Echo density

Thus, in order to simulate a perceptually convincing room reverberation, it is necessary to simulate both the pattern of early echoes, with particular concern for lateral echoes, and the late energy decay relief. The latter can be parameterized as the frequency response envelope and the reverberation time, both of which are functions of frequency. The challenge is to design an artificial reverberator which has sufficient echo density in the time domain, sufficient density of maxima in the frequency domain, and a natural colorless timbre. 

The echo density of the parallel combs is the sum of the echo densities of the individual combs. Each comb filter i outputs one echo per timeT , thus the combined echo density, expressed as the number of echoes per second, is [Jot and Chaigne, 1991] ... 

The higher the order of the system, the longer it takes for the echo density to build up to a pleasing level. 

Figure 3.29 Modification of Schroeder s parallel comb filter to maximize echo density [Jot, 1992b],...

Moorer proposed a slightly different structure, shown in figure 3.7, where the late reverb is driven by the output of the early echo FIR filter [Moorer, 1979], Moorer described this as a way of increasing the echo density of the late reverberation. The delays D and D2 can be adjusted so that the first pulse output from the late reverberator corresponds with the last pulse output from the FIR section. The gain g serves to balance the amount of late reverberation with respect to the early echoes. An important feature of this structure, apart from the early echo modeling, is the control... 

By combining two elementary filters in series, we can dramatically increase the echo density, because every echo generated by the first filter will create a set of echoes in the second. Comb filters are not good candidates for series connection, because the only frequencies that will pass are those that correspond to peaks in both comb filter respones. However, any number of allpass filters can be connected in series, and the combined response will still be allpass. Consequently, series allpass filters are useful for increasing echo density without affecting the magnitude response of the system. 

A parallel combination of comb filters with incommensurate delays is also a useful structure, because the resulting frequency response contains peaks contributed by all of the individual comb filters. Moreover, the combined echo density is the sum of the individual densities. Thus, we can theoretically obtain arbitrary density of frequency peaks and time echoes by combining a sufficient number of comb filters in parallel. 

From equations 3.23 and 3.26, we can derive the number of comb filters required to achieve a given modal density D m and echo density De [Jot and Chaigne, 1991] ... 

Schroeder chose the parameters of his reverberator to have an echo density of 1000 echoes per second, and a frequency density of 0.15 peaks per Hz (one peak per 6.7 Hz). Strictly applying equation 3.27 using these densities would require 12 comb filters with a mean delay of 12 msec. However, this ignores the two series allpass filters, which will increase the echo density by approximately a factor of 10 [Schroeder, 1962], Thus, only 4 comb filters are required with a mean delay of 40 msec. 

The echo density is insufficient, and doesn t increase with time. 

Jot has described many lossless FDN prototypes based on unitary feedback matrices. A particularly useful unitary feedback matrix Ajy, which maximizes echo density while reducing implementation cost, is taken from the class of Householder matrices [Jot, 1992b] ... 

Another Japanese group used a system of 3D endoscopic ultrasonography to analyze the surface, the echo density, and the echo patterns of cross-sectional images of submucosal lesions. They evaluated the quality of the 3D image and cross-sectional images of lipomas, leiomyomas, and cysts. They stated that analysis of 3D EUS images was useful in making a... 

What is addressed by these sources is the ontology of quantal description. Wave functions (and other related quantities, like Green functions or density matrices), far from being mere compendia or short-hand listings of observational data, obtained in the domain of real numbers, possess an actuality of tbeir own. From a knowledge of the wave functions for real values of the variables and by relying on their analytical behavior for complex values, new properties come to the open, in a way that one can perhaps view, echoing the quotations above, as miraculous. ... 

In addition to the above techniques, inverse gas chromatography, swelling experiments, tensile tests, mechanical analyses, and small-angle neutron scattering have been used to determine the cross-link density of cured networks (240—245). Si soHd-state nmr and chemical degradation methods have been used to characterize cured networks stmcturaHy (246). H- and H-nmr and spin echo experiments have been used to study the dynamics of cured sihcone networks (247—250). 

Fig. 2.9.2 Radiofrequency, field gradient and current distributions requires a three-dimen-ionic current pulse sequences for two-dimen- sional imaging sequence [see Figure 2.9.1(a)] sional current density mapping. TE is the Hahn and multiple experiments with the orientation spin-echo time, Tc is the total application time of the sample relative to the magnetic field of ionic currents through the sample. The 180°- incremented until a full 360°-revolution is pulse combined with the z gradient is slice reached. The polarity of the current pulses...

We can perform spatially resolved Carr-Purcell-Meiboom-Gill (CPMG) experiments, and then, for each voxel, use magnetization intensities at the echo times to estimate the corresponding number density function, P(t), which represents the amount of fluid associated with the characteristic relaxation time t. The corresponding intrinsic magnetization for the voxel, M0, is calculated by... 

---