What's up everybody, Jason here for jaspyscasebrakes.com this fills up flawless first off the line picker team number one This random number block includes the Broncos and Redskins So again works like any other random number block break on our website But it's only for these two teams in that specific break of flawless first off the line and good luck guys So let's roll the dice You got a five and a three eight times. Good luck to the customer sir from Stephen K down at SKS one two three Four five six seven and eight Eight times John down at SKS eight times one Two three four five six seven and eight eight times Zero down to seven So John with the zero so any redemptions for any of these two teams would be yours Derek when nine Rob You have eight Steve with five Sean with four Stephen K with two Daniel with three Matt you have one any one of ones would be yours as well as six and SKS with lucky number seven So there you go guys coming up next in a separate video guys first off the line picker team number one of flawless football Appreciate it guys


I'm excited to welcome to the stage Chris Juergens, Director of Impact Investing for Omidyar Network, who's going to announce a new collaborative for Frontier Financing. Thank you, Lindsay, and good morning, Socap. I'm Chris Juergens from Omidyar Network, and it's my pleasure today to announce the launch of the Collaborative for Frontier Finance. The Collaborative is a new multi-stakeholder initiative focused on increasing access to capital for small and growing businesses in emerging markets. Small and growing businesses, or SGBs, are the backbones of emerging economies. They account for about 80% of formal employment opportunities, and they contribute to positive social and environmental impact by driving inclusive economic growth, spurring innovation, and providing underserved populations with access to essential goods and services. SGBs are diverse. They range from a rural coffee cooperative in Peru to a tech startup in Bangalore to an energy access venture in East Africa. They typically seek financing in the range of $20,000 to $2 million, but the vast majority of SGBs struggle to access forms of capital that meet their distinct needs. According to the IFC, SGBs in emerging markets face a $930 billion financing gap. Many are stuck squarely in the missing middle. They're too big for microfinance, too small or risky for traditional bank lending, and lack the growth, return, and exit prospects sought by most venture capitalists. And the missing middle exists for a region. SGBs can be fundamentally hard to finance due to their small size, limited track record of performance, and the transaction costs of serving them. But we know these barriers. We've long known these barriers, and the time is to focus on solutions. And we're heartened to see the growing number of innovative financial service providers that are overcoming these challenges by pioneering new approaches to financing. Fund managers are using a growing array of alternative investment vehicles and financing instruments beyond the typical 2-in-20 closed-end fund. Banks are introducing new SGB-focused lending products that are more tailored to the distinct needs of these kinds of businesses. Local angel investor networks and seed funds are mobilizing local sources of risk capital, and financial technology innovations are making lending to SGBs more efficient and cost-effective than ever before. But despite these positive developments, we're not making progress fast enough. Millions of small businesses are not meeting their full potential to drive impact, and billions of capital are still sitting on the sidelines. That's why a group of like-minded investors, fund managers, and field-building organizations have come together to create the collaborative for frontier finance. We're seeking to close the missing middle financing gap in three ways. First, the collaborative is bringing together stakeholders to find a shared action agenda for SGB finance, defining what it will take to move the needle on this $930 billion financing gap and how working together collaboratively can help get us there faster. Part of this is building the evidence base for SGB finance through a shared research agenda, and I'm pleased to announce today the first flagship report of the collaborative, called The Missing Middles, segmenting small and growing businesses to understand their financing needs. Developed in consultation with over 50 SGB investors, this report defines four distinct families of small and growing businesses and describes why we need distinct financing approaches to meet the distinct needs of each. Second, the collaborative aims to accelerate the adoption and scale up of the most promising financing solutions for missing middle entrepreneurs. We know that there's no single financing solution that's a silver bullet for SGBs. Rather, what's needed is a diverse ecosystem of diverse financial providers that can meet the needs of SGBs at different stages in their growth journey. So what we're doing is developing networks of practitioners that are implementing specific financing solutions that have evidence results, but that are not widely adopted in the marketplace and not recognized by lots of LP investors. For example, we've mobilized a group of fund managers that are using alternative financing instruments, tools like quasi-equity, revenue-based investing, and mezzanine debt to provide growth capital to those SGBs who aren't a fit for traditional VC. These fund managers and practitioners have come together to identify the key challenges that they face and to develop new collaborative initiatives to address them, like working together to share data in order to establish performance benchmarks for these kinds of mezzanine funds, or exploring ways to reduce operational costs through shared services, or training more fund managers in how to use alternative instruments in their work with SGBs. And we have similar networks of practitioners coming together on solutions like local angel investment networks and syndicates and provision of investment readiness technical assistance to SGBs. Finally, and perhaps most importantly, the collaborative will catalyze new capital flows to SGBs. We'll be working with a diverse range of SGB investors from development finance institutions to local corporates to angel investors on ways to address the barriers that keep capital from flowing. And we're particularly focused on how we can mobilize catalytic capital to help unlock new sources of financing. I'd like to thank the founding partners of the collaborative Dutch Good Growth Fund, World Bank Info Dove, the Global Development Incubator, and my organization, Omidia Network, and to recognize our Gideas Foundation, MacArthur Foundation, Small Foundation, and the Australian Department of Foreign Affairs, who've all come together as partners to get this off the ground. Most of all, I'd like to thank the practitioners who are on the ground working with SGBs to get them finance and help them realize their potential. We invite all of those who care about unlocking the potential of SGBs to achieve impact to join us. And I invite you to go to frontierfinance.org to find out more or to come to talk to any of the partner organizations that are involved. Thanks very much.


What if we lived in a world of total transparency? If even what you're thinking right now is something that you could know, that I could know, and that we all could see. The average person, not that any of you are average, thinks about 70,000 thoughts per day.  And those tiny electrical discharges can be measured, visualized, and decoded. And with sophisticated algorithms, we can start to understand what those mean. Whether it's a thought that you're having, whether it's a state of arousal, whether you're drowsy while you're driving. In fact, what you're seeing right now is real-time brain activity of two people who are here in the audience. You're seeing the visualization of literally in real-time brain activity in various regions of the brain being measured through a simple consumer-based EEG device that all of you now have access to. On the right-hand side, or your left, my right, you see a flower, which hopefully will grow soon. There we go. And the way that flower is being grown is, again, through a simple EEG device that one of the participants in the audience is thinking, which means we know that when the flower opens that what he's thinking is grow. And we've been able to decode that now through a simple consumer-based EEG device such that we can literally see what he's thinking. Now, we're not yet at the point where what all of you are thinking right now is something that like a little thought bubble above your head is something we can see, but we're getting there. We're getting to the point where there is transparency of thought, of emotion, of feeling, and that has terrific promise. You heard our last speaker talk about brain health, the ability to have in your very own hands in your very own home, the possibility of seeing your brain activity, decoding it, training it, is something you can do now today. And that has great promise not just for health, but for self-access and self-knowledge, your ability to know your emotions, to get a better handle on them, to understand how you react in various situations and to be able to change it.  That is incredibly exciting. But you might be wondering how it is that you're seeing these on the screen right now. So I'm gonna invite both Tan and Thie to come up on the stage who are wearing these consumer-based devices. Tan is actually the CEO of the company of the device that we're actually looking at, which is Emotives EEG device, and we're seeing their activity in real time, which is really exciting. Now, you might be wondering how we're doing it is because these are communicating via Bluetooth technology to the iPad and to the computer in the back of the room. And one of the things that's remarkable about this technology is Bluetooth is not necessarily the most secure channel, which means a hacker might be able to hack into and gain access to information about what's happening in their brains right now, which means that the NSA and other organizations can spy not just on your email and your cell phones, but soon potentially on your brains as well. Thank you both for letting us spy on your brains for this simple presentation. Any technology in the wrong hands could be problematic, but what these technologies promise us are so extraordinary that I believe that they're gonna become quite mainstream. I believe it's something that we, just like many of you may be wearing fitness trackers right now, that we'll start to use and that we'll start to embrace as a society. And that comes with great promise, but also with ethical and legal perils. It requires that we think about what the world may look like as even the last bastion of freedom and privacy suddenly becomes more transparent. We as a society have to decide whether or not the ability to access and change our brain to something that we want, that we're going to embrace, or something that we're going to put limits on. These devices are one way devices, meaning perhaps you could be hacked into in a way simply by listening to what's happening in your brain, but there won't be any changes to what's happening in your brain. And yet, we can also change our brains right now too.  