 I'm going to be talking about cell phones, signals, and what they have to do with math, so that's good. OK, so have you ever thought about how maybe your cell phone actually works, or like notice that your radio kind of has an issue with the signals that you drive through the mountains? Well, today we're going to talk about what math has to do with these day-to-day objects. So cell phone networks rely on signals. And typically we're going to assign unique signals to each cell phone, but we don't want them to resemble each other or even themselves as they travel through there. And I'll talk what I mean about resembling signals. So as a quick example, when I make a phone call from this room, I want to make sure that the cell phone towers are synchronized well to my phone in order to send that outgoing call. Otherwise the call might get dropped. I'm going to be calling this auto correlation soon, but I'll define that in a few slides. So if another person nearby would make a call at the same time as me, neither of us want the cell phone towers to mix the two of us up, right? So we're going to call that cross correlation. And it's going to minimize confusion. So here's auto correlation. How much your sequence resembles itself as it moves through the air. And cross correlation is just how much your sequence is going to resemble another sequence as it moves through the air. And this is going to be good for synchronization and minimizing confusion. So we're going to assign binary signals to each person and use a dot product to check when the entries of the signals agree or disagree. Longer sequences give us more distinct cell phones so those are beneficial. And shifting the signal is going to model it kind of going through the air like this. So here's an example where we have a sequence against its shifted self. And we see that there's a disagree, an agree, and a disagree. So when we take the dot product of all the overlaying entries, we see that at this shift we have an auto correlation of minus 1. So we're going to want correlation and absolute value to be as close to 0 as possible when the sequences are shifted as this is going to keep the cell phone tower from seeing the signal too early or too late. We get a big spike when it's perfectly overlaid and you guys will see that soon. OK, so this is going to be an example where the cell phone tower is going to check against my signal kind of coming in from the left. And we're going to check the correlation at each shift. So we have minus 1, minus 2, minus 1, 2. So these are all good. And then we have 5. OK, perfect. So now that it's perfectly aligned, it says, oh yeah, that's definitely E like hauling. And it doesn't register it too early or too late. So OK, what's the big deal? Won't the sequence always have the largest dot product when it's fully aligned? I mean, it's based on the length, right? And if it's all going to agree at that aligned shift, it's not a problem. So when would it ever be recognized before that aligned shift? Well, noise is going to be the big issue. Noise can cause a cell phone tower to recognize a signal at a premature shift. And here's an example where I'm going to change just one little entry. That middle one used to be a minus 1. And now it's going to be a 1. And we'll see what happens. OK, so at this penultimate shift right here, we get a correlation of 4 because where that second entry used to be a disagree, it's now an agree. So all the overlaying ones do agree now and we get 4. You can maybe already see what's going to happen if I shift it over one more right here. I get a correlation of only 3. This is because it used to agree everywhere, but now that it disagrees, it shifted and moved it over by a factor of 2 right from 5 to 2, or from 5 to 3 side. So it registered the sequence a little too early and my call gets dropped. OK, so that was bad. We can already see reasons why we don't want to mitigate auto-correlation. And we haven't even started talking about cross-correlation yet, and we don't have time to. So no problem. But wouldn't it be great if we could get both auto and cross-correlation low at the same time? Well, these guys, personally, and Sarvate showed that there is a measure that basically considers auto-correlation and cross-correlation at the same time, some normalized measure of it. And it's bounded below by 1. So we want to try to hit that bound. We want it as low as possible, and we don't really know how low we can get it. I mean, if we take some randomly selected sequence pairs of various lengths, we see that they have a PSC of about 2. So that's not too good. We know it can go lower. And in 2016, Boothby and Katz found a sequence pair of multiple sequence pairs of PSC of about 1.16. So basically, this is where Golay complementary pairs come in. They turn out to be the only ones that have PSC of exactly one. And they were actually discovered in 1949, but nobody ever looked at them like that before. So hopefully, this shows you guys some of the ways math is involved in our day-to-day lives. Even those of us that do math all the time don't really notice that there's math sitting in our pockets. And that's all. So yeah, that's great.