So, I would like to take a little detour, and introduce you to an advanced topic. It's okay, if you don't get most of it, but at least I would like you to get the take away message. So, nitrous sampling has been around for decades. It's the tentative all signal processing. I would like to go back to the SP work through, the pulse oximator example, where we are accurating fast and sensing. Such sensing invariably turns out to be very, very power hungry. It's beyond the scope of many battery operated sensors. So, in this example, I'm showing you the time domain waveform on the top, and the spectrum of the pulse oximeter in the bottom, you can see that there are peaks at 50 hertz, and 100 hertz at the bottom. This is because there is always ambient light, that is flickering as the LEDs are also flickering. So, if we set the F max to be our own 100 hertz then, in practice we would sample the signal at around 250 hertz. So, that means we are fighting up the red, and infrared 250 times a second according to Nyquist. And to get the response from the photo detector. This turns out to be immense power draw on the battery. As much as 20 to 80 millions depending on your design. From the physics of this, we all know that the heart beats roughly 60 beats per minute. The complete range is anywhere from 40 to 180 beats per minute. So, let's say normally 60 beats, 1 beat per second, so that's 1 hertz. So, as you can see, on the left hand side of the bottom pane most of the information is around 1 to 5 hertz. That means people should be sampling at 10 hertz, but we are sampling it to 50 just because the harmonics. This is where comprehending comes. Life below Nyquist. If you new something about this signal, can you take advantage of it? The only restriction Nyquist thereom plays on this signal, is that it is limited to F max. And you take uniform sampling, but what if there are lots of holes in the signal, so the fortia basis that we used to Nyquist we can write the x as spectrum. The w times small x, that's the tie the main signal in matrix notation, so the idea is how can we incorporate the signal priors in compressed sensing. So, in the pulse [INAUDIBLE] kiss, if we know that there only m components that are certain of all certain threshold epsilon. As PPG, photo. Pay for atoximeter. Then, what compressive sampling suggests is that we can make only k measurements, where k is a function of the number of components that raise above, and the total window length, M and N. And that is enough to get back almost all the signal. So, the number K here is the, it represents the sparsity in the signal. If you remember the speech, it was completely populated in the spectral domain, and in PPG we are seeing they launch the and we are going to take advantage of this. So, how do we actually take these measurements on? So, we introduce a sensing cardinal etch which is, in a sense the counterpart of W, which is the basis for reconstructing the signal. The combination of H and W, those letters do designer tailored sensing construction paradigms, the condition is H statistically incoherent with W data presentation basis. And so, in one example, H can be just [INAUDIBLE] that you randomly sample in a time domain, and then, using this optimization that I show at the bottom, we can reconstruct the signal. Here's a graphical illustration of what I have been talking about. The figure on the left is the Nyquist case. Maybe are uniformally sampling. And the top right-hand is, then we are sampling at random locations as shown by the red stem dots. In this case, we are lighting the LEDs 40 times fewer, or lower speed than the Nyquist case. That's the undersampling ratio, and the picture on the bottom right is the same thing but here we actually compute the Nyquist domain samples, and then use random projections for compressed sensing. So, here is the reconstructed signal. The original, and the reconstructing with 40 times fewer samples is shown. If you just acquire fewer samples at uniform locations after doing proper low-pass filtering, as shown by the black line, you miss a lot of information. So, as you see in some of the sensing applications the power of the sensor is exorbitantly high. And there are people working on new paradigms of improving the power of low power sensing. And the research continues.
