The comparison between modeled and measured ITDs can be performed either analytically or subjectively. Meanwhile the analytic evaluation determines the error; the subjective evaluation shows whether this error is audible. For this reason, the current section describes on the one hand, an analytic procedure to
8The algorithm is based on the best fit in a least-squares sense. 9This sinus-extension is introduced by Savioja et al. (1999) (cf. (5.22)).
5.4. Evaluation Standards of Interaural Time Difference Models
determine the ITD error of a whole HRTF data set and on the other hand it deals with a listening test which determines the just noticeable ITD error.
5.4.1. Analytic Evaluation Standards of Interaural Time Difference Models
To evaluate different ITD estimation models, the error between the measured and modeled ITD
ΔITD = ITD𝑀 𝑒𝑎𝑠− ITD𝑀 𝑜𝑑𝑒𝑙 (5.15)
is one possibility, but also the correlation coefficient between the ITD maxima can be investigated.
Since the ITD is direction-dependent, it is difficult to compare different models for every single direction. Therefore, the mean ITD error and its standard deviation can be expressed as a weighted sum of errors
ΔITD𝜃,𝜙 = 𝐾−1 ∑︀ 𝑘=1 𝐿 ∑︀ 𝑙=1 𝛽𝑘,𝑙· ΔITD𝑘,𝑙 𝐾−1 ∑︀ 𝑘=1 𝐿 ∑︀ 𝑙=1 𝛽𝑘,𝑙 (5.16)
where 𝑘 and 𝑙 are the points (Bomhardt et al., 2016b)10 on the measurement sphere and 𝛽 are the surface weights (Leishman et al., 2006). Subsequently, the standard deviation of the overall error is defined as
𝜎ΔITD𝜃,𝜙 = ⎯ ⎸ ⎸ ⎸ ⎸ ⎸ ⎷ 𝐾−1 ∑︀ 𝑘=1 𝐿 ∑︀ 𝑙=1 𝛽𝑘,𝑙· [︀ ΔITD𝑘,𝑙− ΔITD𝜃,𝜙 ]︀2 𝐾−1 ∑︀ 𝑘=1 𝐿 ∑︀ 𝑙=1 𝛽𝑘,𝑙 . (5.17)
5.4.2. Subjective Evaluation Standards of Interaural Time Difference Models
Dependent on the study, the just noticeable ITD error is 16 𝜇s for frontal directions (Aussal et al., 2012) and up to 125 𝜇s for lateral directions (Simon et al., 2016). Due to these large angle-dependent deviations, the error was investigated in a listening experiment for the subjective evaluation of the ITD models. The following experiment design was inspired by the study of Simon et al. (2016) 10The lowest measurement directions are weighted stronger than all others because of the
gap in the lower sphere in the measurements of the database (Bomhardt et al., 2016a). For this reason, these directions are ignored for the mean and standard deviation.
CHAPTER 5. Interaural Time Difference Instruction manual HRTF & HpTF measurement Training: 20 trials Experiment: 6 blocks à 20 trials
Figure 5.4.: The listening experiment consisted of four blocks.
who used different types of alternative forced choice tests to determine the just noticeable ITD mismatch.
Just noticeable ITD error In total, 32 subjects, who were on average 25 ± 4 years old, were tested. Most of them (27) were right-handed, 23 had no previous experience with binaural reproduction techniques and half of them were male. All of the subjects reported normalhearing.
The listening experiment was split into four parts (see Fig. 5.4): Reading the instructions, the measurement of the individual HRTFs and HpTFs, a practice run and the main experiment.
The transfer functions were measured according to the procedure described in Sections 2.6 and 4.1. The measurements and experiment took place in a low-reflection room. Six loudspeakers (Genelec 6010A) in the horizontal plane from 270∘to 345∘and Sennheiser KE3 microphones, which were supported by a dome at the ear canal entrance, were used to measure the HRTFs. To recap, these directions were chosen because of the right ear advantage (cf. Section 2.1). The distance between the loudspeakers and the center of the subject’s head was 1.5 m. According to measured HRTF of the database (Bomhardt et al., 2016a), the measured transfer function was time-windowed and cropped to 256 samples. Afterwards, the HpTFs were measured with KE3 microphones and headphones HD650 by Sennheiser. For this purpose, the subject had to reposition the headphones eight times on the head. An averaged HpTF was calculated using the inverted mean plus twice the standard deviation of these measurements (Masiero and Fels, 2011).
