Part 1
Commuting Operator Modems

Modulation and demodulation are mappings that can be denoted by the operators [M] and [D]. To recover the data correctly, [D][M] = [I] = IDENTITY for any modem. (Note that operators work from right to left, and an overall delay is still considered IDENTITY). If [M][D] = [D][M] over the bandwidth of the channel, then the operators are said to commute. For a perfect Commuting Operator Modem (COM), a bandlimited Gaussian analog input signal, g, can be demodulated and re-modulated at the transmitter since [M][D]g = g. This proves that the modulator of a perfect COM can transmit a bandlimited Guassian, which is a prerequisite for achieving communication at Shannon's Limit.

To actually transmit a Guassian there must be no loss of entropy by [M], since nothing has more entropy than a Guassian (entropy is a measure of randomness). If [M] is a lossless Wavelet Filter-Bank, Shannon showed {Bell Tech, Journal V27, July & Oct 1948} that the entropy-power lost in filters of this type will be proportional to the width (rolloff) of the passband to stopband transition of the two outermost sub-bands. This rolloff transition can be arbitrarily small for Wavelet filters, so INTRA is an optimal transmitter and INTRA receivers can also be optimal as discussed later.

Coordinate Representations

The commuting operators for INTRA can be constructed as digital matrix operators, since any bandlimited signal can be described by digital samples via the Sampling Theorem. Commuting matrix operators can be interpreted as geometric rotations of a vector in some coordinate system. Therefore, Information can be thought of as a vector which can be projected onto DATA or SIGNAL coordinate representations (i.e. axes) by a "rotation" of the axes. An INTRA modem performs an INformation TRAnsformation in this way, hence the name.

Customarily one begins with the DATA and heuristically finds a modulation method that transforms DATA into a bandlimited SIGNAL. In the top-down approach used here, a set of M samples into the D/A of a baseband INTRA modulator define an M-Dimensional vector strictly defined in time and in bandwidth. (This dual constraint on time and bandwidth is possible with a Wavelet Transform). The Synthesizer filter bank transforms the DATA representation of the Information vector into the SIGNAL representation.

Filter sub-bands ---which are the vector coordinates in the DATA representation--- are not actively used for data if they are above or below the channel bandwidth. To design the modem, the sample-rate, dimensionality and rolloff for the filter bank are selected to match the active sub-bands to the channel bandwidth. The number of bits per symbol in each coordinate can be chosen to suit the Signal to Noise Ratio (SNR) in that sub-band.

From the earlier discussions on entropy, the optimum rotation operator is a baseband filter bank---that is, an optimum modem can operate without a carrier. It happens that baseband INTRA can easily function at up to 100 MHz with today's components, and this can serve as the final signal as in Figure 1. With some added complexity, baseband at many GHz is possible owing to the sub-rate polyphase structure (see below) of an INTRA modem. When a carrier is desired, INTRA can be an all digital first IF-stage, for example. INTRA can be used for data over AM or FM carriers and for spread spectrum and wireless communication.

A 2-Dimensional Case

The INTRA requirement for commuting rotation operators is satisfied by Wavelet Theory. A basic Quadrature Mirror Filter (QMF) is shown in Figure 2a and it is the basic building block for Wavelet Transformations and Information Transformations.

Figure 2a

 

Figure 2b

Incoming signals are divided by the Analyzer into high-pass and low-pass branches and then down-sampled by 2, which discards every other sample. At this point, two input samples (x,y) are transformed into two bandlimited samples (a,b), one in each branch. It will soon be apparent that this transformation can be called a rotation. The sequences of (x,y) and (a,b) are both defined in vector spaces of the same dimensions because of the downsampling.

As noted in the introduction, the QMF is a subset of the general case. In fact, the sub-bands don't have to be equal ("Quadrature" means _ of sample rate which is _ of the bandwidth) so long as the information in the input function X(n-1) that is not in the Scaling function V(N) appears in the Residual Function W(N); that is, W(N) = X(n-1) - V(N).

The Synthesizer, or second half of the QMF pair, up-samples each incoming branch by inserting a zero sample. Then the branches are filtered and summed to form the reconstructed signal samples. By designing the filters to obey the equations of a geometric rotation in 2-Dimensions, the reconstructed samples exactly match the incoming Signal with only an overall delay. Evidently, the Synthesizer rotates a sequence of 2-Dimensional vectors from one representation to another, and the Analyzer performs the exact counter-rotation.