How many of you had a cup of coffee today? Okay, that's a cognitive enhancer, right? And it's one that we've come as a society to embrace, but in other contexts, like in sports, we think revving up your body through the use of steroids or other forms of doping is cheating. Consider the fall from grace of Lance Armstrong, who when it was discovered, the types of measures that he took to enhance his performance, went from one of the greatest athletes of our time that we celebrated to one that we questioned to one that we think no longer represents that pinnacle of honed, raw talent. Do we feel that same way about our brains? Is it the same thing that if we're taking different drugs or devices we're somehow less natural? We somehow are less human that we're cheating? Or is that really the point of why we're all here? Is the very idea of being a human honing our abilities, improving our cognition and enabling us to be the best possible version of what we can be? This is Hobby J. Hobby J may not be smarter than you, but he is smarter than the smartest of the Long Evans rats. And the reason Hobby J is so smart is because he's been transgenically modified to increase the receptors in his brain that enable him to remember things better. And that is terrifically promising, not just for Hobby J, but for humans because the targets in Hobby J that turn out to be useful to enhance his memory are also great drug targets for improving memory in humans. And that's incredibly promising for Alzheimer's patients and others who suffer memory loss. But since I saw most of you raise your hand and say that you've lost your keys or forgotten names, it also might be tempting for you to take a memory boosting or enhancing drug. And it's certainly tempting for a college student studying for an exam to be able to remember things better. Is it cheating if they get an advantage relative to their peers by being able to take a drug? If it is, are there limits that we wish to put on it and how would we possibly differentiate between the drugs for improving memory and the coffee that you all drank this morning that improved your ability to be awake and to process information? There's a different drug whose name you probably may be familiar with already and if you're not, you may want to write down because I think it's a fantastic drug. It's called medaffinil, provigil. You might know it as provigil instead. Provigil turns out to be incredibly powerful as a cognitive enhancer. It was first tested on Air Force pilots and there were some remarkable results. They were trying to improve their wakefulness because you might hope that an Air Force pilot is somebody who's able to remain awake and have sophisticated motor skills even when they suffer from sleep deprivation. What we found in Air Force pilots was when they took the drug, even after not having any sleep for 72 hours, they performed better than people who had had seven hours of sleep per night on motor performance IQ tests. But they also did just as well if not better on performance IQ tests, not just motor IQ tests. And yet this is a drug that worldwide is significantly limited because we haven't yet figured out how we actually assess the risks and benefits of enhancement drugs. The traditional model of safety and efficacy doesn't work as we think about these types of drugs and devices and we're gonna have to fundamentally change the way we think about things to enable you to make a choice as to whether or not you want to take the drug. To make the choice as to whether or not we as a society will embrace these types of devices, these types of drugs. But we have to ask as we're able to change our brains, as we're able to access our brains, as we're able to speed them up and slow them down, improve our memory, improve our wakefulness, detect what's happening there, detect what's happening in others' brains if there's some space of mental privacy. Privacy is something that seems to be gone. Big data and then the era of big data, we seem to have lost any sense that there is any aspect of our lives that can't be analyzed and that big data can't be collected about. Is that true for our brains? Does the UN Charter on Human Rights protect something like freedom of thought? Freedom of speech is something we've traditionally protected but what about when we can get to your thoughts? Should we as a global community start to think about protections for our brains or should we embrace a society of truly total transparency? And we have to ask whether or not we're gonna give people access to information about themselves. In the US, the company 23andMe is a direct-to-consumer genetic testing company that was shut down in part for a while when the Food and Drug Administration decided that the information that they were giving to individuals was potentially dangerous and if you look through the reasons why, a lot of it was based on a paternalistic notion of what is dangerous, that we have to go through a gatekeeper to get information about ourselves because we're not sophisticated enough to have access to information about ourselves. Will they decide that? About drugs and devices and consumer-based EEG devices? Worldwide, we're starting to look around regulations of those issues. Will that slow progress down in brain health and access to ourselves? We have to decide, as a global community, how we're gonna approach these issues. The promises of brain science are extraordinary. The promises of being able to access your own brain, change your own brain, and even potentially access the brains of others is extraordinarily exciting. What we can do as we start to harness what has traditionally been in the black box of our brains is incredibly terrific, but it's also a little bit terrifying. Thank you.


The European Union's European Parliament has passed a draft of a law known as the AI Act. Here's some of the highlights in it. The Act bans the use of AI for predictive policing, real-time biometric ID in public like facial recognition, emotion recognition for law enforcement, border management, or workplace uses, and it also bans untargeted scraping of images from the internet or closed circuit television to create a facial recognition database. You can't just do the big wide net thing anymore. It categorizes applications by risk as well. So for example, video games and spam filters are in the lowest risk category. Apps that use AI to evaluate your credit risk or decide whether to issue a loan are examples in the highest risk category and those are where the strictest controls would be applied. Social media recommender systems, as they call it, were added to the high-risk category as well.


All this month, get any smartphone 32-gigand under for 50% off regular price at Appalachian Wireless. Better service, bigger savings. That's today's Appalachian Wireless. While supplies last, two-year agreement required. Several states across the country are passing so-called Blue Lives Matter bills, which classifies some attacks on first responders as hate crimes. The Kentucky House passed a similar bill this week, but not without some controversy. EKB news reporter Sean Allen has the details. 93rd District State Representative Chris Harris has been catching a little heat in recent days since casting his opposing vote for House Bill 14, Blue Lives Matter, a bill that seeks to enhance penalties for violent crimes against police officers. Senator Ray S. Jones II has recently taken to social media saying he doesn't understand how anyone could oppose this bill. Our police officers and first responders should be protected from hate crimes. And we have seen numerous incidents around the country of police officers and even first responders being targeted just simply because of the position that they hold. I think the bill needs to be amended to include police officers' families. This bill is not going to solve the entire issue, but it says to our police officers, first responders, that you matter to us, that we value your sacrifice and if someone attacks you because of the position that you hold, that they're going to be held accountable and they will receive enhanced punishment. So I fully support the bill. I can't see why anyone would vote against an important piece of legislation like this. Representative Harris explained his opposition on KET's Kentucky Tonight, saying that penalties for crimes against police officers are already in place and that the bill is politically motivated. Let me first say today, all lives matter. Not just blue lives, not just black lives matter, but all lives matter. And I felt like this bill was a, it was just a political feel-good bill. I didn't feel like it did anything really to protect our first responders over and above what, you know, we kept hearing stories in the house about so and so, heard this time that somebody was hunting a police officer. And they were hunting, you know, hunting our police officers. Well, if you're hunting a police officer, the full brunt of the law is going to come down on you. You're not ever going to get out of jail.  Because there were advanced penalties on the books already. There are already those in place. The bill passed the House with a 77 to 13 vote and is now in front of the Senate for consideration. Reporting from Pikeville for EKB News, I'm Sean Allen.


The Book of Mormon. My name is Zola. I'm from Swaziland in the Zuluni branch. How the Book of Mormon has blessed me. It has blessed me in so many ways. Whenever I read the Book of Mormon, I feel comforted. I feel the love that Heavenly Father has for us. That He could bring this book in the dispensation in this time where we live. It is the same as the Bible, but the Book of Mormon has got the restored, the whole restored gospel in it. Everything that Heavenly Father wanted and that He wants us to know is in the book. Every question that you have as a person in life, in whatever things you go through, you know you can find the answers in the Book of Mormon. I know I can always rely on the Book of Mormon for advice, for counsel, for courage, for love, for anything that I stand in need of. And it helps that when you read it, you can pray about it. So that's the reason that I know that it is true and it is another testament of Jesus Christ. Not because I've heard someone say it so. Maybe at first I said it because I heard so many people say that it is true. But after I've read it myself and pondered about it and prayed to Heavenly Father, He please manifested that this book is indeed true. And when I did that, I got the answer. It is true. It is another testament of Jesus Christ. And each was written by Asian prophets just like the Bible was. It is in our last dispossession and I love that book. I am grateful for it. I know it's true. Amen.


