So, the last class was quite crucial in several respects and we want to quickly go through some main things that we did there. So, the way to picture that is that you have a vector x and vector y and you have some joint PDF by which these case are related. So, in our case what happens y equals x plus n that is the ideal AWGN type assumption, but in general you can view it to some joint PDF. So, joint PDF could be specified as say, so in many cases this might be possible, maybe you can find all conditional PDFs for all possible y and x. So, that is the way of specifying joint PDF and assuming you also know the PDF for x itself this completely defines the joint PDF. So, the receiver the transmitter has x and the receiver has y. So, from y you have to come up with a function which I called x hat. So, x hat is what the receiver produces which claims is an estimate of x, but what is it exactly it is some function of y. So, one can come up with kind of a joint PDF for x and x hat if you want x y and x hat if you want all of these things together will have joint probability distributions. So, what do I mean by that joint PDF for x and x hat what do I mean by that. If you transmit x the receiver is likely to decide x hat is one of several possibilities from the constellation. So, x belongs to some constellation with some probability. So, basically that is the probability that we are concerned about which is the probability which you want to maximize probability that x hat equals x that is. So, you want the joint PDF to lie on that straight line if you plot the joint PDF of x and x hat you want it to be close to that straight line x equals x hat that is where you want all the mass to be. So, that is how you want to design your function of y. So, that is a very abstract way of viewing the problem. So, in most cases it will simplify to a very easy situation which is not all that difficult to handle. But this itself is called the detection problem in some cases this is the detection problem. It is a very well studied problem and one can argue without major contradictions that the whole of digital communications essentially is a solution to the detection problem. So, there is no you would not be wrong somebody says that. So, that is a very valid argument. So, detection theory is also vast in so many ways. But the way we came up with a good detector we studied two detectors right three the first detector was what the MAP detector which was the optimal detector. What did the MAP detector do? You can succinctly write it down as x hat equals argument of maximum over x in the alphabet of what? Probability that x equals x given y equals y. So, this was my MAP rule. So, you see you do not have to worry too much about the distribution of x hat and all that. You can just use the joint PDF of x and y to define this guy. So, this was optimal in what sense which is the objective function that it maximized or minimized? What was the objective function that I looked at? Probability of correct decision right. So, and that was maximized by this choice of x hat as the function. So, remember this will be a function of y. So, this will be a function of y that also you can see depending on y is x hat will change. So, it is a random variable all those things are clear. So, now there is another detector which is the ML detector which can be written down as argument of maximization over x and x probability that y equals y given well in this case typically you do not write probability you write what? You write some PDF because I am expecting y to be continuous. So, I will write y given x here. So, this is the way I wrote down and this is another detector. So, irrespective of anything else I can choose to run this detector nothing wrong in that. But usually when is this detector optimal? Not usually actually when is this detector optimal? When all the different inputs are equally likely. So, the PDF of x is uniform then ML is also optimal in the sense that it maximizes my objective function which is probability of correct decision. So, that is the thing. So, I derived it based on that. So, please see that. The last thing I saw I called it something else I think the best thing to call that is the minimum distance detector. What did that work out to? It is much simpler argument of maximum x in the alphabet remember my alphabet is a subset of some real space R m. So, all these quantities are nicely defined y minus x squared. So, this is a minimum sorry. So, that is my minimum distance detector. So, I can choose to do such a detector as well if I want to. But when will it be optimal? Several things have to be true first of all ML has to be optimal and then you have to have AWGN all that. So, all these AWGN stuff comes only at the minimum distance level. So, these are general principles and it is important to learn them as general principles and know that you are applying them into a for the specific digital communication problem. So, these are used in so many other areas of communications and signal processing that I do not want you to think that these are digital communication ideas these come from a related area they are used in digital communication as well. And depending on the problem and the joint PDF the minimum distance is not going to be optimal in several cases. But it is useful to simplify the ML condition to as simple a version as possible. So, depending on the noise depending on the way noise is added that version might change ok. So, for the AWGN case it became the very very simple minimum distance criteria ok. So, I know it is still abstract, but it is important to appreciate this a little bit and the power I mean why were we fundamentally able to do this in digital communication ok. The crucial thing was how we transform the waveform channel into the vector channel and how we model the noise how we did this correlation did everything. So, that we got a vector then we got a joint PDF and then you could just directly apply these detection principles without worrying about anything ok. Otherwise you have to think so many get confused with so many other details ok. So, hopefully that the power of this is hopefully clear to you ok. So, the assumptions and the models are very crucial in this ok. Any questions comments ok alright. So, ultimately x hat is a function of y ok. So, that is another thing I want you to I wrote it down once again, but ok. So, it is a function of y and the receiver is assumed to have different types of knowledge. For instance, if the receiver has to implement either the MAP or the ML accurately in the general case what should the receiver know? It should know the joint PDF right. So, if you do not know the joint PDF or the conditional PDFs you cannot do it. But look at the minimum distance detector it seems to be more general you do not have to know much just do not have to know anything. But you may not know if it is optimal or not unless you know the joint PDF, but it is fine I mean you can do it. So, those kind of things are also valuable in so many ways in practice ok. Even if you do not know it is optimal or not you should be able to do something. So, you should have all these detectors at hand ok. So, some getting something to work is also important alright. So, the next thing I did pretty quick without going into too many details is this idea of decision regions I think it is very clear I do