Inverse Matrix Modulation - ITS · modulation, the signal is optionally up-converted for transmission over an RF channel. Several types of linear modulators can be used, resulting

Holtzman Inc. Longmont, CO, USA 1

Inverse Matrix Modulation

By Thomas H. Williams © 2002-2003 Holtzman Inc.

June 2, 2002 Abstract: This paper describes a new digital modulation technique that utilizes a spreading matrix to linearly transform an input symbol sequence into a transmit symbol sequence using a two-dimensional transmission matrix. The transmit symbol sequence is filtered, modulated, and transmitted over a channel. At the receiver, a received symbol sequence is captured, demodulated, equalized, if needed, and again transformed with a recovery matrix that is an inverse of the spreading matrix that was used at the transmitter. Use of an inverse matrix instead of a conventional transposed matrix allows the rows functions of the transmission matrix to be non-orthogonal. This allows the creation of many unique spreading matrices with noise immunity in impaired channels. If a non-square transmission matrix is used, the number of symbols in the transmitted symbol sequence may exceed the number of symbols in the input symbol sequence, creating redundant symbols. The redundant symbols may be used to replace any of the transmit symbols that were damaged by transmission impairments. Alternately, the redundant transmit symbols may be used to reduce the effects of random noise in an output symbol sequence. The transmit symbol sequence may optionally be transformed a second time at the transmitter by an inverse fast Fourier transform (IFTT) prior to transmission, in a technique comparable to OFDM (orthogonal frequency division multiplexing). The second transform process converts time domain symbols into frequency domain symbols. If some of the frequency domain symbols fall into frequency-selective deep channel fades, which are frequently encountered in wireless channels, the redundant transmit symbols can be used to insure error-free reception. Background: The most basic method used to transmit digital information over a band-limited channel is pulse amplitude modulation (PAM). PAM inputs a stream of data to be transmitted and forms the stream into variable-amplitude periodic impulses, which are symbols. The periodic stream of symbols is low pass filtered with a Nyquist filter to limit the bandwidth without creating intersymbol interference (ISI) and sent to a modulator. After modulation, the signal is optionally up-converted for transmission over an RF channel. Several types of linear modulators can be used, resulting in a vestigial sideband signal (VSB), a quadrature amplitude modulated (QAM) signal, or a single sideband signal (SSB) transmission. However, improvements can be gained by performing transforms on the symbols before modulation and transmission to gain an advantage against some channel impairments. Two well-known transform-based modulation techniques are discussed: direct sequence spread spectrum (DSSS) and orthogonal frequency division multiplexing (OFDM). DSSS is also discussed Ref[1] and OFDM is discussed in Ref[2]. Direct Sequence Spread Spectrum:


Direct sequence spread spectrum has been used for many years by military communications systems to prevent reception of a message by unwanted listeners, prevent detection of the transmitted signal by a foe, and offer immunity from jamming by an enemy. DSSS inputs a low speed data stream that occupies a narrow bandwidth, and outputs a wide bandwidth transmission.

Figure 1. Block Diagram of a Spread Spectrum Communications System A DSSS transmission can be created by an exclusive-nor between a low-speed random data stream and a high-speed deterministic bit stream, as shown in Figure 1. A pseudo-noise generator (PN) is running at a fast clock rate, which is also called the chip rate. The PN generator produces the high-speed deterministic bit stream (PN OUT). The PN generator is composed of a set of shift registers and a parity generator, which may be an exclusive-nor device. The number of shift registers and the selection of feedback taps determine the random but deterministic pattern of 1’s and 0’s that is generated. A PN sequence’s random pattern appears noise-like, but is repeated periodically. A second exclusive-or circuit has its inputs connected to the output of the PN generator and a low speed data source. The low speed data source clocks in a new value every p chips, where p is an integer. The output of the second exclusive-or is modulated, amplified, and transmitted. This signal generation process has the effect of spreading the spectrum of the low speed data over a wide bandwidth, making it difficult for a listening enemy to detect, much less decode. At the receive site, the signal is received, demodulated and de-spread using an identical copy of the PN sequence that was used to spread the low speed data source at the


transmitter. The timing of the de-spreading sequence must be accurately aligned with the spreading sequence. Figure 2 is a timing diagram of a DSSS transmitter showing the chip rate clock, the PN sequence, the low speed data input, and an output which is created by an exclusive or function between the PN sequence and the low speed data source. In Fig. 2, the number of chips per low speed data clock, p, is equal to 26.

