There are two basic microphone types, each with a different polar pattern; these are named "pressure" and "pressure-gradient" transducers.
A simple pressure transducer responds to pressure variations in the air adjacent to one side of the diaphragm. We normally use the term "omnidirectional" to describe this sensitivity pattern as it is theoretically equally sensitive to sound sources at all angles of incidence to the microphone. The equation used to describe this pattern is
s = 1
where s is the sensitivity of the transducer. As is evident in the equation, the sensitivity of the microphone is 1 (or 100%), irrespective of the angle of incidence of the sound.
The pressure-gradient microphone responds to the difference in pressure between the two sides of its diaphragm. The resulting polar pattern is usually described as "bidirectional" or "figure-8" and has the sensitivity equation
s = Cos (a)
where s is the sensitivity of the transducer and a is the angle of incidence of the sound (where 0 degrees indicates a sound source directly in front of the diaphragm). The important thing to note about this polar pattern, apart from its basic shape is that one of the two "lobes" (or areas of sensitivity) has a negative polarity relative to the pressure applied to it. That is to say that a positive pressure incident on the diaphragm from some angles will result in a negative voltage output.
First-order gradient microphone polar patterns are determined by the relationship in level between the pressure and pressure gradient components. In effect, we are using an omnidirectional and a bidirectional in some predetermined combination to create a desired sensitivity pattern. Table 2 illustrates the relative levels required for the five most common directional patterns.
Table 2
| Pattern |
Pressure component |
Pressure-gradient Component |
| Omnidirectional |
1 |
0 |
| Subcardioid |
0.75 |
0.25 |
| Cardioid |
0.5 |
0.5 |
| Hypercardioid |
0.25 |
0.75 |
| Bidirectional |
0 |
1 |
As can be seen in the previous table, a cardioid microphone is simply a transducer that has equal pressure and pressure-gradient components. This can either be created by constructing a transducer whose acoustic design includes both components or by electrically mixing the outputs of two transducers (as is the case in many variable-pattern microphones).
This combination of components can be represented mathematically by integrating the table of relative levels with the sensitivity equations for the pressure and pressure-gradient patterns as shown in Table 3.
Table 3
| Polar Sensitivity Pattern |
Sensitivity Equation |
| General Equation |
Pressure component + Pressure-gradient Component |
| Omnidirectional |
1 + 0 Cos (a) |
| Subcardioid |
0.75 + 0.25 Cos (a) |
| Cardioid |
0.5 + 0.5 Cos (a) |
| Hypercardioid |
0.25 + 0.75 Cos (a) |
| Bidirectional |
0 + 1 Cos (a) |
where "a" is the angle of incidence of the sound
These equations are used to find the sensitivity of each microphone type relative to its on-axis sensitivity dependent on the angle of incidence of the sound. For example, if we wished to find the sensitivity of a cardioid microphone to a sound source placed 90 degrees off-axis to the diaphragm, we would use the following steps:
Cardioid Sensitivity = 0.5 + 0.5 Cos (a)
Cardioid Sensitivity = 0.5 + 0.5 Cos (90 degrees)
Cardioid Sensitivity = 0.5 + 0.5 * 0
Cardioid Sensitivity = 0.5
Therefore a cardioid microphone is 50% less sensitive to a sound source 90 degrees off-axis to its diaphragm than it is to a sound source on-axis. This difference can also be expressed in decibels using the following equation:
DdB = 20 * Log (On-axis Sensitivity / Off-axis Sensitivity)
DdB = 20 * Log 0.5
DdB = 20 * -0.301
DdB = -6.02 dB
Therefore a cardioid microphone is 6.02 dB less sensitive to a sound source 90 degrees off-axis as it is to a source on-axis.
By applying this equation to all angles of incidence around a cardioid microphone, a polar pattern can be plotted to represent the different sensitivities of the transducer as is shown in Figure 1. In this graph, the x-axis represents the angle of the sound source and the y-axis shows the sensitivity of the microphone. As is shown in the plot, the microphone has a sensitivity of 1 at 0 degrees (on-axis) and a sensitivity of 0 at 180 degrees. This graph is more commonly represented as a polar plot where 0 degrees is shown at the top of the chart with differing angles represented as the actual angle on the chart. The sensitivity is taken as the distance from the centre of the graph to the plot at a particular angle. An equivalent polar plot to Figure 1 is shown in Figure 2.
Figure 1 - Cartesian plot of the sensitivity pattern of a cardioid microphone.
S = 0.5 + -.5 Cos (a)
Although we have used a cardioid microphone in the previous examples, any polar pattern may be represented similarly through substitution with the appropriate sensitivity equation.
Figure 2 - Polar plot of the sensitivity pattern of a cardioid microphone.
S = 0.5 + -.5 Cos (a)
This investigation of microphone pair characteristics is restricted to configurations which rely on either interchannel intensity differences or interchannel time differences exclusively, and thus concentrates on coincident directional and spaced omnidirectional microphones. Coincident microphone techniques rely solely on intensity differences between the outputs of the two microphone, since both capsules are positioned so as to be equidistant from all sound sources. In theory, both diaphragms occupy the same point in space - as this is not possible, they are usually arranged with one diaphragm directly over the other. In this configuration, the time-of-arrival from a sound source at any angle of incidence to the pair in the horizontal plane is identical. The left-right imaging characteristics of the pair, therefore, depends on the different directional characteristics of the two microphones, as will be discussed to further detail later in the article.
