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The Modern Analyzer, Part 2

May 30, 2014 2:19 PM, By Bob McCarthy

Understanding the transfer function analysis.

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Figure 2: The multi-time windowed FFT. Resolution and display stay constant over the full range of frequencies. See larger image.

A few variable sequences will illustrate this. Let’s put “aaaaaaaa” in. We get “aaaaaaaa” out. Our analysis reveals that the device amplifies the signal to bold face type—easy. It would not matter if we sampled only one “a” or all eight. And it would not matter if we compared the first four letters of the output to the middle four of the input. If we have an endlessly repeating sequence we can fall out of sync and not care.

Now the input sequence is “abcdefgh” and so we get “abcdefgh” at the output. Once again is does not matter if we sample one letter or all letters, as long as we are in sync and therefore compare the same letter. If we are not in sync then “cdef” at the input could yield “abcd” at the output. Now our analyzer is confused about the bold-faced liar it is measuring. So it is with music, at least the kind that is not an endless repeating sequence.

Transfer Function Phase

The technique for reading phase response is literally best described as “connect the dots.” A frequency response is a series of amplitude and phase values. If we solo up one frequency, we get an amplitude value that needs no further explanation. Unity is unity. A gain of +6dB is exactly what you think it is. But that single-phase value gives us nothing to make a conclusion with. We need to get a second frequency and then connect the two phase values together. Now we have a phase slope, which can be decoded into time by discerning the rate of phase shift over frequency. For example, if we have 360 degrees of phase shift over a span of 1kHz, we have fallen 1 millisecond behind (or ahead).

Let’s break it down. First we find the slope direction, read left to right on the frequency axis. A downward slope is delay (output after the input). This is normal and intuitive. An upward slope indicates anti-delay (output before the input). This is also normal and extremely unintuitive. There are two places you are likely to encounter this: anytime we have an internal delay in our analyzer and with filters in our measured circuit. The first is easy. We use an internal delay to line up output and input. If we put in too much delay, we see the phase slope rising. In rooms we see this anytime the temperature rises, since the speed of sound accelerates and gets to our mics quicker. Wrapping our heads around anti-delay in the filter scenario requires a Ph.D. in filter theory and even then it’s controversial. Suffice to say that our rendering of the phase in filters is a simplification of a more complex reality. Nonetheless, if you view the phase response of a filter you will see both a rising slope and falling slope.

The next level is analyzing the slope rate of change. That is the timing information. It is linear, so we clock the rate in Hz, not in octaves. We don’t have to take the entire 2kHz bandwidth into account at once. We can and should break the spectrum down into digestible pieces. Choose a range of frequencies and start there. So if we saw a 90-degree shift between 250Hz and 500Hz we would know we have 1 millisecond of delay. How? 90 degrees is 1/4th of a cycle (360 degrees). If we move 90 degrees in 250Hz, we can extrapolate that to be a rate of 360 degrees per 1000Hz (i.e., 1 millisecond). The same would be true no matter what frequencies start and stop the slope. If we have a 90-degree change in a 250Hz span, we have 1 millisecond. A steeper slope (more phase shift over the same frequency span) indicates more time. A constant rate of phase shift indicates a constant delay over frequency. A variable rate of change indicates a device with frequency dependent delay. Because time is linear and our frequency axis display is log there is a visual trick to remember. A constant delay over frequency will appear as a steepening phase slope, although it is the same linear frequency spacing (and same time).

The Impulse Response

We just saw how the phase response put the frequency domain information in plain sight, but required us to go through a decoding process to convert the data to the time domain. The impulse response reverses this paradigm, giving us a straightforward view of the time domain along with encrypted frequency response information. The impulse response is a wonder of simplicity to put to practical use. It tells us the arrival times of the direct sound and reflections in the units we can most easily understand: milliseconds. While it is immediately apparent how to read the impulse response, it is not at all so to understand how the analyzer derives the response.

The first thing to understand is that the impulse response is a mathematical construction of a hypothetical experiment that we don’t have to actually perform. The display we see is the answer to this question: What would we see on an oscilloscope (amplitude vs. time) if we put a single perfect impulse into the system? Pop goes the pulse into our system and we wait for its arrival at a mic, then stick around to see as it arrives again, and again after reflecting off the floor and other surfaces. That’s the easy part, but now the questions begin: What is a perfect impulse? How can we see the impulse response when we are putting continuous noise or music through the system?

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