not have to define these things formally, but what I thought I will quickly just run through it so that I formally define it and then I talk about it ok. So, in my signal space so why this why belongs to my signal space right well not really yeah it is in the signal space ok, but typically it is not a point on the constellation right on the constellation right ok. So, is this clear? So, if you do BPSK for instance your constellation is minus 1 and plus 1 why can be any real number theoretically any real number practice it will be bounded by something, but it can be any number, but it will be on the space it cannot be somewhere else that is the thing I want to figure out. So, one can think of in an actual reception situation one can think of something called a received constellation what will be the received constellation yeah all the points why that you received if you put a dot next to them you will get a received constellation and you can imagine you know the PDF for why so it will be bunched up around the transmit constellation, but it would not be on top of it will be around that point somewhere ok. So, keep that notion in mind. So, now I want to split the signal space into different regions so that if I am falling in a particular region my x hat is going to be a particular transmit point ok. So, that is the whole notion of decision regions ok. So, you have for instance BPSK 0 plus 1 and minus 1 ok. So, maybe this is bit 0 and bit 1 ok I want to find that subset of the real line in this case my signal space is the real line right that subset of the signal space this is the real line on which my decision will always be 0 ok. So, set of all y in R such that x hat of y equals what plus 1 this will be my decision region for the constellation point plus 1 what will be the decision region for constellation point minus 1 same thing such that x hat of y equals minus 1 ok. So, if you do the minimum distance detector for instance the decision region is very clear and obvious ok. So, every point on the right half of the real line positive part of the real line will be the decision region corresponding to this guy ok. So, this is this decision region and the other decision region will be the negative part of the real line ok. So, I can generalize this definition if I know in general my signal space is R m ok what is my decision region for in general. So, this is general my decision region for some x in my constellation is set of all y in R m assuming my dimension is m ok. So, something I have been doing all along. So, such that what x hat of y equals well I should put a bar here as well equals x ok. So, that is a formal definition and you can see this will this will be a partition of the signal space what do I mean by partition. All these sets will not intersect and together they will encompass the entire space. So, that is the partition ok. So, it will be a partitioning ok one can write down proofs for those things, but I guess it is very very clear if you decide on one thing you do not decide on another thing right. It is very clear that it has to partition the signal space ok. And this decision region ok. So, maybe I will call it b x ok. It is good to write down the probability of error or probability of correct decision in a very succinct form ok do you see why ok. So, the decision region for a particular x can be used to write down the probability of error in a very probability of correct decision in a very clear way. So, for instance what is the probability that x hat equals x given x equals x what is this probability given that you transmitted x small x what is the probability that y lies in dx ok. So, this is nice to write down. So, this decision regions are analogously defined. So, you see where the definition comes from motivation is to write down probability of correct decision in a very compact way. So, once you identify the region. So, you integrate over all y in that region you get your probability ok. So, what is this? This can be nicely written as integrate over y belonging to dx f y y dy ok. So, this is the probability of error for a particular x. So, if you change x this probability might change maybe it does not change ok I do not know, but depending on the decision regions it will work out in this form alright any questions ok. What will be probability of error given x equals x 1 minus this thing. So, it is a lot y not belonging to dx you will take ok. So, it is useful to write it down that is always good to know ok. So, let us do let us do more decision regions for a while just get used to decision regions for minimum distance is very easy to do ok. So, it is no big deal you just look at the Euclidean space and the two dimensions quickly mark it out. We will do that exercise for a while and then we will come back and try to see how some of this probability of error calculations are done and how it will simplify and what are the real parameters of interest is ok. So, that is what we will do next. So, I think what did I do I did I did a few examples I want to know what I got to did I do BPSK think I did BPSK what else did I do just BPSK ok. So, the next example I want to look at is 4 PAM ok see these are examples of decision regions ok. So, I am going to say this is the second example because the first example is did I make a mistake what happened oh yeah yeah oh my goodness yeah yeah yeah so the integral should be over the conditional PDF you are right you are right you are right you are right all my eagerness is forgot about the conditioning ok. So, yeah everything is conditioned ok. So, of course everything is conditioned you are right thanks for pointing it out ok everything is conditioned ok. So, let us go to the examples of decision regions I did I did BPSK let us do 4 PAM next it is the next easiest example ok I have 0 plus 1 plus 3 minus 1 minus 3 ok. So, that is my constellation. So, I will put a X mark around the constellation points ok. So, it is almost trivial right. So, what is the decision region for plus 3 starts at 2 and goes off to the right ok. So, you see that this is the decision region for D3 is going to be 2 infinity ok is that clear. So, what would be D1 yeah it will be exactly only this right this guy is going to be well you might argue about the square bracket and so what do I mean by the square bracket by the including 2 ok. So, you can argue about this square bracket and curve brackets. So, why is it not relevant why should I not care about whether I put a square bracket here or yeah exactly. So, probability that I get y equals 0 is what 0 ok. So, there is no probability since the continuous PDF it will not change any of my probability calculation. So, you can put anything I want it does not matter ok. So, what is D minus 1 ok D minus 1 is going to be this set minus 2 is 0 and then D minus 3 is going to be this guy which is ok maybe I will do minus infinity minus 2 ok. So, it is quite simple. So, the next example I am going to see is 4 QAM ok. So, what is the 4 QAM constellation how many points does it have 4 points what are the 4 points plus or minus 1 plus or minus 1 right. So, those are the 4 points that you have ok. So, you have 4 points ok this is my 4 QAM and what would be the set of all points which will decode to yeah the 4 quadrants right. So, it is easy to write down this is going to be D 1 1 this is going to be D minus 1 1 this is going to be D