Arrangement of bandpass filters

The first term in Eq.(22), $\alpha \ln \omega_k$, depends on how the center frequencies are determined. When we design the arrangement of filters so as to cancel the influence of the center-frequency distribution on the final result, the non-linear phase term is eliminated in Eq.(22). If the center frequencies are determined to have the following relation with arbitrary integer m:

$$\alpha \ln \omega_{k+1} = \alpha \ln \omega_k + 2\pi m,$$ (23)

then Eq.(22) can be rewritten as

$$\Psi_k(\omega)=-\{\alpha \ln \omega_1 + 2\pi(k-1)m + \omega (d_o+\tau_o) - \alpha - 2Q_k + 2v_o \alpha/c + \theta_o \};$$ (24)

and finally we obtain

$$\Psi_k(\omega)=-\{ \omega (d_o+\tau_o) + 2v_o \alpha/c + \Th... ... \ \ \ \Theta_o=\theta_o+\alpha \ln \omega_1 - \alpha -2 Q_k,$$ (25)

where the term $2\pi(k-1)m$ is omitted because k and m are integer values.

If we design each of the band-pass filters to have the same Q value, then the spectral phase given in Eq.(25) is common to all the filters. Thus the summation of the delayed outputs of all the filters has a maximum amplitude at the time $d_o + \tau_o$ with the phase $-(2v_o \alpha/c + \Theta_o)$.

Furthermore, Eq.(23) indicates that the band-pass filters distribute uniformly over the logarithmic frequency: a bank of constant-Q filters is thus constructed. As a result, if we emit an LPM sound, we can estimate the range and speed of a moving object by investigating the phase distribution of the bank of constant-Q filters.