WO2005057817A9 - Multidimensional signal modulation and/or demodulation for data communications - Google Patents
Multidimensional signal modulation and/or demodulation for data communicationsInfo
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- WO2005057817A9 WO2005057817A9 PCT/US2004/040754 US2004040754W WO2005057817A9 WO 2005057817 A9 WO2005057817 A9 WO 2005057817A9 US 2004040754 W US2004040754 W US 2004040754W WO 2005057817 A9 WO2005057817 A9 WO 2005057817A9
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L27/00—Modulated-carrier systems
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04B—TRANSMISSION
- H04B14/00—Transmission systems not characterised by the medium used for transmission
- H04B14/002—Transmission systems not characterised by the medium used for transmission characterised by the use of a carrier modulation
- H04B14/004—Amplitude modulation
-
- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04B—TRANSMISSION
- H04B14/00—Transmission systems not characterised by the medium used for transmission
- H04B14/002—Transmission systems not characterised by the medium used for transmission characterised by the use of a carrier modulation
- H04B14/008—Polarisation modulation
Definitions
- the invention relates generally to the field of communications. More particularly, the invention relates to methods of multidimensional signal modulation and/or demodulation for data communications, and machinery for transmitting and/or receiving such communications.
- n-QAM binary phase-shift keying
- OOK simple on-off keying
- FSK standard frequency-shift keying
- FIG. 24 shows a conventional constellation for 4-QAM with 2 bits per symbol. This same constellation can represent quadrature phase shift keying (QPSK or 4-PSK). Theoretically, higher numbers of states are possible, but the practical constraints of required levels of linearity in the modulating circuitry and realizable signal-to-noise ratios (SNRs) in the transmission links and receiving hardware limit most current-day systems to a maximum density of 256 states (“256-QAM").
- FIG. 25 shows a conventional constellation for 8-PSK.
- a process comprises: modulating a carrier signal in a first domain selected from the group consisting of phase, frequency, amplitude, polarization, and spread; modulating the carrier signal in a second domain selected from the group consisting of phase, frequency, amplitude, polarization, and spread; and modulating the carrier signal in a third domain selected from the group consisting of phase, frequency, amplitude, polarization, and spread.
- a process comprises: demodulating a signal in a first domain selected from the group consisting of phase, frequency, amplitude, polarization, and spread; demodulating the signal in a second domain selected from the group consisting of phase, frequency, amplitude, polarization, and spread; and demodulating the signal in a third domain selected from the group consisting of phase, frequency, amplitude, polarization, and spread.
- FIG. 1 illustrates binary PAM detection probability, representing an embodiment of the invention.
- FIG. 2 illustrates a view of probability density distributions for a 4-QAM signal space, representing an embodiment of the invention.
- FIG. 3 illustrates another view of probability density distributions for a 4-QAM signal space, representing an embodiment of the invention.
- FIG. 4 illustrates a series of shells, where n, the radius of a given shell, is the maximum square root of the energy of any points lying within the shell, representing an embodiment of the invention.
- FIG. 5 illustrates 12 points distributed on the surface of a circumscribing sphere that have a constant and minimal distance between closest neighbors, representing an embodiment of the invention .
- FIG. 1 illustrates binary PAM detection probability, representing an embodiment of the invention.
- FIG. 2 illustrates a view of probability density distributions for a 4-QAM signal space, representing an embodiment of the invention.
- FIG. 3, illustrates another view of probability density distributions for a 4-QAM signal space, representing an embodiment of the invention.
- FIG. 6 illustrates error rate as a function of signal-to-noise ratio per symbol in dB, representing an embodiment of the invention.
- FIG. 7 illustrates percent square QAM leakage error as a ftinction of signal-to- noise ratio per symbol in dB, representing an embodiment of the invention.
- FIG. 8 illustrates percent hex QAM leakage errors as a function of signal-to-noise ratio per symbol in dB, representing an embodiment of the invention.
- FIG. 9 illustrates symbol error rates for hex and square QAMs as a function of signal-to- noise ratio per symbol in dB, representing an embodiment of the invention.
- FIG. 10 illustrates face-centered cubic packing in three dimensions, representing an embodiment of the invention.
- FIG. 11 illustrates the required average energy dB to transmit a symbol as a function of the bits per symbol, representing an embodiment of the invention.
- FIGS. 12A and 12B illustrate symbol error rates (for 6 bits per symbol FIG. 12A and 15 bits per symbol FIG. 12B) as a function of signal-to-noise ratio, representing an embodiment of the invention.
- FIGS. 13 A and 13B illustrate symbol error rates (for 2 bits per symbol FIG. 13 A and 4 bits per symbol FIG. 13B) as a function of signal-to-noise ratio, representing an embodiment of the invention.
- FIG. 14 illustrates required signal-to-noise ratio per bit in 2, 3 and 4 dimensions as a function of the number of bits, representing an embodiment of the invention.
- FIG. 12A and 12B illustrate symbol error rates (for 6 bits per symbol FIG. 12A and 15 bits per symbol FIG. 12B) as a function of signal-to-noise ratio, representing an embodiment of the invention.
- FIGS. 13 A and 13B illustrate symbol error rates (
- FIG. 15 illustrates required signal-to-noise ratio per bit in 4 dimensions as a function of the number of bits, representing an embodiment of the invention.
- FIG. 16 illustrates the normalized power response in dB of matched quadrature filters as a function of channel spacing or frequency difference, representing an embodiment of the invention.
- FIG. 17 illustrates a Walsh-code modulated carrier, representing an embodiment of the invention.
- FIG. 18 illustrates the power spectral density of the signal shown in FIG. 17, representing an embodiment of the invention.
- FIG. 19 illustrates spectral efficiency as a function of signal dimensions, representing an embodiment of the invention.
- FIG. 20 illustrates spectral leakage power in dB as a function of channel spacing, representing an embodiment of the invention.
- FIG. 21 illustrates multipath beats (at 5775 MHz) as a function of path difference in meters, representing an embodiment of the invention.
- FIGS. 22A and 22B illustrate multipath beats (at a frequency difference of 2 MHz in FIG. 22 A and a frequency difference of 10 MHz in FIG. 22B), representing an embodiment of the invention.
- FIG. 23 illustrates 4-dimensional spectral efficiency as a function of QAM amplitude levels, representing an embodiment of the invention.
- FIG. 24 illustrates a conventional QPSK or 4-QAM constellation (2 bits per symbol), appropriately labeled "PRIOR ART.”
- FIG. 25 illustrates a conventional Trellis coded 8-PSK constellation (3 bits per symbol), appropriately labeled "PRIOR ART.”
- Patent Application Ser. No. 10/726,475 (attorney docket number UBAT1430 also known as 2500940.991430), filed December 3, 2003 are hereby expressly incorporated by reference herein for all purposes.
- the invention can include a signal modulation and encoding and/or demodulation and decoding scheme to provide higher rates of data transmission in RF and optical links than previously achievable by utilizing additional dimension (s) of the transmitted signal space to carry the additional information.
- Current standard techniques utilize either one or two signal dimensions to code the data for transmission.
- the invention can incorporate both signal-modulation constellations with one or more additional (i.e., a total of at least 3) dimensions, plus the use of optimized schemes such as Trellis or "turbo" coding adapted to the higher-dimensional signal-modulation spaces to improve overall performance of the link versus noise and/or mutipath or other degradations in signal quality.
- 26 illustrates an embodiment of the invention that includes an additional (third) signal-modulation dimension.
- a two-dimensional 4 x 4 state possesses 16 discrete modulation states (i.e., 16-QAM, with 4 bits per modulation symbol)
- the three- dimensional 4 x 4 x 4 constellation shown in FIG. 3 has 64 states (and thus 6 bits per symbol).
