510, a binary 1532is sent. If the path534does not encircle d0510, a 0536is sent. A bit of information is transmitted with one specified point d0. Power of the transmission can be varied with no impact to a detector that measures rotation about the point to recover the information.

InFIG. 5b, modulation again has n=1, except here path orientation is included. There are three states so 31values may be sent or equivalently log2(3)≅1.6 bits of information. In the oriented one-ternary symbol system550, if the path encircles d0in a counter-clockwise direction580, a binary −1582is sent, if the path does not encircle d0584, a 0586is sent, and if the path encircles d0in a clockwise direction588, a binary 1590is sent.

FIG. 6illustrates modulation with n=2 and no orientation600is provided according to an embodiment. InFIG. 6, the system sends two bits per non-oriented per symbol on the punctured complex plane, C\{a,b}. InFIG. 6,2is punctured at d0=(1,0)610and d1=(−1,0)620. The paths will encircle these points. Thus,FIG. 6shows four possible paths or signal trajectories630that may be transmitted. The data rate and bandwidth are determined by the rate that the closed paths are completed. In another embodiment, different paths may be completed at different rates which, for example, may be used to reduce the bandwidth of the signal. The structure of the topology is shared at the transmitter and receiver, i.e., where the holes are. If synchronized, the topology could be altered as a form of transmission security to protect the information.

FIG. 7illustrates paths for modulation using three symbols700according to an embodiment. InFIG. 7, the system sends three bits non-oriented per symbol on the complex plane, C\{a, b, c}. InFIG. 7,2is punctured at the points (d0710, d1720, d2730), which given in polar coordinates in terms of magnitude and angle are

FIG. 8illustrates a matrix of values800for the oriented paths for d0and d1as {−1, 0, 1} according to an embodiment. InFIG. 8, −1810represents a negative oriented closed path, 0830represents no closed path, and 1820represents a positive oriented closed path. InFIG. 8, the system sends two ternary symbol (3.2 bits) oriented per symbol on the complex plane, C\{a,b}. The space used has n=2 to now provide 32=9 possible trajectories840, i.e., log29=3.17 bits may be sent for each symbol.

FIG. 9illustrates the paths on the complex plane, C\{a,b} when the system sends two bits oriented per symbol900according to an embodiment. InFIG. 9, state 0940represents a path encircling both d0and d1in a clockwise direction930, wherein −1, −1 is sent960. State 1942, a path only encircles d1, wherein −1, 0 is sent962. In state 2944, a path encircles d0in a clockwise direction930and a path encircles d1in a counter-clockwise direction932, wherein −1, 1 is sent964. In State 3946, a path only encircles d0in a clockwise direction930, wherein 0, −1 is sent966. In state 4948, a path does not encircle either d0or d1, wherein 0, 0 is sent968. In state 5950, only d0is encircled in a counter-clockwise direction932, wherein 0, 1 is sent970. In state 6952, d0is encircled in a clockwise direction and d1is encircled in a counter-clockwise direction932, wherein 1, 1 is sent972. In state 7954, only d1is encircled in a counter-clockwise direction932, wherein 1, 0 is sent974. In state 8956, both d0and d1are encircled in a counter-clockwise direction932, wherein 1, 1 is sent976.

FIG. 10illustrates a matrix of values1000for the possible rotations when 3 holes are provided in the topological space with path orientation according to an embodiment. InFIG. 10, the holes are designated d01010, d11012, and d21014, and may have the values {−1, 0, 1}. A negative oriented closed path is represented by −1, 0 represents no closed path around a hole, and 1 represents a positive oriented closed paths. To the right of the values for d01010, d11012, and d21014, the base3values are shown. The equivalent base10values then illustrated. Thus, the three oriented bits provide 33=27 possible oriented signal path trajectories, which provide log2(27)=3 log2(3)=4.75 bits per trajectory.