Figure 3 - A pair of coincident microphones where "A" is the included angle of the pair and "a" is the angle of incidence.
Spaced omnidirectional microphones are arranged to generate different time-of-arrival cues according to different placements of the sound source. The two diaphragms are arranged usually between 30 and 100 cm apart, on a line parallel to the edge of the stage, assuming that a stage exists.
Cardioid Microphone Pairs
If we consider a pair of microphones which has an included angle of 90 degrees (the included angle of a pair of microphones is the angle determined by the two microphones as is shown in Figure 3), one microphone will be aimed 45 degrees to the right of centre-stage and the other to 45 degrees to the left of centre-stage. For the remainder of the paper, we will say that angles to the right are positive and angles to the left, negative. Thus the left microphone is at -45 degrees and the right microphone, 45 degrees. The polar patterns of these two microphones are shown in Figure 4.
Figure 4 - Cartesian plot of the sensitivity pattern of two cardioid microphones with an included angle of 90 degrees.
As was stated previously, the position of images from left to right in the stereo sound stage is entirely dependent on the difference in the intensity of the two channels. Given that we can calculate the sensitivity of the outputs of these two microphones according to different angles to the sound source, we can therefore calculate the difference in their outputs. In terms of the graph in Figure 4, we are calculating the vertical distance between the two plots. Since our information for specific image placements is listed in decibels, we will calculate the difference in the level of the two channels accordingly.
DdB = 20 * Log (Left Sensitivity / Right Sensitivity)
DdB = 20 * Log ((0.5 + 0.5 Cos (A/2 - a)) / (0.5 + 0.5 Cos (A/2 + a)))
DdB = 20 * Log ((0.5 + 0.5 Cos (45 degrees/2 - a)) / (0.5 + 0.5 Cos (45 degrees/2 + a)))
where "A" is the included angle of the microphone pair and "a" is the angle of incidence of the sound source to the pair .
By substituting all angles from -180 degrees to 180 degrees into this equation and plotting the results, we arrive at the graph shown in Figure 5.
Figure 5 - Cartesian plot of the amplitude difference of two cardioid microphones with an included angle of 90 degrees.
This graph shows us the difference in decibels between the outputs of the two microphones according to the angle to the sound source. The horizontal axis again, represents the angle to the sound source and the vertical axis the difference between the outputs. Positive results indicate that the right channel has a higher amplitude than the left, negative numbers show the opposite. The extreme high and low points on the graph correspond to the null points in the two microphones. Since the output of a cardioid microphone at 180 degrees is 0 and the difference in decibels between any signal and no signal is infinite, the graph has very sharp, extreme peaks.
According to Simonsen's findings, a 15 dB difference between the two channels results in a 30 degrees phantom image shift. Any difference greater than 15 dB results in the same image placement, since we are assuming that the image cannot shift to a location outside the loudspeaker. We can therefore reduce the scope of our graph, concentrating on only the first 15 dB of difference between the two channels as is shown in Figure 6.
Figure 6 - Detail of a cartesian plot of the amplitude difference of two cardioid microphones with an included andgle of 90 degrees.
Thus far, the graphs have been restricted to a pair of cardioid microphones with an included angle of 90 degrees. This angle, however, can be varied from 0 degrees to 180 degrees. We can therefore consider Figure 6 to be a "slice" of a more comprehensive plot shown in Figure 7 which shows the same information for all included angles. Again, the x-axis represents the angle to the sound source relative to the pair and the y-axis the difference between the amplitude of the two channels. We now, however, include a z-axis which indicates the included angle of the microphone pair.
Although this plot proves to be of some use in forming an intuitive view of the response of microphone pairs, its format does not provide detailed information. Since our primary concern for the time being is Simonsen's results of 2.5 dB, 5.5 dB and 15 dB, we calculate what is essentially a topographical map of the three-dimensional plot in Figure 7. This new graph, shown in Figure 8, shows 7 curves representing the -15 dB, -5.5 dB, -2.5 dB, 0 dB, 2.5 dB, 5.5 dB and 15 dB differences for all included angles of a pair of cardioid microphones.
This plot demonstrates a number of interesting characteristics of a pair of cardioid microphones.
Figure 7 - Three-dimensional cartesian plot of the amplitude difference of two coincident cardioid microphones with included angles of 0 degrees to 180 degrees.