- the net gain of 2 additional bits per modulation symbol theoretically represents a factor of 2 or 4 improvement in the achievable data rate for the link, compared to the two-dimensional 4 4 state assuming an equivalent signal-to-noise ratio (SNR) characteristic in the additional signal dimension.
- SNR signal-to-noise ratio
- the interior states are shown in gray and only 37 of the 64 states are shown in black on the visible surfaces.
- the density gain improvement that can be effected by the invention is much higher starting with a 64-QAM (8 x 8 state) 2-D case, with 6 bits per symbol.
- the corresponding 3-D 16 x 16 x 16 cubic constellation contains 4096 points, for a density of 12 bits/symbol.
- the mathematical improvement is a factor of 16, or 4 additional bits per modulation symbol.
- the additional signal- modulation dimension of the invention in practical RF and optical data-transmission hardware systems.
- One useful (and totally independent) signal property to exploit for this additional modulation dimension is transmitted electromagnetic polarization orientation, i.e., either vertical, horizontal, and/or a combination thereof.
- polarization orientation i.e., either vertical, horizontal, and/or a combination thereof.
- polarization discriminations of 30 dB are readily achievable; often, optical systems can run better than 40 dB.
- the addition of the polarization modulation can yield a composite rate of 14 bits/symbol, or a 75% increase in the total signaling rate, per individual carrier frequency.
- This embodiment of the invention can be termed 256-QAM/64-PSK (polarization shift keying). It should be noted that starting with a QAM modulation format is only one method of improving the bandwidth efficiency in standard links; other starting formats include m-ary PSK, M-FSK, and [multicarrier] orthogonal frequency-division multiplex (OFDM) techniques.
- QAM is generally considered the most power-efficient bandwidth-limiting coding method, while OFDM is usually thought to exhibit superior robustness to the others in multipath- degraded environments.
- a shortcoming of the r ⁇ -ary PSK (polarization) technique for the signal-polarization modulation described above as an example of the invention is the presence of reflections at virtually all possible polarization angles in a typical mobile radio environment which can drastically impair the fidelity of the polarization components of the received signal, thus at times markedly reducing the corresponding advantage in modulation rates achievable compared to the largely benign (multipath-free) stationary LOS (line-of-sight) link scenario.
- the invention can include utilization of a fourth dimension of frequency, where the individual frequency channels are mutually orthogonal (e.g., as in the case of OFDM schemes).
- each encoded symbol could represent some 22 bits, all within reasonably achievable channel bandwidth and SNR scenarios!
- This document reports on a method of transmitting information at maximum rates on a physically isolated radio-frequency (RF) or optical beam.
- RF radio-frequency
- the main assumption is that a certain frequency band be available and free from external jamming signals. This can be accomplished in a point-to-point situation using high-gain antennas for transmission and reception. Additionally, if the antenna spacing is not too great, the required power can meet the Federal Communications Commission (FCC) regulations for use in the license-free Industrial, Scientific, and Medical (ISM) band limitations.
- FCC Federal Communications Commission
- ISM Industrial, Scientific, and Medical
- Orthogonal frequency channels can be defined and distributed evenly throughout the available band, each channel carrying simultaneous amplitude, phase, and polarization information.
- the invention is materially different from the usual methods of spread- spectrum and code modulation, although it can also make use of these techniques for logically partitioning the band into a large number of virtual users. These virtual channels may be software allocated to any number of physical users.
- the invention is robust and adapts gracefully to changes in ambient channel noise, presence of accidental and/or intentional jammers, and varying partial blockage of the beam between or at the antennas.
- Two schemes are presented for interference adaptation. They are both controlled by software and require minimal communication between transmitter and receiver to establish a set of parameters resulting in an acceptable error rate.
- the four concepts (physical channel, symbol capacity, symbol constellation, and orthogonal symbols) introduced below are not restricted to the central problem of tight- beam communications addressed, but are generically applicable to a broad range of communications applications, both in the RF and optical domains.
- Theoretical Background Three new concepts are introduced to conceptualize a multidimensional transmission system and aid in the design process.
- the first concept introduces physical symbol capacity, contrasted with Shannon information capacity.
- Physical capacity is based on the notion of the physical state of an electromagnetic wave and aids in system design by providing a practical limit on the number of bits per symbol that an information system can profitably employ.
- the physical state provides a concise and physically meaningful definition of "symbol" in a communication system.
- the second concept extends the common two-dimensional signal constellation to three and four (and more) dimensions, while precisely describing the optimal density of two-, three-, and four- dimensional probability distributions to achieve bounded-error detection.
- the third concept defines a composite signal and shows how such entities may be mapped onto constellation points, creating a new type of symbol constellation that extends a system's multiplexing capability as well as provides additional robustness against interference.
- a fourth concept refines the notion of coded-channel orthogonality to allow channel spacing much closer than the standard reciprocal symbol period. INTRODUCTION There are two basic methods of frequency spreading using multiple frequency channels: (1) frequency hopping and (2) simultaneous occupancy. In the first method, one and only one channel is active at any one time.
- the number of channels depends on the available bandwidth and frequency resolution of the receiver. For example, given a 100- MHz bandwidth and a receiver discrimination of 100 kHz, 1000 overlapping channels could be used, one at a time for each of several users.
- all channels are active simultaneously, each one transmitting a particular code based on polarization, amplitude, phase, and direct sequence (DS) if necessary.
- the number of overlapping channels is given by the available bandwidth divided by the channel spacing. The latter is the bit rate or reciprocal dwell time on a channel until a change of code, phase, or other physical parameter is made.
- n simultaneous channels one usually allocates n watts/channel for a nominal total power of 1 watt.
- This allocation produces a system with 100 channels, each transmitting 10 mW.
- the same 1-watt transmitter power in the case of frequency hopping there would always be 1 W/channel or about 20 dB more signal per detection event.
- This enhancement in channel power reduces the bit rate in a one-to-one fashion.
- Spread-spectrum methods frequency hopping and DS spreading or DSS are a means of multiplexing multiple users (including interferers) in the same band. Such methods generally neither provide security nor increase the spectral efficiency. They do, however, grant greater probability of reception by concentrating power into narrower channels, thus avoiding interferers (in a probabilistic view only). The cost is reduced bandwidth per (narrower) channel and reduced aggregate bit rate for the total allocated bandwidth.
- Electromagnetic Waves and Physical Channels A single mode of the electric field of an electromagnetic oscillatory signal may be written as a function of time and frequency as
- the field at the observation point is written as E 0 ⁇ ijLe l ⁇ : (2) A composition of several waves of different polarizations, amplitudes, and frequencies is simply a sum or integral over the relevant parameters (subscripts h and v distinguish horizontal and vertical polarization directions of the electric field). In the discrete case, which is an approximation to the true continuous signal, the combined field at the observation point is (k is now an index)
- This expression may be viewed as a vectorial Fourier decomposition of the electromagnetic field on a set of plane-wave basis functions. There are only two distinct polarizations, even though they may be combined to form a resultant transverse electric field pointing in an arbitrary direction. This may be a useful way to distinguish between different signals but there are only two physical polarization channels. If we examine each of these sums we see that the amplitude terms may be combined for each polarization, resulting in the expression
- the invention can include defining a physical channel as a particular frequency at a particular polarization (later we will combine both polarization states into the same channel for convenience).
- the invention can include defining a symbol on such a channel as a particular choice of phase and amplitude information.
- a state of the electromagnetic wave will be the four-tuple (fold) ⁇ frequency, polarization, amplitude, phase ⁇ or ⁇ v,s,A, ⁇ where there are n v discrete frequencies, 2 discrete polarizations, n ⁇ discrete amplitudes and n ⁇ discrete phases.
- the total number of states n s in the wave is then the product of these four numbers, or 2n v n A n , .
- phase-amplitude state can also be represented by two quadrature amplitudes, we may take n s to be 2 « v « .
