On Saturday, we conducted a demo to see how the proposed three-microphone set up would perform. From different positions throughout the room, we clapped to see how each microphone would pick up the sounds. A note to consider is that one of the microphones was faulty, so my Macbook acted as the central microphone, hence why the sound waves are much more amplified for that respective channel.
From the figure, each clap produced an abrupt and brief waveform. We see that when making a clapping sound from the left side of the room, the left microphone picks up the soundwave first, followed by the central microphone and then the right microphone. From the observed soundwaves we were able to observe the different delays in time of sound as a function of position, which is expected. Originally, our dataset did not demonstrate the different delays in time of the soundwaves based off the respective microphone. We had to calibrate the dataset by placing all three microphones in the same position, and found that we needed to calibrate the dataset by roughly 12 centiseconds. Once we performed this we obtained the three channels of Figure 1.
In Figure 2 below, the set-up we used to collect the data is shown. As mentioned before, we made clapping sounds in different parts of the room to emulate a black box emitting a repeated pulse. Based off the preliminary data collected using this prototype setup, we believe that the project will be feasible.
Our data is stored in a google drive under the link below, which has been made share-accessible.
Files attached:
Sound wave demo pre and post calibration
Clapping video demonstration and set-up
ESPRIT-estimation of signal parameters via rotational invariance techniques R. Roy ; T. Kailath
Summary
The ESPRIT algorithm dramatically reduces computation and storage costs of DOA estimation by requiring the sensor array possess a displacement invariance; the sensors occur in matched paris with identical displacement vectors.
ESPRIT Data Model:
Take m doublets of sensors as shown in Figure 3 below. Assume there are d <= m narrow-band sources centered at frequency w0, and that the sources are far enough away from the receiving array such that the waveforms reaching the array are planar.
The ESPRIT algorithm is translationally invariant, and can be calculated with arbitrary sensor gain and phase patterns.
Discussion
ESPRIT reduces computation times by replacing the long search procedure inherent in other methods and produces signal parameter estimate in terms of generalized eigenvalues. This requires computations of the order d^3.
Professor Raviraj Adve
Department of Electrical and Computer Engineering
University of Toronto
Direction of Arrival Estimation
Summary:
There are a multiple of available algorithms that have been developed for the purpose of solving the DIrection of Arrival (DOA) problem. Depending on the complexities and subsequent capabilities of a given use case, different algorithms may suit best. For the purpose of this paper, the consistent testing framework involves a straight line of sensors (N) along an axis, in order to determine the angle(φ) of an incoming signals (M) from a given source. The 5 techniques detailed in the DOA paper are are correlation, Maximum Likelihood, MUSIC, ESPRIT and Matrix Pencil.
Models:
Cramer-Rao Bound
The Cramer-Rao Bound (CRB) model is used to state the minimum variance of a given algorithm. The idea being, this provides a common metric to determine the merits and drawbacks of a given algorithm.
The CRB theorem: Given a length-N vector of received signals x dependent on a set of P parameters θ = [θ1, θ2, . . . , θP ] T , corrupted by additive noise (n),
x = v(θ) + n,
where v(θ) is a known function of the parameters. (Adve)
For the purpose of processing signals in a given sensor array, the model is represented as:
x = αs(φ) + n
Where s(φ) is the angle of the signal from the received source and “a” assumes the role of a nuisance variable to represent a series of unknowns about the accuracy of an algorithm.
DOA Estimation using Correlation
The Correlation model is the easiest way to measure the effectiveness of a model by determining a series of spikes centered at the predicted angle(s) a signal may be received from a source.
Where M signals are measured and plotted against potential angles (φ) where the maximum of φ=φm.
Where Pcorr(φ) is a non-adaptive estimate of the spectrum of incoming signals. As a result the M largest peaks are the predicted angles.
Algorithms:
The three primary classes of algorithms addressed in this paper are “MUSIC”, “ESPIRIT” and “Matrix Pencil”. MUSIC is followed by sub variants called Root-MUSIC and Smooth-MUSIC.
ESPIRIT was detailed above in the description of the previous paper however, MUSIC needs to still be described.