When the filters have Finite Impulse Response (FIR), the filter coefficients will turn out to be samples of orthogonal functions. A few IIR designs are also possible for QMFs. Computer searches and other constructions can yield QMF designs of practical use without utilizing geometric rotations---these designs are called "near perfect", since the reconstructed sequence is not an exact match to the input sequence. The commuting rotation operator technique can give an exact match even though there is roundoff error.

Rotation Operators

To make a pair of commuting matrix operators for 2-Dimensional vectors one can start with the geometrical coordinate rotation operator [R] and its inverse [C], the counter- rotation through an angle A.

[R] = cos(A) -sin(A)
[C] = cos(A) sin(A)
sin(A) cos(A) -sin(A) cos(A)

For a rotation angle A = 45 degrees [R] is proportional to the familiar wavelet (Walsh/Haar) matrix

[W] = 1 -1
1 1

[W] is a powerful example because it clearly shows that an INTRA modem can code the data in a band-limited way using the same orthogonal function as the "spreading chip-code" for Direct-Sequence Spread-Spectrum Code Division Multiplexing (DS-SS CDMA). This helps explain the interference rejection properties of the correlation receiver in Figure 1 --- obtained by INTRA without spectrum spreading. By combining spread spectrum with INTRA one obtains a "Twice-Coded" modem. (today's DS-SS normally uses a BPSK or QPSK modulator). Incorporating the encrypted modulation explained in PART 2 in a Twice-Coded modem eliminates the known vulnerability of DS-SS in battlefield communications.

Recalling the earlier operator definitions, the choice of [M]=[R] and [D]=[C] for a commuting operator modem (COM), is equivalent to modulation with a one-section, 4-port lattice filter. As is well known, cascading filters ---with different rotation angles--- improves the filter response, if so desired. The following construction for a commuting operator modem is an adaptation to modems of the filter design methods found in the works of Vaidyanathan et al {IEEE Trans ASSP V37-7 Jul 1989} and also of Vitterli et al. {also in IEEE Trans ASSP V37-7 Jul 1989}

To simplify the notation, let [j] denote rotation thru angle Aj( i.e. j is a subscript of A), and define the inverse as [j-1].

Thus [j][j ] = [j ][j] = [I]. Two other commuting matrices are needed; namely,

[Z] = 1 0 and [Z^] = Z-1 0
0 Z-1 0 1

The delay matrices commute except for a scalar delay, since [Z][Z^] = [Z^][Z] = Z-1[I], where Z-1 is the z-transform operator for a one-sample time-delay. Then if [j] takes on the angle subscripts 1,2,3...n , a COM can be constructed by a cascade of these operators since

[M] = [1][Z][2][Z][3]...[n] commutes with [D] = [n ]...[3 ][Z^][2 ][Z^][1 ]

Thus [M] and [D], the modulation and demodulation, commute with an overall delay.

Since a commuting rotation in 3-Dimensions can be decomposed into an ordered sequence of rotations in 2-Dimensions and so forth, the foregoing applies to modems with vectors of any dimensionality. Thus higher dimensions can be derived from the two dimensional case. Furthermore, there can be more than one set of commuting operators for a given dimension. This results in the concept of "Master and Slave sets" of wavelets, which are orthogonal within each set and somewhat orthogonal between sets--- a property that can be exploited for full-duplex communications.

The foregoing construction is one of several ways to provide modem operators for INTRA. The resultant rotation matrices [M] and [D] are also known as polyphase filter matrices. Before defining polyphase matrices and their relation to filter response, some further definitions of historic significance can be made with the aid of the pictorial shorthand notation for a QMF given in Figure 2b and extended in Figure 3.

Transforms

In a Multi-Resolution Analysis (MRA) another QMF pair is placed in the low-pass branch of the first QMF and then a third QMF is placed in the low-pass branch of the second QMF and so on as in Figure 3a. The result is the canonical form of the Discrete Wavelet Transform (DWT). The original signal is projected onto orthogonal basis functions by the filters. Because of the down-sampling, the sample rate is halved in each stage. Figure 3a illustrates MRA in the DWT.

Modems based on Figure 3a might be advantageous for twisted pair and other media where there is a steep change in SNR at low frequencies and a gradual change at higher frequencies. Modems associated with the INTRA encryption technique (see PART 2 on Secure Communications) might also find the configuration in Figure 3a beneficial for digital encryption of voice, for example. INTRA modems can have unequal width sub-bands (including non-octave).