- These states determine the physical capacity of the system producing the electromagnetic wave and the system can be in only one state at any instant in time. Since a symbol represents the state of a physical channel, a system with 6 channels may contain 6 symbols simultaneously in a given, predetermined time interval. This is shown by the following expansion for a system with three frequencies and a total of 6 independent terms E 3 ⁇ , (e itc ° ⁇ A, ⁇ +e it ⁇ z ⁇ .z +e it ⁇ ⁇ A, 3 ) + ⁇ v (e it ⁇ ⁇ A. ⁇ +e it ⁇ z A.. +e it ⁇ 3 Aj).
- the information in the above expression may be recombined in different manners, such as E 3 ⁇ e it ⁇ > ⁇ e h ( ⁇ dress A.3 +e v A, 3 )- (6) to emphasize the three frequency channels.
- the terms in parentheses may be combined to emphasize the angle of polarization at each frequency along with its amplitude and phase.
- the polarization direction of each frequency channel can be selected to point in any direction between o ⁇ ⁇ k ⁇ ⁇ .
- phase of the field ⁇ k at each complex amplitude polarization angle ⁇ k may be independently chosen and detected (by quadrature means); the representation in this case is E 3 ⁇ e, e, e 1 ⁇ e it ⁇ . +e 2 _> z e 1 ⁇ e it ⁇ 2 +e 3 e, j ⁇ Ps e it ⁇ . (7) - - None of this formal manipulation changes the physical nature or separability of the 6 independent channels, each carrying amplitude and phase information, or 3 channels combined in pairs of the same polarization. However, the different representations make different channel states evident and also provide guidance as to optimal receiver design. Physical identification of the various states within each channel is made by signal- strength measurements and quadrature phase detection.
- the polarization states may be separated by defining two orthogonal directions along which two dipole antennas are aligned.
- the frequency separation is made in the usual manner of non-linear mixing, filtering, and envelope detection.
- state information may be conveyed within the discrimination capability of the receiving device, but the number of symbols per channel is limited to a single one at any instant of time.
- the number of bits per symbol is purely an arbitrary matter of convention and convenience, having nothing to do with the number of physical states of the system at a particular time.
- phase modulation or frequency modulation (FM)
- FM frequency modulation
- polarization modulation the polarization direction must remain relatively constant over ⁇ .
- analog information is being transmitted, such as standard AM or FM radio broadcasts
- ⁇ is an approximate notion based on the effective bandwidth of the baseband information.
- ⁇ is a precise concept, being the well-defined time duration of a chip (for DSS systems), a bit for single-pulse amplitude modulation
- ⁇ is the (approximate or precise) period over which the 4-dimensional state of the physical channel is stationary. Since we will be primarily interested in digital modulation, ⁇ is a precise notion intimately connected with the definition of the state of the electromagnetic wave where ⁇ is the time between state transitions.
- the notation ⁇ is used here to distinguish the time duration of a physical state from the closely related concept of a symbol interval, usually denoted by Tin the digital-communications literature (e.g., see Proakis or Peebles). '
- the reciprocal of the coherence time is the symbol rate, and the total symbol rate of a system consisting of n physical channels is c s ⁇ .
- the reciprocity relation is interpreted as an upper bound on the symbol rate or symbol capacity for a system occupying a band S Hz wide as c s ⁇ ⁇ fB . (11) This bound is the physical upper limit to the symbol rate.
- the capacity is finite, unlike the familiar Shannon channel capacity for bit rate. That is, C s represents an objective physical capacity and not an information capacity, which is subjective in nature.
- ⁇ also determines the width of the power spectral density (PSD) in the electromagnetic wave, which is described below in more detail.
- PSD power spectral density
- the width of the PSD can be taken to be the distance between the half -power points on either side of the PSD centered at the carrier frequency; this is about 1.206 ⁇ .
- the spacing of the channels is given by the nulls of the PSD which occur at integer multiples of ⁇ "1 .
- Spectral Efficiency & Shannon's Channel Capacity If the data rate of a signal bandlimited to ⁇ v Hertz is i ⁇ bits per second and the spacing between adjacent wavelengths is ⁇ v Hertz, then the spectral efficiency is 1 ⁇ / ⁇ .v, with units of bits/second/Hertz. This measures how efficiently the bandwidth is used.
- Next-generation commercial optical digital wave-division-multiplexing (DWDM) systems will operate at 40 Gb/s with a channel spacing of 100 GHz, using simple binary
- Shannon's channel capacity can be viewed as providing a limit on the spectral efficiency in bits/second/Hertz of in 2 (l + j ) times the bandwidth S. For a signal-to-noise ratio of 20 dB, the spectral efficiency is then 6.66 bits/s/Hz.
- the physical spectral efficiency of 1 is a measure of the maximum number of physical states of an electromagnetic wave (consistent with orthogonal channels); that is, it specifies a limit on the rate at which symbols, which are objective entities, can be transmitted.
- the Shannon channel capacity is a limit on the rate at which information, measured in bits, can be transmitted without error in the presence of noise.
- the channel capacity in bits per second can be greater or less than the physical capacity in bits per second (for a given number of bits per symbol) and the system's performance is bounded by both measures. That is, if the
- the probability distributions are the same height and width with one centered at the expected value ⁇ ⁇ and the other at ⁇ z.
- the optimal decision boundary is indicated by the vertical black line; that is, any measurement resulting in a ⁇ to the right of this boundary is (optimally) assigned to ⁇ z and any to the left to ⁇ ⁇ .
- the tail of the right distribution to the left of the boundary and the tail of the left distribution to the right of the boundary are measures the probability of error in that the sum of the areas of the left tail under the right curve and the right tail under the left curve represent the probability of confusing one signal for the other.
- the probability of a symbol error is this tail area divided by 2. In the above example, the probability of making an error is about 2.3%. If two quadrature signals with expected strengths at the receiver of ⁇ o and ⁇ 0 are transmitted with equal probability, the probability density is two dimensional, P( ⁇ , ⁇ e- ⁇ - ⁇ .»e- ⁇ -.» (15) where the noise variance is the same as before and affects both components equally.
- the probability densities in the QAM plane for transmitting one of four possible QAM signals might look like four "humps" (with the third dimension representing the probability).
- FIG. 2 illustrates these probability density distributions for a 4-QAM signal space.
- the coordinates are the two amplitudes ⁇ and ⁇ and the probability density, which ranges from 0 to about 0.6.
- a convex hull is defined by the four distributions (the surface lying above or maximal to the distributions). It is somewhat more complicated to compute the tail areas, but the area under the convex hull is a straightforward calculation. Since the total area of the separate distributions is 4 (1 per distribution), the symbol-error rate is 1 minus one fourth of the area under the convex hull. If we view the distributions as a contour plot, the graphic takes on the suggestive form of circles lying in the plane.
- FIG. 3 illustrates a contour representation of the 4-QAM signal constellation.
- the coordinates are the two amplitudes ⁇ and ⁇ , while the probability density is represented by the several hues with the dark centers being the maximum.
- the contours indicate where confusion between any peak and its neighbors arises.
- the confusion boundaries are four vertical planes passing equidistant from each of the four peaks.
- the two dimensions are amplitude and phase ⁇ a h k ⁇ or quadrature amplitudes ⁇ a, h ⁇ o, k i for each of the frequency channels of interest.
- This representation allows a simple visualization in terms of transmitted symbol power, defined as the sum of the squares of the two signal amplitudes. Power may be partitioned into steps defined as the square of the radius from the center of the QAM constellation, which is the origin representing zero transmitted power, to the surface of the 2-D sphere (circle) passing through the centers of the QAM "shells.” If we adjust the lattice spacing such that the minimum distance between adjacent centers is 1, the successive shell radii are the integers, 1, 2, 3,.... Several such consecutive shells in two dimensions are illustrated in FIG. 4.
