# 8 Directional sensitivity of microphones (Polar Patterns)

Each microphone is manufactured with an intended pattern of directional sensitivity. that is, it is variably sensitive to sounds arriving from different directions. Sensitivity is measured in terms of voltage output for a given sound pressure level input. Patterns range from omnidirectional, in which every direction is equally well transduced (at least in theory), to bidirectional, in which sounds from the sides of the microphone are strongly rejected. By changing the acoustical construction of the microphone, the polar pattern can be altered to suit the desired use. Polar patterns can also be adjusted by combining two or more transducers electronically to produce the desired combination.A sealed transducer element is sensitive to the absolute pressure at the front of the diaphragm or plate. When sensitive only to absolute pressure, the microphone is omnidirectional, since it cannot determine where in space the sound pressure originated. If an acoustical pathway for sound to reach the back of the diaphragm is provided, sounds originating from different directions may be forced to interact at the diaphragm of the microphone. By carefully tailoring the pathway, the microphone can be made to cancel sounds arriving from the rear or some other angle. The element is now said to be a pressure-gradient transducer, because it is sensitive to the pressure difference between the front and rear of the sensing element, both of which are then accessible to the sound wave. A pure pressure-gradient microphone responds to the resulting particle velocity and exhibits a bidirectional (figure-eight) polar pattern. Partially pressure-gradient sensitive microphones may exhibit cardioid, hyper-cardioid, and other direction-sensitive polar patterns, while pure pressure microphones are omnidirectional. Some polar patterns are shown in Figure 1 (the upward arrow indicates the front of the microphone):Figure 1: Polar patterns and their polar equations. The upward arrow indicates on-axis direction.

The following are the polar patterns of microphones from the cardioid family: $\rho(\theta)=\alpha+(1-\alpha)cos(\theta)$

The sensitivity of the microphone to sounds from a particular direction is indicated by the length of the radius r from the center of the plot to the perimeter at the angle $\theta$. The radius is expressed in 5 decibels per division, in a 25db scale.

As shown above, each microphone offers a distinct directivity pattern. It is possible to create a soundfield by combining them.

One microphone to rule them all:

In general the invention is applicable, in all cases, where it is desirable to give directional effects to emitted sound" Allen Dower Blumlein. 1414Allen Dower Blumlein. Inventor of the stereo 90^{{̂}} XY ’Blumlein’ pair patented in 1931. https://en.wikipedia.org/wiki/Alan_Blumlein

The stereo pair invented by Blumlein consists of two coincident bidirectional (figure-of-8) microphones at $90^{\circ}$. It was mono compatible and independent of the format of reproduction. By summing the left and right signal, a MID signal can be obtained and by subtracting the signals (left - right), a SIDE signal is obtained.

As shown above in figure 13, the pair can be rotated $45^{\circ}$ to obtain a front-back left-right soundfield that contains MID & SIDE signals as shown in figure 8. The figure-of-8 microphone has the positive lobe to the left and the negative to the right. Selective directivity can be achieved by operating on different microphones’ signals to obtain new patterns. In this case, it is possible to get a LEFT cardioid pattern by summing the signal of an OMNI and a figure-of-8 microphone, and a RIGHT cardioid signal by subtracting, see figure 9. Thus, we can obtain a LEFT-RIGHT cardioid microphone by combining Omni (MID) and a bidirectional (SIDE) patterns. The same applies to FRONT-BACK when rotating the microphones 90 degrees. See figure 10.

Note that the Front-Back becomes the X-axis and the left-right, the Y-axis. W represents the omnidirectional microphone. The array described above becomes an Ambisonic Horizontal B-Format soundfield microphone and can be steered in any horizontal direction in post production. The raw signal of the microphones is called A-Format and cannot be used until processed and converted to B-Format.

The pickup pattern of the Blumlein Omnidirectional and figure-of-eight (native MS) pair for one plane is:

 $g_{x}{}_{y}(\theta)=\begin{bmatrix}1\\ sin(\theta)\end{bmatrix}$

A signal s coming from the angle $\theta$ are recorded as $[X,Y]^{T}g(\theta)s$. At angles $0\textdegree$ (front), $45\textdegree$ (left), and $-45\textdegree$ (right) are represented by the following matrices respectively:

right: $g_{x}{}_{y}(-90\textdegree)=\begin{bmatrix}1\\ -1\end{bmatrix}$

center: $g_{x}{}_{y}(0\textdegree)=\begin{bmatrix}1\\ 0\end{bmatrix}$

left: $g_{x}{}_{y}(90\textdegree)=\begin{bmatrix}1\\ 1\end{bmatrix}$

This solution only carries information on the median and frontal planes and in figure 11 we can see that this 2-D setup works well for "coincidence". But how to position Z to capture information from the horizontal (height) plane ?