Hello my people, my name is Meacham. I'm your college counselor and today we're going to compare five digital SAT practice test services. I'm going to put them against each other and figure out which one is the best value for your money and which ones are going to help you prepare better for the SAT. We're going to look at Score Smart, Test Innovators, SuperTutorTV, The Princeton Review, and thesatcrashcourse.com who I gotta admit is a sponsor of this video. We're going to look at the quality of the tests, how much they simulate the digital SAT environment and like how accurate the questions are, quantity of tests, how much material are you getting and of course the value. What are you getting for your money? Which program offers you the best value? For the purposes of this we're just looking at self-guided programs where you get practice tests, you take them when you want and that's it. I want to start with the Princeton Review because they're one of the biggest names in this space and I feel like it's one of those that people will look for first and then there's the digital SAT self-paced course. That's what was paid for here. I think I got to go to practice up here at the top. Let's take a look at test two. You can choose if we want to be timed, time and a half or double time. People who have you know extra considerations for the digital SAT can get extra time if they have like diagnosed learning difficulties but we'll go ahead and just do the regular timed exam here and I'm going to start doing the English portion. The interface feels pretty blue booky. It looks basically the same. I think that's pretty good. We've got our review page feature down here with all the questions and I can go to the review page. That's cool. Okay and I can go back and I can flag for review. I'm just testing out all these things. Okay interesting that's not working. When I click on mark for review it doesn't work. Okay can I mark it for review now? No? How do I mark it for review? Apparently mark for review doesn't work. As you can see I am clicking and clicking and clicking and clicking. Mark for review is broken. I can zoom in and out like I normally would which I like because I do always encourage you to zoom in and out on these questions. The annotation features are here. Again that's a complete waste of time. Don't ever use those. This hard module introduced some kinds of questions that I didn't see in the first module. For example we have a sentence function question here and we're also looking at a paired passage question here which were not previously in the first module. In the first module we had like six note taking questions at the end. Module two only has one note taking question so that's definitely a good example of how they've been diversifying some of the difficulty here. Overall I would say this second module has been harder. Okay so reading and writing was 660. I got a little bit lazy towards the end. Okay so module one I had three incorrect and on module two I had five. Okay so we have question explanations which I do appreciate on this tool like I'm getting information about why my answer was wrong. I can see why why this would be an addition. Like I think that's a good explanation. Note taking questions here. There were a couple of these that felt pretty like subjective to me. This one says it introduced Maya Darren and her film Meshes of the Afternoon. So because it asked me to introduce both I looked for an answer that had information about both. This one just asked me to introduce the film not its director. So each of these three mentioned the director but it wasn't specifically asking me to introduce the director. It's inconsistent. Like if you're gonna ask me to introduce the film and the director and then I get it right when I do that and then you only ask me to introduce the film and I get it wrong when I do that then there's something off. See this is wrong. Like total languages enunciate words and phrases using pitch change. No they distinguish words and phrase using pitch changes.  I studied language education. This is not right. This is actually wrong. Princeton reviews tests and I've seen this before when they were on paper. They are designed in a way to be unfairly hard which gives you a lower score which encourages you to pay more for preparation. But I will say I like the way that this is presented. This is a good breakdown of your scores. When it comes to quantity Princeton review gives you three full tests. You're gonna spend $300 to get three full tests which makes Princeton review one of the worst values that we're gonna look at today. Considering the SAT itself only costs like $100 to take. Like you could literally just take the damn test instead of going here and doing this and it would cost you about the same. Considering the quality of the Princeton review tests I've given it a 7 out of 10 due to the issues with marking questions for review and some of the issues with the answer explanations and the nitpicky wrong answers that have contradictory explanations. Quantity honestly I'm gonna give it a 4 out of 10. I feel like three full practice tests is okay but you know you could be doing better. This has to be one of the worst values in the entire market. $100 per test makes this one of the most expensive and really that's a rip off. I would never recommend you spend $100 per test just to get three practice tests. Like honestly there's way better alternatives out there. Next up we're gonna look at SuperTutorTV which is also a very popular YouTube channel and one of the few that I actually have a lot of respect for because I think they cover some really important topics and they do it in a very professional way and not the sensationalist way that some other channels like to talk about SAT topics with BS videos like what's gonna be on the March SAT. You don't work for college board. You don't know. They have four tests. They have the one main mock practice test and they said three more by early 2024 and I can confirm that those are there. I'm gonna grab the reading and writing complete section. I see I can choose the different modules if I want to go back and look at like the easy module or the hard module. I think it's nice that they give you that option. Same basic message I see when I open a blue book. We've got our eliminator mode. It works basically the same. Cool. We can mark for review that does seem to work here and then it is shown up here. Okay cool so this looks like a very faithful representation of the blue book app. Zooming works correctly. Look at how it scales the text. Look at how it fills my screen. Look at adjusts. This is great. This is what you want. This one question is a little bit awkward like this is a graph question that's asking you to complete the sentence but they're giving you the whole sentence here as the question instead of just asking you which one completes the text and putting that text down below and for some reason the text is in this odd inverted color box. I'm not quite sure why they're structuring this question this way. I haven't really ever seen one on the digital SAT look like this. The first module is done here and I just want to say this is so far the hardest first module I think I've ever seen in any practice material. I usually can get through a first module in like 15 minutes at Mo's. I can appreciate a test that wants to push you and go harder than maybe the real thing so that you're more well prepared but at the same time I kind of felt like it was unnecessarily difficult in a few cases. Second module was actually easier than the first which concerns me a little bit. It makes me wonder if I got an easy module. Okay so I got 720 so I must have gotten the hard module. They've marked all the ones that I got wrong for review as a way to kind of tell me to go do those. That's nice I guess. They don't really break it down by module here they just have like let me see if I go to reading and writing does it split it up? No so it just gives me all the total questions that I got. It doesn't tell me like where the module ends which is after 27 but still it's like it'd be nice if you could split those up maybe for me. Overall I would say the questions that I missed were hard questions. They were literature questions. Some of them had some small subjective differences but overall I don't really have any major problems with the design of the questions themselves. I feel like these questions were good questions albeit very hard ones. If you're looking to challenge yourself SuperTutorTV's questions are really good. They are strong challenging questions. I would say their quality rating is the best so far. I'm going to give them an 8 out of 10. I only would take a little bit of points off just because I felt like the difficulty was a lot harder than what I'm normally used to seeing for typical digital SAT practice. For quantity I'm going to give them 5 out of 10. Now when talking about the value that you're getting from SuperTutorTV they do include some extra stuff. One, two, maybe three sets of practice for any of these given areas but I don't feel like we're getting a lot of extra material for the money. Average here is about $40 a test if we're getting four practice tests. You're not getting as bad of a deal as something like Princeton Review where you're paying $100 per test but you are still paying a lot of money for these tests and you're not getting a whole lot more for it. So I'm going to put the value down here at 4 out of 10. Next up I want to talk about test innovators. I honestly don't know much about these guys but their website says right here that they have SAT prep for success. Initial impressions like this is pretty close to what I see in Blue Book. Eliminator right next to it could lead to maybe some accidental misclicks because that's why it's nicer to put it on the right. I do got to click the actual letter. I can't just like click the word or the whole area so I got to like pay attention to my clicks here a lot. There is that option to flag it for coming back later. I do like the way they've put this bar up here. I actually think that's a little bit better even zoom in and out but it's not adjusting the way that the text is being presented. So if you notice what's happening here like it's not adjusting where the text is displayed it's not necessarily changing the number of words in a given line. There is some sort of annotation tool here so I can highlight stuff and then click to erase. I'm not seeing like a perfect simulation of Blue Book but it's got most of the things you need. I also like that there's this quick pause in case I want to just come back later. That's kind of a nice feature. This right here is a good example why it gets a little bit annoying when they don't scale stuff like you can see I'm trying to like expand this window so that it should fill more space on the right here and the size of this text is huge so like the question is taking up tons of space on my screen. I got to go all the way down here and now I can't see the notes like I want to be able to see everything all at once. I like that it gives me this clear indication of getting the hard module like that's a really nice touch. I also like that they kind of break things down by easy medium hard difficulty. It says I had seven hard questions and I got one of those wrong and I had 13 medium questions and one of those was wrong. I like this little Gantt chart kind of vibe where they're telling me how much time I spent on them and it's sort of helping me visualize how long it took me to take a look at some of these questions that I got wrong. Oh they even give you the percentages of the questions that people marked. So I got a couple of issues with some of these questions. Not too many. I think for the most part these are pretty good and I would say I would recommend this product. I'm going to say seven also for the quality of the test on test innovators. Questions feel very similar to what I see on the real digital SAT. A lot of data visualizations and they're adding a lot of extra features. Blue book simulation