- FIG. 4 illustrates a series of shells at radii of 1 (green circle 410), 2 (blue circle 420), 3 (orange circle 430), 4 (red circle 440), and 5 (purple circle 450). Note the four red centers 460 inside the orange shell and the empty lattice locations inside the purple shell. Each shell passes through the square represented by the four corners located at ⁇ n,
- n is the radius of the shell or the maximum square root of the energy of any points lying within the shell.
- n is 3, which is the radius of the orange circle passing through the four orange corners of the square that adds the 12 orange centers 470 for that shell.
- four points 460 (red) from the next shell also lie within the orange circle.
- Each new shell fills in vacancies in lower-lying shells, and contains centers at differing distances (root-energies) from the origin.
- a shell is defined as the sphere passing through the 2 d corners of the lattice hypercube (d is the dimension of the signal space).
- the energy level of every center has an integer value.
- the shells are surfaces of 3-spheres, and in four dimensions they are the volumes of 3-spheres (surfaces of 4-s ⁇ heres).
- the invention can thus provide a significant improvement in communications efficiency compared to existing systems via optimum packing of the transmitted symbol constellations.
- the optimal criterion in the case of probability distributions for practical communications is a bit different since the packing problem refers to a space-filling solution whereas signals have finite power and do not fill all space. That is, we need a solution on the surface of the sphere with radius equal to the square root of the power in the signal.
- FIG. 5 illustrates an icosahedron showing the 12 points as vertices of the Platonic solid (distributed on the surface of a circumscribing sphere) that have a constant and minimal distance between closest neighbors.
- ⁇ J2 ⁇ 5erf- 1 - ⁇ .
- ⁇ can be 7.022% larger for hexagonal packing that for square packing.
- the centers can be 6.561% closer for hexagonal packing, thus decreasing the required signaling power by about 13% (about 0.6 dB).
- FIG. 6 shows the exact symbol error rates per symbol for the cases of 4 (green, curve 610), 16 (blue, 620), 36 (purple, 630), and 64 (orange, 640) centers; the right-most curve (red, 650) is for the case above of a space-filling, square-lattice packing.
- FIG. 6 in general illustrates that as the number of constellation points increases, the error rate as a function of signal-to-noise per symbol quickly approaches that of the ideal, space-filling QAM.
- the abscissa, yS is the signal-to-noise ratio per symbol in dB.
- FIG. 7 illustrates the leakage error in replacing the exact QAM by its ideal, spacefilling version. The worst case is the 4-center, square QAM 710 shown in green, reaching about 50% leakage error over the exact expression. As the number of centers increases, the leakage error decreases.
- FIG. 7 illustrates the leakage error in % for a square-lattice
- FIG. 8 is the same calculation as FIG. 7, but for a hexagonal QAM.
- the 6-ary QAM is a special case since all centers have unit distance from the origin and closest neighbors are unit- distances apart.
- the exact hexagonal QAM for the fewest centers 810 behaves a bit differently than the others, since the centers only have neighbors in the same shell or level. Again, as the number of centers increase, the error approaches zero.
- packing fraction is an area measure (in two dimensions), the correction is the square root of the packing fraction. Making this correction ensures that the QAM nearest neighbors are the same distance in both packings.
- FIG. 9 illustrates the symbol-error rate in red (on the right-most or upper curve) for the square lattice and green (on the left/lower curve) for the hexagonal lattice. The difference is about 0.5 dB in yS and results in a factor of 3 lower symbol error rate for the hex QAM at the 10 "5 error level.
- This difference between a square-lattice QAM and the hexagonal-lattice representation will appear in several different guises in the following development.
- Three-dimensional Signal Spaces Since a fully polarized frequency channel is four-dimensional, it may seem pointless to consider a three-dimensional signal space.
- the cell containing a sphere is a dodecahedron — but not the Platonic regular dodecahedron, which has five-fold face symmetry.
- a precise description of the circumscribing dodecahedron is necessary for an exact computation of the leakage from a center to its neighbors. This is a straightforward but complicated task and will not be undertaken since there are satisfactory approximations to the symbol error rates in three dimensions.
- FIG. 10 illustrates fee sphere packing in three dimensions.
- FIG. 10 shows the 12 nearest neighbors of the central sphere in three dimensions. Comparing this graphic with the contour plot shown above for the two-dimensional case gives a visual hint of how much packing density can be gained by going to three dimensions.
- the central sphere 1010 is shown with a red mesh (partially hidden).
- Its 12 nearest neighbors are shown in three layers of four spheres each with the front layer of spheres 1020 indicated by the blue mesh, the middle layer of spheres 1030 by the green mesh (only three are visible), and the back layer of spheres 1040 by the yellow mesh (two are hidden).
- the three layers are each arranged as squares, with the middle square rotated from the other two by 45°.
- Each of the 12 neighboring spheres is the same distance from the center sphere.
- the center sphere can be taken to represent any probability distribution in the 3-D QAM. In particular, consider a center sphere placed at the origin (zero power transmitted). Of course, there is no code corresponding to zero power because interpretation of an absence of energy is ambiguous.
- the particular representations of angles relates the point on the sphere ⁇ s ⁇ , sz, S 3 ⁇ to the polarization ellipse commonly used in optics; this is not required in the present case, but provides a convenient way to think about a three- dimensional QAM.
- ⁇ is related to the eccentricity of the polarization ellipse and - to its orientation along the direction of propagation, quantities that are important for studying the optical properties of crystals, but are merely conveniences for studying data transmission.
- the QAM constellation in three dimensions may then be thought of as lying on the surface of a set of spheres of increasing radii.
- the optimal packing in three dimensions is to place centers on a face-centered cubic lattice as discussed above.
- the origin is at ⁇ 0,0,0 ⁇ and contains a center for counting purposes but not for energy determination.
- the cube grows by increasing the lattice one step at a time (from +1, to +2, to ⁇ 3, and so on).
- the radii of the packed spheres is 1 and must be rescaled to the standard unit distance between nearest neighbors by dividing all distances by Vi .
- the first shell has 12 centers (the kissing number for fee packing in three dimensions), and contains centers at the locations ⁇ 1,1,0 ⁇ , ⁇ 1,0,1 ⁇ , ⁇ 1, 0,-1 ⁇ , 1, -1,0 ⁇ , ⁇ 0,1,1 ⁇ , ⁇ 0,1,-1 ⁇ , ⁇ 0,-1,1 ⁇ , ⁇ 0,-1,-1 ⁇ , ⁇ -1,1,0 ⁇ , ⁇ -1,0,1 ⁇ , ⁇ -1,0,-1 ⁇ , ⁇ -1,0 ⁇ /Vi (22) in rectangular coordinates.
- the centers are ⁇ 2,0,0 ⁇ , ⁇ 0,2,0 ⁇ , ⁇ 0,0,2 ⁇ , ⁇ 0,0,-2 ⁇ , ⁇ 0,-2,0 ⁇ , ⁇ -2,0 ⁇ /Vi (24) in rectangular coordinates and have radius Vi with the angles ⁇ ⁇ , ⁇ ] ⁇ ,o ⁇ , ⁇ i. • ⁇ , ⁇ o, o ⁇ , u, o ⁇ , ⁇ -i ⁇ , ⁇ f, o ⁇ . (25)
- the remaining 4 points are on the equator and are linearly polarized.
- the higher shells consist mostly of elliptically polarized points; for example, there are no circularly or linearly polarized signals out of the 24 centers lying on the third shell.
- the four-dimensional case is a simple generalization of the one in three dimensions.
- the optimal packing in four dimensions is unique in that body- centered cubic (bcc) has precisely half the packing fraction of the ace-centered case ( ⁇ 2 / 32 for bcc and ⁇ 2 / 16 for fee).
- the kissing number is 16 for the bcc and 24 for the fee packings.