minus 1 minus 1 this is going to be D 1 minus 1 ok the 4 quadrants are the decision regions for 4 QAM ok. So, if I did 4 QPSK for instance QPSK the way I defined it which is a rotated version of this right you remember that I did a QPSK where my points were plus or minus 1 plus or minus j. So, I did that also there what would happen you will have to do the x equals y and x equals minus y lines and take the 4 different parts you get from there for that it is quite easy to see ok. It is possible to geometrically define it. So, for the for a general case also it is possible to do it, but I am not going to spend time doing that we will just do for specific cases ok. The next one I will do which is may be marginally more interesting is the 8 PSK ok. So, the 8 PSK constellation it is good to draw a dotted circle. So, I think this thing should do circles and all that very nicely, but I should figure it out sometime anyway. So, so 8 PSK has 8 points ok. So, once again it is I think reasonably clear. So, this angle is what pi by 4 right at angle is pi by 4. So, if you want to write down all the points that would be mapped to say plus 1 what would you do d 1. So, you do from pi by 8 to minus pi by 8 ok. So, everything that falls within that cone. So, to speak a pi by 4 angle around the x axis ok. So, this is this guy is pi by 8 this guy is pi by 8 ok. So, this would be d 1 ok. So, if you want to write down for this guy you will have to again draw a line which is pi by 8 from this axis right and this region would be d e power j pi by 4 ok that would be that region. So, likewise you can do it. So, it is just different type of cones if you will, angular views that are half here ok. So, like I said if somebody gives you an arbitrary set of points you should be able to do this. Can you imagine a systematic way of going about doing this what is the systematic way of doing it? What are these lines? These lines that I drew can be described as some things of the points what are the point what are those? The perpendicular bisectors of lines joining any two points. So, you draw a whole bunch of them and then you can from there you can easily figure out the actual regions that are of interest to you ok. Some of them will nicely overlap. So, given any set of points arbitrary set of points which has a constellation you draw a perpendicular bisectors right and they will intersect somewhere they will not intersect somewhere they will define pretty much all the regions that are of interest to you as far as decision region seconds ok. So, I am not going to do any such thing we will just do examples and we will be happy ok. The next example which is a little bit more interesting is 16 QAM ok. So, so far every single decision region that we have looked at looked exactly the same right except for the 4 PAM case where you had two different types of decision regions right everything else seem to have similar picture, but for 16 QAM you will see you will in fact get three different types of decision regions ok. So, you can get different types depending on how the constellation looks and this is an example to illustrate how you can get three different types ok. So, I have 16 different points 4 in each quadrant ok. So, if I go ahead and draw the perpendicular bisectors you see all these guys share one all these guys share one right and then these guys will share one and these guys will share one and the axis axis themselves upper perpendicular the remaining perpendicular bisectors and that is it once I have drawn all the perpendicular bisectors I should just pick out the regions that are defined by them as decision regions ok. So, you can make a lot of arguments to make your process easier, but anyway I think that is clear enough ok. So, you see essentially there are three types of decision regions one for the interior points one for the corner points and one more for the side points that are on the outside ok. So, you have three different types they all look different ok. So, this part is reasonably clear. So, a very typical standard tutorial and exam question is what? You do something that is not quite square you know when it looks a little bit different then test your skill in being able to draw perpendicular bisectors carefully define decision regions carefully and then do computations with that ok. So, that is a very standard question does not test much, but still helps me great right it is very useful alright. So, let us go through and do a probability of error computation now, but for that I want to stick to M PAM. So, I will do I am going to do for M PAM. So, the final example we will see will be for M PAM ok. So, we will do a general M PAM it is not really a big deal ok. So, you see I have 0 plus 1 plus 3 so on till plus M minus 1 then you have minus 1 minus 3 minus M minus 1 right and then you do decision regions you will have two types of decision regions one for all the interior points which will be what 0 comma 2 comma 4 comma 6 so on till whatever M minus M minus 4 comma M minus 2 ok. And then you have two others for the outermost points which is M minus 2 comma infinity and minus infinity comma minus M minus 2 ok. So, that is the that is the decision region. So, if you want to write down very succinctly you can say d M minus 1 is ok maybe I will do this M minus 2 comma infinity and then d minus M minus 1 is ok.  So, let me just fix this brackets for a while and then get a board ok. And then d I is going to be what? I minus 1 to I plus 1 ok. So, that I can do always this is for other I ok. So, that is M PAM alright. So, it is fine we did the decision regions by themselves like I said they do not mean much, but they give you a nice handle on quickly computing probability of error or writing down the expression which is the first step in going ahead and computing right. So, it is useful to compute probability of error and that is where the real attention should be the point is to compute probability of error ok. So, decision regions are just a tool ok. One can say decision regions can also be used in your decoder right. So, you can then use it in the implementation itself, but the main tool is like I said for probability of error ok. So, so let us do probability of error for M PAM I am not doing M squared QAM in general because M squared QAM will be a simple generalization of 16 QAM ok. So, you just do the perpendicular bisectors and then just look at the squares that it defines. One can write down in fact one can write down general expressions like this for M squared QAM also it is not too difficult alright. So, the thing I want to really concentrate on at least for the rest of this lecture is probability of error for M PAM M PAM under a minimum distance decoder ok. So, remember I have to I have to always specify what decoder I am doing right otherwise it makes no no sense in defining anything ok. So, so I am going to look at probability of error for M PAM under a minimum distance decoder ok. So, you can argue when it will be optimal and all that and that is it is ok for us ok. So, this is the minimum distance decoder that we are going to use ok. So, I want to step back a little bit and point point out some of the initial comments I made about the main parameters of interest in a an additional communication system ok. So, what are the various parameters power of the signal bandwidth of the signal rate at which you are