Article

Time Invariance

by Dr. Howard Johnson. First printed in EDN magazine, November 4, 2010

"According to the Western science you value so highly, time itself eventually comes to an end," argued my friend Breathe. "Nothing can remain invariant for all time."

"Maybe it would help to look at a circuit that is not time-invariant," I replied. "Let's say that the power-supply ripple-filter capacitors in an old tube amplifier were to dry out and start to fail. The dc-power voltage available to the tubes would develop large, 60-cycle undulations. You would probably hear a lot of hum in the output."

"My old keyboard amp did that," said Breathe.

I continued, "Besides the hum, the gain of the amplifier actually varies along with the 60-cycle variations in power voltage, causing a high-speed tremolo effect. A quick experiment to measure the gain on just 10 cycles of a 4000-Hz burst would return different results depending on precisely where within the 60-cycle hum pattern the experiment began.

"In contrast, a time-invariant system behaves the same regardless of when, within a reasonable span of time, you begin your experiment. Even if the time-invariant system already produces a certain amount of inherent delay, if you delay the input by a certain time, you delay the output by that much more."

"Give me an example," Breathe implored.

I said, "A hard-limiting, or clipping, function is perfectly time-invariant. It distorts your signal through an instantaneous process that happens point by point. Time doesn't matter."

"So, a good amplifier with no hum would be time-invariant?" he asked.

"Pretty much," I answered. "Two key properties we look for in any audio system are superposition and time invariance (Reference 1). Given those properties, the system can process any combination of signals, at any time, just as well as an individual signal."

Breathe knit his brow. "You said 'just as well as.' That sounds like an important caveat."

"It is," I agreed. "What processes do you know so far that obey superposition and time invariance?"

"Well, the clipping function you just mentioned is time-invariant but does not obey superposition," he said. "A tremolo circuit probably obeys superposition but varies its gain with time. The only process I know that satisfies both properties is a simple scaling factor."

"One other process," I replied, "satisfies both properties: a time delay. Think about it. Scaling and delay, and linear combinations of different amounts of scaling and delay, are the only things you can do that obey both superposition and time invariance. These simple operations form the basis of almost all forms of equalization."

"Delay and scaling are such simple operations," said Breathe. "Isn't that restrictive?"

"It's wonderfully restrictive," I answered. "The requirements of time invariance, coupled with superposition, weed out a tremendous number of ineffective functions, leaving the rich array of possible system operations generally known as linear operations, or, more precisely, linear-time-invariant operations. For example, take signal x(t) and subtract from it a slightly delayed version of the same signal." On a paper napkin, I wrote y(t)=x(t)-x(t-Δt).

Staring at the napkin, Breathe said, "That looks like the definition of the time-derivative from calculus."

"Yes, it does," I said. "For very small values of Δt (change in time), it looks almost exactly the same. The time-derivative operator from calculus and the act of integration over time are both linear, time-invariant processes. A time-derivative is just a running difference between two values slightly separated in time, then scaled by 1/Δt. Integration over time is just a cumulative, running sum of previous values of a signal. The operations of scaling, delay, differentiation, and integration are the four basic processes that are both linear and time-invariant."

"If I remember my circuit theory correctly, a typical passive, linear circuit comprises a bunch of derivative and integration operations all mixed together," said Breathe. "Does this mean that all passive, linear electronic circuits are linear-time-invariant processes?"

"You got it!" I answered. "All passive, linear circuits do the 'same thing.' They just apply some linear-time-invariant process to the input signal. The big questions are, Which processes do they apply? And how do we characterize what those processes do?"

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.