isn't quite as good as the real thing and it's not quite as good as the other ones I've seen so far. When it comes to quantity this is like a 7 out of 10. You've got 10 full tests but even if you only pay for one test you're going to get access to those thousands of questions that they have in their bank and I'm going to give them a 7 on the value too. I think that this is a pretty solid deal. I would honestly encourage a lot of people to check out that $49 option. Like I think to get one full test and access to all those questions for 50 bucks for a year is a pretty good value. That takes me to thesatcrashcourse.com. This video is sponsored by them and I want to be fully transparent with everybody here. I am not going to rate any company better just because I work with that company or they have sponsored any of these videos. The value and the quality and the quantity like all three things are good across the board and I think that they're probably the most solid option for you. Currently they're focusing on their 10 most popular tests. If you saw my video before I mentioned that finding 20 since I've done a lot of review on their questions they're doing some reworking with those other 10 tests and right now they have a main package of 10. S.A.T. Crash Course has made their first 10 tests even better by making video explanations for the answers. So at any point in your review of those 10 tests you can just click solution video. Why each response is right? Why each one is wrong? You can turn on the transcript subtitles here as well and overall this is a really good way to get an explanation for your answers. I feel like it's much better than just a pile of text at the bottom of the screen that can be a little hard to process. As far as the app design they simulate the blue book environment perfectly. I did not see one missing feature. I've never encountered one bug or glitch quantity. Again we're talking about 10 practice tests here for 200 bucks if you buy them all together and that brings me into the value proposition here. The only other company that can compete on value here is Test Innovators because they're selling 10 tests for $1.99 so it's exactly the same price basically $1 difference. But here's where it gets interesting. If you use the code SCORE you're going to get an extra 20% off your purchase when you're at thesatcrashcourse.com and that definitely makes it the best value that's out there. If you can save 20% off of this it's automatically the best value. You're not going to find a better deal.  Now naturally you would think that I'm going to be a big fan of this one because it's got SCORE in the name that's the name of my business also. Is Score Smart as awesome as Score Test Prep and College Counseling and would I recommend it for you? We're going to find out. They offer one free test that you can take right now which is very cool. I think it's really good for everybody to get a free test so they can see if they like the platform or not. The interface here when you log in. Now I did buy this second digital SAT test. I had opened the first one for like a minute and then I decided I wasn't going to do it and then it basically marked it as done for review. So be advised if you open anything here you better be ready to take it and you better be ready to say goodbye to your $35. Don't clear the browser so yeah don't leave okay. Okay so we have some instructions it looks like there's a few different tips. Eliminator mode works fine. Mark for review seems to be working. Question number two does it show up down there? It does okay. That's features. How's the scaling? Scaling is good. We can zoom in and out. That's what I like to see. I have gotten to the end of the first module here. I feel like the first module was pretty good for a first module. It didn't feel too hard or too easy. However there is no review page. It's kind of like when you get to the end here it's straight up says you better like go back and check until you know until you're super ready to end. Module two is done. Some of the stuff in the dashboard is a little bit awkward to look at. Like I feel like the information here isn't as clear as it could be. It tells me that okay I got only one wrong in the first module and two on the second module. Click on a circle to see more analysis so let's take a look at that report. I feel like this should be easier. Like it'd be nice if it just jumped me straight to the questions and I could look at them the way I do with most things. Okay this is a zip file. One minute 37 seconds later. Back close this out. Review questions. Okay maybe that's where I gotta go. Okay so we missed number 11 on the first module. That's the only one. Let's go to question view. That's gotta be okay finally here we go. Okay so not easy to get to the questions and just go straight to reviewing them. If I'm just looking to find the ones that I made a mistake on like if I go to performance by question right I see that 11 was wrong. I have to remember that 11 was wrong. Like if I go here on question view there's no indication that I missed any of these questions. It just shows the answer that I marked and I can check the answer and it'll tell me if it's okay or not. Okay so it's a green check if it's correct. It's gonna be a little awkward like here I gotta go and click the answer. Okay I was between that one and that one. That seems fair. I don't see any explanations here. I don't see any like reasoning for why those answers are right or wrong so that's kind of a hit against this too. The other thing is I marked some of these questions for review specifically because I kind of wanted to go back and take a second look at them because there were some questions that I wasn't too sure about but that hasn't been saved so like I can't go back and check on some of those ones where maybe I guessed so that kind of brings down the quality ranking for me. I'd put it at like a 6 out of 10. I feel like there's some missing features that need to be added and the interface could be a little cleaner and a little easier to use and that would make this a much better product. So the value proposition for ScoreSmart isn't bad. I mean I'd probably call it a 4 out of 10 because you're talking about $35 per test which is still quite a bit. It's definitely not as bad as Princeton Review which is god awful. And it's certainly better than SuperTutorTV and there's more exams available but there is no extra material and in order to get a better deal on those tests you have to go all the way up to like the big package that gets them for $25 a piece. So I feel like ScoreSmart not a bad deal but definitely not the best option here. Let's just do a quick recap here to close this out. We've got our five services. I'm going to start at the bottom with the Princeton Review. We've got quality issues with the app. Only three tests and an exorbitant price. That one is dead last and it's not even close. After that I'm going to put ScoreSmart in fourth place. I think that while they have a large number of tests in fact more than anybody else that we've looked at today. There's some quality issues, user interface issues and the pricing isn't great compared to some of the other options out there. I'll put SuperTutorTV in third place. I think they have very high quality tests although the difficulty is a little bit broken and while they are more expensive than something like ScoreSmart I think that you're getting more value for your money considering the fact that they do have extra material and overall the interface is an excellent simulation of Bluebook. In second place I have Test Innovators. Their value is tough to beat. You can get 10 tests for $20 a piece if you buy the full package or you can just get one and still get access to tons of extra material. The fact that Test Innovators has all that extra material makes them a really good value as well and I think that the quality of their offering is really good even if it's not a perfect Bluebook SIM. It's decent, it's good enough, gets the job done. In first place I got to put the SAT Crash Course.com. They have just as many exams as Test Innovators but they've also got great video solutions to their questions now. They've got an incredible value if you use the code Score when you check out. You're going to get a better deal there than you will anywhere else. Their Bluebook simulation is simply perfect. There's not a single flaw with it. I can honestly say that SAT Crash Course.com has built a fantastic product. If you're interested in learning more about them you can go to thesatcrashcourse.com. You can use the promo code Score at checkout to get 20% off. Hopefully you found this helpful. If you have any experiences with some of these companies and you want to share them please do in the comments below. I would love to hear what you think about these services and especially if you've used more than one it would be great to get your input. Thank you so much for watching this and I'll see you next time.


Hey everybody, Dr. O here. In this video we're going to introduce the carbohydrates and talk about the monosaccharides. So the word carbohydrate means hydrated carbon because they're made of carbon, hydrogen, and then oxygen. Carbohydrates are called saccharides, another word meaning sugar. So we're going to look at the five monosaccharides, really only three we care about, at least for this video. We have the hexoses, which are called that because they have six carbons in that ring, and those are going to be glucose, fructose, and galactose, come right back to them. And you do have the two pentoses because they have five carbons in this ring. This is going to be deoxyribose and ribose. Maybe you've heard of deoxyribonucleic acid, DNA, and ribonucleic acid, RNA. So we'll cover the pentoses separately because we're primarily looking at the dietary monosaccharides here. Let's take a look at the three, glucose, fructose, and galactose. So carbohydrates like this have four calories per gram. You'll see that fat has nine calories per gram because both carbs and fat have carbon, hydrogen, and oxygen, but fat has a lot less oxygen, so you can cram a lot more hydrocarbon onto a spoonful, I guess. So glucose is the only carbohydrate our body uses. Fructose and galactose are great, but they have to be converted to glucose by the liver before they can be used for fuel. That can be good or bad, depending on what you're looking at. So glucose is, like we talked about blood sugar, we're actually talking about your blood glucose levels. Glucose is the key fuel. Glucose is what your body loves to turn into ATP, especially when you're physically active. Carbohydrates are your primary fuel source during physical activity, but we'll talk later about how fat is your primary fuel source when you're resting. So fructose is what's known as fruit sugar. It is sweeter than glucose, so you probably, well, we can, oops, sorry. So fructose is fruit sugar, and as you'll see there, it's going to be part of sucrose or your table sugar, and then we'll talk about high fructose corn syrup and those kinds of things later. But it is sweeter than glucose, which is why humans love to eat it. And then galactose, you can see like the name implies, is going to be part of lactose or your milk sugar. So glucose, fructose, and galactose are your three key monosaccharides, and we eat all of them, but they need to be converted to glucose before they're used, which is why if you have a diet high in fructose, it doesn't actually raise your blood sugar, which is why you'll see like a lot of diabetic things or some things are touted as being good for your blood sugar control because they have a lot of fructose, but your body will turn into glucose. It's actually going to be taxing your liver, which is why I believe that too much fructose is really, really hard on the liver and potentially even hepatotoxic. So we'll cover all that in much more detail on nutrition. I'm just introducing the monosaccharides right here for now. Okay, so those are your monosaccharides. I hope this helps. Have a wonderful day. Be blessed.


