As I advanced my cart through the check-out line at Walmart, the small child behind me wandered into the impulse zone filled with candy and pop magazines. At once, he commenced wailing and pleading to his mother. Apparently, this strategy had worked for him before. He continued relentlessly until she agreed to a small purchase. About that time, a pair of children in the next aisle moved to a similar position and set up a chorus of wails twice as grating as the first.
"As each child reached a certain point in line," I told my friend Chris "Breathe" Frue, "they became excited in the same way. That reaction forms the impulse response of this system. In reaction to a steady stream of children, the same reaction plays out over and over, in repetitive fashion.
"That must be excruciating for the check-out workers," said Breathe.
"Consider what happens if the children arrive at irregular intervals or in groups," I said, drawing on a paper napkin the axes of a graph showing intensity versus time, with three curves at the top (Figure 1). "Suppose that the first child arrives, making a standard impulse response. That's the curve at the top left. Now, assume that the next two arrive together. If all children make the same standard response, the wails at that time will double, according to the number of participants. That phenomenon is superposition."
"Does the third curve then represent time invariance?" asked Breathe.
"It is certainly a good example of that," I replied. "Assume that each child reacts in turn according to when he or she reaches the candy display. If the last child is delayed, perhaps because she climbs out of the shopping cart and runs off, her ultimate reaction is then delayed by the exact time it takes for her mother to bring her back to the line, but that reaction remains otherwise unchanged. That phenomenon is time invariance." Drawing a final curve at the bottom, I said, "The composite response equals the sum of the three curves at the top."
"Are you saying that linear time-invariant systems react just as simply as children at Walmart?" he asked.
"I'm saying that if you stimulate any linear system with one short, intense pulse, the equivalent in this problem of one standard child, you see a response characteristic of that system. It's called the impulse response. You can then calculate the response to any future series of such short pulses as a sum of delayed and scaled copies of the one true impulse response."
"But real-life signals are not series of short pulses," Breathe said.
"But they are," I replied. "Imagine a train of short pulses, all packed shoulder to shoulder like a picket fence. In the limit, you can approximate any signal to within any degree of accuracy using a train of short enough pulses."
"Let's see if I have this straight," said Breathe. "Think of your input signal as a train of short pulses, each having an amplitude corresponding to the signal value at that point. Each pulse creates its own impulse response, scaled according to the pulse amplitude and delayed according to the pulse position in time. Superimpose all those little impulse responses, and you get the final system output."
"Astonishing, isn't it?" I said. "Any system's impulse-response waveform contains all the information about what a linear system can do. Measure that waveform, and you know all there is to know about the linear behavior of that system."
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.
Other articles in the Basic EE series:
- Linearity -- Linearity is one of two properties essential for good signal fidelity, audio or otherwise. The other property is time-invariance. EDN 9/9/2010
- Superposition -- Linear superposition opens the door to many advanced methods of circuit analysis. EDN 10/7/2010
- Time Invariance -- Hard clipping obeys time-invariance, but not superposition. A tremolo circuit obeys superposition, but varies its gain with time. EDN 11/4/2010
- Impulsive Behavior -- Stimulate any linear system with one short, intense pulse, and you see a response characteristic of that particular system. EDN 12/2/2010
- Undo Machine -- The signal distortion caused by some linear time- invariant processes can be completely un-done. EDN 1/6/2011