sending data noise power that you have and then and then probability of error ok. So, far the first four things we have been doing on and off I have been mentioning ok. So, we have been trying to model it understand it maybe not all the trade offs maybe the trade offs are not clear the thing that brings out all the trade offs in one go is this probability of error computation ok. So, the only thing that may not be apparent is bandwidth we will see soon enough how bandwidth can be brought into the picture ok. So, it is also brought into the picture here. So, the one quantity which ties everything together is probability of error it should right or that is the most important thing do not care about anything else right probability of error has to go to 0 or should be small enough ok. So, and this calculation is what is going to tie everything together ok. So, we wanted to pay attention to it it is not a very difficult computation, but it is a crucial computation because it will tell us the quantity that is of important to us and it will also introduce us to this quantity called signal to noise ratio which becomes so crucial in the entire definition ok. So, that is what we are going to do. So, it is very simple and you might say I am doing this computation for M PAM over minimum distance seems like an abstract calculation, but hopefully you agree that this constellation actually represents actual signals right real pass band signals or real base band signals and I know how to go from constellation to the signal ok energy actually is the same thing ok. Same way noise power is also something which is real which I have modded ok same thing is true for the decoder also something that I can actually implement. So, all these things are very real computations do not imagine them to be some abstract probability exercise alright. So, let us go ahead do that and one argument I made for when I did this PAM was that this root ES or some factor multiplying the minus 1 and plus 1 does not matter ok, but I want to show that it does not matter the probability of error. So, so that is what I am going to do in the definition ok. So, I am going to change my PAM to a general PAM where I introduce an arbitrary distance between minus 1 and plus 1. So, I am going to say my distance between minus 1 and plus 1 is D ok. So, if my distance between minus 1 and plus 1 has to be D what should I say this is minus D by 2 to D by 2 ok. So, my next point would be 3 times D by 2 point on this side would be minus 3 times D by 2. So, until M minus 1 times D by 2. So, until minus M minus 1 times D by 2 ok. So, you will see eventually. So, what really matters is the ratio of D or D square 2 then power of the noise variance. So, the ratio only matters. So, I can fix D to be arbitrary and vary the whole thing by varying the denominator alone. So, that is why in that sense it does not matter ok. So, that is why we did that, but to bring out the point we will introduce this D and this D is called the minimum distance separating any two constellation point right. So, this is the minimum distance in this ok. So, that is a useful thing ok. So, my transmitted signal constellation point is uniform in this alphabet that is my alphabet. Is there a question? Do you have a question to ask or ok? So, this is x and what about my noise? Noise ok. So, before we proceed we should compute first energy in the signal ok. I will denote this E sub s this is what? Expected value of x square ok. So, if you do that computation it will work out to M square minus 1 D square by 12 ok. So, that is the computation that it will work out ok. So, you can play it around and then write down D as what? Or you can do the reverse D is square root of 12 E s by M square minus 1 ok. So, these are the computations that you can play around ok. My noise N is normal with mean 0 and variance and not by 2 ok. So, if I want to think of energy in my noise right. See remember this was what? This was energy of expected value of x of t square right. So, that is how I got to this expected value of x square ok. So, likewise here energy in noise has to be expected value of N square of t ok. So, if you go back and look at the way I did the derivation I will only use N 1 of t which is the projection of N of t on my signal space and I will only worry about the expected value of that and this we showed was equal to N naught by 2. Is that right? Wrong? Maybe yeah I think so ok N naught by 2 this will work out to N naught by 2 ok. We will take it to be N naught by 2 alright ok. So, anyway so even if so there might be a constant missing here, but what happened? Something very bad happened ok. So, this will take to be N naught by 2. So, you can check this you can check the relationship between this guy and this guy and make sure that it works out ok. So, this will work out to N naught by 2 ok. So, so so so so so ok. So, anyway so even if this is not important to us we will take C N to be N naught by 2 ok. So, the next thing is the received value y which is x plus N something that is disturbing me about the way I wrote this down ok. So, maybe I will fix this later, but I will think about this and fix it later anyway. So, y is x plus N this is my received value ok. So, so what is next? Yeah that is what was disturbing me a little bit too, but anyway it does not matter. Yeah let us ok. So, let us not worry about this guy now do not worry about this guy I will fix this later, but I will define some E N to be N naught by 2 ok. So, turns out D squared by N naught is the important parameter ok. So, it has to be N naught. So, it is just wrote it down without thinking too much forget about this N naught by 2 is important. So, I will define E N to be N naught by 2 just take that as a definition ok. So, the received value y is x plus N and what am I doing for the detector ok. My detector is minimum distance decoder ok. So, I have to compute now probability of error. So, I will define as Pe which I will write down as Pe denote as Pe this probability that x hat not equal to x. So, remember how did I define x hat x hat is the minimum distance decoder ok. So, we know all about this alright. So, let us go ahead and do this computation. So, I am going to say I am going to condition on x first ok probability that x hat not equal to x is there a question what happened ok x hat. So, I do not have to write down vectors is all just one number I will just write down in general just so that you get a feel given that x equals x times probability that x equals x ok. So, this I can do ok. So, this quantity now is not too bad to compute ok. So, now I am back to the decision region ok. So, that is a decision region. So, I have to compute this what about this ok what will I do to this I am going to assume uniform I said that uniform. So, this will be 1 by m so that is fine alright. So, that is that is how I am breaking down my probability of error ok. So, go back and look at the way I did it did for deriving the optimal detector deriving the optimal detector what did I do to probability of error or probability of correct decision. I did something which was different from this yeah. So, you conditioned on y ok. So, it is a slightly different way of looking at probability of error ok. So, once you do the optimal detector and fix your detector this is a much easier way of computing probability of error ok. So, when you want to