This is lecture 8, okay, and we saw a whole bunch of very important things in the last lecture. I want to go through roughly the crux of what we saw in the last lecture was how to go from a waveform type channel which is y of t equals x of t plus n of t, okay I want to remind you once again that since the bits are unknown and random, x of t itself becomes a random process, n of t anyway is the noise process about which you can't claim to know and then y of t in effect becomes a random process, so you can't expect any deterministic type reduction. So if at all you reduce it, you will get a random vector type of solution. So we saw how to reduce this. What is the main, what is the most important tool from a purely mathematical point of view, what did we use? Yeah, just simple orthogonalization based linear algebra, right? So we saw if you have a finite number of, you can think of these x of t as vectors, okay, once you fix a basis and how do you find the basis from all possible signals that are going out, you do orthogonalization, find a basis. Once you find a basis, everything becomes very easy, everything becomes a vector, okay. So you went from here using I'll say linear algebra to a large extent, okay, that too very simple linear algebra and actually if you do it rigorously, there's lots more areas involved, okay. So you have to define what happens to random processes when they go through correlators, all these things carefully, you can do all that and you can meticulously make your way towards a vector channel, okay. So the vector channel then becomes y is x plus n, okay. So what's the dimension of this vector space in which this vector channel gets defined, okay. Where does the dimension come from? Number of basis vectors that you got from the orthonormalization, okay. So if you have a whole bunch of x of t's, say you're sending n bits, then you'll have 2 power n different x of t's, okay. So you start from there and reduce it to an orthogonal basis, you might get a certain number of basis vectors, okay. So that is the dimension here, okay. So each of these guys now is a random vector, okay. So from random process, well continuous time random process, we've gone to random vectors, okay. And once you have random vectors, it's also possible to quickly figure out what the joint PDF and joint PMFS, okay. One of the typical assumptions you know, you don't have to make this assumption from in the derivation we saw that n would be distributed as what, assuming this is a white Gaussian process, okay. And you take a PSD to be n naught by 2, okay. So PSD n naught by 2 in all frequencies, you take that, then n becomes what? It becomes IID, normal, zero mean variance, n naught by 2, okay. So the PDF for n is very clear and also clear that x and n will be independent. How do you find the PDF for x? Joint PDF for x. First of all, in most cases, what will be the type of distribution for x? Will it be continuous? Will it be, for instance, n is continuous, right? So will x be continuous or will it be discrete? Yeah, in most cases it will be discrete, right? So you have only one of 2 power n possibilities, okay. And typically you don't think of the 2 power n as a very large number. So usually when you do modulation, you do modulation over very small number of bits together. It's also possible to do it for a large number of bits. Maybe we'll see such a system later on. But for now you can imagine just a small number of bits at a time, okay. So x would be discrete, okay. And it's not, you should know how to find it, okay. So it's usually assumed to be IID over the number of discrete signal vectors that you got, okay. So that's the distribution. But this will be typically discrete, okay. So this x is typically denoted on a signal space, okay. So that's called a constellation, okay. So that's pretty important. What's the constellation? Okay, I'm doing it in 2D here. So in general, it will be m dimensional signal space, okay, where this m came from the reduction that you did, or the normalization that you did. And each possible x, you put a kind of a star here. Each of these guys are signal points now. They can be converted into a signal if you want, okay, using the orthonormal basis. I also made a remark later saying typically we can also do the reverse, okay, to design signals themselves. You can say I start with an orthonormal basis first and define points on that orthonormal signal space, on that signal space, using my orthonormal basis vectors to go back to the signals. That's another thing we did, okay. So in this conversion, there is one thing that is preserved, okay. One aspect of that signal that's very important to you from a constrained point of view, one thing is preserved. What is that? What do you preserve? So from this vector itself, without knowing the basis, you can compute something about the signal. What can you compute? The energy, right? The norm of the vector is going to equal the energy of the actual signal, okay. So the power or energy in the signal is faithfully represented in the vector as well. What is not easily represented without knowing the basis? The bandwidth, right? You know the bandwidth, you should know what the basis is, what the actual signal is, okay. So other than that, the energy itself is nicely represented in this vector itself, okay. So since norm of vector x equals energy of x of t, so I should put small here, doing x of t, okay. So typically the assumption is by the time you come to the constellation, you've already figured your bandwidth out, okay. You've figured out your bandwidth and you've chosen the basis suitably so that it fits nicely into your bandwidth. So you don't have to worry too much about it. So if you don't do that, no point in coming to the constellation, okay. So the constellation doesn't make much sense if you don't do that, right. Because your x of t is what? It's the signal that is received at the receiver. It's already gone through the channel and all that. You've arranged for all those things the way we did that, okay. If you're not suddenly saying bandwidth is not enough, then you have to go back and fix that first, okay. So once you do that, energy is pretty much the only thing you need, right. That's very good. And it's also nice to note that in the noise process, the noise random vector that we got, the PSD of the noise process is nicely represented, right. The variance of your noise is N0 by 2 which is the height of your power spectral density for your noise, okay. Like I said, in practical systems, it's possible to measure this power of the hub, I mean, height of the power spectral density of noise, okay. So this model nicely captures many of the important complications that will arise in the real communication system, okay. So that's what's so nice about this model, okay. There was also another part of the noise which we rejected. The reason why we rejected is, okay. Given this vector y, that part of the noise is irrelevant to x, okay. So it's independent of x. So you don't have to look at it at all. It won't affect any of your detection process, okay. So how did our receiver look now finally? Okay, so you have a bit B. You're doing transmitters doing a modulation to get some vector, some signal x of t, okay. Noise is getting added to it, right. You get r of t. What are you going to do with the receiver? What's the first step?  So you do correlate with phi i of t, i going from 1 to m, okay. So you do correlation to get a vector y, and that's enough. What do you do next?  So at this point, what you have is a detection problem. We'll precisely define it later on. But at this point, once you get your y, you have a detection problem in the sense that y has a certain joint distribution with, if you correlate the x of t, what do you get? Correlate with phi i. What will you get here? You'll get a certain vector x, right. So this x and y have a joint distribution, okay. And the joint distribution also involves n, which is the correlated version of n of t, right. So given a joint distribution, you're allowed to observe only one part of it. You're not allowed to observe x, right, at the receiver. You can observe only y, which has a certain joint distribution with x. Given that, you have to find the best guess of x, okay. So that is the detector. That is a detection problem. We'll see later on how it's properly framed. And it's very easy to do it with the vector formulation. At least the formulation is easy. Solution might be difficult in certain cases, but formulation is typically easy, okay. So this is the picture. So once you have this picture in mind, you can nicely move to a vector picture, okay, where you say, from a bit b, I go to what? Okay, I do a certain modulation. I go to what? The vector x, okay, to which what gets added is what? A vector n, okay, and you get a vector y, which directly goes into my detector. To produce, well, you would like b hat, but you might actually produce even x hat. It doesn't matter, right. So b hat or x hat, whatever you produce. Okay, because b to x is a, hopefully a one-to-one more transformation. You don't want to do anything other than one-to-one, okay. So that's what you're doing, okay. So this is the picture which we will use pretty much, but you should know that where it comes from is this correlation picture, okay. If you don't do these correlators, this picture makes no sense, okay. And auto normalization plays a very important role in the way we got to the vector of presentation, okay. Any questions? People have time to think about this. Any comments, questions about what's going on? Sir, in the sending, what is actually being sent? Well, it's an interesting question. What do you think the answer is? Vector x, in reality, in the real system, you know. I think you should think of it as x of t. Okay, in the real system, x of t is the actual signal that is being sent, okay. So in reality, signals will be continuous time, right. You can't think of real, I mean, see, you're probably used to thinking of signals as vectors. It's okay. It's not a big, it's not a bad approximation given that you have sampling and all that. But in reality here, x of t has to be sent because