of a Matrix.) We drew random matrices B and then calculated the condition numbers of all relevant combinations of four columns of C = BA. We chose a matrix B where none of the condition numbers was too large. Even after looking for and finding a "good" B matrix, all was not sweetness and light. We had an unexpected problem. Certain frequencies showed a very moderate tendency to get confused with one another. Eventually, we traced the problem to an interesting effect related to the condition number. Suppose that there are two 2-sparse vectors you know you may want to measure, x1and x2, and each vector is associated with a different frequency. Let D be the matrix whose four columns are the four columns of C associated with the two frequencies, and suppose that D's condition number is relatively large. Now the difference of the two vectors of inter consider est, d = x1 − x2. Suppose that this vector is associated with the smallest eigenvalue of D, λsmallest. Then, the values you will need will be Dx1 , Dx2, and their difference will be to measure Dd = λsmallest d -which is very small. Despite x1 and x2 being rather different, if D has a relatively large condition number, even a small amount of noise could confuse us when measur ing Dx1 and Dx2-because these vectors are very similar. "Our" Matrices After careful searches and some trial and error, we found that making use of five samples of sines and cosines at the frequencies 0, 400, 666, and 975 Hz worked well. (In the case of the 0 Hz signal, we only stored the samples of the cosine-of the nonzero constant signal.) The B matrix we used was: 1 0 B= 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 0 1 (Thus, the first element of b was the first element of y; the second element was the sum of the second and fourth elements of y, the third element was the sum of the second and third elements of y, and the last element of b was the fifth element of y.) We checked that any four columns of C = BA were linearly independent and that the square matrices they defined had condition numbers. From the four elements of reasonable b = By = BAx, we were able to calculate x, and from x we could proceed to y = Ax-the vector of measured values. From Math to E.E. If we were interested in a "pencil and paper design," we might have been able to say that we were finished. At this stage, we know how to compress five measurements into four values- and we could sample these four values below the Nyquist rate and still retain all of the information about the original measurements. We, however, wanted to see sub-Nyquist sampling in all its glory. To that end, we built the system described schematically in Fig. 1. This system is composed of a sample and hold subsystem, summing circuits that implement B, a second, simpler sample-and-hold subsystem, an ADuC841 microcontroller [10], and a PC. A signal enters the system (Fig. 1) from the "left" and is sampled at a rate of 2 kSamp/sec by the analog shift register. At any given time, five "sampled and held" samples are available. In the "adder" block, the five samples are combined into four linear combinations- by analog circuits that implement the operations "specified" by B. The output of the analog circuit is "sampled and held" every fifth clock for five clock periods-for 2.5 ms. The microcontroller samples the held values every (2.5/4) ms-at a rate of 1.6 kSamp/ sec. Thus, the microcontroller's sampling rate is sub-Nyquist for any signal whose frequency is above 800 Hz, and the microcontroller can make use of an A/D designed to work at this lower sampling frequency. The sample and hold units were implemented using the design shown in Fig. 2-although the inverted clock is actually provided to the circuit by the ADuC841 that controls the system. Because of the simple nature of our B matrix, the only summing circuits we needed to implement in order to "implement" the matrix were two simple adders. The design we used is given in Fig. 3. Next, the output of the summers is sampled every fifth sampling period and stored for five sampling periods. Fig. 1. The System we designed. 98 IEEE Instrumentation & Measurement Magazine April 2020

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