We implemented the 3D simulation code for DAS in Matlab. In order to create 3D radiation patterns, we made some modifications to the array factor from Eq. 3.3.
DAS_AF (θ, φ) = of the target with maximum radiation. R is the radius of the microphone array and κ is the wave number as defined in Eq. 3.4. For 3D simulation, we assume all the microphones are omnidirectional and the weights wn are the same among all the microphones, and are set to be 1.
The 3D patterns are obtained based on the following setup: the radius of the circular microphone array is 5.5 cm, and the target direction is set to (90◦, 135◦), which is equivalent to−
√2x,
√2y, 0 in the Cartesian coordinates. The sampling rate of input speech signals
is assumed as 16k samples/second. Thus, we simulated the 3D beam patterns at different frequencies, starting from 400Hz to 8kHz, according to the Nyquist theorem. The power pattern is then converted into the logarithmic domain by the following equation:
Radiation pattern (in dB) = 10 log10|AF |2, (3.28) where AF could be DAS_AF from Eq. 3.27 or MVDR_AF from Eq. 3.29. Fig. 3.5 be-low provides the DAS radiation responses at selected frequencies 400Hz, 1400Hz, 2000Hz, 4400Hz, 5600Hz and 8000Hz. Fig. 3.6 illustrates the comparison of different beamforming patterns at the corresponding frequencies in units of dB. We scaled the pattern to the range from 0 dB down to -40 dB, and the colormap was scaled to [−40, 0] dB as well. In both figures, we use the jet colormap (see Fig. 3.4) to define the strength of the radiation in a specific direction. The rectangular coordinate system (x, y, z) is adopted, and every axis is normalized to [−1, 1]. The intersection of the dotted lines represents the origin of the coordinate system, and the blue dot at−
√2 2 ,
√2
2 , 0on each sub-image indicates one point in the target direction, where the maximum radiation should occur. The red dot that ap-peared in the opposite direction represents the interference direction that will be explicitly suppressed in MVDR-2C.
The power plot of the DAS beamforming pattern in linear scale (Fig. 3.5) shows the array pattern at different frequencies. Each point in the graph has a reference in the colormap, which helps to distinguish the level of pattern intensity. The radial distance from the origin also corresponds to the strength of the pattern in a specific direction. As shown in the figure, the radiation patterns at all the frequencies have the largest lobe towards the target
−
√2 2 x,
√2
2 y, 0 and reach the maximum value at that point, whose color is maroon in the colormap. The lobe oriented in the desired direction is usually called the ‘main lobe’. From the plot, the radiation pattern behaves less directional at low frequencies, e.g. 400Hz. In Fig. 3.5b, the color at the origin shown as navy blue indicates that it has relatively weaker pattern at the origin compared to the target direction. As frequency increases, side lobes arise in some unwanted directions and are separated by several nulls. At 2000Hz, a ‘back lobe’ occurs away from the target direction. This plot supports the theoretical predictions that the low-frequency pattern tends to be omnidirectional, whereas the array directivity gets stronger as frequency increases.
We also implemented the 3-dimensional radiation pattern for the conventional MVDR beamformer, whose weights can be obtained through Algorithm 1. At any specific frequency, 8 weights are returned to steer the array beam. MVDR weights are not generic but input-dependent, since the weights are computed from the steering vector of the target direction and the covariance matrix of the input signals. In other words, different inputs will end up with a different set of weights. Below is the equation for the conventional MVDR 3D
(a) 400Hz (b) 1400Hz
(c) 2000Hz (d) 4400Hz
(e) 5600Hz (f) 8000Hz
Figure 3.5: 3D radiation patterns generated by Delay-and-Sum beamformer at frequency 400Hz, 1400Hz, 2000Hz, 4400Hz, 5600Hz and 8000Hz. The blue dot at−
√ 2 2 ,
√ 2
2 , 0 repre-sents the desired sound source with distance to the origin equals to 1, and the line connected the origin and the red dot at
√2 2 ,
√2
2 , 0shows the direction of interference.
pattern: where (θ, φ) is the angle distributed over the entire space, R is the radius of microphone array and κ is the wave number. Other than Eq. 3.27, where wnare the amplitude weights without phase shift, the steering vector of MVDR is already included in weights wn in conventional MVDR, as shown in Eq. 3.20. Accordingly, Eq. 3.29 does not include a redundant steering vector in the exponential part. For the same reason, the radiation response for MVDR-2C can be implemented using Eq. 3.29 with different weights wn calculated by Eq. 3.26.
Fig. 3.6 demonstrates the simulated array patterns of different beamformers in the units of decibel. We compared the behavior of DAS, MVDR and the proposed two-constraints MVDR, at frequencies of 400Hz, 1400Hz, 2000Hz, 4400Hz and 5600Hz. From the figure, the array patterns of all three beamformers are almost equivalent in all directions when frequency is 400Hz. At the same time, the side lobes arise naturally with the growth of frequency. Starting from 1400Hz, MVDR-2C shows a sharp ‘null’ response towards the direction of interference. On the other hand, DAS and MVDR patterns do not exhibit such strong attenuation in the direction of interference.
To further examine whether MVDR-2C maintains the desired signal with maximum gain, while providing better suppression towards interference, the energy differences between target and interference were computed, based on Fig. 3.6. The signal-to-interference ratio (SIR) in Fig. 3.7 is calculated by subtracting the radiation gain (in dB) at interference direction from the gain towards target. This plot illustrates SIR for DAS, MVDR and MVDR-2C within the frequency range of [400, 8000] Hz. From the figure, MVDR-2C has the highest SIR value at most frequencies, and has an overwhelming advantage over DAS and MVDR, especially when the frequency is between 3000Hz and 6000HZ. A particular case occurs at 1200Hz where DAS outperforms both MVDR and MVDR-2C. The area under the curve computed by trapezoidal numerical integration indicates that MVDR-2C performs better interference attenuation subject to the same target signal than DAS and MVDR.