we have a certain bandwidth limitation, power limitation and all that. All that is only satisfied by x of t. If you say x vector, what about bandwidth for it? I mean, what do you think of that? Okay, so all those things are not true. So what's actually being sent physically through the pair of wires is x of t, okay. So this is a model which nicely fits that reality, okay. Everything about that reality is nicely captured in this model, okay. So it's called the ideal AWGN model, okay. So that's an important question because, so the reason why that question is important is, so see, as you keep learning digital communications a lot, you'll pretty much only do vectors, okay. So in fact, it's also become to a stage where all your signal processing algorithms sit on processors and computers on these boards that you have. So I don't know if you're familiar with building of systems and all that. Today people make boards, okay. What are those boards? Who do they make them? What's that called? It's a nice, it's called PCBs, right. So you always do PCBs, okay. Printed circuit boards on which all your circuits are now mostly chips, right. So there's no resistors and capacitors of course there, but most of the important things that are done inside chips, okay. So it's all fabricated chips. And it's just a question of when you build a system going there and getting the right processor, getting the right A to D converter, D to I converter, getting this, putting them together on a PCB, making the layout, maybe so many layers, you send it out, PCB comes back, you populate the board and you test it out, right. That's how you make systems today, okay. So what is the communication engineer see mostly? You write pretty much C programs, right. Which get compiled into some processors, assembly code and it's running there, right. So for you, you can even think that the only thing you're worried about is this vector X. Whose problem is it to convert that vector X into an actual signal that's going out? Somebody else's problem. Maybe somebody who's doing analog or RF or BLSA or something. It's not your problem. If you can make some system work with this vector model, it will convert faithfully into a continuous time model, okay. I can say that up to certain approximations, right. So obviously some problems there, okay. There's so many other issues, but those are not very important for communication engineers, okay. So it's enough if you work with this model. So you can even think of it as what's going out is X, okay. And say, once it leaves my processor, I don't care. Whatever happened to it can happen to it. I don't care, okay. The reason is even the energy of the signal is accurately converted is converted in this X, okay. So it's okay. It's not too bad to think about it. But if you want to be a complete systems kind of guy, you want to know everything that's happening in your board, you better know how the whole process is and how it's happening, okay. But yeah, I mean, in reality today, most communication engineers works with this model. Nobody really goes into the actual conversion, okay. So there are some real problems in that conversion. For instance, before you put it out on the pair of wires, these signals will have to be driven into the wires, right. So you have to drive up something, okay. Some part of the signal, like the current or voltage, you have to drive up, okay. So when you drive it up, you have to use some power amplifiers and all that. And this power amplifiers are not linear, okay. So what's the problem? Suddenly you have a non-linear component somewhere. Yeah, your whole assumptions will go for a toss, okay. So all your basis function you chose carefully for an X of t will not be basis functions anymore, will not be orthogonal anymore, okay. So there are several ways of dealing with it, okay. You'll see when you learn, people try to minimize peak power as opposed to average power. See, the X, vector X will represent only average power, right. So you might want to reduce peak power. At that time, you'll have to worry about the X of t, or figure out how to change X, the vector so that the X of t can change. So all those things are real problems when you convert them into practice. So I would say, don't ignore the X of t aspect, okay. So keep that also in mind, even though mostly what you deal with is the vector. Okay, any other question, comments? So if you're not thinking about this seriously, it's a good thing to think about, okay. So it's the foundation, right. Once you accept this, everything else will seem like tutorial problem, okay. But this is the foundation. So you should think more about these things. Okay, all right, so let's move on now. From here then, we saw a few examples of specific modulation schemes. And I can now describe everything in signal space. I think I first give you an example of how it happened from X of t, how you went back. So maybe we'll do that a little bit more, but I'll describe the first thing we saw, which is BPSK in signal space. So it's very easy in signal space, right. So what do you do in signal space? Simply say it's a one-dimensional signaling scheme with my bit zero being plus one and my bit one being minus one, okay. So that picture nicely represents everything that my modulator is doing, okay. It might actually be some complicated PCB with several layers, all that is relevant to me. What do I care about? What it's actually doing in terms of the detection problem. Okay, so ultimately the detection problem is what's important to you when you try to decode, right. So what it's doing is only this, okay. I have bit zero and bit one. I take bit zero and my vector or in this case scalar corresponding to that is plus one. And if I'd have bit one, scalar corresponding to that is minus one. I might have used root ES or something and it's okay. Plus one and minus one is general enough, okay. You'll see the reason is as long as you keep N naught by two as general, you can take this to be the same, okay. It's not a big deal. All right, so from this, the important step that you have to do in the model before we go to the detection problem is what? This is X, right. You have to go from X to Y and then figure out what the PDF or the joint PDF for X and Y is, right. In this case, since X is discrete, Y is continuous. Typically the conditional PDF will nicely capture the joint PDF, okay. So what are you only about the conditional PDF? What will be the conditional PDF? Y given X is plus one will be normal mean plus one variance and not by two, okay.  So conditional PDF, okay. I'm not writing everything here, okay. So I think I used a different notation, right. So I did this, I'll stick to that. This becomes what? One by root two pi sigma, what's sigma? Y minus one square by two sigma square. What's sigma now? Yeah, root N naught by two, okay. You can write a similar expression for Y given minus one. So now your detection problem is all set. You can say it's a question of just doing detection. Okay, so we'll do it. We'll see how to do that. That's also very easy. I mean, even the answer there, we could guess, right, very easily. How will you detect an arbitrary Y? If I say Y is 0.5, what will you say X was? Plus one, right. So it's just based on some very simple intuitive notions, symmetry of the noise, et cetera. You can say it has to be the closest point, okay. So we'll justify that later on, rigorously. Okay, so that's BPSK. The next example we saw where we kind of, I mean, we were halfway through that example. So I'll pick it up from the beginning. What's this, FSK, okay. Frequency shift k-ing, okay. So I'll do it once again like I did before. I said in this case, we choose N equals one, okay. So we do one bit at a time. So my bit B becomes zero or one, okay. And corresponding to bit zero, I had this signal to send out, okay. So root two ES by T cos two pi M0 by T, okay. This is between zero and T, okay. I said F0, I will let as M0 by two, M0 will be a positive integer, okay. And then I said X1 of T is going to be root two ES by T cosine two pi M1 T by T. Again, the same time interval. Never let F1 be M1 by T, okay. So I asked you to figure out something. If you want to even plot this, KM0 and M1 will be integers because of that. In fact, these two signals themselves will be orthogonal, okay. So if I integrate from zero to two T and multiply, you'll get zero, okay. So if you do basis vectors, you'll get two different basis, okay. So you can think of it as cos two pi M0 T and cos two pi M1 T by T, okay. So you'll get two different, so it's a two dimensional signaling scheme. What is the thing that I asked you to figure out? Yeah, so the complex envelope for X0 of T and X1 of T, okay. So what's the answer? What's the complex envelope of X0 of T? Give me a description if you don't have an exact answer. It's okay, I mean, if at all you've tried it, you can give the answer. We'll see if it works out or not. What's the first choice you have to make when you figure out something called a complex envelope? What is involved when you go from a pass-band real signal to a complex envelope? There's one quantity which you have to fix. The center frequency, right? So what's a good choice for center frequency for X0 of T? F0 itself, right? So you maybe pick center frequency as F0, okay. So what does this fix? When I say center frequency is fixed, what does it mean? Okay, my up conversion and down conversion circuits are fixed now. So what is the whole point of doing complex envelope? I want to process my signals in baseband, right? So if I want to process it in baseband, then I have to down convert with this F0. If I down convert with some other F, then I'll get something else, okay. Then all my processing might have to be different, okay. So I'm fixing F0 as center frequency for X0 of T. So it has a real meaning, okay.  