derive the detector you have to condition on y it seems like a strange tool, but it is useful ok. So, that is probability of error and now this each of these case can be easily computed. So, how do I compute probability of x hat not equal to x given x equals x ok. So, how do I compute this? There are two cases. First case I will consider is x is plus or minus m minus 1 d by 2 ok. In this case what will be the probability? What will this be? Integrate from minus what? Yeah. So, you have to integrate from. So, for instance let me fix x to be plus m minus 1 d by 2 ok. So, somehow suppose I fix this to be m minus 1 d by 2 ok. So, what should I do here? Minus infinity to m minus 2 2 times d by 2 ok. This always a times d by 2 floating around. Integrate what? I am sorry minus no I am taking x to be plus m minus 1 d by 2. Is that fine? I am looking at the right most point ok. So, I have to integrate this out. What? What is my f y? Ok, give me some numbers. The limit is correct or not? It is correct. It is correct? Ok. So, what should I put inside? The normal pdf right. So, it will be some normal pdf. What will be the mean and I will write down simply normal pdf. What will be the mean? Minus 1 d by 2 and the variance will be n naught by 2 ok. So, that will be the integral ok. So, you have to put the d whatever I did and write it carefully. So, this is the this is the integral ok. So, what about the minus case? x equals minus m minus 1 d by 2. The same thing will be equal to integral from minus m minus 2 d by 2 to infinity normal with mean minus m minus 1 d by 2 and naught by 2 ok. So, I want you to write give me these two expressions in terms of q functions. What is q? q is q x is what? q t I will write so that I do not, there is too many x's floating around. q t is what? 1 by root 2 pi integral t to infinity e power minus t square by 2 d t. So, this is q. I want you to give me these two expressions, simplify them and give me an expression in terms of q. I am going to claim these two are the same. Do you believe me? They have to be equal. So, what will what will it be in terms of q? q. So, the claim is what he is saying is this is equal to q I am sorry minus d by ok. Just give me q of positive numbers 1 do not give me minus. He gets scared if we say q of minus ok. So, let me write this down completely this probability that x hat not equal to x given is equal to x ok. Why do I get confused? Why do I get scared if you say q of minus something? q of minus anything will be greater than 0.5 and all that. It is not a very nice thing to have. q of d by integral ok. So, you should get something like this d by 2 by root and not by 2 ok. How many of you think you can get to this expression? So, it is very simple not very difficult, but you have to be able to get to the get to that expression ok. So, this guy is probably a much easier guy to work with than the other one the other one you have to do a little bit of a who is turning around to understand ok. So, this is how it works ok. So, it is q of d by 2 divided by root of and not by 2 ok. So, that is this probability. What about interior points? 2 times that do you believe me ok. So, for interior points x equals plus or minus i d by 2 i equals what 1, 3 so on till m minus 3 probability that x hat not equal to x given x equals x is 2 times q of d by 2 divided by root and not by 2 ok. So, we will stop here I think I am running out of time. So, we will pick up from here on Monday and try to finish it.


I'm Lance Corporal Clayton Phil Povage from the Corps Report, and I'm here with the gear. Now I'm at Fort Devin Naughty Base, and I'm about to learn about this beast. Now I'm joined by Kevin Brinkensporen. Sir, can you tell me exactly what you do here and what we're looking at? Sure, Clayton. This is the Legand Squad Support System, it's really known as LS3, and I'm the Chief Engineer on the LS3 project. And what you're basically looking at is a robotic pack fuel. So the real goal for this machine is to go where the dismantled warfighter goes and to help carry their load. So how does it stand up to like mountains, rivers, different terrain? Yeah, that's a great question, and this is really why we developed this machine is so that it would be all terrain. So we've tested stream crossings, we've gone to the mountains in 29 Palms, California. We've tested a lot of east coast woods and mountains, so we really are trying to go to as many different terrains as we can and push the mobility and the autonomy of the machine so it can keep up with the warfighter. So I'm really excited to actually see this thing in action. So can you gear up and give me a demo? Yeah, let's take it for a walk. All right, great, let's do it. What can you tell me about the different functions and modes that this machine has? Well, it's a good time to ask because we're in one of them right now. So this is a leader follow mode, and this is kind of the bread and butter for how LS3 is really going to operate. The idea is that, you know, you can see I'm not really distracted by the robot right now. We're able to have a conversation or I could be doing something else. But effectively, the robot's following the stripes on my backpack and using me as a guide for working through the terrain. If you get in a tight pinch, you can also directly control it with a joystick. And then we have a fully autonomous mode, which is go to wave point. So you can tell the robot, you know, go 100 meters that way and the robot without any driver or leader will autonomously work its way to that goal point. So how can this be effective for Marines? Well, the main idea for Marines right now is as an offloading capability. So, you know, dismounted Marines typically carry around 100 pounds of gear between body armor, backpack, food, water, ammo. And the idea is for us to get that off the warfighter and really improve the warfighter's mobility, reduce injury rates, and generally improve their capabilities. So, you know, ideally you could take a lot of the equipment that the warfighters carry through long foot patrols and put it on the robot and have it carry the gear for them. The LS3 is still on development stages to make it more suitable for Marines. Thanks for watching, guys. We'll see you on the next installment of With a Gear.


The objective of this study was to monitor the frequency of different adverse transfusion reactions and to assess the compliance of clinical staff with the process of sending proper transfusion reaction workup within the specified time. Methods, a retrospective audit was conducted at the Blood Bank of Kikutkana Memorial Cancer Hospital and Research Center, Lahore, Pakistan, and included all transfusion reaction forms received between July 1, 2017 and June 30, 2018. The forms were analyzed for type of blood component, time in which they were received by the Blood Bank, whether or not the form was completely filled, whether or not all required samples were provided, and the type of reaction. Results, a total of 12,787 units were dispensed and transfused during the study period, resulting in 50-0.39% transfusion reactions. Allergic was the most common type, 24%, followed by hemolytic, 18%, febrile non-hemolytic, 15%, and other types, 31%. Red cells accounted for. This article was authored by Mohamed Hassan, Imran Ahmed Siddiqui, Zahid Kumar, and others. We are article.tv. Links in the description below.


