Click here to purchase the entire book in PDF format. Do-It-Yourself Polar Patterns If you go to your local microphone store and buy a ``normal'' single-diaphragm cardioid microphone (don't worry if you're surprised that there might be something other than a microphone with a single diaphragm... we'll talk about that later) the manufacturer has built the device so that it's the appropriate mixture of Pressure and Pressure Gradient. Consider, however, that if you have a perfect omnidirectional microphone and a perfect bidirectional microphone, then you could strap them together, mix their outputs electrically in a run-of-the-mill mixing console, and, assuming that everything was perfectly aligned, you'd be able to make your own cardioid. In fact, if the two real microhpones were exactly matched, you could make any polar pattern you wanted just by modifying the relative levels of the two signals. Mathematically speaking, the output of the omnidirectional microphone is the Pressure component and the output of the Bidirectional Microphone is the Pressure Gradient component. The two are just added in the mixer so you're fulfilling the standard sensitivity equation: $$S = P + G * cos(\alpha)$$ (7.13) where P is the gain applied to the omnidirectional microphone and G is the gain applied to the bidirectional microphone. Also, let's say that you have two cardioid microphones, but that you put them in a back-to-back configuration where the two are pointing 180$^{\circ }$ away from each other. Let's look at this pair mathematically. We'll call microphone 1 the one pointing ``forwards'' and microphone 2 the second microphone pointing 180$^{\circ }$ away. Note that we're also assuming for a moment that the gain applied to both microphones is the same. $$S_{TOTAL} = (0.5 + 0.5 * cos(\alpha)) + (0.5 + 0.5 * cos(\alpha + 180^{\circ}))$$ (7.14) $$= 0.5 + 0.5 + 0.5 * (cos(\alpha) + cos(\alpha + 180^{\circ}))$$ (7.15) $$= 1 + 0.5 * (cos(\alpha) + cos(\alpha + 180^{\circ}))$$ (7.16) Now, consider that the cosine of every angle is the opposite polarity to the cosine of the same angle + 180$^{\circ }$. In other words: $$cos(\alpha) = -1 * cos(\alpha + 180^{\circ})$$ (7.17) Therefore the cosine of any angle added to the cosine of the same angle + 180$^{\circ }$ will equal 0. In other words: $$cos(\alpha) + cos(\alpha + 180^{\circ}) = 0$$ (7.18) Let's go back to the equation that describes the back to back cardioids: $$S_{TOTAL} = 1 + 0.5 * (cos(\alpha) + cos(\alpha + 180))$$ (7.19) We now know that the two cosines cancel each other, therefore the equation simplifies to: $$S_{TOTAL} = 1 + 0.5 * (0)$$ (7.20) $$= 1$$ (7.21) Therefore, the result is an omnidirectional microphone. This result is possibly easier to understand intuitively if we look at graphs of the sensitivity patterns as is shown in Figures 6.112 and 6.113. Figure 6.112: Cartesian plot of the sensitivity patterns of two cardioid microphones aimed 180$^{\circ }$ apart. The blue plot is the forward-facing cardioid, the green is the rear-facing cardioid. Note that, if summed, the resulting output would be 1 for any angle. Figure 6.113: Polar plot of the sensitivity patterns of two cardioid microphones aimed 180$^{\circ }$ apart. Note that, if summed, the resulting output would be 1 for any angle. Similarly, if we inverted the polarity of the rear-facing microphone, the resulting mixed output (if the gains applied to the two cardioids were equal) would be a bidirectional microphone. The equation for this mixture would be: $$S_{TOTAL} = (0.5 + 0.5 * cos(\alpha)) + -1 * (0.5 + 0.5 * cos(\alpha + 180^{\circ}))$$ (7.22) $$= 0.5 + 0.5 * cos(\alpha) - 0.5 - 0.5 * cos(\alpha + 180^{\circ}))$$ (7.23) $$= 0.5 - 0.5 + 0.5 * cos(\alpha) - 0.5 * cos(\alpha + 180^{\circ})$$ (7.24) $$= 0.5 * cos(\alpha) - 0.5 * cos(\alpha + 180^{\circ})$$ (7.25) $$= cos(\alpha)$$ (7.26) So, as you can see, not only is it possible to create any microphone polar pattern using the summed outputs of a bidirectional and an omnidirectional microphone, it can be accomplished using two back-to-back cardioids as well. Of course, we're still assuming at this point that we're living in a perfect world where all transducers are matched - but we'll stay in that world for now...