So once you down convert, you'll get two signals, the real part of X0 tilde of T and the imaginary part of X0 tilde of T, which you can process in baseband in your receiver, okay. So what's, suppose I choose it as F0, what is my X0 tilde? I'm sorry, it's not of T, you're giving me the formula, okay. No, I mean, this may not be that difficult to figure out, right? So in this case, it's really simple. Go back and think about what X of T is in terms of the complex envelope, okay. Right, what is it? It's XI of T times cos 2 pi F0 T minus XQ of T times sin 2 pi F0 T. But that cos 2 pi F0 T and sin 2 pi F0 T are for all time, okay. So here, something else is slightly different. So how will you translate this? What's the only thing you have to do? What do you have to do? Okay, take some time, right. Put pen on paper, figure it out. Tell me what XI of T is. It's not really that difficult. Any answers, any ideas? If somebody has any ideas, it's also okay. Yeah, so what is it? You're right. I mean, so what's, yeah, you're right. I mean, X0 tilde of T can be real, okay. It's not something to be so often. Delta of T, there's no delta of T. What will it have to be? Route 2 is S by T between zero and T, do you agree? Does it seem fair? Okay, so can I say this is just simply 2S by T between and say maybe a suitably shifted rect, right? So I'll have to say T minus T by 2. Is that fine? That's not fine. It's not? Yeah, why? Oh, where is the band limitation coming from? I don't care about, you see, first of all, X0 of T. I made a lot of remarks about band width of X0 of T. What did I say band width of X0 of T is going to be? Theoretically, it's going to be infinite, okay? But in practice, you can approximate and say if I allow about F0 and what F0, if I allow, say 10 times 1 by T on, say 5 by 5, 5 by T or 6 by T or 7 by T, so enough lobes of the sink. If I allow, then I will get this signal almost, okay? So that's the assumption. So my bandwidth is supposed to be very large. So if we take such a large signal and do baseband equivalent, we'll once again not get a band limited thing, right? Okay, so it'll be once again for a large band, but assuming you have a large enough baseband processing bandwidth, so it's okay. So it'll work out, okay? Are you able to make sense of this in the frequency domain? Are you happy with this, the Hilbert transform type picture? Is it working out? Okay, right? Yes or no? No? I don't know. Think about it, okay? So try to make sense of it in the frequency domain and see if it makes sense, okay? So I think this is enough, like he points out. Yeah, yeah. So you'll have to think of some approximations and all that. So this is a good thing, a good way of doing it. But what about X1 of T? How will you deal with X1 of T? Okay, one way of doing it is, say I'll change my center frequency now. I'll say I'll change my center frequency to F1, but is that a fair thing to do? Do you think that's, at least from practical point of view, is that reasonable? What will it mean if you do think of X1 of T now with a different central frequency? First of all, in practice, then you'll have to have another down converter or up converter with a different frequency. Maybe that's not too bad. You'll say that's just a cost I'll add. I have enough money. What else is a problem? Yeah, that's a problem, right? Signal is random. You don't know whether it's being F0 or F1, okay?  Okay, you can't keep doing both and trying to figure out what it actually was later, okay? It'll be disaster. You might as well do just one, okay? So the whole philosophy of the thing will change once you realize the signal is random. So you don't know what you have to fix some frequency. So a good choice is to fix F0 for both, okay? You fix F0 as the center frequency for both. If you do that, what will happen for X1 tilde? Okay, same center frequency. What will happen to X1 tilde? Okay, anyway, I'll put a question mark next to it since enough people have questions.  So make sure the mathematics works out. Read up something about the complex envelope. See if it works out like you've seen it before. So what will you do for X1 tilde of T? So I think the zero to T is what's confusing people. So if it was just delta for all time, then yes, answers will come out, okay? So that's a good way to start. I'm assuming T is very large, okay? So you have the whole cosine. Then what would you get if you take cos 2 pi f1 T and downconvert it to the center frequency of zero? What will you get? You'll get E part J, 2 pi f1 minus f0 T, okay? That's what you would get. So here, what can you expect change to be? It's going to be defined for, multiplied by the rect, okay? So pretty much that's what will happen, okay? But once again, go back and make sure all the mathematics of the definition of upconversion and downconversion are working out properly and everything that you can do, you can do that, okay? There'll of course be a problem with this band limitation. So you approximate the band limitation to, say you band limited to some large enough thing and make sure it works out, okay? So it has to work out to something like this, E part J, 2 pi f1 minus f0 T between T and T. Okay, so you see the complex envelope works out to different types of numbers, all right? So you can also pick other center frequencies and make both of them complex if you want, okay? So you can make, pick something like f0 plus f1 by 2, okay? Maybe then both of them become complex. So that's just a change of basis type thing. So it's not a big deal, okay? I'm sorry? Times? Oh, the root 2 ES by T. You're talking about the root 2 ES by T? Yeah, yeah, that'll be there. That'll be there. I didn't write it. Yeah, yeah, you'll have the root 2 ES by T. Sorry. So I mean, once you think of going back to practice, right? These constants are not too crucial, okay? Ultimately, there'll be some variable resistor or capacitor which you'll tune to get whatever constant you want, okay? So don't worry so much about the constant. Yeah, you have to worry about these things when you do the detection, okay? And the PDF, when I write down the signals, it's quite irrelevant what these constants are, okay? Of course in the exam, it's very relevant. So can't say changing by constant, okay? All right, so convince yourself that this is right or if you have more thoughts or if you think the answer is wrong, something else is right, figure it out and come back and convince me. But I think this is a good enough baseband signal to process. If you don't convert, you'll get something like this with large enough bandwidth, okay? All right, so that's the FSK part of it. And let's go back now to the signal space. How did it look in signal space now? So the basis we picked where? The basis you can pick is root 2 by T cos 2 pi F naught 2 pi F naught T. And again between the rect multiplication is always there and F1T is root 2 by T cos 2 pi F1T, okay? Between T and T, okay? You have to pick F0 and F1 such that, I mean, I'm assuming F0 and F1 have been picked so that these two are orthogonal, okay? Or orthonormal, okay? So you have a integer difference in terms of frequency, so that'll work out, okay? So once you do this, your signals become what? Your vector X0 becomes, right? It becomes a two-dimensional vector. What is a two-dimensional vector? Root ES and? Zero. Okay, and X1 becomes again a two-dimensional vector, zero and root ES, okay? So this is nicely plotted in a 2D signal constellation. Okay, maybe I'll do it right here, okay? So this is the FSK constellation, root ES here, root ES here, say maybe this is zero and this is one, okay? So bit zero goes to root ES and bit one goes to the other root ES, okay? So if I define my random vector X, okay? How will I define it? Equals X0 with probability half and X1 with probability half, I assume my bits are equally likely, right? Zero and one are equally likely with probability half, okay? So I can also think of this X as a random vector with two components, X1 and X2, okay? What's the PMF for X1? Root ES with probability? Half. Half? Zero. And zero with probability? Half. X2 also is identically distributed as X1, okay? But are they independent or X1 and X2 independent? Clearly not independent, okay? So together they take only, X takes only two possible values, everything else is zero. So it's clearly very, very dependent, okay? In fact, totally dependent, okay? But they are identically distributed, okay? So if you wanna compute energy, what will it work out to? Okay, you can compute energy in this form, right? Expected value of mod X squared, okay? This will work out to ES, okay? So that's the motivation for this notation, root ES by T, two ES by T and all that, okay? So usually it's very common to fix ES as one, okay? Like I said, those constants don't matter, but you have to keep N0 as variable to make sure the detection problem is being faithfully captured. All right, so that's FSK. And so once again, so if you go towards the detection problem, so I wanna spend some time on that because this is a strange situation. So what do you do with the receiver once again? Your R of T, okay?  One correlator is with respect to cos two pi F0T, another one with respect to cos two pi F1T, okay? So what will be match filter and all that? So if you want to match filter, so you should wanna implement the correlation as a match filter, you want a filter with response, impulse response cos two pi F0T, can you think of a very simple circuit which will have that impulse response? What kind of circuit will have that? Very simple RLC circuit. So LC, yeah, that's the LC oscillator will have that, okay? So you have an LC and make sure you drain off all the, there's a thing and then turn it on, it'll go to that. So an LC is a nice way of implementing this, okay? So, but you have to be careful with the timing, okay?  So if you can manage to do that, you can do these correlations with a simple LC. So you'll see some very, very simple systems, these RFID tag communicators, I don't know if you're familiar with RFID tags, they'll use systems like that. So you'll see in those readers and all, people will use simple LC stuff to do reception, it's possible to do such things, okay? So you have, strictly you have correlators with, I don't know what I called it. They call it F0, F1 or F1, F2, F0, F1, okay? F0, T and the other correlator is with F1, T, okay?  I've been doing F1, F2, I think in the rest, so I'm sorry for the 01 conversion, okay?  