Firstly, given an included angle of 90 degrees for example, it can be seen that the 10 degrees, 20 degrees and 30 degrees positions in the stereo sound stage (2.5 dB, 5.5 dB and 15 dB respectively) are the result of sound sources at 20 degrees, 41 degrees and 89 degrees on stage. Therefore a musician sitting 20 degrees off-centre on stage will appear 10 degrees off-centre in the stereo sound stage and so on. Consider that 10 degrees, 20 degrees and 30 degrees in the sound stage are all equal spacing separated by 10 degrees - yet these positions correspond to unequal distances on stage. Therefore in a "normal" situation (for example, when recording a symphony orchestra) more of the group will be compressed into the outside of the sound stage, since 0 degrees - 20 degrees on stage occupies 0 degrees - 10 degrees during the playback, while 41 degrees - 89 degrees on stage fill the same width, albeit in a different location, between the loudspeakers.) This phenomenon, dubbed "angular distortion" has been extensively evaluated and discussed by Michael Williams (Williams 1990).
Figure 8 - Contour plot for a pair of cardioid microphones showing various interchannel intensity differences in dB as indicated.
We must also consider that the contours continue at the "rear" of the microphone pair. Again using the 90 degrees included angle example, the 10 degrees, 20 degrees, and 30 degrees points occur at 176 degrees, 173 degrees, and 161 degrees respectively. This fact has two implications. It tells us the maximum angle for a sound source whose image is located in one loudspeaker. (Since all differences greater than 15 dB result in images at 30 degrees, all sound sources, in the case of 90 degrees cardioids, situated between 89 degrees and 161 degrees inclusive will appear to be positioned in one loudspeaker.) The second point is that the spread of the 10 degrees, 20 degrees, and 30 degrees phantom positions are, as is the case with the front of the pair, unevenly distributed.
Finally, the graph allows us to compare the imaging characteristics of different included angles. By extracting the appropriate data from a number of configurations (in the case of Table 4, every 5 degrees from 50 degrees to 135 degrees inclusive) we can create a table which tells us the sound source angles of incidence which result in 10 degrees, 20 degrees, and 30 degrees image placements in the stereo sound stage. Given these results, listed in Table 4, a comparison of actual positions on the stage which result in similar phantom image placements is greatly simplified. For example, in order to achieve a phantom image placement of 20 degrees with a pair of coincident cardioids with an included angle of 90 degrees, the instrument will be positioned 41 degrees off-centre in front of the microphone pair. To achieve the same image placement with an included angle of 120 degrees, the performer must be re-positioned to 30 degrees off-centre.
Table 4
Coincident Cardioid Microphone Pairs
Included angles of 50 degrees - 135 degrees showing one side of the pair. The opposite side is symmetrical.
| Included Angle |
10 degrees |
20 degrees |
30 degrees |
30 degrees |
20 degrees |
10 degrees |
|
2.5 dB |
5.5 dB |
15 dB |
15 dB |
5.5 dB |
2.5 dB |
| 50 degrees |
36 degrees |
70 degrees |
123 degrees |
170 degrees |
176 degrees |
178 degrees |
| 55 degrees |
33 degrees |
65 degrees |
118 degrees |
169 degrees |
175 degrees |
178 degrees |
| 60 degrees |
30 degrees |
61 degrees |
113 degrees |
168 degrees |
175 degrees |
178 degrees |
| 65 degrees |
28 degrees |
56 degrees |
109 degrees |
166 degrees |
175 degrees |
177 degrees |
| 70 degrees |
25 degrees |
53 degrees |
104 degrees |
165 degrees |
175 degrees |
177 degrees |
| 75 degrees |
24 degrees |
50 degrees |
100 degrees |
165 degrees |
174 degrees |
177 degrees |
| 80 degrees |
23 degrees |
46 degrees |
96 degrees |
164 degrees |
174 degrees |
177 degrees |
| 85 degrees |
21 degrees |
44 degrees |
92 degrees |
162 degrees |
173 degrees |
176 degrees |
| 90 degrees |
20 degrees |
41 degrees |
89 degrees |
161 degrees |
173 degrees |
176 degrees |
| 95 degrees |
18 degrees |
40 degrees |
85 degrees |
160 degrees |
172 degrees |
176 degrees |
| 100 degrees |
17 degrees |
37 degrees |
82 degrees |
158 degrees |
172 degrees |
176 degrees |
| 105 degrees |
16 degrees |
35 degrees |
79 degrees |
157 degrees |
171 degrees |
176 degrees |
| 110 degrees |
15 degrees |
33 degrees |
76 degrees |
156 degrees |
171 degrees |
175 degrees |
| 115 degrees |
15 degrees |
32 degrees |
74 degrees |
155 degrees |
170 degrees |
175 degrees |
| 120 degrees |
14 degrees |
30 degrees |
70 degrees |
153 degrees |
170 degrees |
175 degrees |
| 125 degrees |
14 degrees |
29 degrees |
68 degrees |
152 degrees |
169 degrees |
175 degrees |
| 130 degrees |
13 degrees |
27 degrees |
65 degrees |
151 degrees |
168 degrees |
175 degrees |
| 135 degrees |
12 degrees |
26 degrees |
63 degrees |
150 degrees |
168 degrees |
174 degrees |
The data in Table 4 can be used to generate a form of "polar plot" for the microphone pairs in question, showing the angles of incidence required to produce amplitude differences of 2.5 dB, 5.5 dB and 15 dB. These plots, shown in Figure 9, show the six locations on either side of centre which produce the appropriate amplitude differences. The lightly shaded sections of the graph indicate positions that will result in phantom image placements of 30 degrees since the interchannel intensity differences of sound sources in these locations are greater than 15 dB. The diagrams show that a pair of coincident cardioid microphones with an included angle of 80 degrees presents a wider portion of the stage between the loudspeakers than a pair with an included angle of 110 degrees. It is, however, interesting to note that, while increasing the included angle of the microphones decreases the frontal angle located between the -30 degrees and 30 degrees extremes of the reproduced sound stage, it increases the rear angle. As a result, larger included angles of cardioid microphone pairs will place more of the ambient hall and audience sound between the loudspeakers rather than in them while conversely pushing more of the stage area into the loudspeakers rather than between them.