Since the localization performance in the rear hemisphere is comparable to the frontal hemisphere, only the frontal directions were tested (cf. Section 2.1). Furthermore, localization performance decreases out of the horizontal plane. Subsequently, it was assumed that the noticeable ITD error will increase towards these directions.
The stimuli were generated using pink noise with three pulses with a length of 0.3 s and a pause of 0.1 s which were convolved with the measured HRIRs and headphone impulse responses. The main experiment consisted of six rounds 48
5.4. Evaluation Standards of Interaural Time Difference Models
Playback
2-AFC Adaption JND
ref ±Δ𝜏 of Δ𝜏
Figure 5.5.: Each block of the listening experiment consisted of the first three
steps to determine the JND of the ITD. The first three steps were repeated 20 times in each block.
respectively to the measured HRTF directions. In each round the ITD was manipulated: Either a delay was added or subtracted. If the delay is subtracted and the difference is noticeable, the subject is able to identify that the sound is shifted towards the front. If the delay is added, it will be shifted towards the interaural axis. So, the task for the subject was to identify whether the stimuli was shifted to the front or to the side compared to the stimuli convolved with the original HRTF.
The procedure in each experiment block is illustrated in Fig. 5.5: In the first trial of each block the subject heard the reference and the manipulated stimulus with an initial delay of 100 𝜇s. Then the subject had to decide by a two-alternative forced choice (2-AFC) whether the second stimulus was more to the front or back11. Depending on whether the answer was right or wrong, the delay is
adapted by the QUEST method (Watson and Pelli, 1983). A correct answer resulted in a shorter delay in the next trail and a wrong answer in a longer delay. The QUEST method is an adaptive psychometric procedure which starts with an initial probability distribution (see Fig. 5.6). The mean of this distribution was 100 𝜇s and its standard deviation 100 𝜇s to cover large variations. This distribution was multiplied by the psychometric function which took the guessing probability 𝛿 = 0.1, the false alarm rate 𝛾 = 0.5 and additionally a slope 𝛽 = 3 into account. Furthermore, this psychometric function depended on the answer. If the answer was correct, it was the mirrored version of the false one (see Fig. 5.6). Finally, the resulting mean distribution was shifted to lower or higher test values. In this experiment the mean value of the current distribution was used to determine the next delay. After 20 trials the tested delays varied only slightly and can be assumed as converged. Therefore, the last tested delay was the desired just noticeable ITD error.
The adaption by the QUEST method has the advantage that it is faster and more 11The direction of the shift was chosen randomly.
CHAPTER 5. Interaural Time Difference
initial distribution psychometric function resulting distribution S(x) F(x) success failure
Figure 5.6.: The adaptive QUEST method is a psychometric procedure to deter-
mine subjective threshold levels. It calculates the testing threshold from a probability distribution and adapts this distribution using the psychometric function based on the answers of the subject.
precise than testing a set of discrete thresholds. The disadvantage is that the psychometric function and initial distribution have to be estimated beforehand. This was done by pretests.
345 330 315 300 285 270 0 20 40 60 80 100 Azimuth [∘] Δ ITD [𝜇 s]
Figure 5.7.: The just noticeable ITD error is plotted direction-dependently by
the median of all subjects (black line). The grey area marks the interquartile range. The subject-averaged just noticeable ITD error is marked by a diamond.
The results of 24 subjects, who performed all test conditions12, are depicted in Fig. 5.12. The JND is almost stable between 300∘ and 345∘ and grows faster for lateral angles. In the present experiment no significant gender effects are detected. Only a slight advantage of females is observed.
The study by Simon et al. (2016) investigated the JND of the ITD in the horizontal 12The remaining eight subjects were tested six times at 𝜙 = 345∘.