Article

Superposition

by Dr. Howard Johnson. First printed in EDN magazine, October 7, 2010

"Imagine a soccer ball," I said to my friend, Breathe. "When you view the ball as a dynamic mechanical system, your kick is a system input. How far the ball goes is a system output."

"The harder I kick, the farther it goes. Is that what you mean?" asked Breathe.

"Right," I replied. "Let's quantify that relation. Suppose that you kick the ball and it goes 50 yards. Then I kick it, and it goes only 40 yards. If we both kick the ball at the same exact moment, what happens?"

Breathe paused. "If we avoid kicking each other and this is not one of your trick questions, then it should go a lot farther than 50 yards, shouldn't it?" he asked.

"Exactly—much farther," I answered. "In fact, if we ignore air resistance and some other technical factors, I expect it should go about 90 yards. When we superimpose our two inputs, a perfect soccer-ball system superimposes its outputs, creating one big output equal to the sum of the two individual outputs it otherwise would have created. That property is called superposition."

"I still have some questions," Breathe said. "Why can't you play bass and I play flute through the same loudspeaker?"

"If the loudspeaker works properly— and most do at low volumes—it clearly reproduces both sounds," I said. "That's the key idea behind superposition. When superposition holds, you can't tell the difference between two sounds played in consolidated form through a single speaker or the same two sounds played independently through two separate speakers placed side by side. What goes wrong in a practical system is the processing of high-volume signals."

"Like what happens when you turn up your bass until it distorts?" he asked.

"Especially then," I replied. "A bass waveform at high volumes can drive the speaker hard against its end stops. That technique results in a ratty sound, called overdrive, that some people like. A flute superimposed onto a speaker experiencing bass overdrive comes out horribly distorted."

"Give me a concrete example of a system that does obey superposition," Breathe ordered.

"Try the zero function. No matter the input, it puts out zero. Of course, I'm being trivial," I said.

"You are never trivial," he replied. "It seems to me that if I pass the signal X plus Y through your function, the answer comes out zero. By definition, that result equals the result from X, which is zero, plus the result from Y, another zero. Technically, the output signals superimpose, but, if they are all just zero, what's the point?"

"Try a linear scaling function," I responded. "For input X, let the output be KX for some arbitrary constant, K. If you add two signals into the input, you get K times X plus Y. That result is the same as adding together the two results, KX and KY."

"That concept sounds familiar," said Breathe. "Didn't I learn that in fifth grade?"

"Yes. It's the distributive property of multiplication over addition," I said. "Here's another example. Let's say the input to your system is an audio signal—a voltage waveform, X, which varies with time. You can multiply that input signal by any other known function, K, even if K also varies with time, and the result still obeys superposition. That approach works because, at each point in time, the process operates just like a simple scaling function, only the scaling factor, K, changes with time. An RF mixer, for example, might multiply its input by a square-wave carrier. The output appears all chopped up, with many sudden, discontinuous shifts, but, overall, the process still obeys superposition."

"Why does it matter?" Breathe asked.

"Superposition opens the door to the method of analysis by decomposition," I answered. "If I can decompose a complete waveform into a sum of many small parts and if I can calculate the response to each little part, then, in a superimposing system, I can sum all the partial responses to understand how the whole system behaves. This process will make more sense after I describe to you the concept of time invariance."

Chris "Breathe" Frue is a talented musician and audio technician who wants to learn more about equalizers, a subject pertaining to both audio and high-speed digital systems.