Figure 9 - "Polar patterns" of various pairs of coincident cardioid microphones showing the angles of incidence required to result in -15 dB, -5.5 dB, -2.5 dB, 0 dB, 2.5 dB, 5.5 dB, and 15 dB interchannel amplitude differences.
Bidirectional Microphone Pairs
The same procedures that were used for coincident cardioid microphones can be applied to pairs of bidirectional transducers. The primary difference lies in the equation, since, as was previously discussed, the mathematical representation of a pressure gr adient transducer contains no pressure component. Thus, our equation is as follows :
DdB = 20 * Log | Cos (A/2 - a) / Cos (A/2 + a) |
where "A" is the included angle of the microphone pair and "a" is the angle to the sound source relative to the pair.
Note that we are now performing the calculation using the absolute value of the ratio of the two amplitudes. This is due to the negative rear lobes in the microphone polar patterns. When using a pair of coincident bidirectional or hypercardioid capsules, there are a number of angles of incidence which result in the two channels having an opposite polarity. Such a signal difference generates an error when we attempt to calculate the interchannel amplitude ratio in decibels, since we cannot calculate the logarithm of a negative number. As a result, we must first convert the ratio to a positive number before finding its logarithm. The results of this equation will show us only the amplitude difference between the two channels irrespective of their relative polarities. This omission of data must be considered in any later analysis of the various pairs' response patterns.
Figure 10 - Three-dimensional cartesian plot of the amplitude difference of two coincident bidirectional microphones with included angles of 0 degrees to 180 degrees.
When we apply this equation to included angles between 0 degrees and 180 degrees, we obtain the three-dimensional Cartesian plot shown in Figure 10. The primary characteristic of this graph to be noted is the left-right placement of the images. As in previous plots, a positive result on the vertical axis indicates that the right channel is stronger than the left. In Figure 10, we can see that sound source placements in the 0 degrees - 90 degrees quadrant in front of the microphones result in a stronger right channel. Sources placed between 90 degrees and 180 degrees will have a greater intensity in the left channel. Thus, sound sources on the rear right of the microphone pair will image on the left. This is not surprising, since we would expect that the predominant signal from such a source would occur in the rear lobe of the left microphone.
It is well known by any who have experimented with opposite polarity channels in stereo reproduction that such signals result in a very unstable phantom image position. High frequencies tend to locate somewhere between the loudspeakers, the mid range information appears to originate from locations outside the ±30 degrees defined by the loudspeakers (going so far as to appear behind the listener) and the low frequency information disappears completely. However, this effect happens when the two channels are relatively equal in intensity. Large amplitude differences will result in the phantom image being located at 30 degrees in the louder of the two loudspeakers.
Following a rather informal experiment using various types of full-bandwidth pre-recorded music as program material and myself as the sole test subject, I found that the minimum interchannel intensity difference possible for a 30 degrees phantom image placement when the channels are exactly opposite in polarity is approximately 11 dB. Any smaller amplitude difference results in the unstable image position described previously. Differences of 11 dB or more will place the phantom image in the louder of the two loudspeakers. This threshold can be considered to be equivalent to the 15 dB interchannel difference found by Simonsen for phase-correlated stereo channels.
The contour plot for pairs of bidirectional microphones therefore, contains two extra curves, showing the positions required to produce an interchannel amplitude difference of 11 dB or -11 dB when the channels are of opposite polarity (denoted "11 dBø" and "-11 dBø"). Any sound source located between the angles which generate the 15 dB and 11 dBø difference will image at 30 degrees in the louder loudspeaker. Sources in positions between the 11 dBø and -11 dBø positions will have very unstable images with maximum instability at the point where the two channels are of opposite polarity but equal amplitude (marked "0 dBø" on the plots).
As is apparent from Figure 10 and its corresponding contour plot shown in Figure 11, the 0 dBø positions for all configurations of coincident bidirectional microphones are at 90 degrees and 270 degrees.
As was the case with cardioid microphone pairs, the frontal angle between the 15 dB and -15 dB interchannel amplitude differences becomes more narrow as the included angle increases. Unlike coincident cardioids, however, the rear angle also decreases as the included angle increases. The lightly shaded regions in the polar plots in Figure 12 indicate sound source locations that result in a phantom image placement of ±30 degrees as indicated on the graphs. The darker regions are areas that result in interchannel amplitude differences between -11 dB and 11 dB with opposite polarity and thus have unstable and therefore unpredictable phantom image locations.