So what does the detector need to know once again? The joint PDF of Y and X, okay? So how will you do that? Like I said, since X is discrete, what can be readily described as the conditional PDF, Y given X, okay? So what is F, Y given X, Y given, let's say, okay? So I'll specify B here, okay? B, specifying B is the same as doing, okay? Anyway, I'll do the whole thing, it's not a big deal. I'll say root ES0, okay? What is this distribution? Gaussian, 2D Gaussian with mean? Root ES0 and variance or covariance function, which is, which we'll work out to? Yeah, N0 by 2, identity matrix, okay? So that's how it will work out. So it'll be circularly symmetric about root ES0. So that's how the PDF will look. So if you plot this, if you do a 2D plot of this PDF in MATLAB or something, it'll look circularly symmetric about, it'll die down in all directions about root ES0, so it'll look, okay? What about the PDF for Y itself? Half of this plus half of the other normal PDF, 2D PDF, centered at zero root ES0, okay? So that's how the joint PDF will look, okay? So this is the information that the detector cares about. We don't need anything more for the detector, okay? So you have this, you can do a detection. Okay, what do you think the detector will do? Yeah, so you'll split about Y equals X and if you fall on this side, you say it is that point, if you fall on that side. Since it's all circularly symmetric, it has to work out eventually, okay? So once again, we'll see a rigorous form of this as we go along later, okay? So this is a good reason why I'm repeating myself in these things. I'm going much slower than I usually should maybe, but I think it's good to spend time with these things. These are very basics, very important things. You shouldn't go back and question yourself on these things later on, okay? So make sure you understand, actually the complex envelope picture is also reasonably important, okay? So make sure you understand that stuff very well in your head. But finite bandwidth, yes, it'll be much simpler, but even in this case, you should maybe know what the idea is behind that, okay? Think about that. Okay, so let's go to the next example that we'll see, which I will call four PAM, okay? Four PAM, okay? So here, we'll start doing the, we'll start doing the, what? Just give me one minute. Okay, we'll start doing the reverse. So I'll pick my basis first, and then I'll give you a constellation, okay?  So just to give you more to it. So I'll say N equals two is what we pick. So my vector B is going to be two bits, B1, B2. So there are four possibilities, all with equal probability, I'm assuming, okay? 0, 0, 0, 1, 1, 1, 1. I'll say my basis is like we did for BPSK before. So BPSK can also be called two PAM, okay? It's also called two PAM. So here, we're doing four PAM. So the basis we pick will be just like the baseband type basis, one by root T between zero and T, okay? So my constellation with respect to this basis will be, I'll have zero here, but zero is not a signal point. I'll have four different points, okay? So I'll drop my root ES stuff from now on, okay? So get used to this root ES being one from now on, okay? So the reason I stuck on to that root ES is most books that you see will always do ES, okay? So I don't want you to get confused by that. But I will drop my root ES, I'll say ES is one from now on, okay? So we'll see later on in the detection problem that that's good enough or maybe in the detection problem we'll do root ES and then show you why ES can be one without any problem, okay? So we'll say this is plus one, this is plus three, this is minus one, this is minus three, okay? Obviously this picture is not to scale, okay? Why is this not to scale? Yeah, right? So plus three is suddenly much compressor. Maybe I should make this to scale, okay? We'll do that, okay? It's not too bad, so it's good to do this to scale. This is not too bad, plus three, it's not too bad, minus three, okay? So basically four equidistant points on the real axis, symmetric about zero because typically you might want your mean to be zero for your random process that you put up, okay? So this constellation and the basis completely defined all my actual signals. So how will my signals look? I'll have four possible signals, one corresponding to each choice of bits, zero, zero, zero, one, one, zero and one, one. I haven't put out what that is. So maybe one way of doing that is to say zero, zero, zero, one, one, one, zero. Maybe this is the way I do it, okay? So what will be the signal corresponding to zero, zero, okay? So if you want to plot, say for instance, X zero, zero of T, what will it be? It will be root one by T between zero and T, okay? What about X zero, one of T? It will be minus root one by T between zero and T. What about X one, one, three times or minus three times, and then X one, zero will be plus three times, okay? So it's amplitude modulation in the sense that you're changing the amplitude of the signal between a certain time interval to figure out, to send different bits. So it's amplitude modulation pulse because some reason we'll come to it later, okay? All right, so my random variable X that is going out, random vector X is just one variable, right? You have only one basis vector. So it's just one, it takes four different values all with equal probability, minus three, minus one, plus one, and plus three, all these things with probability one by four. Okay, so that's my, the distribution for my input random variable, okay? You have only one basis vector. So at the receiver, R of T should go through only one correlator, okay? So it's just one correlator, you'll get only one Y, okay? Okay, that's there. What about N? You have only one N, okay? So the Y becomes simply X plus N where N is normal with zero mean and variance and not by two, okay? So what would be the conditional PDF of Y given X is plus one, be normal with mean plus one variance and not by two. If X is plus three, it'll be normal with mean plus three and variance and not by two, okay? So it's just a simple reputation of how that happens. If you want to do the joint PDF of Y, how will that look? There'll be a mixture Gaussian, right? Mixture Gaussian, there'll be four Gaussians, one fourth weighted with center at minus three, minus one, plus one, and plus three. Okay, that's how it looks. It's a very simple extension. Based on this also, you can do a detection process. What do you think the detection process will be? You'll have several different thresholds. Closest point is what you'll pick as your perceived point. For instance, if you receive 2.5 plus 2.5, what do you think you'll say? Plus three, minus 2.5, it's gonna be minus 1.5, it'll be something close to one. So it's a very simple detection process that you can do, okay? So even though I receive an R of T, which could be a complicated signal, I'm gonna correlate it with phi one of T, get only one Y, and I can make a decision based on that. So I went over this point before, okay? So if you transmit this X, zero, one of T, you will get what receive vector sum. Squiggles will be there, right, around this, okay? So you don't have to worry about this value of that receive signal at every time. What can you do? You can simply correlate with phi one of T, all that information about X. The bit is there and that after the correlation, okay? So that's good enough, that's the motivation for this. Okay, so that's four PAM. Now I can also do a more general version, okay? So one more thing we have to do, sorry, I forgot that. The power of the signal, we have to compute. What is the expected value of X squared? Five by four, sounds too low, man. Between plus three, minus three. Five. Five, right, five, okay? So it's not really normalized. Maybe you want to normalize it. How will you normalize it? Divide everything by root five, you'll normalize. If you want, you can do that, but it's okay. So yes, we can live with five, okay? Really doesn't matter, it's just a number, okay? All right, so this is some computation you can do, so I'll quickly introduce the most general version of four PAM, what's the most general version of four PAM? You can say M PAM, okay? Where M is a power of two, okay? So four PAM, you can start with two PAM. Two PAM is the simplest case. Then you have four PAM, eight PAM, 16 PAM, 32 PAM, so on, okay? So again, I'll do a very simple definition for my set of transmitted signals, okay? So if M become 64, I'm not going to write down all the 64 transmitted signals. I'll say basis and a constellation, okay? So I'll give a general definition for how that constellation will look. My N, which is the number of bits I'm using will be log M base two, okay? Does that make sense? Should it be log M base two? Yeah, you can map five points, one, two, a 32 PAM constellation, okay? So you have 32 points, you can do that, okay? So my vector B is going to be B1, B2, so on till B log M base two, okay? Right? And my basis is the exact same as before. Just one basis element root one by T between zero and T, okay? And my constellation point is going to be plus one, plus three, plus five, so on till M minus one. On the negative side, minus one, minus three, so on till minus M minus one. That's my constellation. So I've defined my signaling completely. If you go ahead, I'll assume all my bits are with equal probability. Do this computation, this will be useful. You can compute it to be M squared minus one by three. And if you do Y equals X plus N, you will get mixture Gaussian with all of these guys. What's one more thing to figure out? There's one more thing you have to figure out, which is how do the set of bits map to each point, right? I have a set of say, for instance, if I pick M equals 16, I have to map four bits to 16 points, okay? How will I do that mapping? There are several ways of doing it. One way of doing it is to simply start at the left and go from all zeros to all ones. You can do that. There are some people who studied such things and said maybe that's not the best thing to do. There are all kinds of other mappings out there. But we'll ignore that for now. Come back to it later, maybe. Okay, so you can do it in any way you like. All right, so it's top here now. You'll have to do your...