- a unit hypersphere at the center of the hypercube of side 2 just touches all 16 unit hyperspheres at each of the corners of the hypercube. For dimensions lower than
- the radius of this center sphere is less than 1 and for dimensions higher, it is greater than 1.
- the problem of the shapes of the 24 hyperplanes (three-dimensional solids) that bound the spheres is just too complicated to be worthwhile for understanding, so we will rely instead on approximations to compute the symbol-error rates.
- a four-dimensional arrangement of 24 hyperspheres surrounding a central one projected onto a two-dimensional surface would take more explanation for less insight than the projection in the previous section. The main idea is clear — there are 24 nearest neighbors and hence twice the density of the three- dimensional case and 4 or 6 times that of the two-dimensional cases.
- the lattice is specified by ⁇ i,j,k,l ⁇ e Z
- the centers are located at even values i+j+k+l.
- the origin is at ⁇ 0,0,0,0 ⁇ and again contains a center for counting purposes but not for energy determination.
- the hypercube grows by increasing the lattice one step at a time on all sides (from ⁇ 1, to ⁇ 2, to ⁇ 3, and so on).
- the enclosing energy hypersphere that passes through all 16 comers of the growing cube has radius where -l -is-the lattice growth index, so the cube volume increases as ⁇ « and the enclosing sphere has volume x z 16£ 4 .
- the close-packed spheres have unit radius on this lattice and are rescaled to give a unit distance between closest neighbors by dividing all lengths by Vi.
- the centers on the first energy surface are located at the 24 points ⁇ 1,1,0,0 ⁇ , ⁇ 1,0,1,0 ⁇ , ⁇ 1,0,0,1 ⁇ , ⁇ 1,0,0,-1 ⁇ , ⁇ 1,0,-1,0 ⁇ , ⁇ 1,-1,0,0 ⁇ , ⁇ 0,1,1,0 ⁇ , ⁇ 0,1,0 ⁇ , ⁇ 0,1,0 ⁇ , ⁇ 0,1,0 ⁇ , ⁇ 0,1,0 ⁇ , ⁇ 0,1,0,1 ⁇ , ⁇ 0,1,-1 ⁇ , ⁇ 0,0,1,1 ⁇ , ⁇ 0,0,1,-1 ⁇ , ⁇ 0,0,-1,1 ⁇ , ⁇ 0,0,-1,-1 ⁇ , ⁇ 0,-1,1,0 ⁇ , ⁇ ,0,1 ⁇ ,
- the second sphere (radius Vi ), however, also has 24 centers with one at each of the poles ⁇ 0,0,0,+ Vi ⁇ ; six on the equatorial plane (linear polarization), and the remaining 16 represent elliptical polarization. Note that the coordinates above can be transformed to phase differences and power differences that should make for a simpler receiver architecture.
- the spreading codes are random but consist of 16 chips each [Shannon found random codes to be as good as carefully designed codes for most purposes] but known to both transmitter and receiver.
- a bit is then represented by a particular set of 16 random phase "chunks" each lasting about 1 ⁇ s.
- the bit rate is simply one-sixteenth of the chip rate, or 62.5 kb/s. Now take precisely the same physical system, but reinterpret the received signal as 16 different bits detected independently (each chip is now a bit).
- each symbol rides on a carrier having a specific phase and amplitude.
- the usual representation of such a signal is by a QAM constellation consisting of a number of points, each occupying the same frequency band.
- the QAM constellation is 16 by 16, consisting of
- the bit rate is back to 1 MHz even though nothing physical in the channel has changed.
- An objective, numerical value for the two interpretations can be assigned by inverting the channel capacity to give an expression of the minimum noise level needed to achieve an error-free transmission (as was implicitly done above) as N ⁇ P (Z % -X (28) where 9Vis the noise power at the receiver, Pthe signal power at the receiver, is the Shannon channel capacity in bits per second, and $ is the bandwidth in Hertz.
- the noise limit is ⁇ r dB ⁇ PdB -101og(2*-l) (29)
- f ⁇ B ⁇ Pd B - 24.06 dB for the knowledgeable receiver and a much larger 3 & B ⁇ P dB for the unaware one. Since both cases are assumed to take place at the Shannon limit, the transmissions are essentially error free.
- the first observation is that the physical situation is identical in the two cases, so the extra 24 dB of signal-to-noise can only be a subjective numerical measure of the knowledge required to interpret the physical signal as one of 256 possible states of the electromagnetic wave.
- the next-higher multidimensional QAM with a number of centers greater or equal to 2" can be chosen and the extra centers, if any, can either be ignored or assigned 30 for auxiliary communications, perhaps at much lower bit rates, for conveying control or status information between transmitter and receiver.
- Peak & Average Powers Since the number of centers and the total power in the 2-, 3-, and 4-dimensional QAMs can be calculated exactly for a given minimum distance between QAM centers, the following graph showing the behavior of the average QAM energy as a function of the number of bits represented by the constellation is accurate. Note that, at 15 bits, the 2-D fee QAM requires 21 dB more power than the 4-D case.
- FIG. 11 illustrates the required average energy to transmit a symbol as a function of the bits per symbol. These curves were obtained by computing the positions of all QAM points as a function of lattice growth; the number of centers were then counted and the energies (distances squared) were computed, allowing the average energy per constellation to be found. If the QAM in each case is confined to lie inside a sphere of radius ⁇ , where ⁇ is the distance to that center farthest from the origin, then ⁇ 2 is a measure of the peak power.
- FIGS. 12A and 12B illustrate five QAM types in the context of 6-bit (FIG. 12A) and 15-bit (FIG. 12B) symbols, including the two-dimensional fee case (2-D sq in red) and hexagonal packing case (2-D hex, orange), the three-dimensional fee case (3-D, purple), the four-dimensional product of two 2-D hexagonal QAMs (2D , blue), and the four-dimensional fee case (4-D, green).
- the minimal case of 2 bits per symbol all error rates converge at about 9 dB for an SER of 10 " .
- FIGS. 13A and 13B illustrate five QAM types in the context of 2-bit (FIG. 13A) and 4-bit (FIG. 13B) symbols, including the two-dimensional fee case (2-D sq in red) and hexagonal packing case (2-D hex, orange), the three-dimensional fee case (3-D, purple), the four-dimensional product of two 2-D hexagonal QAMs (2D , blue), and the four- dimensional fee case (4-D, green).
- FIG. 14 illustrates the required SNR per bit in dB for 2 (2D 2 & 2D Hex), 3 (3D), and 4 (4D) dimensions.
- FIG. 14 shows the required SNR per bit for 2 dimensions (square in red and hexagonal in orange), 3 dimensions (purple), and 4 dimensions (blue).
- FIG. 15 illustrates the required SNR per bit for fee packing in four dimensions. The four-dimensional fee case is shown in FIG. 15 alone for clarity.
- orthogonality is dependent on the method of detection.
- a matched filter provides an optimal detector, consider the signal/(t,v); it has the matched filter represented by the signal /( ⁇ -t,v).
- the normalized response of the matched filter to a signal on an adjacent channel ⁇ v Hertz away, assuming the phase is known to the receiver, is /(r,v + ⁇ v)/( ⁇ r-r-f) ⁇ r sinc( ⁇ r ⁇ v) -sinc( ⁇ r( ⁇ v + 2v)).
- the concept of multidimensional signals provides a commonly used coding scheme wherein orthogonal sets of functions comprise the constellation points of a signal space.
- a set of N orthogonal signals can always be constructed by choosing orthogonal functions as the signals (orthogonalization procedures are available for sets of arbitrary signals) then assigning a single signal to each coordinate in the iV-dimensional space.
- the signal points form a constellation in a space of N-l dimensions where each point lies on the vertex of an (iV-1)- dimensional simplex.
- M-ary pulse-position modulation M-PPM
- M is usually a power of 2 and allow biorthogonal signals (essentially the signal and its negative), providing two states for each component of the -dimensional vector.