Figure 11 - Contour plot for a pair of bidirectional microphones showing various interchannel intensity differences in dB as indicated.
Table 5
Coincident Bidirectional Microphone Pairs
Included angles of 50 degrees - 135 degrees showing one side of the pair. The opposite side is symmetrical.
| Included Angle |
10 degrees |
20 degrees |
30 degrees |
-30 degrees |
-20 degrees |
-10 degrees |
30 degrees ø |
-30 degrees ø |
|
2.5 dB |
5.5 dB |
15 dB |
15 dB |
5.5 dB |
2.5 dB |
11 dBø |
-11 dBø |
|
50 degrees |
17 degrees |
33 degrees |
56 degrees |
124 degrees |
147 degrees |
163 degrees |
75 degrees |
105 degrees |
| 55 degrees |
15 degrees |
30 degrees |
53 degrees |
127 degrees |
150 degrees |
165 degrees |
74 degrees |
106 degrees |
| 60 degrees |
13 degrees |
27 degrees |
50 degrees |
130 degrees |
153 degrees |
167 degrees |
72 degrees |
108 degrees |
| 65 degrees |
13 degrees |
26 degrees |
47 degrees |
133 degrees |
154 degrees |
167 degrees |
70 degrees |
110 degrees |
| 70 degrees |
11 degrees |
24 degrees |
44 degrees |
136 degrees |
156 degrees |
169 degrees |
68 degrees |
112 degrees |
| 75 degrees |
10 degrees |
22 degrees |
42 degrees |
138 degrees |
158 degrees |
170 degrees |
67 degrees |
113 degrees |
| 80 degrees |
10 degrees |
20 degrees |
39 degrees |
141 degrees |
160 degrees |
170 degrees |
65 degrees |
115 degrees |
| 85 degrees |
9 degrees |
19 degrees |
37 degrees |
143 degrees |
161 degrees |
171 degrees |
63 degrees |
117 degrees |
| 90 degrees |
8 degrees |
17 degrees |
34 degrees |
146 degrees |
163 degrees |
172 degrees |
60 degrees |
120 degrees |
| 95 degrees |
7 degrees |
15 degrees |
32 degrees |
148 degrees |
165 degrees |
173 degrees |
58 degrees |
122 degrees |
| 100 degrees |
6 degrees |
14 degrees |
30 degrees |
150 degrees |
166 degrees |
174 degrees |
56 degrees |
124 degrees |
| 105 degrees |
5 degrees |
13 degrees |
27 degrees |
153 degrees |
167 degrees |
175 degrees |
54 degrees |
126 degrees |
| 110 degrees |
5 degrees |
12 degrees |
26 degrees |
154 degrees |
168 degrees |
175 degrees |
51 degrees |
129 degrees |
| 115 degrees |
5 degrees |
11 degrees |
24 degrees |
156 degrees |
169 degrees |
175 degrees |
49 degrees |
131 degrees |
| 120 degrees |
4 degrees |
10 degrees |
21 degrees |
159 degrees |
170 degrees |
176 degrees |
46 degrees |
134 degrees |
| 125 degrees |
4 degrees |
9 degrees |
20 degrees |
160 degrees |
171 degrees |
176 degrees |
43 degrees |
137 degrees |
| 130 degrees |
3 degrees |
8 degrees |
18 degrees |
162 degrees |
172 degrees |
177 degrees |
40 degrees |
140 degrees |
| 135 degrees |
3 degrees |
7 degrees |
15 degrees |
165 degrees |
173 degrees |
177 degrees |
37 degrees |
143 degrees |
Figure 12 - "Polar patterns" of various pairs of coincident bidirectional microphones showing the angles of incidence required to result in -15 dB, -11 dBø, -5.5 dB, -2.5 dB, 0 dB, 2.5 dB, 5.5 dB, 11 dBø and 15 dB interchannel amplitude differences.
Hypercardioid Microphone Pairs
With some minor exceptions, the procedure and analysis of hypercardioid microphone pairs is almost identical to that of pairs of bidirectional transducers. The first and most significant difference, of course, lies in the equation used to create the relevant graphs and table. It is as follows.
DdB = 20 * Log | (0.25 + 0.75 Cos (A/2 - a)) / (0.25 + 0.75 Cos (A/2 + a)) |
where "A" is the included angle of the microphone pair and "a" is the angle to the sound source relative to the pair.
The resulting appropriate data are presented in the graphs shown in Figures 13, 14 and 15 and Table 6. There are two interesting characteristics to note using the three-dimensional plot and the contour representation.
Figure 13 - Three-dimensional cartesian plot of the amplitude difference of two coincident hypercardioid microphones with included angles of 0 degrees to 180 degrees.
Firstly, the maximum included angle for hypercardioid microphone pairs which results in equal interchannel amplitude differences but opposite polarities is approximately 140 degrees. This is in contrast to pairs of bidirectional microphones which have such positions with every included angle except 0 degrees. (The previous statement should be qualified with a note that an included angle of 180 degrees with a pair of bidirectional microphones will result in all positions generating equal intensity, opposite polarity information in the two stereo channels.)