Figure 2 Timing Diagram of a Spread Spectrum Transmission If there is only one direct sequence spread spectrum transmission, DSSS has a characteristic of being very spectrally inefficient: a lot of bandwidth is expended for one very low data rate stream. This can be remedied by adding other data streams (or other users) to the same bandwidth. However, if multiple users in a channel are all using the same bandwidth at the same time, their signals will look like random noise to each other. When a second user begins transmitting in the same channel, the second user’s signal appears to the first user to be an increase in the background random noise. One solution that has been used extensively is to use multiple digital codes that are orthogonal to each other. Walsh codes are one example of orthogonal codes. With orthogonal codes, the different users with different codes will not experience interference from each other. To maintain the codes in an orthogonal state at the receive site, the channel must be equalized to have a flat frequency response and low group delay. Likewise, the timing of the orthogonal PN sequences that the different users are running must be exceedingly accurate. If the codes are all coming from one transmitter, this is not typically a problem, but if multiple users are all transmitting at the same time, the transmissions’ timing must be very accurately aligned. Orthogonal Frequency Division Multiplexing: Although OFDM was invented in the late 1960’s, it has been made a practical transmission method with the coming of the digital signal processor (DSP), which can perform a discrete fast Fourier transform (FFT) operation very quickly. OFDM is a block transmission method that is in wide use for a variety of services, such as digital terrestrial


television and audio broadcasting in Europe, coaxial telephony systems, and high speed DSL telephone modems. A transmitted OFDM data block is made up of many harmonic carriers (HC’s) at different frequencies that can be accurately distinguished from each other at the receive site because the individual HC’s (which are cosine waves) comprising the composite signal are integer multiples of a fundamental frequency, and therefore orthogonal to each other. Assigning different discrete values to the magnitudes and phases of the individual HC’s conveys information. For example, if E(t) is an OFDM transmission with only four HC’s, it may be represented as:

)4cos()3cos()2cos()cos()( 44332211 φωφωφωφω +⋅++⋅++⋅++⋅= tAtAtAtAtE (1) The magnitude of An may take on values such as 1.33 or 0.75, and the phase φn may take on values such as 45, 135, -45, or –135 degrees. The index variable n is the HC number. The magnitude and phase angle comprise the coefficient of a HC. In practice, tens, hundreds, or even thousands of individual HC’s make up an OFDM transmission.

Figure 3 A Time Domain Plot of an OFDM Signal Showing Four Component Harmonic Carriers Figure 3 is a time domain plot of a 4-HC waveform, as well as a plot of the sum of the 4 individual HC’s. Figure 3 has another feature: a guard interval (GI) has been formed by copying a number of microseconds of data from the end of the transmission block and attaching the samples onto the beginning. The guard interval is also described in Ref[2] as a ‘cyclic extension’. If the time duration of the guard interval is slightly longer than


the duration of the longest echo that afflicts a channel, the echo can be completely cancelled in a noise-free channel. With conventional OFDM, an equalizer is still needed to cancel the effects of an echo, but it only needs to perform a single complex multiplication on each received HC’s coefficient to correct the effects of linear distortion. Figure 4 is a frequency domain (spectral) plot showing the 4 HC’s of Figure 3 as four vertical spectral lines. The HC’s magnitudes can be seen, and the HC’s phases are printed above the HC’s spectral lines.

Figure 4 A Spectral Plot of an OFDM Signal Showing Magnitudes and Phases of Harmonic Carriers Common Elements - Viewing Well-Known Modulation Techniques as Matrices Direct sequence spread spectrum and OFDM transmissions may both be viewed as signals that can be formed by multiplying a 1 dimensional matrix (or vector) containing an input symbol sequence (symbols to be transmitted) by a 2 dimensional transmission matrix that is comprised of rows that are orthogonal basis functions. Let a data block input symbol sequence to be transmitted be given by:

[ ]jeeeeeE ......,,, 4321= (2) Where the elements en are the symbols to be transmitted, and j is the number of samples in the data block. The elements may be real or complex numbers. A transmission matrix C is given by:

•••••••••••••••

••••••••••

=

),()2,()1,(

),3()1,3(),2()2,2()1,2(),1()2,1()1,1(

kjcjcjc

kcckccckccc

C (3)


where c(j,k) values are the coefficients, which may also be real or complex, j is the row index, and k is the column index. Remembering the rules of matrix multiplication, the first product term is formed by multiplying each of the terms in the first row of the E matrix by the corresponding element number in the first column of C, and adding the products together. If the input symbol sequence is multiplied by the two-dimensional transmission matrix, a resulting transmit symbol sequence (to be transmitted) is given by:

[ ]jfffffCEF ......,,, 4321=⋅= (4) If the DSSS transmission uses multiple orthogonal codes, each row in the C matrix contains a PN sequence spreading code that is orthogonal to the codes in other rows. At a receive site, the input symbol sequence is recovered by multiplying the received data sequence by another recovery matrix. Simply transposing the matrix C that was used at the transmitter for spreading creates this de-spreading recovery matrix. For OFDM, the basis functions located in rows are sampled cosine functions, where the top row may be the fundamental, the second row the 2nd harmonic, the third row the 3rd harmonic, etc. At the receive site, the input data sequence may be recovered by multiplying the received symbol sequence by a transpose of the matrix C that was used to create the OFDM block. As a practical matter, however, it is faster to create the F vector (transmit symbol sequence) by performing an IFFT and recover the E input symbol sequence at the receive site with a FFT. With direct sequence spread spectrum, the matrix C may be square (k=j), but with OFDM the matrix is frequently rectangular (k>j), thereby creating a guard interval. The guard interval is normally discarded at the receive site. It is possible to employ only a subset of the total number of orthogonal rows for signal modulation, and then demodulate with a matrix created by a transpose of the rows that were employed. Wavelet modulation follows a similar modulation and demodulation process since wavelets have orthogonal signal sets like OFDM and DSSS. Pulse amplitude modulation can even be viewed as being multiplied by an identity matrix (1s on the diagonal and 0s elsewhere). Of course, the identity matrix makes no change between the input symbols sequence and the transmit symbol sequence. Rows as Orthogonal Basis Functions The rows of the C matrix are typically created of j linearly independent functions, called basis functions. These basis functions define a j-dimensional orthogonal space. The property of orthogonal rows in a C transmission matrix is shared by OFDM, DSSS and wavelet modulation. The orthogonality property between basis functions implies that:


∑ == =⋅kn

nnycnxc

10),(),( for any row x that is not equal to row y (5)

∑ == =⋅kn

n yxKnycnxc1 ,),(),( for any row x that is equal to row y (6)

n is the column index, and k is the total number of columns, and Kx,y is a constant that depends on the row numbers chosen. Ref[8] describes orthogonal basis functions in greater detail. Channel Impairments on DSSS and OFDM There are many channel impairments that afflict different types of channels. Figure 7 is a diagram of time vs. frequency showing common impairments in both time and frequency. Assume that a transmission uses the full channel bandwidth for the total allocated transmission time. The four impairments illustrated in Fig. 7 show random noise, a deep frequency-selective fade, a narrowband interference, and a shot of burst noise. Random noise occurs throughout the transmission as a rectangular grid. Random noise is typically uniform both in time and frequency. Neither DSSS nor OFDM have an inherent relative advantage against random noise, since random noise is pervasive in both time and frequency.

Figure 7 Impairments affecting a transmission of a block of data In wireless applications, the presence of multipath distortion can create deep frequency-selective fades. The deep frequency-selective fades presents two problems. The first


problem is that when there is random noise, and the signal fades into the noise at some frequency, the energy of the signal in the fade is obscured. The second problem is with adaptive equalizers. With a deep fade, a stable inverse channel solution may not be realized to program an adaptive equalizer. Another common impairment is continuous wave (CW) or other narrowband interference, such as microwave oven leakage. OFDM, with a guard interval that is longer than the longest echo, does very well against deep channel fades and narrowband interference when it is used in conjunction with a forward error correcting (FEC) code. The erred symbols, which typically are HCs attenuated down into the noise floor, may be corrected by the FEC code. If the location of the erred symbols are known in advance, they may be erased, giving additional error correction ability to the FEC code. A Reed-Solomon code is an example of a linear block code that is used in OFDM systems. OFDM also can do well against narrowband interference since only a small number of HCs may be corrupted, and they can also be corrected by the FEC. Conventional DSSS spreads all of the impairments in a channel to create a uniform impairment over the entire block. Non-transformed PAM, when combined with a forward error correcting code, also does well against impulsive burst noise, provided the duration of the noise burst does not exceed the correction ability of the FEC. Burst noise is a channel impairment that presents problems for both OFDM transmissions and DSSS. In essence, the burst noise impairment corrupts the channel for a short period of time, reducing the perfect orthogonality between the orthogonal functions. If the burst noise is short and very severe, it might be advantageous to blank out both the noise and the data transmission for the duration of the burst; however, the perfect orthogonality between basis functions (rows) will still be degraded. Some engineers argue that DSSS and/or OFDM perform well against burst noise, since the energy in the burst is spread over many symbols, doing slight harm to all symbols instead of fatal harm to a few symbols. However, this claim depends on the duration and energy of the burst, as well as the receiver’s ability to perform noise blanking. Description of Inverse Matrix Modulation: If C is a two-dimensional square transmission matrix, an inverse recovery matrix K can be created from C:

1−= CK (7) using, for example, the Gauss-Jordan Elimination method. Assume an input symbol sequence, E, is multiplied by a two-dimensional transmission matrix, C, creating a transmit symbol sequence, F.


