|Phrequency, Phase and Philters
Prophessor Doug Jones
This month, I’m going to try to explain some of the most fundamental aspects of audio. This is part one of a two-part series that will culminate in a discussion of FIR filters. It did not seem right to me to discuss FIR filters if we are a little fuzzy on “regular” filters, so here goes!
Let's start with the concept of frequency. The term frequency is defined as Events Per Unit Time. Years ago, a friend of mine had an antique, vacuum tube based analyzer called an EPUT meter. It was a rudimentary frequency counter! The frequency
of a full moon is one event per month. The frequency
of Christmas is one event per year. Before 1960 the official unit of frequency was the cycle per second. We now call the unit ‘Hertz’ to honor the German physicist Heinrich Hertz.
We are all familiar with the sine wave shown here. We like to use the sine wave in these explanations because it is the result of simple harmonic motion. We like simple! Imagine you had a pendulum and attached a marker to the weight, and set it in motion above a sheet of paper such that the marker was drawing a line on the paper. If you pulled the paper under the marker at a steady rate it would draw for you a nice sine wave. A sine wave is really a graph of something where the X axis is time.
If this graph represented the action of the pendulum, the vertical axis of the graph or the Y axis would be the displacement of the pendulum in some length units, like inches or centimeters. If this graph represented an audio signal, the Y axis might be volts. If this were a representation of an acoustic wave, the Y axis would be pressure. So in an acoustic wave, the events that we count are oscillations of pressure, and the time unit is the second. The graph in figure 1 shows 2.5 Hertz or 2.5 cycles in a second. Generally speaking, we perceive frequency as pitch, and our ears, at least young undamaged ones, respond to frequencies between roughly 20 Hz and 20,000 Hz, with us older, shall we say “seasoned” folks maybe making it out to 12,000 Hz.
Ok, onward to phase! Phase is related to, but not equal to, time. It is most often expressed in degrees. One cycle of a sine wave can be thought of as a 360-degree event. The beginning of the cycle is 0 degrees, ¼ of the way through is 90 degrees, ½ of the way through is 180 degrees etc. Here is an imperfect analogy that might help. Imagine a racecourse, say an oval. A blue car and a red car are driving around the oval at exactly the same speed. You could say that they are in phase because no matter where they are on the track, they are side by side. Now imagine that the red car hits a patch of mud on the track and slows down a bit. Now the blue car is ahead. We could measure the difference in degrees. If the red car is behind by a ¼ of the track we might say that red is 90 degrees behind blue. It gets really interesting when red is delayed by exactly 360 degrees. Now the cars are back in “phase” but the blue car is winning!
Most of the time when we talk about phase we are comparing something to something else; the red car to the blue car, the phase of a signal at the input of a circuit to the phase of the signal at the output of the circuit and so forth.
The question of how we perceive phase is a very loaded one and deserves a better treatment than what I can do here, but phase shift can be perceived as a cancellation or even as a sort of time smear that messes up the perception of a stereo image.
Ok. Now on to filters! One of the very cool things that smart people discovered a while back is that there are things that behave differently depending on frequency. A wire is not one of them, right? Wires (at audio frequencies) don't care what signal is going through them 20Hz, 90Hz, 15,762Hz… it’s all good. Until you take that wire and make it into a coil. It then becomes an inductor, and guess what? Low frequencies go through it just peachy, but high frequencies do not. The inductor belongs to a class of devices known as reactive devices. The capacitor is the other member of this exclusive club. It behaves as precisely the inverse of the inductor. A capacitor will block low frequencies but allow high-frequency signals to pass through. In the analog world, all filters are built using reactive devices. Now I’m not going to try to explain how these reactive devices work, but like all real world devices, they take time to do what they do, and this time is manifest as a phase shift!
Of course, filters are used in all forms of audio processors from crossovers to equalizers to the humble tone control.
It has taken me 879 words to get to the point of this article… analog filters always produce a phase shift. They have to. It's the law! Not a suggestion. . . not optional compliance! Next time, the mighty FIR filter!