Hello, my fellow snippers and welcome to Parent Tip Tuesday. I knew once I started talking she would leave, that's just what she does. My name is Marlene McCohen and this is Jersey, my umbrella cockatoo. And here is my lovely mustache parakeet Picasso today for this episode in light of Halloween. I am going to show you guys how to make Jersey's sock costume. Now, I thought this would be a great video for today because if you want, you could not only make this sock costume for plucking issues or so your bird is not bare, but also you can dress your bird up for Halloween. So I thought I would show you guys exactly how I make this today. Jersey must know that this video is about her and like it because I've never seen her stay here. Usually she just jumps off. Oh, look at that. I swear I had the camera on a break for like a minute and she just sat there the whole time. And once I started talking, she was like, I'm out. Hey Jersey, we're going to talk about you today. Actually, I have a surprise for you guys. Before we do the Engage Knot cage parrot, let me do a couple of shout outs. Now these shout outs are going to be from Parrot Station because I saw these two posts today and they really made me feel something and I thought that I should give you guys a shout out. So here it goes. The first shout out is to Brandy Metcalf on Parrot Station. You were thinking about rescuing George the McCaw. He's 50 years old. I was just so impressed. I really hope you're able to rescue George. You guys know I have an insane sensitivity to Parrots named George. The picture of George and your story just really got to my heart. I hope that you do bring home George and keep me updated on that. It's really nice to have Parrot Station and be able to see everything that you all do for birds. It really makes me feel good to be able to see into the lives of all the Parrots around the world. So thank you Brandy for considering getting George. I really hope you do that. Tag me on Parrot Station. If you do keep us all updated because I think we are looking forward to it. And my next shout out goes to Alexis Lukowitz. I loved your post today about Rue. I saw that you went to the pet store and you adopted Rue. I know that you recently lost your bird willow and I'm really sorry about that. But I hope that Rue brings you a lot of love and a lot more joy back into your life because I know that when I lost George that feeling was there so strong until I got Cody. So I totally understand you. And I hope that Rue does bring you the happiness that you deserve. And thank you for saving another bird. And now for the Engage Not Caged feature. These, by the way, guys, I choose from Instagram only. So if you follow me on Instagram and tag me and also follow Engage Not Caged and hashtag Engage Not Caged, do whatever you can to get my attention. This is where I choose the birds. By the way, I have 4,999 Instagram followers. It's like one more. And they'll have 5,000, which is kind of an exciting number, I think, right? So today's Engage Not Caged bird is Shabadu. Oh my God, you guys. Shabadu's profile says he is a tiny charismatic cockatiel from New Jersey. OK, I had to choose Shabadu. Let me tell you why. One, gorgeous bird. Two, I'm from New Jersey and my first bird was a yellow cockatiel named Dooley. What a cute name. I love that name. So Shabadu and his profile just it just Shabadu stole my heart. And he has cute videos up there and the whole profile is about Shabadu. So check out Shabadu the bird. You will love it. I will put the link below. My goal is to give you other cool birds to follow on Instagram. And I hope you guys do go follow them because birdies deserve our support and love in a world of dogs and cats. Don't you guys agree? So I hope you really enjoy following Shabadu.   I will agree. I have seen some of Shabadu's videos. I'm in love. So go follow Shabadu and now we are going to move on to our video. First, let's talk about the items that you are going to need to make your bird sock sweater. First things first is the sock itself. Now this is a good quality sock. I used to make them out of any kind of sock. I like the ones with the one inch neck because it's good for Jersey to cover up her neck. This is a Ralph Lauren sock. The quality of the sock does make a difference. It's super soft. So it's very comfortable for her. When I used to use other socks that weren't Ralph Lauren, just something I got at Target, she was able to pull them apart and stretch them a lot and then get under there and chew her feathers. So I suggest a really good quality sock. I got this from TJ Maxx. If you feel like your bird is a plucker, then the one inch neck so it will cover your bird's neck.  Right, Jersey? Also, you'll want to make sure that the sock is approximate size of your bird. It will stretch, but you want your bird to be comfortable. This is about Jersey's size. She's making that noise because she's like, this better not be one of those days where we change my sock. So that's the first item you'll need. Second item, this is Jay. This Jay is for Jersey. You could put anything on your bird. This is an iron on. I actually used to make her wear Superman and Batman logos, which is a lot of fun and really cute. I didn't find any iron on logos like that. So what I used to do was buy a boy's underwear.  But this is a lot easier. I'm going to show you today how to iron this onto the sock and where you would want to iron it on. So that is the other item you'll need. The next item you will need is a scissors. Self-explanatory, you're going to have to cut the sock. And here's another thing you have to have on hand. To get this sock on Jersey, I have to have a nice soft blanket. Make her relax and they're petter because I don't want to traumatize her when I'm taking on or off the sock. That's why I don't do it often. Fun fact about me, one of my compulsions is collecting soft blankets. So if I go into a blanket aisle anywhere, I have to really try to help myself not to buy the blanket. And I like all different colors. I choose at night which one I'm going to sleep with or which three. And then this one I just opened last night so I'm really excited about it still bouncing. I don't know, the colors make me feel different. Anyway, moving on. We start with our sock. Now the sock is flat this way, but you are actually going to flatten it the opposite way, right? And here's something to note, if you have a large cockatoo especially, this part, the heel of the sock, is going to fit your bird's crop perfectly. So when your bird gets full, this is going to get filled out a little bit more. So first thing you're