What is Transpositional Modulation? How does it work?

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Transpositional Modulation (TM) is a patented radio frequency waveform technology invented by Richard Gerdes for TM Technologies, Inc. (TMT).

TM offers dramatic bandwidth increases for existing wireless and wired networks. TM accomplishes this by enabling the simultaneous transmission of two or more distinct data paths on a single carrier signal. This effectively doubles the data rate.

This data rate increase is implemented transparently (with insignificant noise effects).

All types of RF modulation modify an electromagnetic wave (radio, television, WiFi, light, etc.) to carry information. Current methods of modulation change the amplitude (strength) of a wave (AM), its frequency (FM), or phase (PM).


TM is fundamentally different from those basic methods and from the various hybrid modulations, coding and data compression techniques that have been developed to increase the amount of information that a single carrier wave can carry (QAM etc).

Once information has modulated (encoded) a carrier wave, and has been transmitted, the original information must be extracted at the receiving end in a process known as demodulation.

Demodulation techniques must carefully match the transmitted signal to reduce noise or data errors.

How does TM work?

Transpositional Modulation encodes information into an electromagnetic wave by a process that creates new inflections in the waveform.

In the mathematical representation of electromagnetic waves, an inflection point is an infinitely small point of time when a waveform changes from convex (like a cup with the top up) to concave (top down).



The inflection point is also the point in time at which the voltage of the wave is zero. In the waveform represented above, there is no change in the cycles which means the wave is unmodulated and no information is present.

All values above represent positive voltage, those below, negative.

Most diagrams of electromagnetic waves show the inflection point occurring midway between the wave crest (most positive amplitude) and the trough (most negative).

However, that holds only for idealized waves.

Further, one of the preliminary steps in the TM process works by creating new inflection points then using those new points as the backbone for the modulation process.

Such a line for a TM modulated wave would be very irregular.

The diagram, below, may offer a valuable, familiar point of reference for people accustomed to seeing the familiar horizontal line in standard sine waveform drawings.


Indeed, this waveform illustrates an interim modulation step in TM. However, if a wave in this form were to be transmitted directly, the result would be highly unsatisfactory.

Converting this to a usable form explains the reasons for the follow-on steps described below.

The TM modulation, step-by-step

The diagrams below show:

  • a carrier wave (with quarter cycles numbered 1-4 for cycles A-C),
  • a modulating waveform and, finally,
  • the modulating wave superimposed on the carrier wave.



The frequency of a carrier wave is also referred to as the “fundamental frequency” of a particular system.

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In this case, the modulation information is contained in a quantized analog signal having periods of a value representing a digital number. The period of constant value is to enable the process described below.

As with all modulation schemes, specialized electronic circuitry combines the two waves as seen below.

The inflections are clearly shown in the diagram below to illustrate their modulated positions.



It must be noted that transpositional modulation inflections are not points as represented on traditional waveform diagrams. Instead, TM inflections are trigonometric functions that are determined by the processes described below.

The TM modulation value encodes data according to the position of the inflection along the slopes of the sinusoidal waveform of the carrier signal.

Each quarter cycle (QC) piece is unique and represents a mathematical value which is determined by the information encoded.

There are a number of details to discuss in this waveform, presented here with the new inflection points marked and showing the changes in length of the four quarter-cycle segments of each cycle.

Compare the unmodulated waveform to the modulated one to see how the inflection points have been changed and the quarter cycles have been modified.

If we color each cycle of the carrier wave as illustrated below, we can see how the modulation process has cut each wave into four pieces and re-assembled them in a different order and shape.





Figure 5 shows how the QCs are transposed just like the inflections in the original carrier wave. These transpositions underpin the name: Transpositional Modulation.

The vital details

Vital details of TM are revealed when we take a closer look at the modulated wave (below).


This diagram above omits the familiar horizontal timeline that normally passes through the infection points of AM, FM or PM waves. That line has been left out of this diagram so that the following points can be more clearly observed:

(1) The first carrier cycle (a) shows an inflection midway from the most negative value of the carrier to the most positive value.

This pattern is repeated on the second half of the wave. This value is arbitrarily called a zero TM modulation value.

(2) Wave cycle (b) shows a different location of the inflection. This one is close to the most negative value of the carrier cycle.

In addition, the second half cycle of (b) shows a repeat which locates the inflection close to the positive carrier value.

The positions of inflections such as these (and the one mentioned in (3) below) are precisely determined by mathematical functions contained in the TM modulator system.

(3) Cycle (c) shows a pattern that is similar to (a) but not exactly the same. That means the modulation has encoded different information in that cycle.

(4) Figure 4 also shows that the negative and positive peaks of the TM modulated signal have not changed with the modulation in amplitude or time.

The absence of any variation in the crest-to-trough voltage of the wave cycles (amplitude) makes it transparent to AM demodulators.

Also, there is no shift in the time position of the negative or positive crests and troughs. That makes it transparent to FM and PM demodulators.

(5) A final characteristic is vital to avoid distortion in a Transpositionally Modulated signal: the area under the curve of each cycle of the modulated signal is constant.

Next steps and details

In an intermediate process not diagrammed above,(for the sake of simplicity and concept clarity), the modulation process begins by generating half-waveforms at twice the carrier frequency. Those then become the quarter cycles for the fundamental waveform construction diagrammed above.

The result of the modulation process generates a modulated third harmonic (a wave with 3 times the frequency as the carrier).

Harmonics are often unwanted side effects. However, for TM, they have become essential elements in the modulation process.

Carrier frequency restored

Once the quarter cycles are stitched back together, they form a fundamental carrier signal.

The resulting third harmonic is frequency shifted back to the original carrier frequency by mixing (heterodyning) with the second harmonic which was created during the initial modulation process.

This illustrates that the initial frequency doubling is a technique which makes it efficient to produce a new modulated carrier wave. This is because heterodyning a second  harmonic (the doubled wave) with the third harmonic creates the transmittable carrier wave fundamental frequency.

Once all of these steps are completed, the Transpositionally Modulated wave is broadcast.

Because of the way transpositional modulation encodes information in a wave, the final signal that is broadcast displays only minute differences from the pre-TM modulation. Those differences are invisible to all other forms of modulation. This characteristic is key to allowing TM signals to be broadcast on the same carrier wave as an existing modulation.

Many paths to TM

TM modulation can be achieved by a variety of methods and circuits, many of which are described in the main patent filed by inventor Richard Gerdes.

The Quarter Cycle method described above can be implemented in a variety of different ways including a Look Up Table (LUT) based design.

In addition, the TM process can proceed directly from the initial combination of the carrier and modulating signal all the way to the creation of the third harmonic using advanced DSP (Digital Signal Processing) circuity and a math-based algorithm embedded in a powerful microprocessor.

The diagram, below, from the main patent, outlines two possible methods.

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