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Random-Energy Response (RER)

Think about an omnidirectional microphone in a diffuse field (the concept of a diffuse field is explained in Section 3.1.21). The omni is equally sensitive to all sounds coming from all directions, giving it some output level. If you put a cardioid microphone in exactly the same place, you wouldn't get as much output from it because, although it's as sensitive to on-axis sounds as the omni, all other directions will be attenuated in comparison.

Since a diffuse field is comprised of random signals coming from random directions, we call the theoretical power output of a microphone in a diffuse field the Random-Energy Response or RER. Note that this measurement is of the power output of the microphone.

The easiest way to get an intuitive understanding of the RER of a given polar pattern is that it is simply the square of the surface area of a three-dimensional plot of the pattern. The reason we square the surface area is that we are looking at the power of the output which, as we saw in Section 2.2, is the square of the signal.

The RER of any polar pattern can be calculated using Equation 6.27.

$$RER = \int_{0}^{\pi} \int_{0}^{2 \pi} S^{2} \sin \alpha d \phi d \alpha$$ (7.27)

where S is the sensitivity of the microphone, $\alpha$ is the angle of rotation around the microphone's ``equator'' and $\phi$ is the angle of rotation around the microphones axis. These two angles are shown in the explanation of spherical coordinates later in the book in Section 10.4.2.

If you're having some difficulties grasping the intricacies of Equation 6.27, don't panic. Double integrals aren't something we see every day. We know from Section 1.9 that, because we're dealing with integrals, then we must be looking for the area of some shape. So far so good. (The area we're looking for is the surface area of the three-dimensional plot of the polar pattern.)

FINISH THIS OFF

There are a couple of good rules of thumb to remember when it comes to RER.

1. An omni has the greatest sum of sensitivities to sounds from all directions, therefore it has the highest RER of all polar patterns.
2. A cardioid and a bidirectional both have the same RER.
3. A hypercardioid has the lowest RER of all first-order gradient polar patterns.

COMMENT HERE ABOUT REVERBERATION AND DIRECT TO REVERBERANT RATIOS

Table 6.8: Random Energy Responses for various microphone polar patterns.

Polar Pattern (P : G)	RER	RER (decimal)
Omnidirectional (1 : 0)	$4 \pi$	12.57
Subcardioid (0.75 : 0.25)	$\frac{7 \pi}{3}$	7.33
Cardioid (0.5 : 0.5)	$\frac{4 \pi}{3}$	4.19
Supercardioid (0.375 : 0.625)	$\frac{13 \pi}{12}$	3.40
Hypercardioid (0.25 : 0.75)	$\pi$	3.14
Bidirectional (0 : 1)	$\frac{4 \pi}{3}$	4.19

Figure 6.121: Random Energy Response vs. the Pressure component, $P$ in the microphone.