10

A beamformer is basically a spatial filter. It can be passive, just like a temporal filter. Instead of samples separated by time, they are separated by space. A passive temporal filter can be a bandpass that is "aimed" or "steered" at a particular frequency. For passive spatial filters (i.e. beamformers), the filter can be steered towards a particular ...


6

The equations relate to how the directivity of a uniformly spaced linear array of sensors works. Let's start with an array: where $\theta$ is our "look" direction (where we want to steer the array), $d$ is the sensor spacing, and, for our purposes, the angles of each segment of the array are all 0 (i.e. $\theta_1 = \theta_2 = \ldots = 0$) so the array is ...


5

The work by Darren Ward, Rod Kennedy, and Bob Williamson investigated how to design filters applied to a delay-and-sum beamformer that allowed broadband signal acquisition. As you can see from their figure 2, a narrow-band beamformer does not perform well as the frequency changes from the design frequency. Suitable choice of filters in the paths of the ...


5

If you are dealing with say 8 kHz for a nominal speech bandwidth of of 4kHz i.e. 0 Hz - 4 kHz, then the speech is essentially a wideband signal. Therefore narrowband beamforming won't work very well. Your beam pattern will be okay for the particular frequency of interest - but once you start moving away from that frequency your beam patterns will deteriorate....


4

For now I will focus on the conceptual, pending feedback from you: What I'm interested in to know if it is possible to take two adjacent beams, and do another step of beamforming, The answer to this is yes, this is possible to do, and this is done in many beamforming applications. Think about it like this: All beamforming is really doing, is undoing ...


4

8+ high quality inputs for beamfroming and "...beginner friendly..." are competing requirements. There will soon be the audioinjector which attaches on Raspberry Pi and would be a relatively pain-free option for what you mention. You can program the Raspberry in a number of different ways including "low-level" C to high level Python or other. Another ...


3

The MVDR beamformer is an adaptive technique - the response is dependent on the the interfering signals i.e. their locations, their SNRs and their correlations. This makes it difficult to say what the resolution is in traditional 3dB width type of terms. Essentially the MVDR will try to place nulls in the beampattern where the interferers are - so a 3dB ...


3

MVDR is a narrowband beamformer. For broadband signals it is usually applied for each frequency bin. That means that $\mathbf{R}_{xx}$ is frequency dependent. In other words, for each time you should have $M$ matrices, each one is $3\times 3$. Now, since you usually cannot compute $\mathbf{R}_{xx}$ exactly, you perform covariance estimation $\tilde{\mathbf{...


3

As I said in the comment, you sound like you are doing things correctly. Below are two plots generated by the scilabcode at the end. The first plot shows two different weightings applied to the sensors. The second plot shows the associated beam patterns. While the $\tt sinc$ one is not "uniform" it is a fair approximation. EDIT Changing the code below ...


3

In your second link, the time and space dependence of a simple plane wave is given by Ψ ( x , t ) = A sin ( ω t ± k x ) Just as a Fourier transform of the equation above can be taken from the time domain $t$ to the frequency domain $\omega$, so also can a Fourier transform be taken from the spatial $x$ domain to the wave vector $k$ domain. For the very ...


2

I've not worked on the design of such systems before, but I think your notions are on the money. Specifically, yes, beamforming arrays do have RF front ends that are replicated many times. The complexity of contemporary phased array radars is astounding in this regard; there are designs that have hundreds of individual antenna elements in them with ...


2

Have a look at the scilab code where I answered your other question. That question is about a discrete aperture, but it is analogous to the question you are asking here. For that question, the beam pattern is just given by: $$ D(\theta) = \sum_{n=0}^{N-1} w_n \exp\left(j\frac{2\pi n d}{\lambda} \sin(\theta) \right). $$ With judicious choice of parameters, ...


