Click here to purchase the entire book in PDF format. Diffuse field Now we have to talk about what a diffuse field is. If we get into the official definition of a diffuse field, then we have to have a talk about things like infinity, plane waves, phase relationships and probability distribution... maybe some other time... Instead, let's think about a diffuse field in a couple of different, equally acceptable ways. One way is to think that you have sound coming from everywhere simultaneously. Another way is that you have sound coming from different directions in succession with no time inbetween their arrival. If we think of reverberation as a very, very big number of reflections coming from all directions in fast succession, then we can start to think of what a diffuse field is like. Typically, we like to think of reveberation as a diffuse field - this is particularly true for the people that make digital reverb units because it's much easier to create random messes that sort of sound like reverb than it is to calculate everything that happens to sound as it bounces around a room for a couple of seconds. We need to pay a lot of attention to the correlation coefficient of the diffuse component of the recorded signal. This can be used as a rough guide to the overall sense of ``spaciousness'' (or whatever word you wish to use - this area creates a lot of discussion) in your recording. If you have a correlation coefficient of 1, this will probably mean that you have a reverberant sound that is completely clumped into one location between the two loudspeakers. The only possible exception to this is if your signals are going to the adjacent pair of front and surround loudspeakers (i.e. Left and Left Surround) where you'll find it very difficult to obtain a single phantom location. If your correlation coefficient is -1, then you have what most people call two ``out of phase'' signals, but what they really are is identical signals with opposite polarity. If your correlation coefficient is 0, then there could be a number of different explanations behind the result. For example, a pair of coincident bidirectionals with an included angle of 90$^\circ$ will have a correlation coefficient of 0. If we broke the signals hitting the two diaphragms into individual sounds from an infinite number of sources, then each one would have a correlation coefficient of either 1 or -1, but since there are as many 1's as -1's, the whole diffuse field averages out to a total correlation of 0. Although the two signals appear to be completely uncorrelated according to the math, there will be an even distribution of sound between the speakers (because there are some components in there that have a correlation of 1, remember...) On the other hand, if we take two omnidirectional microphones and put them very, very far apart - let's put them in completely different rooms to start, then the two signals are completely unrelated, therefore the correlation coefficient will be 0 and you'll get an image with no phantom sources at all - just two loudspeakers producing a pocket of sound. The same is true if you place the omni's very far apart in the same concert hall (you'll sometimes see engineers doing this for their ambience microphones). The resulting correlation coefficient, as we'll see below, will also be 0 because the sound fields at the two locations will sound similar, but they'll be completely unrelated. The result is a hall with a very large hole in the middle - because there are no correlated components in the two signals, there cannot be an even spread of energy between the loudspeakers. The moral of the story here is that, in order to keep a ``spacious'' sound for your reverb, you have to keep your correlation coefficient close or equal to 0, but you can't just rely on that one number to tell you everything. Spacious isn't necessarily pretty, or believable... Coincident pairs Calculating the correlation of the outputs of a pair of coincident microphones is somewhat less than simple. In fact, at the moment, I have to confess that I really don't know the correct equation for doing this. I've searched for this piece of information in all of my books, and I've asked everyone that I think would know the answer, and I haven't found it yet. So, I wrote some MATLAB code to model the situation instead of doing the math the right way. In other words, I did a numerical calculation to produce the plots in Figures 10.137 and 10.138, but this should give us the right answer. Some of the characteristics see in Figure 10.137 should be intuitive. For example, if you have a pair of coincident omnidirectional microphones in a diffuse field, then the correlation coefficient of their outputs will be 1 regardless of their included angle. This is because the outputs of the two mic's will be identical no matter what the angle. Also, if you have any matched pair of microphones with an included angle of 0$^\circ$, then their outputs will also be identical and the correlation coefficient will be 1. Finally, if you have a pair of matched bidirectional microphones with an included angle of 180$^\circ$, then their outputs will be identical but opposite in polarity, therefore their correlation coefficient will be -1. Everything else on that plot will be less obvious. Just in case you're wondering, here's how I calculated the two graphs in Figures 10.137 and 10.138. If you have a pair of coincident bidirectional microphones with an included angle of 90$^\circ$ in a diffuse field, then the correlation coefficient of their outputs will be 0. This is somewhat intuitive if we think that half of the field entering the microphone pair will have a correlation of 1 (all of the sound coming from the front and rear of the pair where the lobes have matched polarities) while the other half of the sound field results in a correlation of -1 (because it's entering the sides of the pair where you have opposite polarity lobes.) Since the area producing the correlation of 1 is identical to the area producing the correlation of -1, then the two cancel each other out and produce a correlation of 0 for the whole. Similarly, if we have a coincident bidirectional and an omnidirectional in a diffuse field, then the correlation coefficient of their outputs will also be 0 for the same reason. As we'll see in Section 10.5, if you have a coincident trio of microphones consisting of two bidirectionals at 90$^\circ$ and an omni, then you can create a microphone pointing in any direction in the horizontal plane with any polar pattern you wish - you just need to know what the relative mix of the three mic's should be. Using MATLAB, I produced three uncorrelated vectors containing a bunch of 10000 random numbers, each vector representing the output of each of the three microphones in that magic array described in the previous paragraph sitting in a noisy diffuse field. I then made two mixes of the three vectors to produce a simulation of a given pair of microphones. I then simply asked MATLAB to give me the correlation coefficient of these two simulated outputs. If someone could give me the appropriate equation to do this the right way, I would be very grateful. Figure 10.137: Correlation coefficients in the horizontal plane of a diffuse field for coincident omnidirectionals (top), subcardioids, cardioids, hypercardioids and bidirectionals (bottom) with included angles from 0$^\circ$ through 180$^\circ$. Figure 10.138: Correlation coefficients in the horizontal plane of a diffuse field for coincident microphones with an included angle of 180$^\circ$ and various values of P (remember that G = 1 - P). Spaced omnidirectionals If we have a pair of omnidirectionals spaced apart in a diffuse field, then we can intuitively get an idea of what their correlation coefficient will be. At 0 Hz, the pressure at the two locations of the microphones will be the same. This is because the sound pressure variations in the room are all varying the day's barometric pressure which is, for our purposes, 0 Hz. At very low frequencies, the wavelengths of the sound waves going past the microphones will be longer than the distance between the mic's. As a result, the two signals will be very correlated because the phase difference between the mic's is small. As we go higher and higher in frequency, then the correlation should be less and less, until, at some high frequency, the wavelengths are much smaller than the microphone separation. This means that the two signals will be completely unrelated and the correlation coefficient goes to 0. In fact, the relationship is a little more complicated than that, but not much. According to Kutruff [Kutruff, 1991], the correlation coefficient of two spaced locations in a theoretical diffuse field can be calculated using Equation 10.20. $$r = \frac{sin(kd)}{kd}$$ (11.20) where $k$ is the ``wave number.'' This is a slightly different way of saying ``frequency'' as can be seen in Equation 10.21 below (also see Equation 10.18). $$k = \frac{2 \pi f}{c}$$ (11.21) Note that $k$ is proportional to frequency and therefore inversely proporational to wavelength. If we were to calculate the correlation coefficient for a given microphone separation and all frequencies, the plot would look like Figure 10.139. Note that changes in the distance between the mic's will only change the frequency scale of the plot - the closer the mic's are to each other, the higher the frequency range of the area of high correlation. Figure 10.139: Correlation coefficient vs. Frequency for a pair of omnidirectional microphones with a separation of 30 cm in a diffuse field.