Figure 14 - Contour plot for a pair of hypercardioid microphones showing various interchannel intensity differences in dB as indicated.
Secondly, included angles of less than approximately 145 degrees result in an imaging characteristic analogous to that of a pair of bidirectional transducers in that sound sources in the front right image on the right, whereas sources in the rear right image on the left due to the predominance of the rear lobe of the left microphone. At included angles of more than approximately 145 degrees, this general characteristic changes to be more similar to a pair of cardioid microphones since the effect of the relatively weak rear lobes of the transducers is negated by the overlapping front lobes. This is evident in Figure 13 as the small sections in the "rear corners" of the plot and in Figure 14 as the contour lines in the top corners of the graph.
Table 5
Coincident Hypercardioid Microphone Pairs
Included angles of 50 degrees - 135 degrees showing one side of the pair. The opposite side is symmetrical.
| Included Angle |
10 degrees |
20 degrees |
30 degrees |
-30 degrees |
-20 degrees |
-10 degrees |
30 degrees ø |
-30 degrees ø |
|
2.5 dB |
5.5 dB |
15 dB |
15 dB |
5.5 dB |
2.5 dB |
11 dBø |
11 dBø |
| 50 degrees |
23 degrees |
45 degrees |
74 degrees |
142 degrees |
158 degrees |
169 degrees |
96 degrees |
125 degrees |
| 55 degrees |
21 degrees |
41 degrees |
70 degrees |
144 degrees |
160 degrees |
170 degrees |
95 degrees |
127 degrees |
| 60 degrees |
19 degrees |
38 degrees |
67 degrees |
147 degrees |
163 degrees |
171 degrees |
94 degrees |
120 degrees |
| 65 degrees |
18 degrees |
35 degrees |
64 degrees |
150 degrees |
164 degrees |
172 degrees |
92 degrees |
131 degrees |
| 70 degrees |
16 degrees |
33 degrees |
61 degrees |
152 degrees |
166 degrees |
173 degrees |
91 degrees |
133 degrees |
| 75 degrees |
15 degrees |
31 degrees |
58 degrees |
154 degrees |
167 degrees |
174 degrees |
89 degrees |
135 degrees |
| 80 degrees |
14 degrees |
28 degrees |
56 degrees |
157 degrees |
168 degrees |
175 degrees |
88 degrees |
138 degrees |
| 85 degrees |
13 degrees |
26 degrees |
53 degrees |
159 degrees |
170 degrees |
175 degrees |
86 degrees |
140 degrees |
| 90 degrees |
12 degrees |
25 degrees |
50 degrees |
161 degrees |
171 degrees |
176 degrees |
85 degrees |
144 degrees |
| 95 degrees |
11 degrees |
24 degrees |
48 degrees |
163 degrees |
172 degrees |
176 degrees |
84 degrees |
146 degrees |
| 100 degrees |
10 degrees |
22 degrees |
45 degrees |
165 degrees |
173 degrees |
177 degrees |
82 degrees |
149 degrees |
| 105 degrees |
10 degrees |
20 degrees |
43 degrees |
167 degrees |
174 degrees |
177 degrees |
80 degrees |
152 degrees |
| 110 degrees |
9 degrees |
19 degrees |
40 degrees |
169 degrees |
175 degrees |
178 degrees |
78 degrees |
155 degrees |
| 115 degrees |
8 degrees |
18 degrees |
38 degrees |
172 degrees |
176 degrees |
178 degrees |
76 degrees |
158 degrees |
| 120 degrees |
8 degrees |
16 degrees |
36 degrees |
174 degrees |
177 degrees |
178 degrees |
75 degrees |
162 degrees |
| 125 degrees |
7 degrees |
15 degrees |
34 degrees |
175 degrees |
178 degrees |
179 degrees |
73 degrees |
166 degrees |
| 130 degrees |
6 degrees |
14 degrees |
32 degrees |
NA |
NA |
NA |
70 degrees |
168 degrees |
| 135 degrees |
6 degrees |
13 degrees |
30 degrees |
NA |
NA |
NA |
68 degrees |
NA |
Figure 15 - "Polar patterns" of various pairs of coincident hypercardioid microphones showing the angles of incidence required to result in -15 dB, -11 dBø, -5.5 dB, -2.5 dB, 0 dB, 2.5 dB, 5.5 dB, 11 dBø and 15 dB interchannel amplitude differences.
Omnidirectional Microphone Pairs
Unlike coincident directional microphones, spaced omnidirectional capsules rely on different times of arrival to achieve the desired stereo image locations. Since the separation between the diaphragms is rarely more than approximately 1 m, and the sound source, particularly in the case of classical music, is comparatively much farther from the pair, we assume that the amplitudes of the two channels are identical. Given this assumption, we can concentrate on Simonsen's findings for stereo locations as they relate to delay times, with a 0 dB interchannel amplitude difference. These delays, found in Table 1, are 0.2 ms, 0.44 ms and 1.12 ms for 10 degrees, 20 degrees and 30 degrees positions respectively. In order to calculate the delay times for different sound source locations around the microphone pair, the following equation is used.
