8 of the 9 phase change vectors are shown below. The 9th case is the "no change" case. The solid, curved, arrows represent the phase change vectors. The dashed, straight, arrows represent an initial phase (arrow pointing horizontally to the right) and the phase, after the corresponding "phase change" has taken place.
The plot below shows the angle changes corresponding to each of the 9 different phase change step sizes. Except for the "no change" (0) case, the absolute value of the angle spends the first sixth of the symbol period increasing, then the next sixth of the symbol period decreasing. Thus, the "edge" of each phase change step takes a third of the symbol period. The remaining two thirds of the symbol period is spend at the "new" angle (indicated by the 0 value during the last two thirds of the symbol period, for each of the phase change step sizes).
The vertical grid lines are one symbol time apart. The sequence of phase change steps represented below is: -4, -3, -2, -1, 0, +1, +2, +3, and +4.
The angle, starting from 0, for all 9 possible phase change steps is shown below. The vertical grid lines are one symbol period apart. The plot shows what the angle would be, starting from 0, for phase change steps of: -4, -3, -2, -1, 0, +1, +2, +3, and +4 units.
Each edge takes one third of a symbol period, with the remaining two thirds of the symbol period being spent at the "new" angle.
The block diagram below shows how the phase change steps from the inner encoder are used to produce the corresponding modulated subcarrier. The operations represented by the diagram below are done for each of the 8 subcarriers.
The 1400 Hz signal is a dummy subcarrier. It is used internally, and is not present in the modulated signal.
The first step in the modulation process is to take the cos(modulation_angle + 1400Hz_angle). A typical spectrum of the signal at this point in the processing is shown below.
The next two steps in the modulation process are isolating the energy near 1400 Hz, due to the modulation, and translating this energy to the proper subcarrier frequency. The structure published by Weaver, to effect his 3rd method of single sideband generation, is used to accomplish these two steps. The corresponding typical spectrum after these two steps is shown below.
Fsc, in the diagram below, represents the frequency of the audio subcarrier, around which the modulated energy is placed. The nominal value of each of these audio subcarriers in the "pm7b" case is:
Below is a typical spectrum of the cos(modulation_angle + 1400Hz_angle) signal.
Below is a typical spectrum, after the lower frequency energy due to the modulation, has been isolated and translated to the desired subcarrier frequency.
Portions of the previous two plots are reproduced below, at an expanded scale. The red trace is before filtering and translation; the blue trace is after filtering and translation. Note that the details of the pattern near 1400 Hz, shown in red, are effectively moved to be near 805 Hz., as shown in blue.
After the modulation operations are done for all 8 subcarriers, and the modulated signals added together, the resultant spectrum looks like the typical example below.
In order to increase the RMS power, for a given peak power limit, the actual symbol period boundaries for the various subcarriers are staggered from each other in time. The lowest frequency subcarrier begins changing phase first. Successively higher frequency subcarriers begin changing phase at successively later times, so that the highest frequency subcarrier begins changing phase near the end of the the symbol period for the lowest frequency subcarrier.
Not show on the block diagram is the clipping that is done, after all 8 subcarriers are added together. This clipping is done to increase the RMS power of the signal, constrained by a peak power limit.
Below is a plot of the error in one of the demodulated subcarriers, due to clipping that preserves the middle 98% of the sample values and clips the most positive 1% and the most negative 1% of the sample values.
The scale ranges from -0.5 to + 0.5 phase change units. Thus, an error whose magnitude is 0.25 phase change units uses up half of the noise margin. The 0.25 phase change unit error takes the signal half way to the boundary between making the correct decision about which phase change actually occurred and the adjacent phase change.
What actually matters is the error at the time when a decision is made about the size of the phase change. Not all of the larger errors show below happen at a time that would influence a decision about which phase change occurred.
Below is a plot, expanded in time, near the relatively large error just after sample number 20,000 on the plot above. The vertical scale, on the plot below, ranges from -4.5 to +1 phase change units.
The red trace, on the plot below, shows the demodulated subcarrier, without clipping. The blue trace shows the same demodulated subcarrier, after clipping. The magenta trace shows the difference between the blue and red traces.
The important part of this plot is in the neighborhood of Sample Number 20,983. This is where there is a relatively large error, for a significant portion of a symbol period (vertical grid lines are spaced one symbol period apart), during the time when a decision about which phase change occurred should be made. Thus, external influences that would tend to make the phase change appear more negative than it was, would more easily make the blue (clipped) trace go below -1.5 (and thus be received as -2) than they would the red (unclipped) trace.
On the Australia to United States test, a clipping level that kept the middle 97% of the sample values intact was used.
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