CEF ⋅= (8)

Assume F is transmitted, and received in a channel with no distortions or additive impairments. A received symbol sequence, F can be multiplied an inverse recovery matrix (which is the inverse of the C matrix) to recover the original input symbol sequence, E:

KFE ⋅= (9) This is because a matrix multiplied by its inverse makes an identity matrix, I:

••••••

•••

==⋅

1000

010000100001

ICK (10)

This idea of using an inverse matrix (as opposed to a transposed matrix) removes a requirement that the rows in the C matrix be orthogonal to each other. However, a new requirement is established: the C matrix must have an inverse. That is, the matrix must not be singular. Mathematically, if the determinant of a matrix is zero, the matrix is singular. A square matrix may be viewed as a method to represent j equations in j unknowns, where j is the number of rows and columns in the matrix. If one or more of the equations comprising the matrix are essentially linear combinations of other equations, the matrix is singular. If a number of columns in the C matrix is greater than a number of rows, additional redundant information can be placed in the transmitted signal, allowing selective signal processing to be done at the receive site. If the matrix C represents k equations with only j unknowns, where k > j, the set of equations may be said to be “over-determined”. The over-determined state may be exploited to remove the effects channel impairments. Example: Let an input symbol sequence be a 1 by 5 matrix: E=[0,0,1,0,0] (11) And let an over-determined transmission matrix without orthogonal rows be a 5 by 6 matrix:


−−−−

−−−−

−−

=

310232231023323102232310

023231

C (12)

This matrix was created by a simple circular shift of one row, but this is not a requirement. If the input symbol sequence E is multiplied by the transmission matrix C, a transmit symbol sequence is created: F=E*C = [-2, 0, 1, –3, 2,3] (13) This F vector is transmitted. The received data sequence (again F, assuming no channel distortions or noise) is received and multiplied by the inverse of a subset of the C matrix. Note that in multiplying two matrices, the ‘inner’ dimensions must agree. Therefore, we will truncate the last column from the F matrix, making: F6=[-2,0,1, -3,2] (14) Likewise, we truncate the last column from the C transmission matrix, making a square 5 x 5 matrix.

−−−

−−−

−−

=

1023231023

231023231023231

6C

Now take the inverse of the C6 matrix. (To take the inverse of a matrix, it must be square.) This gives a first inverse recovery matrix K6:

K6=

−−−−−−−

2585.00127.04322.01780.03093.00254.02119.02034.00339.01780.03475.02288.02203.02034.04322.02161.03008.02288.02119.00127.03941.02161.03475.00254.02585.0

(15)

If the received vector F6 is multiplied by K6, the result is: E=F6*K6=[0,0,1,0,0] (16)


Which is exactly the original input symbol sequence. Now, lets create another K recovery matrix by omitting another column from the C matrix, and taking the inverse. This time we remove the third column from the left.

C3=

−−−

−−−

−−

31032231233230223210

02331

(17)

This gives another inverse recovery K matrix:

K3=

−−−−

−−−−−−−

−

8039.05294.05098.04706.00000.14118.0882.05294.00882.05000.09216.04118.08039.05882.00000.1000.15000.00000.15000.05000.1

5882.00882.04706.00882.05000.0

(18)

The received vector, F3, is also processed to remove the third term from the left: F3=[-2, 0, –3, 2,3] (19) If F3 is multiplied by K3 the result is: E=F3*K3=[0,0,1,0,0] (20) which is also the originally transmitted vector without error. By multiplying an F vector that has had a term (row) removed with the inverse of a C vector that had a corresponding number row removed, the original input symbol sequence is recovered. A practical application is that a received vector may have some symbols corrupted in transmission. This technique allows the damaged symbols to not be used in determining the input data block. To reiterate, it is not necessary that the rows of the C spreading matrix be orthogonal, as long as an inverse exists. For the case of orthogonal signal sets, the transpose matrix is the same as the inverse matrix. Ref[9] Frequency Domain Modulation with Inverse Matrices


The de-spreading inverse recovery matrix idea developed above can be extended into the frequency domain. Prior to transmission, an IFFT may be performed on the F matrix, followed by the addition of an optional guard interval. This creates a ‘spread’ transmission comprised of frequency domain symbols. At the receiver, the guard interval is deleted, an FFT is performed on the received data sequence, and then de-spreading is done with an inverse recovery matrix. In essence, the IFFT process converts spread time domain symbols into frequency domain symbols, or HC’s. The FFT process converts the frequency domain symbols back into the time domain. If the spread transmit symbol sequence F is transmitted directly, you have “Time Domain Inverse Matrix Modulation” or TDIMM. If the spread transmit symbol sequence is again transformed with an IFFT prior to transmission, you have “Frequency Domain Inverse Matrix Modulation”, or FDIMM. If FDIMM is being employed with a transmission matrix, the original C matrix may be combined with a frequency transforming matrix, creating a new combined two-dimensional transmission matrix. This combined matrix eliminates one matrix multiply. The same matrix combining operation can be done at the receiver for a single recovery matrix that combines the FFT and de-spreading operations. If an over-determined matrix is employed in a TDIMM transmission, the additional time-domain symbols may be exploited to remove the effects of channel impairments such as impulse noise or signal clipping. If an over-determined matrix is employed in a FDIMM transmission, the additional frequency domain symbols may be used to eliminate the effects of deep frequency-selective fades. Wireless Channels Figure 5 is a spectral plot of a wireless channel showing the channel’s magnitude response, H(f), and a noise floor. Also shown are the magnitudes of 37 HCs, assuming that they all were transmitted with the same magnitude but unequally attenuated by a channel. Note the channel has a deep frequency-selective fade around HC0 to HC4. In Figure 6, the channel has been equalized, and the magnitudes of all HCs are now equal. The process of equalization unfortunately increased the noise floor underneath the weak HCs, and the signal to noise ratios of the 5 HCs in the deep fade is especially low. These deeply faded symbols can be omitted in the selection of which inverse recovery matrix to use. One might simply view this technique as another method to accomplish the same result as conventional OFDM with forward error correcting codes, but another advantage can be obtained.


