PEST/ABX Hybrid Method

In order to home in on the subject’s threshold of detection between the reference and variable sample, the PEST (Parameter Estimation by Sequential Testing) metho- dology (Taylor and Creelman, 1967; Taylor et al., 1983) was considered. This is an adaptive method, similar to the 2-alternative forced choice, which forces a response to a simple question, “is there a difference between the reference and test stimuli?”. The PEST produces results directly through its adaptive method. It is maximally efficient, resulting in fewer trials per listener to output the threshold. Each reference and variable sample pairing is known as a trial, with a set of trials completing a PEST run until the rules state that the PEST sequence is to be terminated.

An initial difference between the reference and variable volume was set which could be easily discernible by the subject, and following this first trial, a set of PEST ‘rules’ govern the determination of new variable room volume levels. In this test, the initial volume was 100m3, a relatively low modal density at each of the three

test frequencies, where differences due to the discrete modes are clearly audible. Wherever a trial has been completed, a Wald sequential likelihood test (Wald, 1947) determines whether a new level should be set or the previous repeated. If a new level is to be set, the rules can be summarised as follows:

1. If a reversal is made, halve the step size.

2. A second step in the same direction requires the same step size as the first. 3. Fourth and further steps in the same direction require a doubling of the step

size.

4. Step size on the third trial in the same direction depends on the step prior to the last reversal. If, on this step, no doubling occurred, the step size should now double, while if a doubling did occur, no doubling should take place here. The original PEST specification suggests a logarithmic change to step size, al- though after initial experimentation, a linear change proved most appropriate. The end of a PEST run occurs when a new step size is required which falls below a defined minimum. The selection of this minimum step size is key to the success of the PEST run, as it determines the accuracy by which the final threshold can be defined. It is therefore necessary to make this small enough to output thresholds of

reasonable accuracy, but not so small that it becomes difficult for the PEST process to converge. The selection of this value also directly affects the time taken to test, and therefore has a bearing on factors such as subject fatigue and boredom, which can be a significant source of error (see Goldberg (2006)). Pilot testing revealed that a reasonable minimum step was 100m3.

In such a test, it remains a possibility that the subject simply responds that they can hear a difference between the reference and variable samples even if they may perhaps not (most likely due to the belief that they are performing better if they can consistently hear a difference). To ensure that they could not simply claim to hear a difference, the PEST process was augmented with a standard ABX procedure whose output informed the PEST routine of success or failure.

At each room volume a maximum of three comparisons could be made. Sample A was the reference, B the varying volume (dependent on the PEST routine) and X either A or B, randomly chosen. If the X sample was correctly identified three times consecutively, the room volume and hence modal density was increased. However, a single incorrect answer would immediately register a failure to detect a difference and therefore the volume would decrease. An incorrect answer reveals that there is no difference evident between the current varying modal density and the reference case, an indication that a sufficient density has been reached in order to smooth out any audible artifacts arising from the modal response. The requirement of three consecutive correct answers reduces the probability of the subject guessing to 12.5%, and while this is not at the typical statistical threshold (<5%), it was considered sufficient given the association with the PEST methodology, which would bring the volume back down at the next comparison unless another three guesses were correct - totalling six consecutive guesses - a probability of just 1.6%.

5.4.4 Results

Figure 5.5 shows the mean room volume threshold and standard deviation where no detectable difference existed between that volume and the reference. In practice, the results provide the volume threshold for a particular frequency. However, to extract thedensity threshold, a modal bandwidth for the corresponding frequency has to be obtained. Modal density is therefore calculated using Bolt’s equation, with modal bandwidth obtained using:

Bw= 2.2

0 50 100 150 200 250 300 0 500 1000 1500 2000 2500 Frequency (Hz) Room Volume (m 3 )

Mean Threshold Volume

Figure 5.5: Mean room volume threshold for the detection of difference over three test frequencies

Frequency (Hz) 63 125 250 Modal decay (seconds) 1.01 0.84 0.58 Required Volume (m3) 1529 803 433

Required Modal Density (modes per bandwidth) 4.1 10.3 31.6

Table 5.1: Modal density according to bandwidth taken from reverb conditions in modal decomposition model

”N = 8.8fiF

2_V

c3RTmodal

(5.2) where N is the modal density, F the frequency,V room volume, c the speed of

sound in air and RT is taken from Figure 5.4 at any given frequency. This density

is indicated in Table 5.1.

The results show that at 63Hz a subject would require around four modes per modal bandwidth to even out degrading effects. Furthermore, under these test conditions, subjects require an increasing modal density as frequency rises. This is shown in Table 5.1, where each volume is associated with a density, giving the required density threshold for perceiving a smooth response at a given frequency. Consequently, no definition of a generic modal density across frequency is possible from these results. It is also seen that the application of Schroeder’s transition to a statistical region is not replicated subjectively, and it would be unwise to do so. Although at the very low frequencies a modal density of about four is sufficient and

Figure 5.6: Schroeder and subjective ‘cut on’ frequencies across room volume in accordance with the definition of three for the Schroeder Frequency, as frequency increases, subjects require a much higher density if degradation of the stimuli is to be inaudible.

Another way of representing these results is shown in Figure 5.6. A subjective ‘cut-on’ frequency above which modal effects are negligible is indicated both with regards to these subjective tests and the Schroeder Frequency. It is clear that, for smaller rooms the Schroeder Frequency underestimates the subjective ‘cut-on’ frequency - subjects still detect differences in modal sound fields above it. For larger volume rooms, the subjective results converge to towards it. For example, at433m3, fc is 66Hz, while subjectively, this was the required volume at 250Hz.

At lower room volumes, the Schroeder Frequency may well accurately predict where a statistical soundfield begins, but we do not begin to perceive a uniform, smooth response until we reach a much higher frequency. Therefore, at low frequen- cies, even in rooms large enough for us to expect that we won’t suffer from audible modal effects, it has been shown that we can perceive differences from the ideal smooth case.