that y = Ax, and we know that x is s-sparse. It should be possible to compress y. One way would be to find the linear combina tion of columns of A that gives y-to find x-and to save the location of the columns and the "amount" of each that goes into y. That, however, is not generally a simple problem. Often it is possible to "design" an additional M × K , M K matrix, B, that we can use to compress our measurements by calculating b = By with only M elements. Thus, each a compressed vector b is a linear combination of the elements of y, and element of b does not have many elements. This is the "compression" of compressive sensing. At this point, we find that b = By = BAx ≡ Cx where C = BA. Given that x is sparse, b is a combination of a few columns of C. As we have seen, in order to be able to find a sparse vector x given b, the matrix C must satisfy certain conditions. If x is s-sparse, then at a minimum we need to know that any 2s columns of C are linearly independent. It is easy to show that this implies that any 2s columns of A must be linearly independent. The matrix A is derived from the signals being sampled. Once it is chosen, the matrix B must be designed in such a way that C is appropriate for use with one of the three types of algorithms described above. One way of designing the matrix B is to draw its elements randomly. (For more about this possibility see, for example, [8].) Another method, and the one we use, is to draw matrices from a set of matrices with desirable properties and then check to see whether the matrix is actually appropriate. Having found an appropriate matrix B, let C = AB. The matrix C is now our dictionary, our words are the columns of C, and the vector b is a linear combination of a small number of columns of C. To recover y, we calculate the s-sparse solution of Cx = b us ing one of the three methods described above. Having found x, we then recover y by calculating y = Ax. The Signal and How We Sample It We designed and built a system that uses compressive sensing to take samples at a rate lower than the Nyquist rate, sends the samples to a PC running MATLAB, and then uses MATLAB to un-compress the samples. The measurement system has several stages and involved doing a bit of interesting analog electronics work. As we have discussed, compressive sensing can be used when you know that your signal is a linear combination of a few signals from a larger set of signals. To keep things simple, we assumed that our signal was either a constant or one of three frequencies-all of which were lower than 1 kHz, and one of which was close to 1 kHz. Because the sinusoid's phase was not known, we had to assume that the sinusoid was a linear combination of a sine and a cosine at one of "our" frequencies. Thus, our A matrix has seven columns. We then took five samples of the signal and stored them using a simple analog delay line of our own design. Thus, our A matrix has five rows. At this stage, we sampled at 2 kHz-but these samples were not transmitted to the "outside" world, and this is not the rate at which our microcontroller's A/D April 2020 sampled. These five samples are "our" y-and the next stage is to compress them. To summarize, at this stage we know that y = Ax where the matrix A has seven columns and five rows, the vector x has seven elements, and the vector y has five elements. One column of A is a constant vector, and the other six columns are samples of sines and cosines at the three frequencies we are using. As we either send one of "our" sinusoids or a constant, we know that y = Ax and x is 2-sparse. We want to take advantage of the fact that the vector x never has more than two non-zero elements to compress the vector y. Because, other than the microcontroller we used, we built everything, we wanted to compress the samples in the simplest possible way. To do this, we "designed" the matrix B to have four rows and five columns and to have all of its elements be zeros or ones. In this way, B takes the five samples in y and compresses them into the four elements of b. The design considerations turn out to be quite interesting. Designing the "Compression Matrix" Despite the fact that for large problems the brute force method is not practical, for relatively small ones it is practical, and it gives you somewhat more freedom when choosing the matrix B. For this reason, we decided to use the brute force method to un-compress our data. We drew potential B matrices at random from the set of matrices all of whose elements were ones and zeros and tried to determine how well the brute force method would perform for each matrix-to what extent it would be noise-resistant. In order to make this determination, we made use of the condition number. (See the Sidebar: The Condition Number Sidebar: The Condition Number of a Matrix Let A be a "generic" invertible matrix, and let x and b be vectors that satisfy Ax = b. Then you can calculate x −1 as A b. Suppose that the vector b is something that was measured and that the measurement was noisy-that rather than measuring the exact value, you measured b + n. Then, when you try to calculate x, you actually calculate A −1 b + n = x + A −1n. We need to know how −1 to know how big an annoyance is A n may be. We need −1 A b . large this term may be relative to In the worst case, A −1b is "amplified" very little while A −1n is amplified a lot. The condition number is designed to measure how bad the worst case here can be. For symmetric matrices, the condition number is the ratio of the absolute value of the largest eigenvalue (in absolute value) of the matrix to the absolute value of the smallest eigenvalue (in absolute value) of the matrix. (For more information about the condition number, see, for example, [9].) ( ) IEEE Instrumentation & Measurement Magazine 97

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