Figure 5 A Spectral Plot of a Channel with a Deep Fade Combining HC Terms in the F Vector with FDIMM If the fade is deep, the HC symbols very deep in the bottom of the fade may be eliminated in constructing an inverse matrix. If the fade is not too deep, the multiple terms associated with HCs in the fade can be combined (summed) to produce a single sum HC with a better signal to noise ratio than any of the individual HCs comprising the single sum HC. This is because the individual HC’s will add on a 20-log n basis, while the random noise on the HC will add on a 10-log n basis, where n is the number of terms that are combined. In Figure 6, it may be an advantage to combine HC0 and HC1, eliminate HC2, and combine HC3 and HC4. If two carriers with equal signal to noise ratios are combined, the new carrier will have, on average, a 3 dB better signal to noise ratio. This technique reduces the number of required redundant symbols. Note that if you sum two or more terms in the F matrix, the corresponding two or more columns in the C matrix must also be summed together before computing an inverse to produce a new K recovery matrix. If a matrix has more equations than unknowns, it is said to be over-determined. Another method to solve an over-determined matrix is to use the least squares method Ref[3]. This is a valid method for channels with relatively uniform signal to noise ratios over the entire channel. However, in the case of HCs that are highly contaminated with noise relative to most other HCs, a better strategy is to excise the corrupt terms rather than letting them contaminate the entire transform. Another technique to solve an over-determined matrix (without making it square by dropping terms) is to create a pseudo-inverse matrix.


Figure 6 A Spectral Plot of a Signal after Equalization Showing an Elevated Noise Floor Tricks with Inverse Matrix Modulation With the orthogonality limitations removed from the matrix construction process, many unusual matrices can be created, provided that an inverse matrix exists. For example, a TDIMM matrix could be constructed that has a probability of low power for some symbols in the ‘F’ transmit symbol sequence and higher power for other symbols in the sequence. This variable power feature can be accomplished by making the sum of the elements in the individual rows of the C matrix higher or lower. Likewise, some of the symbols in a transmitted symbol sequence can carry fewer bits of data than other symbols, while all symbols are transmitted at a statistically equal power. For example, by prearranged convention, symbols 1-16 could represent a QPSK signal, while symbols 17-32 could represent a 256 QAM signal. Nearby receivers can receive all of the symbols, while distant receivers receive half of the symbols. In constructing a input symbol sequence from a random input data block and a transmission matrix created by selecting random terms, the transmitted signal will have a probability distribution function resembling a two-dimensional Gaussian random variable. This implies that the transmitted signal will have a high crest factor with infrequently occurring peaks. With the ability to abandon arbitrary symbols, a transmitter could clip the high-powered symbols below their natural crest factors, and a receiver could detect the high amplitude symbols that were clipped and flag the clipped symbols for removal. This creates a type of peak limiting without an attendant distortion penalty. One could also envision a matrix that implemented a type of frequency hopping spread spectrum modulation implemented in the selection of a single C matrix where both the


instantaneous frequencies and power levels could be established by setting the coefficients of the C matrix. Another technique that can be deployed on two-way channels is to download a set of transmission matrix coefficients for a transmitter to use on a return channel for future transmissions. This gives the ability to customize the return signal for the channel impairments that are being experienced at the time. In general, matrices can be constructed to fulfill particular needs, which may be a highest channel capacity for a given transmit power, detection avoidance, interference avoidance, jamming avoidance, or security from hackers In general, for highest channel capacity in a channel that has a uniform signal to noise ratio, a matrix that uses the entire available spectrum all of the available time will produce the highest data throughput.