Dt = (Sin (a) * S) / 34
where "t" is the interchannel time difference in ms, "a" is the angle of the sound source to the pair, and "S" is the distance between the microphone capsules in cm.
Repeating this equation for all angles of incidence between 0 degrees and 180 degrees, and microphone separations from 0 cm to 100 cm we can generate the three-dimensional plot shown in Figure 14. Analogous to coincident directional transducers, this graph can be used to generate the contour plot shown in Figure 16, outlining the angles of incidence which produce the desired interchannel time differences at various microphone separations. Table 6 shows a list of such angles derived from the resulting contour plot shown in Figure 17.
Figure 16 - Three-dimensional plot of the time difference of two spaced omnidirectional microphones with separations of 0 cm to 100 cm.
The resulting data shows a number of points worth discussing. Firstly, as is seen most clearly in the contours of Figure 17 and the data in Table 6, smaller microphone separations simply do not produce adequate interchannel delay times to achieve a stereo sound stage which fills the entire ±30 degrees width. This is due to the fact that the capsules are not far enough apart to generate the 1.12 ms time difference required to place a phantom image at the 30 degrees position, given any sound source location. In the case of a microphone separation of 10 cm, it is not possible even to generate a 0.44 ms interchannel time difference.
Figure 17 - Contour plot for a pair of omnidirectional microphones showing various interchannel time differences in ms as indicated.
Secondly, it can be said that the general imaging characteristics of a pair of spaced omnidirectional microphones is similar to a pair of coincident bidirectional capsules in that both are symmetrical along two axes. Although a bidirectional pair, unlike the spaced omnidirectionals, will flip the apparent side of sound sources located in the rear of the pair, the distance from the centre point between the loudspeakers is identical. This is most evident when comparing Figures 12 and 18. This characteristic can be exploited in situations where it is possible to place the microphone pair in the center of an ensemble, with musicians positioned both to the left and right as well as the front and rear of the pair. In the case of bidirectional capsules, this arrangement precludes maintaining absolute polarity of signals emanating from sources positioned in the rear of the microphone pair.
Table 6
Spaced Omnidirectional Microphone Pairs
Microphone separations of 10 cm - 110 cm showing one side of the pair. The opposite side is symmetrical.
| Microphone Separation |
10 degrees |
20 degrees |
30 degrees |
30 degrees |
20 degrees |
10 degrees |
|
0.2 ms |
0.44 ms |
1.12 ms |
1.12 ms |
0.44 ms |
0.2 ms |
| 10 cm |
43 degrees |
NA |
NA |
NA |
NA |
137 degrees |
| 15 cm |
27 degrees |
86 degrees |
NA |
NA |
94 degrees |
153 degrees |
| 20 cm |
19 degrees |
48 degrees |
NA |
NA |
132 degrees |
161 degrees |
| 25 cm |
16 degrees |
37 degrees |
NA |
NA |
143 degrees |
164 degrees |
| 30 cm |
13 degrees |
30 degrees |
NA |
NA |
150 degrees |
167 degrees |
| 35 cm |
11 degrees |
25 degrees |
NA |
NA |
155 degrees |
169 degrees |
| 40 cm |
10 degrees |
22 degrees |
72 degrees |
108 degrees |
158 degrees |
170 degrees |
| 45 cm |
9 degrees |
19 degrees |
58 degrees |
122 degrees |
161 degrees |
171 degrees |
| 50 cm |
8 degrees |
17 degrees |
49 degrees |
131 degrees |
163 degrees |
172 degrees |
| 55 cm |
7 degrees |
16 degrees |
44 degrees |
136 degrees |
164 degrees |
173 degrees |
| 60 cm |
6 degrees |
15 degrees |
40 degrees |
140 degrees |
165 degrees |
174 degrees |
| 65 cm |
6 degrees |
13 degrees |
36 degrees |
144 degrees |
167 degrees |
174 degrees |
| 70 cm |
6 degrees |
12 degrees |
33 degrees |
147 degrees |
168 degrees |
174 degrees |
| 75 cm |
5 degrees |
11 degrees |
31 degrees |
149 degrees |
169 degrees |
175 degrees |
| 80 cm |
5 degrees |
11 degrees |
28 degrees |
152 degrees |
169 degrees |
175 degrees |
| 85 cm |
5 degrees |
10 degrees |
26 degrees |
154 degrees |
170 degrees |
175 degrees |
| 90 cm |
5 degrees |
10 degrees |
25 degrees |
155 degrees |
170 degrees |
175 degrees |
| 95 cm |
4 degrees |
9 degrees |
24 degrees |
156 degrees |
171 degrees |
176 degrees |
| 100 cm |
4 degrees |
9 degrees |
22 degrees |
158 degrees |
171 degrees |
176 degrees |
Figure 18 - "Polar patterns" of various pairs of spaced omnidirectional microphones showing the angles of incidence required to result in -1.12 ms, -0.44 ms, -0.2 ms, 0.0 ms, 0.2 ms, 0.44 ms, and 1.12 ms interchannel time differences.
Thus far our analysis has relied on theoretical models of microphones. This view must be supplemented with some comments on the effects of various parameters that are not taken into account using mathematically perfect examples.
