# How to compute beam pattern given a delay-and-sum filter in spectral domain?

Given a two-mic system (the mic spacing $d$ is known), how can I compute the beam pattern (see the Figure below) over frequency and angle $g(f,\theta)$ for a pair of spectral filters ($H_0$ for channel 0, and $H_1$ for channel 1)?

Both $H_0$ and $H_1$ are complex vectors.

• If your $H$ are really just single complex numbers, then you have a very simple filter; it's just a multiplication with a constant. Do you mean something like "$H_i$ are complex tap delay line model vectors"? – Marcus Müller Aug 24 '17 at 7:45
• Use a steering vector for your array (see array signal processing literature) to generate directional signals from -90 to 90. Spatially filter you received signal with H. Compute the power of the signal. Repeat the process by varying f. – learner Aug 24 '17 at 9:05
• @MarcusMüller $H_{0}$ and $H_{1}$ are vectors. For example, if we use 512-point FFT, then $H_{0}$ has 257 dimensions – read Read Aug 24 '17 at 19:54

## 1 Answer

One of the method for calculating beam pattern could be, you simulate/capture audio from different directions on array and filter these captures with filters($H$'s) generated. the audio coming from direction of beam pattern will be passed as it is, but the audio coming from other directions will be attenuated. the amount of attenuation with respect to spatial location is nothing but the beam pattern .
for implementation, Take audio in time domain, window it, take n-point DFT of this windowed time domain audio after this do element wise multiplication with filter coefficients to get beamformed output.

• Should I simulate waves of different frequencies and different directions? How is the dB in the above Fig. calculated. – read Read Aug 24 '17 at 20:11
• Yes, because axes for beampattern are frequency and direction. dB is nothing but logarithmic scaling of relative attenuation, ie no attenuation is zero dB and others are negative. – arpit jain Aug 25 '17 at 3:38
• How would you represent audio of different frequencies and different directions? Can you give an example? I would like to have these audio samples defined in spectral domain (i.e. in complex vector) so that the beamformed signal can be quickly calculated by element-wise multiplication with filters and summation across channels. Thanks. – read Read Aug 25 '17 at 3:55
• For a two-channel and n-point DFT, if I have $H_{0} = [a_{0}+jb_{0}, a_{1}+jb_{1}, ..., a_{n-1}+jb_{n-1}]$ and $H_{1}$. How would you represent an audio sample with frequency $f$ and angle $\theta$ in complex vector. – read Read Aug 25 '17 at 4:02
• Take audio in time domain, window it, take n-point DFT of this windowed time domain audio after this do element wise multiplication with filter coefficients to get beamformed output. I hope this clears out your doubt and you might accept the answer. – arpit jain Aug 25 '17 at 6:44