FIG. 11illustrates the paths on the punctured complex plane, C\{a, b, c}, when the system sends three bits oriented per symbol1100according to an embodiment. As described above with reference toFIG. 10, the three oriented bits provide 33=27 possible oriented signal path trajectories. InFIG. 11,2is punctured at the points (d0, d1, d2), which given in polar coordinates in terms of magnitude and angle are

(1⁢∠0,1⁢∠⁢2⁢π3,1⁢∠⁢4⁢π3).
Thus, when the path circles d0, d1, and d2, in a clockwise pattern1130, the values −1, −1, −1 are sent1140. The other paths and corresponding patterns are described below:When the path only circles d0and d1in a clockwise pattern1130, the values −1, −1, 0 are sent1142.When the path circles d0, and d1in a clockwise pattern1130and circles d2in a counter-clockwise direction1132, the values −1, −1, 1 are sent1144.When the path only circles d0and d2in a clockwise pattern1130, the values −1, 0, −1 are sent1146.When the path only circles d0in a clockwise pattern1130, the values −1, 0, 0 are sent1148.When the path circles d0in a clockwise direction and circles d2in a counter-clockwise pattern1132, the values −1, 0, 1 are sent1150.When the path circles d0and d2in a clockwise pattern1130and circles d1in a counter-clockwise direction1132, the values −1, 1, −1 are sent1152.When the path circles d0in a clockwise direction and circles d1in a counter-clockwise pattern1132, the values −1, 1, 0 are sent1154.When the path circles d1and d2in a counter-clockwise pattern1132and circles d0in a clockwise direction, the values −1, 1, 1 are sent1156.When the path only circles d1and d2in a clockwise pattern1130, the values 0, −1, −1 are sent1158.When the path only circles d1in a clockwise pattern1130, the values 0, −1, 0 are sent1160.When the path circles d1in a clockwise direction and circles d2in a counter-clockwise pattern1132, the values 0, −1, 1 are sent1162.When the path only circles d2in a clockwise pattern1130, the values 0, 0, −1 are sent1164.When the path does not circle d0, d1, or d2, the values 0, 0, 0 are sent1166.When the path only circles d2in a counter-clockwise pattern1132, the values 0, 0, 1 are sent1168.When the path circles d1in a counter-clockwise direction1132and circles d2in a clockwise pattern1130, the values 0, 1, −1 are sent1170.When the path only circles d1in a counter-clockwise pattern1132, the values 0, 1, 0 are sent1172.When the path only circles d1 and d2in a counter-clockwise pattern1132, the values 0, 1, 1 are sent1174.When the path circles d1and d2in a clockwise pattern1130and circles d0in a counter-clockwise direction1132, the values 1, −1, −1 are sent1176.When the path circles d0in a counter-clockwise direction1132and circles d1in a clockwise pattern1130, the values 1, −1, 0 are sent1178.When the path circles d0and d2in a counter-clockwise pattern1132and circles d1in a clockwise direction, the values 1, −1, 1 are sent1180.When the path circles d0in a counter-clockwise direction1132and circles d2in a clockwise pattern1130, the values 1, 0, −1 are sent1182.When the path only circles d0in a counter-clockwise pattern1132, the values 1, 0, 0 are sent1184.When the path only circles d0and d2in a counter-clockwise pattern1132, the values 1, 0, 1 are sent1186.When the path circles d0, and d1in a counter-clockwise pattern1132and circles d2in a clockwise direction, the values 1, 1, −1 are sent1188.When the path only circles d0and d1in a counter-clockwise pattern1132, the values 1, 1, 0 are sent1190.When the path circles d0, d1, and d2, in a counter-clockwise pattern1132, the values 1, 1, 1 are sent1192.

Accordingly, the term “module” is understood to encompass a tangible entity, be that an entity that is physically constructed, specifically configured (e.g., hardwired), or temporarily (e.g., transitorily) configured (e.g., programmed) to operate in a specified manner or to perform at least part of any operation described herein. Considering examples in which modules are temporarily configured, a module need not be instantiated at any one moment in time. For example, where the modules comprise a general-purpose hardware processor1202configured using software; the general-purpose hardware processor may be configured as respective different modules at different times. Software may accordingly configure a hardware processor, for example, to constitute a particular module at one instance of time and to constitute a different module at a different instance of time. The term “application,” or variants thereof, is used expansively herein to include routines, program modules, programs, components, and the like, and may be implemented on various system configurations, including single-processor or multiprocessor systems, microprocessor-based electronics, single-core or multi-core systems, combinations thereof, and the like. Thus, the term application may be used to refer to an embodiment of software or to hardware arranged to perform at least part of any operation described herein.