going to do is fold the sock. And the next thing you're going to do is cut the sock. Look, you'll see it's on the side. And don't worry too much about where you cut it right now because you can improve upon it when your bird is in the sock. First thing we're going to do is cut across here. So now I have two pieces, this part and this. We don't need this. This is not going to do anything. We can put our sock back this way. The next thing we're going to do is make wing holes. Now here's something very important to know about the sock. If you want this neck part to go up really high on your bird, let me show you Jersey. I feel like this time around, Jersey's neck is a little too low. I could totally make it higher. And in order to do that, you want to make the wing holes lower on the sock. So let me show you what I mean by that. That means that last time I probably made the wing holes high, whereas now I got to make them a little lower and then the neck part will go up higher. So how do you make the wing holes? Very simple. So simple. You just cut a little cut like that and then you can do it. I always think I'm doing things lower than I am. And I really think I already made that too high. I think I made this too high because if you see it's in the middle of the crop, I need to make it lower. So let's do another one. I make this mistake all the time and then I'm like, damn, I should have made that higher. All right. So I made a wing hole here and here. These are very, very small right now and they're temporary. The next thing we're going to do is open up our emblem. Why don't we do the blue one for Jersey right now? I have to iron this. Where are you, mom? I need you to iron this for me. The dog got a little excited when I did that. This is a job for a mother. Okay. So how do you iron these things on?  So I have the iron ready. It literally smells like something's burning.  Oh no, Jersey has been alerted of something.  What's the matter, Jersey? That might be a little bit low on Jersey right now. So I'm going to put it higher. I'm going to put it right on the crop. And now I am going to iron it. Now the measurements may come out a little bit off. That happens. And that's why the great thing is you only need one sock out of a pack of, let's say 12, because you know, let's say it's a six pack. You get 12 socks. That's what's good about it. So you get a lot of tries to figure it out. So now we are going to iron this on. Wish me luck. The last time I ironed anything, I was probably about 10 when my mom was obsessed with teaching us things.  I should have read the directions. Jersey, stay over there. This is an iron. Why is it not hot? I put it on cotton. Those of you who use irons, you probably know this part better than me. Do not use glues and things like that because there's a great chance that your bird is going to chew it. I'm very surprised that Jersey didn't chew her J. She's chewed a lot of other things that I've put on, but I think she's very proud. I think she knows this is J for Jersey. And I think she likes it because her red J is holding up. But you wouldn't believe red is a hard color to get. So now we're doing blue. Oh, well, this blanket is getting ironed. Don't try this at home the way I'm doing it. I bet you have to let it cool because things like this, you always have to let them cool before you move on. I'll give it one more iron. So I found the instructions and it said to also iron the back. I think that's what it meant. So I'm gonna iron the back. I put Picasso in a safe place. What do you think, Jersey? By the way, my iron is on a cotton setting. I put the iron in a safe place where none of the birds can get at it. I have ironed this and let's turn it inside out and see what it looks like. It looks pretty good. So this is how you make the costume. You're gonna wanna make these holes a lot bigger. Where this hole is, you're gonna cut up towards the neck a little and then on the other side. And believe me, this is going to stretch when you put it on the bird and it's gonna become bigger as you put it on the bird. So I'm gonna show you how to do that and I'm gonna show you how to tweak it so you don't give your bird too tight of a fabric feeling over here because they can get blisters and stuff from that. So it's gotta be really nice and soft and perfect. Let's see if Jersey will let me do this. Usually my sister helps me with this actually. Let me ask her if she wants to help me. You have a tension because she's gonna get sick. I don't know if she's gonna get sick. Harry, come here. Harry, come here.  She's not gonna come. Oh, dude. Marla, where is he? Literally turned around and was like, hey. Picasso, you scared Jenna? She didn't know you were in here the whole time. I had no idea. I was wondering, I heard him go, kush, kush, kush, and I was like, when is she gonna notice? You're like, Marla, what is that? Oh, please be the commander. Come. Okay, so Jersey is cuddling me and I think the last time I put her sweater on, she was like this. Yeah, cause we wanna make it like a nice, cool, calm environment. Just make her think that she's getting cuddles.  It's like she needs to get used to it. And I just kinda like trick her and pet her and make her think it's like, you know, a little love time. And then like, I'll slowly take a wing out. I think she knows the sweatshirt switching is happening. So she's a little overwhelmed. There's some birds that just take to it right away. And the best thing to do is to like, get them in a comfortable position. Right, baby? I don't even think we'll need the blanket with the way you do it. Sometimes if I cuddle her and massage her head the right way and then like, I'll put my hand over, I'll show you guys. Step one was making it. Step two, if you already made one, is to take it off. And then step three is putting it on.  Fun fact, yeah. Harry might have the sock, cause I can't find it. Like it hit me. Okay. Where, is that it? No. That's like when I made it up. Oh, there it is. I found it, I found it. Oh, this is perfect. Okay, so let's see. The first thing you guys need to know is that hole doesn't just go around the wing, it goes around the foot. So let's see if Jenna's able to get this off. Oh, let's put this, okay, nothing's happening. What are you? I'll keep petting her. You know, whatever you gotta do to make your bird feel comfortable. Oh, wow. Cause these kind of things can be traumatic for birds, harnesses, outfits, all that. See, she got one leg out. One foot. Dang, your leg looks like Angelina Jolie in that one dress. Oh, good girl, relax. You know what? I'll tell you what, she's not looking right? Right. I'm gonna get the other foot on the other side. You get the other foot? Oh, I see. And then I'll just cut it off. You're not gonna even know. Make sure if you do this, you don't cut any of their feathers. Be really careful with the scissors. This is like the least amount of resistance. So, you know, it's just gonna come off easier.  