Figure 7 A Diagram Showing Symbol Interleaving on a Transmitted Symbol Sequence-HCs with the same shape belong to the same matrix Practical Considerations If the inverse matrix needs to be recomputed for every block, depending on which symbols were to be discarded, one might assume that there would be a significant computational burden. This is particularly true if the matrix dimensions are large. One way around the problem would be to store all possible inverse matrices, but again, a computational problem would be replaced by a storage problem. One solution to this problem is to use a relatively small inverse matrix, such an 8 x 8 or 16 x 16. A large transmit symbol sequence F could be comprised of interleaved terms from many small matrices, providing protection from burst errors in the time domain or frequency selective fades in the frequency domain. Figure 7 is a diagram showing interleaved symbols, where the interleaving depth is 5 symbols. An “A” transmit symbol sequence is comprised of symbols A1, A2, A3, A4, A5, A6, … . , while a “B” transmit symbol sequence is comprised of symbols B1, B2, B3, B4, B5, B6, … . After equalization, the A symbols and B symbols are grouped together


for multiplication by a recovery matrix. The symbols may be either in the time or the frequency domains, depending on whether TDIMM or FDIMM is being employed. While a distinction was made between frequency domain inverse matrix modulation and time domain inverse matrix modulation, they really must be viewed as two sides of the same coin, since the transmission matrix C can include both a spreading function and a time-to-frequency transform. Likewise, a guard interval can benefit both FDIMM and TDIMM transmit symbol sequences. Selection of Matrix Values One technique that can be used to construct a C transmission matrix is to generate a set of random numbers with a zero mean. The zero mean characteristic would prevent the creation of a large DC (zero frequency) term, which can be difficult to transmit on RF channels. If FDIMM is used, the DC term would generate an undesirable impulse in the time domain transmission. When choosing a matrix for FDIMM, keep in mind that after the IFFT is performed you want a uniform transmit power as a function of time to prevent a high crest factor and avoid clipping the power amplifier. The condition number of a matrix that can be used to determine if a matrix is close to being singular from its condition number. A return value of 1.0 is ideal, while a very large return value is undesirable. A value of 1.0 is returned for orthogonal signal sets. Ref[6] describes how the condition number of a matrix is determined, and also lists many C-language algorithms for efficient and accurate matrix processing. As mentioned above, if two matrices are with the same rank multiplied together, the two matrices can be combined, resulting in just one multiply. If one matrix is used for spreading, and another is performing the IFFT operation, both matrices can be combined. Likewise, if there is a need to equalize the channel because of multipath distortion, an equalization matrix can also be included in the recovery matrix. Performance in Noise Extensive modeling of random noise has not yet been performed. However, a quick simulation revealed that the signal to noise ratio in the recovered input symbol sequence was the same as the signal to noise ratio on the RF channel. Security One of the big problems currently with wireless communications is unintended listeners. Because most transmission formats are standardized, attackers have an easier time deciphering transmissions. When using non-orthogonal matrices and their inverses, the possible alphabet of non-orthogonal codes that can be used is much greater than the possible alphabet of orthogonal codes. This is because the row vectors are not constrained to be orthogonal signal sets. With inverse matrix modulation, it is also possible to change or hop C matrices frequently. Establishing the Channel Response


Channel response, H(f), may be established in a conventional manner by sending a reference (or training) signal as often as needed. The channel response is needed to perform equalization. The channel response, along with an estimation of the noise floor, can determine which HCs may be combined, which should be left unused, and which should be used without combining. A better method of channel characterization is to send a first burst of data followed by a second burst of data with a frequency domain reciprocal relationship to the first burst. This technique is called frequency domain reciprocal modulation (FDRM). This frequency domain modulation technique provides a method to characterize a channel, while at the same time transferring data with a high immunity to dynamic multipath. FDRM is discussed in Ref[7]. Finding Corrupt Symbols As mentioned above, a number of terms created by an over-determined C matrix can be dropped when making an inverse recover matrix. With FDIMM encountering multipath distortion, the terms to drop can be readily determined by HC’s signal to noise ratios. With TDIMM encountering random impulsive bursts of noise, the problem is not as easy, since the corrupt terms cannot be identified in advance. If the noise burst is severe, corrupt symbols can be easily detected when the instantaneous power level crosses a high threshold level. If the noise burst is not severe, one method to eliminate terms is to use small matrices, small signal constellations, and match patterns on the equalized F vector for valid transmission patterns. Better techniques will be developed. One technique used by statisticians to find erred points is called “leverage”. An Example of FDIMM Modulation Implemented in Hardware Appendix A contains an example of FDIMM that was implemented in hardware. Composite Transmissions from Several Users It is possible to have several users transmit simultaneously with each user using a subset of the total number of rows in the transmission matrix. All of rows that an individual user was not using would have zero-valued elements. However, strict control still needs to be maintained between users on timing and transmit power for this multiple access system to work. Summary of Advantages of Inverse Matrix Modulation Compared to OFDM, over-determined FDIMM has the ability to combine weak HCs, rather than discarding them. Combined HCs have a better signal to noise ratio. For wireless applications, this combining ability allows portions of the frequency band with low signal to noise ratios to be utilized, thereby improving data throughput. The HCs may be interleaved in frequency to survive deep fades that degrade several adjacent HCs. Compared to DSSS, over-determined TDIMM has the ability to survive noise bursts without a penalty, provided that the noise burst is shorter than the number of over-