2

If you consider the beam pattern for a linear array - so I'm looking at just the magnitude (or magnitude square) i.e. $|B(\theta)|$. Then the $\sin(\theta)$ is telling you how the response changes to different directions of arrival. There are few things to keep in mind: There is the direction to which you steer the beam - these are the weights you use on ...


2

You choose the weights the steer the main beam of the antenna in the particular direction you want - let's call this $\theta_s$ i.e. steering angle. Now, given that you are steering the antenna in a given direction you are also interested in the array's response to incoming signals from other directions. This is the directivity pattern and is a function of ...


2

In MATLAB the default direction of orientation of Planar array is y axis, which can be seen as ula.getElementPosition ans = 0 0 0 0 0 0 0 0 0 0 -22.4844 -17.4879 -12.4914 -7.4948 -2.4983 2.4983 7.4948 12.4914 17.4879 22.4844 0 0 0 ...


2

Some examples would be the null to null bandwidth(s) as well as the sidelobe levels of the first/second sidelobes. Example vector of 5 numbers: [mainlobe gain, first null location, 1st sidelobe gain, second null, 2nd sidelobe gain] From this you could produce an estimate for the 3 dB points of each lobe using some "reasonable" formula, i.e. halfway ...


2

They're essentially the same, with the steering vector typically representing phase shifts (==delays) using complex numbers from the unity circle, and the weight vector often containing real values to scale the samples. In practice, both can be unified into one and hence, the terms are sometimes used interchangeably. Also note that for many algorithms, the ...


2

Because we are dealing with only a single frequency $f,$ the 2-d spatial coordinates are more conveniently represented by 2-d phase coordinates where a distance of $2\pi$ represents one wavelength. The microphones are located on a square grid at phase coordinates $(x\Delta, y\Delta)$ for every combination of $x$ and $y$ each in $\{-N, -N+1, \dots, N-1, N\}$, ...


2

In this context $\Phi_y$ often describes the (estimated) power spectral density matrix of $y(f)$, which is $$\Phi_y(f) = E\{y(f) y^H(f)\} = \begin{bmatrix} y_1 \cdot y_1^*(f),& y_1 \cdot y_2^*(f),& \dots \\ \vdots & \ddots & \\ y_M \cdot y_1^*(f),& & y_M \cdot y_M^*(f) \\ \end{bmatrix},$$ ...


2

Assume first that the medium is homogeneous - this implies that the time delay of arrival between the sensor is constant for all frequencies of interest (over the bandwidth of interest). Now to beamform these signals, you need to implement a true time-delay beamformer. These can be implemented in either the time-domian and frequency domain. A common example ...


2

First, this question is probably better for MATH.SE, but I'll give it a shot. It's been a long long time since I did this stuff. If $N > M$: 1) $C^*$ has rank M. 2) $R_x^{-1}$ has rank M. (or it wouldn't exist as an inverse) 3) Therefore $C^*R_x^{-1}$ has rank M, since $R_x^{-1}$ is a full rank square matrix. 4) $C$ has rank M 5) $C^*R_x^{-1}C$ ...


1

Not in the way you think, no. Digital Beamforming works because the digital receiver is able to sense the phase of the incident radio wave; that phase is retained. Photodiodes don't sense the phase – they only sense the intensity. Also, this requires the whole incident radio or light wave to be phase-coherent across its front – that's the case for laser ...


1

The spatial sampling frequency of a linear array with equidistant sensor (antenna) spacing on the $x$-axis is simply given by $$f_x=\frac{1}{d}\tag{1}$$ where $d$ is the distance between the sensors (cf. Eq. $(35)$ in the document you linked to). What is meant by the equation in your question is the spatial frequency of the incoming wave along the $x$-axis,...


1

The answer will be NO, but to be precise especially in case of conventional beam forming. Theoretical Derivation The antenna electric field pattern of array antenna can be given by where N is the number of antenna elements d is the spacing between antenna elements In order to find the beamwidth (3 dB), the above equation should be equated to and ...


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