MUSIC at its core relies on matrix algebra and the assumption that the incoming signals are uncorrelated. By taking this assumption, the signal covariance can be taken and used to determine the eigenvectors corresponding to the zero eigenvalue. Once this pseudospectrum is graphed, the highest M peaks detail the approximate incoming angles of the source signals.
Matrix Pencil is different than MUSIC or ESPIRIT in the regards that it does not rely on a correlation matrix R. In addition, those algorithms require a large number of samples in order to be effective while Matrix Pencil was created with the purpose of use cases where the scenario is rapidly changing.
Maximum Likelihood Estimator (MLE) for Direction-of-Arrival Estimation
When you utilize maximum likelihood methods, you are essentially searching for the value of your parameter which has maximum likelihood. Based off the observations, its a way of estimating the parameters of a model. We have two unknown parameters, which are DOA and magnitude. To estimate ϕ, the MLE is given by the following equation, where
is the is the pdf of the data vector x given the parameters α, ϕ.
To solve for the DOA estimate, we must find the maximum of the following function below (the maximum likelihood estimate of the spectrum of the incoming data).
Something to note is that when only one user exists and the interference covariance matrix
Rn = 𝞼 ^2*I we are performing the DOA Estimation using Correlation method, which was described above. There are downfalls with this algorithm however. It's requires a lot of computational resources and it assumes we obtained the interference covariance matrix. In reality, this is not commonly the case. Moreover, estimating the covariance of the interference by itself is nearly impossible as noted in the paper.
Conclusion:
This paper ultimately concludes that each method has their merits depending on the use case. While all having similar levels of accuracy, MUSIC and ESPIRIT both require N sensors to return the data on N-1 signals. Meanwhile Matrix Pencil can only determine N/2 signals from a given array. However, Matrix Pencil does not involved the calculation of a covariance matrix.
As a result, Matrix Pencil ultimately serves as the model with the greatest advantage. This is due to its increase speed, but also increased accuracy because of the removal of the inconsistencies incorporated by calculating a covariance matrix.
Maximum Likelihood Methods for Direction-of-Arrival Estimation
P. Stoica & K.C. Sharman
This 1990 paper introduces five methods that use Maximum Likelihood method (MLM) to distinguish multiple signals received at multiple sources.
Detailed mathematical explanation of each method would involve large amount of notational clarification, hence is omitted. Below are an summary of the methods, along with their pros and cons:
1.
Name: MLM (deterministic or conditional maximum likelihood method (MLM)
Description: Application of the ML principle to the statistics of the observed raw data.
Pro:
Con: computationally intensive, and not statistically efficient for practical cases.
2.
Name: MUSIC-1 (multiple signal classification 1)
Description: a brute force approximation to the MLM.
Pro: computationally much simpler than the MLM
Con: provide significantly less accurate estimates
3.
Name: MUSIC-2 (multiple signal classification 2)
Description: an improved version of MUSIC-1, obtained by applying the ML principle to the statistics of certain linear combinations of the sample noise space eigenvectors.
Pro: computationally much simpler than the MLM
Con: provide significantly less accurate estimates
4.
Name: MODE-1 (method of direction estimation 1)
Description: a large sample realization of the ML estimator, a compromise between statistical performance of MLM and computational simplicity of MUSIC.
Pro: computationally simpler than the MLM
Con:
5.
Name: MODE-2 (method of direction estimation 2)
Description: obtained using the ML principle on the statistics of certain linear combinations of the
sample eigenvectors.
Pro: computationally simpler than the MLM
Con: and statistically more efficient than MLM
The author regards MODE-2 method as the significant new result in the field (1990), and introduces multiple advantages over other methods (although MODE-1 and MODE-2 are close in terms of computations).
Comment: For our use now, we are only looking at a single source beacon. Although there would be interferences from neighboring competition spots, but we believe those interference can be filtered out before we start direction-of-arrival analysis. Even if we were to introduce multiple other noises or interferences, the apparent goal would be to filter out irrelevant signals, rather than recognizing all of them. Therefore, the methods are not useful to us for now.
That being said, if we later move on into exploring underwater localization using multiple source beacons, where we will try and separate the signals, the methods introduced in this paper will be of great value.