Regularisation

The introduction of regularisation was originally proposed in Kirkeby et al. (1998c) as a means to address the problem of time-aliasing that arises from the computation of the inverse in the frequency-domain (see also §2.3.4). This is done by modifying the exact inverse modelling problem of equations (1-1)–(1-3) to the optimisation problem of minimising the quadratic cost function of equation (2-18) which is now formulated using the frequency domain expression of the quantities defined in §1.220.

( )

( ) ( )

( ) ( )

J e ω =e eω e eω +βy eω y eω (2-18)

where the real valued and positive regularisation parameter β is introduced. In this cost function the minimisation of the squared error eHe is penalised by the term βyHy, which is proportionate to the magnitude of the control effort represented by the source input signals y. The unique minimum of the cost function of equation (2-18) can be shown (Nelson 1994) to be obtained when the frequency response of the inverse filter matrix H becomes equal to:

( )

( ) ( )

( )

j j j j

eω =⎡_⎢ e ω eω +β ⎤_⎥− eω

⎣ ⎦

H C C I C (2-19)

As is readily seen in equation (2-19), the introduction of regularisation effectively penalises the power of the control effort y that is needed for the exact inversion. That is, for a value of βequal to zero the optimal solution of equation (2-19) coincides with the exact solution of equation (1-4) while for positive values of the regularisation parameter β a trade-off is effected between the accuracy in the inversion and the required control effort. As is shown by Kirkeby et al. (1996), this translates to the replacement of each pole in the responses Hij(z) (i.e. each zero in the response det[C(z)]) with one zero and a pair of poles placed on either side of the unit circle in the z-plane21. These poles are further away form the unit circle than the original (non- regularised) pole of Hij(z) and consequently the impulse responses hij(n) will decay faster in time, thus reducing the required length of the inverse DFTs needed to

20 _{The superscript} H _{denotes the conjugate transpose.}

21 _{Obviously, this would also address the hypothetical case whereby a zero of det[C(z)] lies exactly on} the unit circle.

suppress the time-aliasing effect. This was the basic principle for the original introduction of regularisation as described Kirkeby et al. (1998c).

On the other hand, as is demonstrated in further detail with the results of Chapter 4, by moderating the applied control power the introduction of regularisation effectively increases the dynamic range of the inversion. As will become apparent in Chapter 4 this is a more desirable property of the introduction of regularisation, since the dynamic range of the non-regularised inversion can be prohibitively low. When used with this objective, however, an extension to the formulation of regularisation seems appropriate. This is because, as was also identified by Kirkeby and Nelson (1999), the control power needed for the exact inversion becomes excessively high at the low and high end of the spectrum where the application of control is either prohibitively inefficient or of no interest. The increase at the low end of the spectrum in the magnitude response of the inverse is due to the roll-off of the loudspeakers’ response22 at this region and also due to the acoustical properties of the reproduction23. In this low frequency region, the control is bound to be ineffective even when no regularisation is used. Similarly, when the plant matrix comprises measured responses, it will have to exhibit a roll-off at the high end of the spectrum (close to the Nyquist frequency) because of the antialiasing filters used for the measurement. Obviously, the correction of this roll-off to a flat response is of no interest. Hence it would be desirable to be able to apply a stricter optimisation penalty in these two frequency regions and penalise less (or even not at all) the control in the region in-between which we both wish to and are able to apply control more efficiently. This can be straightforwardly achieved by replacing the regularisation parameter with the frequency variable parameter β(ω) in equation (2-18) which then results in the form of the inverse matrix described by equation (2-20)24.

( )

( ) ( )

( )

( )

eω =⎡_⎢ e ω eω +β ω ⎤_⎥− eω

⎣ ⎦

H C C I C (2-20)

22 _{It should be noted that this is not true for the case where the plant matrix is modelled with HRTFs} which by definition should converge to unity at the DC limit (Algazi et al. 2001). It is however true in the case where the plant matrix also contains the responses of the loudspeakers used for the

reproduction, in which case the inverse filtering will have to correct these responses also.

23 _{As is discussed in detail in (Takeuchi 2001), the inverse control at the low frequency end of the} spectrum becomes ill-conditioned, a characteristic that is further amplified by small loudspeaker-span angles as is the case with the Stereo Dipole.

24 _{Another approach, formulated in the time-domain is taken by Kirkeby and Nelson (1999). We} believe the method proposed in equation (2-20) to be simpler.

The application of this form of frequency varying regularisation is further discussed in §4.3.