Phase variabilities and zeros in a reverberant transfer function

The phase trend of a transfer function (TF) has a nearly constant group delay in a reverberant sound field under high modal overlap conditions. This phase trend is predictable from the number of nonminimum phase zeros of the TF. This paper investigates the phase variability from such a phase trend in a reverberant TF. These phase fluctuations, which are generally due to random occurrences of both poles and zeros, can be expressed using the group delay, which is the first derivative of the accumulated phase. Theoretical analysis of the group delay implies that its fluctuations have a ‘‘1/f’’ type of spectrum and that the variance of the group delay is independent of the frequency band and decreases as the damping of the system increases. The results are confirmed by the power spectrum and correlation analysis using measured TF records. Accumulated phase variances are also investigated on the basis of the Δ statistic [F. Dyson and M. Mehta, ‘‘Statistical Theory of the Energy Levels of Complex Systems. IV,’’ J. Math. Phys. 4, 701 (1963)]. The variances increase in linear proportion to the frequency interval of interest, as the frequency interval increases; however, they increase in proportion to the square of the frequency interval when the frequency interval is short.