determined symbols. A TDIMM transmission can survive longer noise bursts if the symbols are interleaved in the time domain. Patent pending. Conclusion In this paper a comparison was made between direct sequence spread spectrum and orthogonal frequency division multiplexing. It was shown that both modulation techniques can be viewed as matrix multiplication techniques that have orthogonal rows, and are received (demodulated) by a transpose of the matrix that was used at the transmitter. If the inverse of the transmission matrix is used as a recovery matrix, the requirement for orthogonality between rows is removed, provided that an inverse matrix exists. If an over-determined matrix is used at the transmitter, corrupted terms can be discarded at the receiver, and an inverse matrix is selected to match the discarded terms. Relative freedom in choosing the transmission matrix enables many novel and unique methods to allow data to occupy the time, frequency, and power limits of a channel.


Appendix A An Example of a FDIMM Implementation using Hardware

To better illustrate the technique of frequency domain inverse matrix modulation (FDIMM) a hardware implementation is provided to show one technique for implementing the invention. This example uses an input symbol sequence (“E”) of 9 symbols and a transmit symbol sequence (“F”) of 10 symbols, so the system is over-determined by one symbol. Figure A1 is a block diagram showing a PROM used to generate the signal, an 8-bit digital to analog converter, a low pass filter, a signal path without impairments, and a digital signal acquisition device. A 10 MHz clock / counter increments the PROM address and clocks a new value into the digital to analog converter every 100 microseconds. A digital oscilloscope is used as the signal acquisition device to capture the signal with a sampling rate of 10 Msamples per second, matching the PROM’s clock. 2500 time samples are captured and downloaded into a personal computer (PC) over a GPIB (general purpose interface bus). The PC processes the burst to recover 54 input symbol sequences (“E”) with 9 symbols each. 27 symbol sequences are located in the upper sideband and 27 are located in the lower sideband. The single repetitive burst transmission is generated as follows:

1. Pseudo random data is created as a series of 1’s and 0’s. 2. The data is mapped into two 9 rows x 54 columns data matrices, one for real

values and one for imaginary values. Each row of the 9 x 54 matrix is an “E” input symbol sequence.

3. The matrix of Figure A2 is used as a spreading transmission matrix. The condition number of this matrix is 4.18, so it is not singular and the rows are not orthogonal.

4. Each of the 9 x 54 data matrices is multiplied by same spreading matrix to create two 10 rows x 54 column matrices. Each row of the resulting 10 x 54 matrix is an “F” transmit symbol sequence.

5. The transmit symbol sequences are mapped into a 2048 point data structure to perform an IFFT. Every 11th term is made into a pilot signal. Figure A3 is spectral plot showing the mapping of pilots and data HC’s to put the HC’s energy at the desired frequencies. Each “F” transmit symbol sequence is located between two pilots. Zeroes are inserted for frequencies where no energy is desired. Note that HC number 2047 is adjacent in frequency to HC number 0, which is a pilot.

6. A 2048 point IFFT is performed. This gives 243 active complex harmonic carriers in the lower sideband and 243 complex harmonic carriers in the upper sideband, with each HC transporting 2 bits of data. 55 pilots are interspersed to assist with equalization.

7. A guard interval of 128 samples is added giving a total of 2176 samples to be transmitted in the burst.

8. The sequence is up-converted from DC to a center frequency of 2.5 MHz with a bandwidth from 1 to 4 MHz. Up-conversion is done by multiplying the real and imaginary time domain terms by cosine and sine wave functions respectively.

9. The sequence is scaled into the 256 levels (8 bit output) of the PROM.


10. The transmitted signal is burned into a PROM, clocked out through a digital to analog converter, lowpass filtered, sent through the signal path, and captured on a digital oscilloscope. The clock rate is 10 Msamples per second for both the PROM and the digital oscilloscope.

11. The sequence is transmitted and appears in Figure A4, which is a time domain plot.

The captured time domain transmission is captured and sent to a PC for processing as follows:

12. The time sequence is converted back to DC . 13. The time sequence is converted back into the frequency domain using a FFT. 14. The pilots are used to perform a frequency domain equalization using

interpolation. 15. The de-ghosted data is mapped back into real and imaginary 10 row x 54 column

matrices. 16. The pseudo-inverse matrix of Figure A4 is used to multiply each row in the 10 x

54 matrix to make a 9 row x 54 column matrix. Each row of the 9 x 54 matrix is a recovered input symbol sequence.

