Properties of Audio Summation
Bob McCarthy

   By Guest Contributor   Categories: Recording

Audio summation defined
Summation occurs when two or more audio signals are combined together and create a new waveform. The summation could be only a momentary event, in which case there is little we can do to manage it. As long as certain criteria are met, the summation will be stable over time and the outcome of the combination is predictable and manageable.

Summation criteria
Stable summation occurs only when the signals maintain a consistent level and phase relationship. This is not to say they must be matched in level and phase. They may be drastically mismatched. But whatever the relationship, it must be constant. The necessary conditions for stable summation are matched sources and overlapped duration at the summing junction.

FIGURE 2.1 Summation flow block. Top: The input signals may be simple or complex, but must have common origins. Middle: The sources must have originated from a related waveform for stable summation. Bottom: Unrelated sources will have a random summation relationship.

SOURCE MATCHING
Stable summation is evaluated on a frequency-specific basis (Fig. 2.1). Stable summation at a single frequency requires that the input signals have a fixed differential at that frequency. To expand this over the full frequency range requires that the input signals have a stable differential at all frequencies. The signals must, then, have related waveforms, i.e., they must come from the same original source waveform. In genetic terms, they must be the children of the same parental waveform. This occurs in our audio system in two forms: electrical copies and acoustical copies. These copies are found all over our audio world with mixing consoles (electrical), speaker arrays, and reflections (acoustical) being obvious examples. If we add perfect clones of the source signal, the summation will behave like simple mathematical addition. If the cloning is unsuccessful the copies combine with the original in complex form. The outcome can be predictable and stable, but not necessarily addition.
The monaural signal arriving from two displaced speakers will create a steady-state summation at a given point in the room. While this summation may have a different response at each frequency, the response is stable over time. For this reason such a response is measurable and has the possibility of being treatable with delay, equalization, and other alignment procedures. By contrast the summation of Beethoven’s Ninth Symphony with Black Sabbath’s Iron Man would produce an unstable summation, since it would only be by random chance that the two pieces would contain a moment of source matching. A hybrid case between these extremes is stereo. A stereo feed from two speakers will provide a semi-stable summation. The portions of the mix that appear in both the left and right channels would create a stable response, while those which appear in only one side would not. Because the two signals are not fully correlated, the summed response will change over time at a rate proportional to the degree of difference between the two channels. Therefore, the summation of such a system is not treatable by equalization since the combined frequency response is in constant flux. This unstable summation property of stereo music is easily demonstrated by a simple listening test: perform an electrical summation of the left and right signals at the console. The resulting unstable electrical summation is then reproduced in the sound system and can be confused for acoustical summation in the space.

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