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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 1: The limitation of a single time window on the FFT analyzer. Frequency resolution is linear no matter what the time record length is or how it is displayed. See larger image.

In part one, we explored how the modern FFT analyzer creates a composite high-resolution log frequency response out of a series of linear captures. Now we will build on that foundation by adding a second analyzer channel, which allows us to make the comparison measurements known as “transfer function” analysis.

Before we dive back inside the analyzer, let’s think about the desired function of the transfer function: an aid to sound system optimization. We would like it to help in five main categories: speaker positioning (aiming, spacing, splay angle, etc.), acoustical modifications, equalization, delay, and level setting. All of these, except delay setting, require high-resolution frequency response data. Positioning, acoustical issues and delay require time data. We need to be able to identify the direct sound, reflected sound, and noise unrelated to our signal.

Transfer Function Amplitude

We use the term “transfer function” to describe the behavior of linear devices in our signal path. They can be passive like a wire or active, such as an equalizer. It is a scientific sounding term that boils down to the overall effects on frequency and phase between the input and output of the device; the difference between the gozinta and the gozoutta.

The math of the transfer function is complicated under the hood but extremely simple to understand from the outside. For level analysis, it can be modeled as division: output over input. For temporal analysis (time and phase), it can be modeled as subtraction: output minus input.

Let’s start simple. If the output is 2V when the input is 1V we get a transfer function amplitude ratio of 2:1 (a.k.a. +6dB). If the same device had an output arrival of 2 milliseconds after an input arrival of 1 milliseconds, then we have a transfer time of 2-1=1 milliseconds. If the device is linear (i.e., it maintains a constant transfer level and transfer time), then we can predict its output behavior relative to its input. If we now put 4V at 10 milliseconds into it, we will see 8V arrive at the 11 milliseconds mark.

Now let’s take the 48 point/per/octave data we studied in part one and put it to work. One FFT channel captures the output and the other the input. Now we have the transfer amplitude and transfer time (phase) at every frequency.

The beauty of transfer function analysis is portability. We can pick any starting point and stopping point in the signal path and see the difference in level and time. It can span from the input of the console all the way to the sound at the last row, a single device or as even just a single resistor inside a circuit. Physically, this amounts to two probe points (in and out) that are each “Y’d” into the analyzer, allowing the signal to flow through the system as normal while it feeds us data in parallel.

Let’s put the concept of linear relationship to bed with some examples. One kHz alone goes in, 1kHz alone goes out: linear. One kHz goes in, 1kHz and 2kHz show up at the output: non-linear (harmonic distortion). Pink noise in and pink noise out: linear. Punk noise in and (the same) punk noise out: linear. Good jazz at the input and Kenny G at the output: non-linear. We can get the transfer function of a device using a random source because to us, it is not random (since we have a copy of what arrived at the input). In practice, even the best system we measure has some non-linear behavior. It has a noise floor and some distortion. If a system has too much noise, or too much distortion, the transfer function data will not be reliable and repeatable.

The advantage to source independence, (i.e., being able to use random signals such as music), is extremely obvious, especially when the lights go down. There is a catch though, if we use a random source and the input and output are offset in time (such as music through a DSP, or sound traveling through the air). Our analyzer needs a little help to get an accurate frequency and phase response. We must synchronize the two channels inside the analyzer before we crunch the numbers, without affecting the actual signal flow.

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