The Modern Analyzer, Part 2
May 30, 2014 2:19 PM, By Bob McCarthy
Understanding the transfer function analysis.
Mathematicians don’t throw words like “perfect” around lightly. If you guessed that this is another one of those infinity constructions, you win a prize. A perfect impulse is an audio stimulus signal created by the following recipe: all frequencies, equal level, in phase for the duration of a single cycle each. It is like a one lap race for all frequencies with everyone ready to run once the starter pistol fires. In fact a version of a starter pistol is used as an impulse generator by acousticians for room analysis. The perfect impulse with its infinite frequency range is an infinitely thin single transient spike that rises and returns to normal. In the practical world we need the impulse to have the same frequency range as we are measuring (e.g., up to 20kHz) so the pulse width becomes finite and visible. A system with a perfectly flat amplitude and phase response over frequency will return an impulse that rises and returns in the minimum time with no additional features such as overshoots, undershoots, or ringing. That is the only way we can ever see a perfect impulse response. If there are any peaks or dips in the amplitude and/or differences in phase slope, the impulse will show overshoots, undershoots, or ringing.
If there is a time offset between the output and input (recall that this is a transfer function measurement) then the impulse will shift its position on the x-axis to the left (output precedes the input) or the right (output is after the input). Of course the output cannot precede the input in physical reality, but our analyzer is just responding to two signals, which can come from anywhere and we also have internal delay in our analyzer so this is not an exotic occurrence.
Let’s study the impulse response and see what it reveals. The standard features are a dead zone before the arrival, the shape of the impulse and any secondary arrivals that follow. The dead zone is the transfer time, i.e., latency; for example, the time it takes to get through a digital device or travel through the air. The analyzer finds this by recognizing the content of the input and output signals as similar but offset in time. The next thing we notice is the orientation of the impulse, which reveals the polarity of the system (positive or negative). The third feature is the rise and fall of the pulse. A steep vertical line with no ringing indicates a flat amplitude and phase response. A rounded impulse indicates the system has high frequency loss and/or frequency dependent phase shift. Ringing (before or after) is the hallmark of filtration (electronic or acoustic filters). Beyond the first arrival we will see extra copies of the input signal arriving later. These are recognized as the children of the input and are shown as secondary transient peaks lining up behind the first arrival. Reflections are identified this way.
How does our analyzer give us a display of transient peaks that look like snare drum hits when we put in the music of 101 strings? This is where the math magic comes in. Recall that the original FFT transform took our time domain signal and converted it to the frequency domain. Then we took two channels to create a transfer function phase and amplitude over frequency. The impulse response we see is the result of a second generation of FFT calculations, in effect, an FFT of the amplitude and phase over frequency, that converts us back to the time domain. This is termed the Inverse Fourier Transform (IFT). The product of the IFT process is a simulation of the waveform that would result from a perfect impulse passing through the system that had the measured amplitude and phase. Because the transfer function amplitude and phase are source independent, we are able to build our impulse simulation by listening to pop music (not the sound of a single pop).
Coherence Over Frequency
Coherence is the answer to the question: “Hey analyzer, do you have any idea what you’re talking about?” Coherence values range from 1 (yes, I mean it) to 0 (I am just making it up), with educated guess in the middle. The real math is beyond the scope, but we can comprehend the principals pretty easily. Coherence is statistically derived; it requires multiple samples, which we average together. What we are looking for is agreement between the samples. If they are all providing the same transfer function values then we have confidence in the averaged value. If the answers are all over the map then we lose confidence. Why would there be disagreement? Noise. Any transfer measurement contains a noise component, i.e., uncorrelated signal at the output that was not present at the input. The question is simply, how much noise compared to the signal we are sending through the device. The answer is “coherence.” In the case of a line level device, the signal to noise ratio should be very high. Once we get into the acoustic world everything moves in noise’s favor. We have HVAC, forklifts, reflections, and much more.
Coherence is evaluated on a frequency-by-frequency basis, so we can see which ranges are faring better than others. We use coherence to make decisions, and most important, to not make decisions. Very high coherence tells us we can make adjustments with confidence that the changes will have a predictable effect. Low coherence means that there is more going on here than meets the ear, and we could find ourselves turning knobs on the equalizer and seeing little or no improvement. Adjustments such as adding absorption, adjusting splay angle, and setting delays can produce great improvements in coherence. Many audio engineers will differ on whether a particular equalizer setting is an improvement. Coherence, on the other hand, correlates highly with quality of experience. If we make an adjustment that improves coherence there is usually widespread agreement.
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