- M-PPM M-ary pulse-position modulation
- each signal is represented as the corner of a hypercube.
- the hypercube is the ordinary cube with comers at ⁇ ⁇ , ⁇ J ⁇ , ⁇ V ⁇ / -J ⁇ , representing 8 orthogonal signals; here, eis the energy in a symbol (signal).
- Such signal structures are convenient for binary systems where each signal or comer of the hypercube represents a binary number with M digits.
- a transmitter based on an -dimensional hypercube can transmit an average of M bits per symbol.
- the peak transmitter power (the energy per symbol divided by the symbol time, or e ⁇ ) is also the average power and the system is inefficient unless probability hyperspheres centered on the hypercube corners just touch their nearest neighbors (as illustrated in FIGS. 3 and 10). Since there must also be a hypersphere at the origin having the same radius as the other probability hyperspheres to represent the case of zero power transmitted, the most efficient packing is where the 2 hyperspheres just touch the zero-energy hypersphere. This is impossible except in four dimensions.
- the zero-energy sphere (the region where it is impossible to distinguish between the cases of no signal being transmitted and the presence of an actual transmission) has a radius larger than the spheres touching it, meaning that the comer hyperspheres cannot touch, leaving an unused volume of the space between corner distributions that cannot be occupied by any signal volume; this unused volume is a measure of the inefficient usage of the signal space.
- the usual way of representing the symbol error for the binary hypercube is as a function of the SNR per bit. Since the comers of the hypercube move away from each other as the signal strength increases, the noise level can also increase by the same amount for the same level of error. This is possible since there are no probability distributions filling the volume of the hypercube other than those at the 2 comers.
- each component of a signal occupies a different time slot, with the Mi component present only during the Mi period.
- Transmission of an -dimensional signal then requires a period of M ⁇ , where ⁇ is the symbol interval required to detect a single component.
- ⁇ is the symbol interval required to detect a single component.
- each constellation signal requires a time period 20 ⁇ and an energy ⁇ .
- Each channel can carry 1 bit of information by antipodally phase-modulating its carrier [or 2 bits with quadrature phase modulation, though we have assumed that the latter is more generally expressed as a 4-QAM modulation] .
- Each bit in the code defines a chip of duration ⁇ , which is the same as the symbol period since the state of the electromagnetic wave is stationary for the duration of the chip.
- Composite States & Direct-Sequence Codes Define a composite state as a concatenation of physical states with known transitions separating them.
- a composite state may consist of a carrier modulated by a 16-bit Walsh code, or a multidimensional signal as used in M-dimensional frequency-shift key ( -FSK) or M-PPM modulation techniques.
- the quadrature amplitudes could be represented by the amplitudes of the Walsh codes, one on each of the quadrature phases; alternatively, the constellation points or QAM states can each be represented by a Walsh code, all having the same amplitude as long as the number of codes is greater than or equal to the number of QAM states.
- the analogous phase distinction carried over from the QAM concept would be to consider different codes, each quadrature phase carrying its own code at a particular amplitude, this is a more efficient mapping of codes onto QAM states.
- each quadrature amplitude can independently carry one of the 16 codes, so the combined DSQAM (direct-sequence [coded] quadrature amplitude- modulation system) symbol corresponds to 8 bits.
- the number of states is (2x16) , corresponding to 10 bits.
- the DSQAM idea is a new concept in modulation.
- the DSQAM idea combines standard QAM techniques with direct-sequence codes. In essence, the DS code is assigned to a state of the modulation constellation and not to a bit state (0 or 1) as is usually done. Standard CDMA communications can be enhanced by this reinterpretation.
- Each user would have several DS codes, defining a constellation unique to that user.
- the composite state be represented by a sequence of m pulses each with an arbitrary phase; the state can be written as where the sum is over the code sequence of length m, U.
- Each polarization direction may carry such a composite state. If the receiver is coherent with the transmitter, the state S c is distinguishable from the state - * S C (the carrier phase can differ from one state to the next).
- FIG. 17 illustrates chip time for a Walsh-Code modulated carrier.
- the power spectral density of the signal shown in KG. 17 is given by the absolute value squared of the spectrum; it is shown in FIG. 18.
- FIG. 18 The power spectral density of the signal shown in KG. 17 is given by the absolute value squared of the spectrum; it is shown in FIG. 18.
- the abscissa is the frequency difference from the carrier frequency in units of the chip rate.
- the remaining Walsh codes have spectra differing considerably from this one.
- the notable constant features are the lobes with envelope period ⁇ " (some codes have periods at half -integer values), the high-frequency component indicative of the carrier frequency, and the overall envelope falling off from nominally 0 dB at the carrier frequency to -24 dB at the maxima of the 4th lobes.
- Distinction between DSQAM and M-ary Frequency-Shift Keying The particular example uniting the different threads in this discussion is that of a multi-frequency system where the allotted bandwidth is partitioned into several subcarriers on orthogonal channels.
- (composite) symbol representing a point in the constellation (usually the comers of a hypercube) is Mx ⁇ where ⁇ is the transition time between states of the carrier.
- the time to transmit the (composite) symbol representing a point in the DSQAM constellation is ZVx ⁇ , where the DS code used has length N.
- lfN M, then the times are the same for both methods.
- the number of states for a single-amplitude DSQAM with a code-word of length N is (2xN) n for a physical symbol having n dimensions.
- the number of states for the standard modulation methods is 2 N Xn p where n p is the number of polarization states (1 or 2) and N is the dimensionality of the signals being transmitted.
- the logarithm, to the base 2, of the number of states is the number of bits in the signal constellation. For a symbol interval of
- bit rates are given by dividing the number of bits per constellation point by the time to transmit that information (Vx ⁇ in all three cases).
- the rate for the DS-QAM is logarithmic and must eventually fall below that of the other two methods. This behavior is illustrated in FIG. 19.
- FIG. 19 illustrates a bit-rate or spectral efficiency comparison for the DSQAM (red 1910) and the PPM and FSK methods (green 1920).
- the abscissa is the code length for the DSQAM and the signal dimensionality for the other multidimensional signals.
- the DS-QAM format shows a bit-rate advantage out to about 21 dimensions where it slowly drops below the other two in performance.
- the spectral efficiency is in bits per second per Hertz of bandwidth. This analysis assumes that the channel spacing remains at ⁇ "1 Hertz. If we make use of the orthogonality property of the codes, the number of channels per Hertz can approach a factor of 4 ⁇ times this amount and the spectral efficiency would not start to drop until about 12 signal dimensions were employed.
- FIG. 19 clearly shows that the DSQAM has a superior spectral efficiency out to about M ⁇ 21 dimensions. However, this is only half the story as the performance in the presence of interferes has not yet been presented. It will be shown [later] that the noise performance of the DSQAM is far superior to M-FSK and certainly to the less efficient
- CDMA code-division multiple access
- TDMA time-division multiple access
- FDMA frequency-division multiple access
- Such systems have the capability of "spreading" information over a wider frequency band that those in which the carrier is analog modulated (frequency or amplitude) or controlled via simple digital modulation schemes such as PAM or on-off keying (OOK).
- CDMA methods automatically provide spectral spreading due to the access code, which also serves as a spreading code.
- a code would merely provide security to the particular user, as it would not, on average, increase the spectral bandwidth by merely swapping time slots.
- FDMA can also make use of codes to provide better interference immunity and spread the information over a wider bandwidth.
- the code components or chips may be of several types such as those describing frequency-hopping codes in fast-hopping systems or merely simple state changes (amplitude, phase, polarization) within a single frequency channel.
- a "chip” In DSS spectrum (DSSS) parlance, a "chip” is usually thought of as one of a sequence of stationary states wherein each chip in the sequence can have a different phase; normally, the phases differ by jt radians.
