All Pole Model Performance

−150 −100 −50 0 50 100 150 −50 −40 −30 −20 −10 0 Angle (Deg) NMSE (dB)

Figure 5.1.2: NMSE of all-pole model order 15

Figure 5.1.2 shows the normalised mean squared error calculated between the measured spec- tra and the spectra modelled as a 15thorder All Pole filter using the linear prediction method. The figure shows a range of a little over 15dB between the angles at which maximum and minimum NMSE occurs. The worst performance of the model occurs in the contralateral hemisphere, with the maximum error arising at -107◦, however the model exhibits a similar level of error in the angles close to the ipsilateral position, and as such exhibits an approxi- mate symmetry that is centred not around 0◦ but seemingly centred around∼25◦. A fairly flat region of minimum NMSE exists between approximately 0◦ and 50◦, covering the centre of the apparent symmetry in the error.

The distribution of the NMSE over angle, shown in figure 5.1.2 appears to be a direct characterisation of the inherent weakness in the all-pole approximation of the HRTF. Notches are a significant feature of the HRTF, at both ipsilateral and contralateral angles sharp notches arise due to destructive interference caused by pinna reflections, as the all-pole filter has no non-zero zeros it is unable to capture the notches in the frequency response of the HRTF.

J. Sinker Compact HRTFs CHAPTER 5. PARAM. APPR. 102 103 104 −70 −60 −50 −40 −30 −20 −10 0 10 20 Frequency (Hz) Magnitude (dB) Measured Modelled

(a) Maximum NMSE Case: -107◦

102 103 104 −70 −60 −50 −40 −30 −20 −10 0 10 20 Frequency (Hz) Magnitude (dB) Measured Modelled

(b) Minimum NMSE Case: 29◦

Figure 5.1.3: Maximum and minimum error HRTFs : All-pole order 15

Figure 5.1.3 illustrates the measured and modelled HRTF at the two angles which yield the maximum and minimum NMSE respectively when using a 15th _{order all-pole filter. The}

minimum NMSE case occurs at 29◦, in the frontal region of the measurement circle, whereas the maximum NMSE case occurs at -107◦, towards the rear of the head in the contralateral hemisphere. The increased error of the modelled spectra observed at the maximum NMSE case shown in figure 5.1.3a is clearly a result of the significant notches present in the measured HRTF at this angle, such notches are not present in the frontal HRTFs like the one shown in figure 5.1.3b. For relatively low orders of all-pole filter, the model is unable to capture the spectral notches in a given frequency response and as such is incapable of reproducing the measured spectra with sufficient detail to maintain a consistent level of model error. However it is possible that given a sufficient, comparatively high, number of poles to be allocated, the linear prediction method will seek to approximate notches in the frequency spectra using clustered pole placements.

J. Sinker Compact HRTFs CHAPTER 5. PARAM. APPR. 0 20 40 60 80 100 −30 −25 −20 −15 −10 −5 0 Model Order MNMSE (dB)

Figure 5.1.4: MNMSE of all-pole model orders 1 to 100

Figure 5.1.4 shows the mean normalised mean squared error of the all-pole linear prediction model as a function of the model order; that is the NMSE of each model order averaged over all angles to obtain a single value per model order expressed in decibels. The data was generated by analysing the performance of a series of linear prediction all-pole filters used to model the Tu Berlin measured HRTF at each angle for all-pole model orders ranging from 1 to 100. It is clear that with the addition of poles the model is better able to capture the spectral detail of the measured HRTF, and based upon the already well performing fit of the peaks in the spectra shown for relatively low orders in figure 5.1.3, it can be reasoned that the increased performance is a result of the improved ability to capture detail surrounding notches that comes with a surplus of available poles. This can be seen directly by observing the increased accuracy of the modelled spectra for the same angles that yield the minimum and maximum NMSE for the order 15 all-pole case.

J. Sinker Compact HRTFs CHAPTER 5. PARAM. APPR. 102 103 104 −70 −60 −50 −40 −30 −20 −10 0 10 20 Frequency (Hz) Magnitude (dB) Measured Modelled

(a) Prior Maximum NMSE: -107◦

102 103 104 −70 −60 −50 −40 −30 −20 −10 0 10 20 Frequency (Hz) Magnitude (dB) Measured Modelled

(b) Prior Minimum NMSE: 29◦

Figure 5.1.5: Prior maximum and minimum error HRTFs of all-pole order 15 modelled with all-pole order 100

Figure 5.1.5 shows the improved performance of the 100thorder all-pole model for the angles which correspond to the maximum and minimum NMSE in the 15th _{order case respectively.}

Note that the order 100 model still performs worse for the more notched frequency response shown in 5.1.5a, however for both angles it can be seen that the increased number of poles allows the linear prediction method to better approximate the frequency notches.

The overall trend in figure 5.1.4 appears to be similar to that of an exponential decay; the MNMSE reduces rapidly as the number of poles increases from 1 to approximately 7, this is followed by a small region of noticeably shallower gradient, at approximately model order 11 the roughly exponential shape resumes. The curve exhibits slight fluctuations deviating from the ideal exponential curve at lower model order, but exhibits increasing smoothness approaching the higher model orders in the plot. These fluctuations are likely caused at lower model orders due to the number of poles available being less than optimal for the modelling of specific measured HRTF spectra, as such the linear prediction method may focus all available poles on significant peak details, possibly leaving too few poles to capture other lesser peaks, differently depending on the relative optimality of the number of poles used to each of the measured HRTFs. The decrease in NMSE continues with a progressively

J. Sinker Compact HRTFs CHAPTER 5. PARAM. APPR. shallower gradient as the model order is increased, between model order 80 and 100 the curve exhibits a near smoothness and very shallow negative gradient, seemingly suggesting that the NMNSE is approaching convergence with a lower limit between -25dB and -30dB. If the trend continues beyond model order 100 it is likely that the exponential decay characteristic of the curve will yield an asymptote at approximately -26dB, meaning that the addition of poles after approximately 80 will yield little to no further increase in performance according to the MNMSE criterion.