17. The terms of the de-spread matrix are displayed in the X-Y plot of Figure A5. The clusters of points are spread somewhat due to noise in the setup, as well as a quantizing operations for the PROM, and the digital oscilloscope. If implementation errors are eliminated by demodulation from an ideal unimpaired time-trace file, the constellation spread goes to virtually zero. Instead of using a pseudo-inverse matrix, any of the columns in the 10 x 54 matrix could have been eliminated, and a K matrix, which depended on which row was eliminated, could be used for the de-spreading. Note that interleaving of the transmit symbol sequences was not done, which would have been useful for combating frequency selective fades. That is, all of the 10 terms in a single F transmit symbol sequence were placed between two adjacent pilots. Frequency selective fades typically “knock-out” several adjacent HCs.


Figure 1A. A Block Diagram of the Hardware to Generate, Transmit, Receive, and Process a FDIMM Burst Transmission.

0 1 -1 1 0 -1 -1 1 0 1 1 0 1 -1 1 0 -1 -1 1 0 0 1 0 1 -1 1 0 -1 -1 1 1 0 1 0 1 -1 1 0 -1 -1 -1 1 0 1 0 1 -1 1 0 -1 -1 -1 1 0 1 0 1 -1 1 0 0 -1 -1 1 0 1 0 1 -1 1 1 0 -1 -1 1 0 1 0 1 -1 -1 1 0 -1 -1 1 0 1 0 1

Figure A2. A Transmission Spreading Matrix (“C”) Used In This Example


Figure A3. The spectral plot for this example showing pilot HCs , data HCs, and zero energy frequencies.

Transmitted FDIMM Burst

-150

-100

-50

0

50

100

150

0 500 1000 1500 2000 2500

Time

Vol

tage

Figure A4. The FDIMM burst transmitted burst signal as voltage vs. time


-0.0223 0.1967 0.0940 0.0949 -0.0916 -0.2323 0.1121 0.1239 -0.0769 0.2352 0.0342 0.2752 0.1741 0.1372 0.0468 -0.1975 0.1978 0.1988 -0.0693 0.1712 -0.0393 0.2444 0.0571 0.0989 -0.0211 -0.2651 0.1489 0.1533 -0.0859 0.1656 0.0048 0.1538 0.1502 0.0565 -0.0131 -0.2467 0.2323 0.2550 0.0405 0.2518 0.1478 0.2881 0.2405 0.1822 0.0869 -0.2164 0.1760 0.1781 -0.0339 0.2033 0.0207 0.2508 0.1578 0.1109 0.0186 -0.2433 0.1488 0.1796 -0.0999 0.2160 -0.0149 0.2294 0.1459 0.0843 -0.0077 -0.2647 0.1704 0.0918 -0.0478 0.1717 -0.0259 0.2298 0.1155 0.0403 -0.0455 -0.2363 0.0324 0.1637 -0.1183 0.1494 -0.0299 0.2601 0.1633 0.1115 0.0409 -0.2223 0.1863 0.1842 -0.0367 0.2238 Figure A5. The De-Spreading (Recovery) “K” Matrix which is the Pseudo-Inverse of the Spreading Matrix of Figure A2.

Constellation Plot for FDIMM

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-2 -1 0 1 2

Real Value Voltage

Imag

inar

y V

alue

Vol

tage

Figure A6. The constellation plot after de-spreading with the recovery “K” matrix


References [1] Principles of Communications Systems 2nd ed. Herbert Taub and Donald Schilling, Ch. 17 [2] Digital Communication 2nd ed. Kulwer Academic Publisher by Edward A. Lee and David G. Messerschmitt pp. 256-260 [3] Mastering Matlab 6: a Comprehensive Tutorial and Reference by Duane Hanselman and Bruce Lifflefield, Prentice Hall 2001. p234 [4] Digital Communication 2nd ed. Kulwer Academic Publisher by Edward A. Lee and David G. Messerschmitt pp. 489-510 [6] Numerical Recipies in C The Art of Scientific Computing, 2nd ed. Cambridge University Press by W. Press, S. Teukolsky, W. Vetterling, and B. Flannery p. 61 [7] 1999 NAB Broadcast Engineering Conference Proceedings, Frequency Domain Reciprocal Modulation (FDRM) for Bandwidth-Efficient Data Transmission Over Channels with Dynamic Multipath, by Thomas H. Williams pp. 71-82 [8] Digital Communications Fundamentals and Applications, Prentice Hall, by Bernard Sklar pp.120-123 [9] Fundamentals of Digital Image Processing , Prentice Hall, by Anil K. Jain p.26 Important terms used in this paper: spreading matrix input symbol sequence E two-dimensional transmission matrix C transmit symbol sequence F transmit symbols received data sequence F inverse recovery matrix K redundant transmit symbols low speed data source (DSSS)

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