- the concept of a chip as a generic change-of-state, extending the direct-sequence modulation or coding to amplitude states and polarization states is new; in particular, the idea of a chip as a change of state of the polarization is introduced here for the first time. Since concept of a chip is extended to be equivalent to a state or symbol, the spectmm of a single chip of duration ⁇ is precisely that shown in FIG. 16.
- the channel spacing is ⁇ " , chips on adjacent channels are orthogonal and hence non- interfering. If the multiple-channel system is not coherent in that chips may start at different instances on different channels, then orthogonality may be lost as shown above.
- FIG. 20 illustrates this case, showing spectral leakage from adjacent channels carrying orthogonal codes.
- FIG. 20 illustrates spectral leakage into a center channel from two adjacent channels of equal power spaced either side in units of ⁇ " .
- the blue trace 2010 is for symbol leakage
- the red trace 2020 is for orthogonal code leakage.
- the neighboring channels are coded with orthogonal sequences (16-bit Walsh codes with a time span of 16 ⁇ were used here), as the channels start to merge, the code orthogonality begins to dominate and the filter output drops as shown by the red trace. Codes with lengths longer than 16 would have correspondingly less spectral leakage than shown above. Additive noise would have the effect of filling in the spectral nulls in both cases above. Note that the physical-symbol capacity limit prevents the channel spacing from approaching zero closer than the reciprocal of 4 ⁇ if we really are to make full use of multiple frequency channels. Basing channel spacing on the bit period has two drawbacks. The first drawback is that the channel spacing is limited in any case by the reciprocity relation of Eq.
- FIG. 20 shows spectral contamination at a level of 20 to 29 dB below the in-channel signal. If this equivalent noise is acceptable, then the channels can indeed be moved closer together, but not quite to the hoped-for goal of the reciprocal bit time.
- APPLICATIONS Introduction Above we discussed physical channel capacity, probabilistic and geometrical interpretation of error rates, and channel orthogonality and overlap. The goal for this section is to apply the general discussions on capacity, error rates, and coding symbols to specific instances of n-QAMs and n-DSQAMs in two, three, and four dimensions. Practical devices are always bandlimited and always require a finite time to complete a measurement. These are the only restrictions — in principle — for a practical device.
- Another advantage of subband division is the possibility of overcoming multipath losses by redundantly transmitting information on different subchannels such that the multipath spectral nulls overlap at several chosen frequencies assuring that at least one of the channels carrying the redundant information is of sufficient strength to be reliably detected.
- a third advantage is that orthogonal DS codes can be used on each of the subbands, providing additional processing gain (at the expense of hardware complexity and decreased bandwidth) .
- Orthogonal Channels Let the band ® Hertz wide be divided into n channels of width ⁇ v Hertz each.
- the overlap integral between two adjacent channels separated by ⁇ v averaged over the symbol-determination period ⁇ is given by
- the second term is bounded by which is about 14 parts per million for the case of 1 ⁇ s observation times and 5.775 GHz; the first term is bounded by ⁇ .
- ⁇ y ⁇ r is an integer, the first sine function is zero.
- the channels are orthogonal to within 14 parts per million in the worst case if we take ⁇ v ⁇ rto be an integer.
- a method to ensure orthogonality is to offset each channel in phase by 90° from its two neighbors. In this case, the cosine term is zero and adjacent channels are orthogonal.
- the envelope of the field intensity for a 5775 MHz carrier interfering with a single reflection of equal intensity appears as a sequence of nulls and maxima as a function of the path difference with a complete oscillation about every 100 cm. Travelling through such a pattern at a rate of a few meters per second would create an intensity oscillation of a few tens of Hertz, resulting in data loss or highly objectionable audio reception. If the same information is transmitted on another frequency channel, the maxima of one channel can be made to appear at the nulls of the other, depending on the multipath difference and the channel separation.
- 22A and 22B illustrate two examples where the fading of one channel (shown in blue 2210, 2230) is partially compensated by the presence of the other (purple trace 2220, 2240).
- a difference of about 443 meters can be overcome by transmitting on two channels separated by 2 MHz.
- FIG. 22B if the path difference were about 32 meters, a 10 MHz separation would be required.
- the same information can be transmitted on three channels instead of two, allowing the gaps between the peaks shown above to be filled in.
- An agile system that can adapt to changing path differences by adjusting the channel spacing, perhaps by using a pilot channel for the receiver to inform the transmitter of the current status, can largely compensate for multipath losses.
- Chips, Bits, and Codes Introduction Suppose we now want to interpret our symbols as groups of chips — nothing changes in the physical signals being transmitted, but we have a new possibility. Since a certain grouping of chips is now interpreted as a comprising a bit, the idea of joint detection offers the possibility of increased robustness against noise-induced errors.
- the main goal in this section is to devise a means of partitioning the several chips optimally amongst the constellation points. Of course, the effective bit rate obtained by reinterpreting bits as consisting of m chips is reduced by a factor of m. From an error standpoint and a spectral-spreading standpoint, the trade-off might be worthwhile in particular cases.
- Process gain from spreading codes can provide some immunity from interferers as well as help the system meet certain FCC legal requirements.
- 2. In the case of using both polarizations to achieve a full 3-D or 4-D system, different members of an orthogonal set can be used on each of the four quadrature channels, thus increasing the robustness against slight antenna misalignments. 3. If the codes are varied so that adjacent frequency channels do not contain the same codes (at least at the same times), the effective noise from overlapping channel interference can be reduced (this is similar to the usual process-gain argument). 4. If 4 different codes are used for each of the four different quadrature amplitudes on a particular frequency channel, the system will have greater robustness against quadrature phase errors in the receiver. The contribution from each of these benefits can be calculated in a particular case.
- the detector identifies both the power or amplitude in a Walsh code as well as the code itself on each of the quadrature amplitudes. Take a particular frequency channel: there are two quadrature amplitudes belonging to each polarization and four independent 'Walsh codes can be extracted; each polarization can transmit two Walsh codes each at a different amplitude. This preserves the 4-D nature of each frequency channel. Since we are transmitting and detecting codes, the concept of SNR per symbol and not per bit is the valid method of comparison. The price that must be paid for sending codes is that the (composite) symbol rate drops by a factor n c .
- the uncoded bit rate is 2000 Mb/s, just below the Shannon channel capacity.
- the cost of using the code is then a factor of 8.9 in bit rate, or about 9.5 dBb.
- the processing gain of about 12 dB due to the 16-bit code words more than makes up for this loss of bit rate, but it is only meaningful in a high-noise environment.
- the use of orthogonal codes makes sense only where the signal-to-noise ratio is less favorable than for the low-noise case and bit rate must be reduced significantly to compensate for the increased noise.
- Design Example: Power Budget Specification For this example, the transmitter operates in the ISM band in the range between 5725 and 5825 MHz with a bandwidth of 100 MHz. The total power at the transmitter is 1 watt. The antennas are 3-meter parabolic dishes spaced 35 km apart. What is the maximum bit rate that can be transmitted at a symbol-error rate of 10 " ? How much link margin is available? Finally, how gracefully can the system adjust to increasing noise and decreased power at the receiver?
- the ratio of the energy in the received bit to the spectral density of the noise at the receiver is where T b is the time for a single bit and T R the power; the product is Sb, is the energy in the received bit, and ⁇ s the reciprocal time per bit or bit rate.
- the power spectral density of noise in the receiver is Boltzmann's constant times the effective temperature.
- Channel Capacity The maximum bit rate must conform to Shannon's channel capacity in the following sense. First, consider the power at the receiver determined by the transmitted power, the free-space losses, and the antenna gains. Since the atmosphere is quite transparent at the frequencies of interest, absorption due to atmospheric losses and possible obstructions are not included in this initial capacity calculation.
- the noise at the receiver is essentially the spectral power density due to the effective temperature.
