ITD deviations of smoothed transfer functions

If the transfer function E(ω, φ, θ) in Equation 3.4 is not minimum phase the inverse transfer function E⁻¹(ω, φ, θ) can be unstable. In this case the calculation of A(ω, φ, θ) should be restricted to the absolute spectrum of the HRTF and an appropriated phase can be applied to the spectrum.

It has been shown by Mehrgardt and Mellert (1977) and Kulkarni et al. (1999) that the empirical HRTF phase is almost minimum phase plus an frequency independent group delay.

A minimum phase can be obtained from the absolute HRTF spectrum by

P_min(ω, φ, θ) = Ξ(−ln(|A(ω, φ, θ)|)) (3.6) where Ξ is the Hilbert transform. The complex HRTF is then given by

A(ω, φ, θ) =|A(ω, φ, θ)| ×e^{−iPmin(ω,φ,θ)}. (3.7) An important property of minimum phase transfer functions is that they have a minimal energy delay ((Oppenheim and Schafer, 1975)). As a consequence, the group delay of the minimum phase impulse response is always nearly zero. Therefore, if both the left and the right ear HRTFs are minimum phase, the ITD is nearly zero independent on source location. To apply an appropriate ITD an frequency independent group delay is introduced to one ear that matches the ITD obtained from the empirical impulse re-sponses. However, the ITD of the pure minimum phase HRTFs is not equal to zero for all source positions. Therefore, the frequency independent group delay that is applied to the minimum phase HRTFs has to be corrected for the inherent time delay of the minimum phase HRTFs. This correction term has to be subtracted from the ITD that is introduced to the minimum phase impulse responses.

Only the low frequency range of the ITD is perceptually relevant, because for high fre-quencies the phase differences at the two ears are ambiguous. Thus, it is important that the low frequency ITD of the minimum phase plus frequency independent delay HRTFs is consistent with the empirical ITD. Hence, in this study the group delay is calculated in a way that the low frequency ITDs of the minimum phase plus delay HRTFs and the empirical HRTFs are equal.

As pointed out in Section 3.3 the ITD of minimum phase plus delay HRIRs is directly related to the spectrum. Therefore, the smoothed HRTF spectra have to be taken into account when the group delay, that is introduced to the minimum phase HRTFs, is cal-culated.

Taken together, three different ITDs have to be calculated for introducing an low fre-quency ITD to minimum phase impulse responses that matches the low frefre-quency ITD of the empirical HRTFs. They are obtained as follows:

τEmp = argmax(Γ(hl, f)⊗Γ(hr, f)

τCorr = argmax(Γ(hl,min,S, f)⊗Γ(hr,min,S, f) τM in = τEmp−τCorr

(3.8)

τEmp is the ITD of the empirical HRTFs calculated as the time shift of the maximum of the cross-correlation function (marked by the symbol ⊗) of the left (hl) and right (h_r) ear HRIRs. ’argmax’ denotes the time shift of the maximum of the cross cor-relation function. The function Γ(h, f) is the low-pass filtered impulse response h(t) with edge frequency f. It is applied to extract the low frequency ITD (f = 500 Hz).

The correction termτCorr is calculated from the minimum phase HRIRs with smoothed spectra (hr/l,min,S). The indexS denotes the degree of smoothing either for 1/N octave or cepstral smoothing. Then the frequency-independent group delay introduced to the minimum phase impulse responses is given byτM in.

Based on this calculation, the low frequency ITD of the minimum phase plus frequency independent group delay HRIRs should match the empirical low frequency ITD τEmp, independent of spectral smoothing. To verify this, the ITD τReCalc is re-calculated from the minimum phase plus frequency independent group delay HRIRs by

τReCalc=max(Γ(hl,min,S(t+τM in), f)⊗Γ(hr,min,S(t), f)) (3.9) The ITD error between the minimum phase plus frequency independent group delay HRIRs and the empirical HRIRs is then given by

τ_Err=τ_Emp−τ_ReCalc (3.10)

In Figure 3.13(a) the ITD error τ_Err (averaged across 10 subjects) is plotted as a func-tion of azimuth for four different degrees of smoothing. The error is small for sound incidence out of the median plane ('5−8µs) and increases at lateral angles. Further-more, the ITD error is not independent from smoothing and is varying in a range of approx. 10µs. The results for 1/N octave smoothing are comparable and not shown here. It is important to note, that the perceptually relevant low frequency ITD τReCalc

only matches the empirical low frequency ITD τEmp if the minimum phase correction term τM in is computed from low pass filtered HRIRs. In a study of Kulkarni et al.

(where the correction term τCorr was introduced) the sensitivity of subjects to HRTFs phases was investigated by discrimination experiments. It was shown, that minimum phase plus frequency independent group delay HRTFs were distinguishable from em-pirical HRTFs at lateral source positions. At these positions the low frequency ITD of the minimum phase plus frequency independent group delay HRTFs deviated from the empirical ITD. It was concluded that these deviations served as a cues for the subjects.

0 15 30 45 60 75 90 105120135150165180 0

5 10 15 20 25 30

Azimuth [°]

ITD deviation [µs]

64 32 16 8

(a)|τemp−τReCalc|

0 15 30 45 60 75 90 105120135150165180 0

50 100 150 200 250

Azimuth [°]

|τ Corr−τ Corr,LP| [µs]

64 32 16 8

(b)|τCorr−τCorr,LP|

Figure 3.13: Left Panel: The differences between the low frequency ITDs of the empirical HRTFs (τ_Emp) and the minimum phase plus frequency independent group delay models (τReCalc) are shown for different degrees of cepstral smoothing as a function of azimuth.

Right panel: The difference of the correction term τmin calculated from unfiltered and low-pass filtered minimum phase impulse responses for different degrees of smoothing are shown for source positions in azimuth. The data is averaged across 10 subjects for both plots.

However, in their study the ITDs were computed from unfiltered impulses responses (i.e.

by removing Γ in equation 3.8). ITDs calculated in this way represent the group delay of the broadband signal. If the correction term τCorr is also computed from broadband impulses, the low frequency ITD deviates from the empirical low frequency ITD. This is illustrated in Figure 3.13(b). The absolute differences between the correction term τ_Corr computed from unfiltered and low-pass filtered minimum phase HRIRs are plotted as a function of azimuth for different degrees of cepstral smoothing. It can be seen from this figure that the low frequency group delay of the minimum phase HRIRs is clearly dif-ferent from the overall group delay of the unfiltered minimum phase impulse responses.

The range of the differences strongly depend on spectral smoothing, whereas the general shape of the curve is conserved.

The differences between the low frequency ITD of the empirical HRTFs and the ITD obtained from minimum phase plus frequency independent group delay HRIRs shown in Figure 3.13(a) are below the detection threshold (Durlach and Colburn, 1979;

Kinkel, 1990) and are, therefore, perceptually irrelevant. However, the differences of the minimum phase correction term shown in Figure 3.13(b)are above the detection thresh-old for lateral sound incidence, both the absolute differences between the empirical and the minimum phase plus delay ITDs and the differences between different degrees of

smoothing.

It can be concluded from this investigation, that only if τCorr is calculated from low pass filtered minimum phase impulse responses the smoothed minimum phase plus fre-quency independent group delay HRTFs are perceptually indistinguishable from empir-ical HRTFs.