Figure 3a

 

Figure 3b

The Analyzer decomposes the signal as an expansion in orthogonal basis functions, but unlike the Fourier decomposition, the generating functions have finite support. That is, Wavelets do not extend to infinity like Sinusoids.

X(t) = Wavelet Expansion
X(t) = Fourier Expansion

Note that there is only one generator of Fourier functions; namely, the complex exponential , which extends to infinity. But there are many examples of Wavelet generators, , such as the Spline functions. And in both expansions the generator function's argument is shifted by k and scaled in frequency to create the basis functions. As shown, wavelets usually scale the frequency by powers of two to represent downsampling In this example, the expansion coefficients, , are now called the Discrete Wavelet Transform and are the projection of X onto the basis.

If an MRA is performed in the high-pass branches as well as the low-pass branches, then the result is M sub-bands all at the same down-sampled rate as shown in Figure 3b, which has been arranged as an 8-Dimensional modem. As already noted, a multi-rate modem based on Figure 3a is also useful. Rather than use 2-Dimensional QMFs, the M-Band bank in Figure 3b ---for any value of M--- can be more efficiently constructed from commuting M-dimensional rotations (using ordered sets of 2-D rotations). These M-band filter banks perform a convolutional rotation in M-Dimensions directly. The equivalent QMF has M bandpass filters with a sample rate change by a factor of M. Figure 4a is a shorthand notation for the 8-D rotation operator. Rotations are further defined in the section on Vector-Filters shown in Figure 4b.

The Polyphase Matrix

From the foregoing, a vector coordinate rotation is used to implement an INTRA modem. The rotation can be viewed as "vector-filtering" by factoring the polyphase matrix for a QMF filter bank. This terminology and its relation to hardware implementations for modems is briefly explained below so the modem designer can utilize the literature for multi-rate filter-banks. Polyphase matrices are well known in sub-band coded speech compression, although the vector-filter concept appears to be a new way of expressing the matrices that gives added insight for modem applications.

A polyphase matrix is a computational simplification of the filtering process due to the change in sampling rates in a QMF bank. Consider the Synthesizer in the right half of Figure 2a. Each FIR filter performs a weighted sum of its delayed input stream. But because of up-sampling, every other sample into the filter delay line is zero. That means that only half the tap weights ---the even numbered filter coefficients--- contribute to the even numbered output samples. Likewise, only the odd numbered weights contribute to the odd output samples. When computing output samples it is therefore possible to use half-length filters if each computes at half the rate of the output. This feature extends from a factor of _ to a factor of 1/M for an M-Dimensional (M-band) filter bank, and is responsible for the exceptionally low computational complexity of QMFs and INTRA.

Likewise, these observations are applied to the modem receiver by factoring the filter coefficients of the 2-Dimensional receiver's filters into even and odd powers of z. Thus the response of the receiver's high-pass Analyzer filter (upper right in Figure 2a) is

H1(z) = az0 + bz-1 + cz-2 + dz-3 + ...

= (az0 +cz-2 +...) + z-1 ( bz-1 +dz-3 + ...)

= h00 (z2 ) + z-1 h01 (z2 )

Since there are two Analyzer filters --- high-pass, H1, and low-pass, H0--- the pair of filters can be described by a vector H. Thus H(z)= [h(z )]d(z) , where

H= H0(z) d= 1 and [H]= h00(Z2) h01(Z2)
H1(z) z-1 h10(Z2) h11(Z2)

And d is called a delay vector. The more general case of an M-Dimensional filter bank can be written as

H=[h(zzM)]d where the transpose of d is z0 +z-1 +z-2 +z-3 + ... z-(M-1)

But Figure 2a also includes a change in sample-rate. Therefore, to complete the description of the INTRA modem, the upsampling and downsampling operators are denoted by [up] and [dn]. Then the modem receiver is [dn]H and the transmitter is G[up]. A simplification known as the "Noble Identities" in multi-rate filter theory gives [dn][h(zM)] = [h(z)][dn]. Thus [dn]H = [h(z)][dn]d = [h(z)][sp], where [dn]d makes a serial-to-parallel converter, [sp] positioned after the demodulator's A/D. Similar mathematics applies to the transmitter yielding a parallel-to-serial conversion prior to the D/A converter. In other words, the modem operates at the downsampled rate on non-overlapping frames of digital samples, which are the SIGNAL or DATA vectors described above.