Look at this two ply. Look at this two ply, okay. So she's fine. You see, and she's fine right now. That's easy, and I can even do the other leg. We have the whole thing off? Yeah, we have the whole thing off. Let's compare. Old one, definitely needed a new one. You'll see if you look up close how tattered it was, but she still left that J intact, which I'm pretty impressed about. Yep, show the new one. Yeah. I'm gonna be matching. Yeah, you're gonna match Jenna. Oh, she sees the sock. Nothing's happening. There's no sock. There's no sock. There's no sock. Keep boo boo's. So what I like doing, so while she's cuddling here, she's like, she can't see. I like putting my hand, what, the J would go on her chest. What are you doing? Put my hand through it like this, okay?  Okay. So let me do this. Oh, that's nice. And then I'll pet her. Yeah, and then I go through like that. Yeah, I'm petting her. And then from there, it's pretty easy. It's really stretchy material. I know this looks like not the best thing, but I mean, I think most birds actually tend to be okay with it.  You wanna get the wing through, but here's the other thing you guys need to know after the wing. You gotta get the leg through. Get your sexy leg through there. And then we put this side through. Oh, she's naturally putting her foot through.  Oh, here. And then you get the leg through. Oh, baby. So all in all, not too bad, but we're gonna show you now how to tweak it because there's something important that you need to know here. What you need to do is, let's show them this tightness. Actually, this side is good, but this side's tight. So, okay, let's show them that tightness if we can. You always wanna check this section of the wing. There's too much material there. I'm gonna have to cut that material. You don't want it tight on the wing or it could cause some blisters here. So we're just gonna adjust. I'm gonna adjust it by simply cutting it right here. It's okay, Jersey. Make sure you don't get any feathers. Just cutting it so she has some room. And then just tweak it all around. Make sure it's not tight. She looks traumatized. Happy, let's see you. Let's see this side. And then you can cut off extra material. She actually never minds me doing the cuts. You could cut them off just so it's a little bit smoother.  I know, it's just for extra attention. So like, literally it would like have a child who is scared of shirts and then forgets it's wearing a shirt. And then when you put a new shirt on it, huh, you look so pretty one. Oh, this one looks a little awkward. But isn't it the same brand? What do you do? And then she just gotta get adjusted. It always looks awkward at first though, I feel like. Yeah, it does. Okay, let's see. Jersey, let's see how we could tweak this. Is the neck thing too long? But isn't it the same brand? Yeah, but if I made the hole too low then what I gotta do is cut up here. You see, like here, so that I could pull the neck down a little. Oh, I see, yeah. Okay. So you guys see, I'm making the neck a little shorter by giving the wing hole more room. That means we can just pull it down a little bit more, make her more comfortable. Let's see if you're pretty. She's not gonna wanna show this to the camera. I know her. Oh, you look beautiful. And guess what? I think it's just her, cause she's ducking, it makes her, like she's ducking for me. Oh, it's so cute. So that is the making of this costume. There's a few things to note about this. If you wanna make it for a bird that plucks, especially a bird that plucks this area, it can be a great idea, but you have to keep your eye out for, if then they start plucking the rest of their feathers, that's something that could happen. So in that case, you might wanna talk to a vet and look into getting a cone or some better options.  If you want a Halloween costume for your bird, also a great option. One time I attached kind of like, I gave her a black sock and put a Batman logo so you could do all different color socks of all different kinds. I personally think she looks like Simon from Elvin and the Chipmunks. You know the one with the, the nerd with the, the, yeah. Aw, you're so cute. I can't even, we're just like bird upsetters. So she's gonna be like this, like all cuddly and like feel like something dramatic happened to her. When in fact nothing happened, we just switched to shirt. We usually love on her and then at night, you know, we put her in the cage and she acts like she can't stand and needs more cuddles but she gets over that pretty quick. Right? Right? You're fine now. Are you fine already? Yeah, you're fine already. Yeah. Do you see your new shirt? It's beautiful, it's blue now. You match Jenna's shirt. So that is it guys. Jenna's always the best at these things. Jenna's the one that pulls the blood feathers, things like that. If I need to keep Jersey calm, trim the nails of the birds, things like that, I always bring in Jenna because two is better than one and also she's really, really good at it. Then together, we really know how to get down and do those things that nobody else wants to do. And we love sniffing our birds, huh? You see what's new? With Picasso, it's really easy to forget that we're on camera. Yeah, I'm so sorry. No, I tell them that all the time. I'm like, guys, I just cut five minutes out of this video because I was sniffing. So that is it. Thank you guys so much for watching. As always, follow Jenna on Snapchat and Instagram at JennaSara. Link is below. If you guys want your bird featured on Engage Not Cage, follow me at Marlene McCown and at Engage Not Cage. Follow both of those accounts, tag me, hashtag me, whatever you've got to do, message me. And also, come join Parrot Station. Who's your favorite bird on Parrot Station? It's bird that played the xylophone. The eclectus? The eclectus. I know, that was so cute. Didn't we play that video like crazy? It was, it was pretty amazing. That's one of our favorites. So join Parrot Station on Facebook. We love having you. We love looking at all your birds. Thank you guys so much for watching this video. Please like, share, comment, subscribe. Right, Brett? Subscribe. You must subscribe. Yeah, yeah, yeah. Yep, yep. And sniff. It's crazy, crazy. Okay. Okay, say bye. Bye. Bye. She's like, give me out of here.