- the system With this many bits per symbol, the system, under favorable conditions, can achieve a transmission rate arbitrarily close to the information capacity of 2024 Mb/s. Note that it would be a waste of power and a useless expense of design effort and hardware cost to build a system with more bits per symbol than inferred from the ratio of C b to C s .
- Link Margin & Graceful Degradation If we are interested in budgeting for the data rate of 2 Gb/s (93.0 dBb) which is close to the channel capacity, know from the calculations above that the power at the receiver is -62.9 dBW and from the error-rate calculations that ⁇ req for a .s ⁇ iftof 10 " is 26.7 dB, then the excess capacity of the system available for atmospheric losses, receiver inefficiencies, and in-band interferers (i.e., the "link margin”) is given by ML ⁇ ⁇ P R -N 0 - ⁇ ai - R (57) or, substituting the particular values, the link margin turns out to be a healthy 24.5 dB.
- bit rate will correspondingly drop as can be computed from the channel capacity. What are the preferred methods to lower the bit rate? The goal is to achieve graceful degradation of the transmission service. If the measurement time is increased, there is wasted spectrum between channels unless their number is increased, which increase would necessitate an additional hardware reserve for such a situation. For example, doubling the time from 1 ⁇ s to 2 ⁇ s and decreasing ⁇ v from 1 MHz to 0.5 MHz increases the number of channels from 100 to 200. The symbol rate remains constant at
- FIG. 23 illustrates the spectral efficiency of the 4-D QAM as a function of amplitude levels. A spectral efficiency of 20 bits per second per Hertz requires about 18 discrete amplitude levels.
- FIG. 23 shows spectral efficiency in bits per second per Hertz of a 4-D QAM as a function the number of levels required in each of the four quadrature-detector sets. Restating the above results, graceful system degradation in the presence of noise is a desirable feature of any communications system.
- the system can operate from 2000 Mb/s (near the Shannon limit) down to any desired rate by reconfiguring amplitude-partition maps.
- Suboptimal Channel Spacing Suppose the chip rate is 1 MHz so there is a symbol on each channel every 16 ⁇ s. Every 16 ⁇ s there are 2400 bits sent out for a total of 150 Mb/s, which is essentially the same as above. The way to get larger throughput is to increase the number of amplitude states to 8, say. If this is done, and the same chip rate of 1 MHz is kept, the total rate is 175 Mb/s. This is equivalent to an 8x8 QAM in the quadrature states. Going to 16 amplitude states gives 32 bits per symbol or 200 Mb/s.
- the question is then one of the effects of additive Gaussian noise at the receiver, which must be estimated for the particular design. Supposing that the signals are reasonably noise-free so that the reciprocity limit can be approached, the number of channels increases from ⁇ / ⁇ v to a maximum of approximately "max ⁇ -T— (58) ⁇ r In the case of DSQAM modulation with 16-bit Walsh codes reserving orthogonal sets of 8 each for adjacent channel, the spectral efficiency of a single channel is 4+4 ln 2 n s where n s is the number of amplitude states for the codes.
- the bit rate in dB is approximately 91. If the required rate is 1 Gb/s (90 dB), then there is only 1 dB of margin to expend on the limit not being quite 4 ⁇ , bringing it closer to 3.2 ⁇ »10.
- the canonical channel width is 40 MHz and close spacing would produce a maximum of about 31 overlapping channels spaced some 3.2 MHz apart.
- a 1 Gb/s rate could then be achieved with about 25 channels having a slightly larger spacing and allowing for a few MHz guard bands at each end of the system bandwidth. This set of parameters seems not unreasonable for a realistic system.
- HYPERCUBES AND THE OPTIMUM DIMENSION— PROOF OF 4-D OPTIMALITY The comers of an n-dimensional hypercube are located at ⁇ 1, ..., ⁇ 1 ⁇ where there are n components in the vector.
- the distance from the origin to any comer is obtained from the usual expression as (60) Place unit-radius hyperspheres at each of the 2" corners. At the origin, place a hypersphere with radius VT-i. This central hypersphere touches all 2" comer hyperspheres on the side towards the center of the hypercube.
- preferred embodiments of the invention can be identified one at a time by testing for the presence of low error rates.
- Preferred embodiments of the invention can also be identified one at a time by testing for robustness against (RF) interference.
- Preferred embodiments of the invention can also be identified one at a time by testing for resistance against cross talk between the carriers.
- the test(s) for the presence of low error rates, robustness against (RF) interference and/or resistance to crosstalk can be carried out without undue experimentation by the use of a simple and conventional error rate experiment.
- the invention can also be included in a kit.
- the kit can include some, or all, of the components that compose the invention.
- the kit can be an in-the-field retrofit kit to improve existing systems that are capable of incorporating the invention.
- the kit can include software, firmware and/or hardware for carrying out the invention.
- the kit can also contain instmctions for practicing the invention. Unless otherwise specified, the components, software, firmware, hardware and/or instmctions of the kit can be the same as those used in the invention.
- the term spread, as used herein in the context of a domain, is defined as a time expanded concatenation or simultaneous grouping of a fundamental modulation technique of a classical physical dimension such as amplitude, frequency, phase and/or polarization and which represents bits and/or signal information states (more generically constellations) by a sequence or grouping of modulation states.
- a or an, as used herein, are defined as one or more than one.
- the term plurality, as used herein, is defined as two or more than two.
- the term another, as used herein, is defined as at least a second or more.
- the terms including and/or having, as used herein, are defined as comprising (i.e., open language).
- the term coupled, as used herein, is defined as connected, although not necessarily directly, and not necessarily mechanically.
- the term approximately, as used herein, is defined as at least close to a given value (e.g., preferably within 10% of, more preferably within 1% of, and most preferably within 0.1% of).
- the term substantially, as used herein, is defined as largely but not necessarily wholly that which is specified.
- the term generally, as used herein, is defined as at least approaching a given state.
- the term deploying, as used herein, is defined as designing, building, shipping, installing and/or operating.
- the term means, as used herein, is defined as hardware, firmware and/or software for achieving a result.
- the term program or phrase computer program, as used herein, is defined as a sequence of instmctions designed for execution on a computer system.
- a program, or computer program may include a subroutine, a function, a procedure, an object method, an object implementation, an executable application, an applet, a servlet, a source code, an object code, a shared library/dynamic load library and/or other sequence of instmctions designed for execution on a computer or computer system.
- transmitter and/or receiver described herein can be a separate module, it will be manifest that the transmitter and/or receiver may be integrated into the system with which it is (they are) associated. Furthermore, all the disclosed elements and features of each disclosed embodiment can be combined with, or substituted for, the disclosed elements and features of every other disclosed embodiment except where such elements or features are mutually exclusive. It will be manifest that various substitutions, modifications, additions and/or rearrangements of the features of the invention may be made without deviating from the spirit and/or scope of the underlying inventive concept. It is deemed that the spirit and/or scope of the underlying inventive concept as defined by the appended claims and their equivalents cover all such substitutions, modifications, additions and/or rearrangements.
Abstract
Description
Claims
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EP04813125A EP1695469A1 (en) | 2003-12-03 | 2004-12-03 | Multidimensional signal modulation and/or demodulation for data communications |
CA002579749A CA2579749A1 (en) | 2003-12-03 | 2004-12-03 | Multidimensional signal modulation and/or demodulation for data communications |
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US10/726,446 US7340001B2 (en) | 2003-12-03 | 2003-12-03 | Multidimensional signal modulation and/or demodulation for data communications |
US10/726,446 | 2003-12-03 |
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- 2004-12-03 CA CA002579749A patent/CA2579749A1/en not_active Abandoned
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US20050123061A1 (en) | 2005-06-09 |
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US7340001B2 (en) | 2008-03-04 |
EP1695469A1 (en) | 2006-08-30 |
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