The matrix [h(z)] is called the polyphase filter matrix. It is obviously a square matrix with each of its MxM matrix elements a filter, hjk(z). These sub-rate filters can be represented as the scalar product of a coefficient vector, vjk with a delay vector, z. That is, hjk= vjk* z , where z can have any storage length as needed for sharp band rolloff.

Of course there are two polyphase matrices, one for the transmitter and one for the receiver. Ideally the transmit and receive polyphase matrices for INTRA should commute with a delay, that is [h(z)][g(z)] = [g(z)][h(z)] = z [I]. In filter theory this result is described as a Perfect Reconstruction filter-bank and can be designed for any number of ports using the commuting lattice filter method described earlier. A Near-Perfect Reconstruction is often used for filter banks, wherein the Perfect design is subsequently computer optimized to improve the stopband attenuation or other filter design tradeoffs.

The Vector-Filter

By assembling all the polyphase terms with like powers in z, the polyphase filter matrix can be factored into the form of a vector-filter as shown below and in Figure 4b. That is,

[h]=[c0]z0+[c1]z -1+[c2]z-2+[c3 ]z-3+ ...[cn-1]z -(n-1)

Figure 4a

 

 

Figure 4b

Figure 4b looks like an ordinary transversal filter except that the tap-weights [ck] are MxM scalar coefficient matrices and the delay-line contains vectors --- the current (i.e. z0) and n-1 prior vector inputs. For the transmitter the input vectors are the DATA vectors, and for the receiver the input vectors are the SIGNAL vectors, i.e. contiguous frames of A/D samples. The number of tap matrices, n, can be 1 but is typically around 5 for a modem and depends on the desired filter response. The transmitter and receiver vector-filters are matched; that is, the coefficient matrices are time-reversed, so the design of one vector-filter determines the other. The rows and columns of the coefficient matrices are in fact segments of wavelet functions. The more taps, the longer the tails of the wavelet and the sharper the rolloff of each sub-band. Very little energy is in the tails.

The vector-filter concept highlights the duality of the matrix and Wavelet approaches to INTRA modem operation. In the transmitter, for instance, the M components of the output vector become M samples into the D/A converter. We see that these transmitted SIGNAL samples are computed from the current and past DATA vectors by vector addition after mapping with matrices [ck]. Thus vector-filters perform "rotations" and counter-rotations that preserve Information during the modem transmission. The vector filter also demonstrates the equivalence of Rotations versus the Synthesizer/Analyzer description.

The vector-filter modem can also be understood in terms of orthogonal functions; where every wavelet is orthogonal to any other wavelet that is shifted by any multiple of M samples. By superposition, a sequence of DATA vectors can be analyzed by examining a single symbol impulse on one DATA axis. As a single component of a DATA vector "impulse" proceeds thru the vector-filter's shift register on subsequent null symbols, it creates samples for segments of an entire wavelet for the impulse symbol. There are M samples per segment and the number of segments equals the number of tap-matrices, n. Since the overlapping wavelets for each symbol are orthogonal when the superposition principle is applied, there is no ISI or ACI for a complete modem system as described in the opening section.

The receiver's matrices contain the time-reversed wavelets so that the receiver's vector-filter computes the correlation between each orthogonal wavelet and the received signal to recover the DATA vectors. This is in fact the textbook form of an optimal MAP (Maximum Apriori Probability) receiver. Not only is INTRA an optimum receiver in this MAP sense, but it also has the following six very powerful features:

1. It can be self-equalizing by applying any adaptive equalization algorithm (i.e. LMS) to the vector filter matrices in the same manner as FSE.

2. Interference is suppressed, since the symbols are recovered by correlation.

3. FIR vector filtering (Tx or Rx) can be done in the analog domain with SAWs or CCDs, so no D/A or A/D converter or DSP is required at high rates.

4. Fractional bits per vector coordinate can be assigned according to SNR.

5. Since vector-filtering is a convolutional modulation, a receiver Viterbi Algorithm might provide error-correction without sending parity at the transmitter. (This is a potential theoretical breakthrough since conventional Trellis Coded 6.Modulation (TCM) sends parity, which wastes transmitted entropy and therefore lowers the potential data rate near the Shannon Limit.)

6. Vector-filters can be used for compression-less networking, a new concept described in PART 2.