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- Written by Bjørn Kolbrek
Evaluating the accuracy of low frequency reproduction isn't as straightforward as one may think. There is talk about "fast bass", and PRAT (Pace, Rythm And Timing) is sometimes mentioned as a description. In such a discussion someone will invariably point out that a 20Hz sine wave isn't "fast", and that the issue of fast or slow bass is meaningless. It may seem that way using these simple arguments, until you experience accurate, detailed bass reproduction. But how do we quantify it? Obviously not by measuring the rise time of a 20Hz sine wave!
This is actually a very real problem. For the mix engineer, perhaps the biggest problem is how a loudspeaker with poor temporal response alters the perceived balance between rhythm section instruments like kick drum and bass guitar. Though not exclusively the case, loudspeakers with ported cabinets (bass reflex loading) are more likely to exhibit this type of problematic behaviour, due to the use of resonant elements to increase bass extension. An incorrect mix made on loudspeakers like this may not transfer well to other sound reproduction systems, and can't be corrected later during mastering, since the two instruments occupy the same part of the audio spectrum.
Lara Harris, Keith Holland and Philip Newell have done extensive work on quantifying bass reproduction accuracy, resulting in Lara's PhD work and the Bass Transmission Index (BTI). The BTI is an objective measure of a loudspeaker's ability to accurately reproduce low-frequency musical content. The metric aims to describe how well a loudspeaker reproduces the temporal envelope of a dynamic and harmonically-complex signal, something that cannot easily be evaluated from the usual frequency response (magnitude plots).
Like the STI (Speech Transmission Index), as used in speech intelligibility measurements, the BTI is based on a metric called the Modulation Transfer Function (MTF). MTF-based methods such as these pass an amplitude-modulated signal through the system, varying the rate of the temporal fluctuations across a range of values that are likely to feature in the real-world signals that the system will encounter. The preservation of modulation depth between input and output is used as an indicator of how well the system can reproduce these temporal variations. The figure below illustrates the concept, showing modulation depth for input (mi) and output (mo) signal envelopes.
The BTI evaluates the MTF in 10 frequency bands from 16-160Hz, with 7 modulation frequencies from 0.8-11.7Hz. The algorithm computes a matrix of modulation index scores between 0 and 1 for each combination of frequency band and modulation frequency. This grid of numbers can be used to calculate an overall average score, and is visualised as a grayscale intensity image where white = 1.00 (perfect reproduction). As a rule of thumb for interpreting these BTI intensity images:
- Inconsistent shading in the horizontal direction (left-right) indicates variation in the frequency response magnitude (this is variation across frequency bands).
- Inconsistent shading in the vertical direction (up-down) indicates variation in the faithfulness of envelope transmission (this is variation across modulation frequencies).
The figure below shows an example BTI intensity image and mean score for a bass-reflex studio monitor. Frequency bands (centre frequencies) are on the x-axis, and modulation frequencies on the y-axis.
Recently I have worked with Lara to make her BTI research code production ready, and it has now been officially made available on GitHub. It runs in Matlab and Octave (although not all the advanced options are available in Octave, and some requires certain Matlab tooboxes), and demonstrates a reference implementation. Here are the links:
Lara's LinkedIn post about BTI
The measurements made it obvious what the problem was: the output fell below 70Hz, and apart from a peak at 50Hz, the response was more like a midbass horn than a 30Hz bass horn. What was going on? 