Coincident Pairs of Directional Microphones
The most obvious, albeit relatively minor issue to consider in a more practical analysis of microphone pairs is the effect of the actual sensitivity response of the transducer rather than the mathematically perfect model. Although many microphones have polar sensitivity patterns which are quite close to their theoretical ideal, none are perfect. Small deviations from the mathematical model will result in inconsistencies in the imaging characteristics of the pair. It is true, however, that in high quality microphones, this is a minor problem and will result only in very small departures from the calculated data.
The second, and more significant consideration is off-axis frequency response variations of individual microphones. The polar pattern of a transducer is categorized by the sensitivity response at a nominal frequency, usually 1 kHz. It is common to see the polar responses at other frequencies deviating from the mathematical model, following a general trend of increasing directionality with increasing frequency for all transducers. This tendency is commonly known as the off-axis frequency response of the microphone. Tables 7 through 10 show measurements of the frequency response of various microphones at multiple angles of incidence. All measurements, performed using a DRA Laboratories MLSSA system, are pseudo-anechoic and normalized to the on-axis frequency response of each capsule. Details on the measurement techniques and equipment used are given in Appendix 1.
As can be seen from the comparison of the measured sensitivities relative to the calculated theoretical response included in the top row of each of the four charts, there are considerable variations in the polar response patterns of these particular units at different frequencies. The result of these differences is most noticeable in coincident pairs of directional microphones as a "smearing" of the image location. Since the capsules have different sensitivities to different frequencies, a given angle of incidence will produce various interchannel amplitude differences depending on the frequency in question.
For example, Figure 19 shows the calculated polar response of a pair of bidirectional microphones in a traditional "Blumlein" configuration of an included angle of 90 degrees. Figure 20 shows the same graph, at 1/3 octave frequency centres from 98 Hz to 20.02 kHz, calculated using the off-axis frequency response measurements of a Sennheiser MKH30 RF-condenser microphone presented in Table 8. It should be noted that, although there are obvious deviations in the polar response of this microphone, particularly at high frequencies, is has the most uniform and accurate sensitivity response of the units measured.
Figure 19 - Calculated polar response pattern of a pair of bidirectional microphones with an included angle of 90 degrees.
Figure 20 - Polar response pattern of a pair of Sennheiser MKH30 microphones with an included angle of 90 degrees, calculated from the off-axis frequency response measurements in 1/3 octaves from 98 Hz to 20 kHz.
The off-axis frequency response of omnidirectional microphones will have little effect on the imaging characteristics of a spaced pair, assuming that the two microphones have an included angle of 0 degrees. Should another included angle be employed, then the pair should be considered to be spaced directional microphones at higher frequencies.
A third issue to be questioned in a real-world analysis is the consistency of Simonsen's and my data among various individuals and playback environments. Different listeners in different acoustic spaces will find that the 10 degrees, 20 degrees and 30 degrees phantom image locations do not necessarily correspond to the 2.5 dB, 5.5 dB, 15 dB and 11 dBø "magic numbers." It is, however, safe to assume that each listener is consistent. That is to say that, an individual may locate the image of a sound with an interchannel amplitude difference of 5.5 dB at a position other than 20 degrees but that position will be consistent with the amplitude difference. It can also be assumed that variations in the acoustic characteristics of playback environments will result in unpredictable inconsistencies in the location of phantom images.
One last consideration in our analysis is that of unpredictable amplitudes in the two channels due to strong early reflections received by the microphones. It is well known that such reflections will cause comb filtering at various frequencies dependent on reflection time delays. The magnitude of this effect is dependent not only on the absorptive coefficient of the reflecting surface and the distance to it, but also the directional characteristic of the transducer. If the reflection is on-axis to a directional microphone and the direct sound off-axis, and given a highly reflective surface, and comparable total distances between the direct sound and the reflected sound, it is likely that the resulting comb-filtering effect will cause drastic changes in the apparent frequency content of the sound source. Since a pair of microphones will have differing sensitivities to both the direct and reflected sounds, the resulting image location will prove to be quite unpredictable. It should be added, however that this is only a minor point to be considered and will have noticeable effect only in specific situations.
Spaced Omnidirectional Pairs
There are fewer issues to contemplate in the analysis of spaced omnidirectional microphones in practical situations as opposed to theoretical calculations.
The first issue is the off-axis frequency response of the transducers. As was mentioned previously, this is only a consideration if the pair has an included angle other than 0 degrees, which, although common, is beyond the scope of this investigation.
The second consideration is the frequency content of program material. Since spaced omnidirectional microphones rely on interchannel time differences, there are as a result interchannel phase differences of increasing magnitude with increasing frequency. In Simonsen's case of full-bandwidth program material, a 1 ms interchannel time difference will result in a phantom image positioned nearly 30 degrees off centre. It must be remembered, however, that a 1 ms interchannel time delay can also be considered as a 360 degrees interchannel phase difference at 1 kHz but a 180 degrees difference at 500 Hz. Using narrow band noise as a sound source, it is readily evident that a fixed interchannel time delay generates very different phantom image locations at different frequencies.
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