Machine (e.g., computer system)1200may include a hardware processor1202(e.g., a central processing unit (CPU), a graphics processing unit (GPU), a hardware processor core, or any combination thereof), a main memory1204and a static memory1206, at least some of which may communicate with others via an interlink (e.g., bus)1208. The machine1200may further include a display unit1210, an alphanumeric input device1212(e.g., a keyboard), and a user interface (UI) navigation device1214(e.g., a mouse). In an example, the display unit1210, input device1212and UI navigation device1214may be a touch screen display. The machine1200may additionally include a storage device (e.g., drive unit)1216, a signal generation device1218(e.g., a speaker), a network interface device1220, and one or more sensors1221, such as a global positioning system (GPS) sensor, compass, accelerometer, or other sensor. The machine1200may include an output controller1228, such as a serial (e.g., universal serial bus (USB), parallel, or other wired or wireless (e g, infrared (IR)) connection to communicate or control one or more peripheral devices (e.g., a printer, card reader, etc.).

The storage device1216may include at least one machine readable medium1222on which is stored one or more sets of data structures or instructions1224(e.g., software) embodying or utilized by any one or more of the techniques or functions described herein. The instructions1224may also reside, at least partially, additional machine readable memories such as main memory1204, static memory1206, or within the hardware processor1202during execution thereof by the machine1200. In an example, one or any combination of the hardware processor1202, the main memory1204, the static memory1206, or the storage device1216may constitute machine readable media.

While the machine readable medium1222is illustrated as a single medium, the term “machine readable medium” may include a single medium or multiple media (e.g., a centralized or distributed database, and/or associated caches and servers) that configured to store the one or more instructions1224.

The instructions1224may further be transmitted or received over a communications network1226using a transmission medium via the network interface device1220utilizing any one of a number of transfer protocols. Example communication networks may include a local area network (LAN), a wide area network (WAN), a packet data network (e.g., the Internet), mobile telephone networks or other networks now known or later developed.

For example, the network interface device1220may include one or more physical jacks (e.g., Ethernet, coaxial, or phone jacks) or one or more antennas to connect to the communications network1226. In an example, the network interface device1220may include a plurality of antennas to wirelessly communicate using at least one of single-input multiple-output (SIMO), multiple-input multiple-output (MIMO), or multiple-input single-output (MISO) techniques. The term “transmission medium” shall be taken to include any intangible medium that is capable of storing, encoding or carrying instructions for execution by the machine1200, and includes digital or analog communications signals or other intangible medium to facilitate communication of such software.

Table 1 provides a comparison of proposed fundamental group modulation with traditional linear and non-linear modulation methods. Table 1 demonstrates the significant differences between common linear and non-linear modulation methods compared to fundamental group modulation. The primary difference is that a topology is assumed at the modulator and elements of the fundamental group, as defined later, bear the information. The channel is viewed as an operator on the original topological space and the group elements are invariant between transmit and receive spaces when the channel has a local (homeomorphic) mathematical property. Therefore, it is feasible to detect information in the channel's image space without inverting the effects of the channel, i.e., without equalization.

Fundamental group modulation may be used to reduce complexity, provide greater data rates, and increase resilience for many systems through a scintillating channel or for terrestrial communications through multipath. Fundamental group modulation may be applied to wired or wireless communications. A single user may modulate with the space or subsets of the space may be assigned to multiple users. For example, for n=3, each of d1, d2, d3may be assigned to different users and the information from each coded in either a non-distinguished or distinguished orientation. Subgroups may be similarly applied. Data sequences are associated with fundamental groups. Hence signal paths may vary within these equivalence classes. While examples herein have shown modulations with paths in2, paths in other topological spaces may also be achieved. For example, if two orthogonal carriers are used we can modulate on a space in4=S1×S1(S1is a circle in2, x is the product operator on a space). S1×S1is referred to as a torus. Hence, the system may apply to modulation on higher dimensional topological spaces. This notion also includes the use of multiple transmitters that produces a path on some space, which may be useful for transmission through saturated amplifiers or operation in a linear region of an amplifier. Further, as mentioned above, information may be assigned to path that respect orientation or may not respect orientation, symbol directions and rate of movement may be varied in a given implementation and the topological space may be varied over time, wherein nodes communicating share an understanding of the underlying space associated with the paths used to modulate the information. In the case where the channel effects the holes, e.g., morphs the topology, the holes may be identified over the medium by sending such information to identify at the receiver where holes are in the topological space.