1 00:00:00,012 --> 00:00:05,407 So in this video we are now going to use the knowledge we have about how channels 2 00:00:05,407 --> 00:00:11,134 behave, to develop what I'm going to call the basic channel model, that applies to 3 00:00:11,134 --> 00:00:17,356 both wireless and wireline situations. As we know interference and noise are 4 00:00:17,356 --> 00:00:20,688 going to be a problem with wireless channels. 5 00:00:20,688 --> 00:00:23,922 Channels not so much with wireline channels. 6 00:00:23,922 --> 00:00:28,503 That makes from a communication and engineering view point. 7 00:00:28,503 --> 00:00:34,233 Wireline channels are fairly boring. They don't have much interest or design 8 00:00:34,233 --> 00:00:38,006 flexibility. Whereas in wireless channels because of 9 00:00:38,006 --> 00:00:43,433 the interference and the noise makes it a much more challenging and interesting 10 00:00:43,433 --> 00:00:46,320 problem. I'm going to give an example on how to 11 00:00:46,320 --> 00:00:51,613 apply this model of channels to what's called baseband communication, which is 12 00:00:51,613 --> 00:00:55,261 perhaps the simplest kind of communication possible. 13 00:00:55,261 --> 00:00:59,127 Baseband basically means you send the message directly. 14 00:00:59,127 --> 00:01:03,099 But let's develop that basic channel model. 15 00:01:03,099 --> 00:01:08,917 And as you know, this is our block diagram model of communications. 16 00:01:08,917 --> 00:01:13,064 And we're concentrating, of course, on the channel. 17 00:01:13,064 --> 00:01:18,750 And what we've seen that for both wireless and wire line situations. 18 00:01:18,750 --> 00:01:25,131 This is a very good model for the channel. So what I'm doing is I'm putting the 19 00:01:25,131 --> 00:01:29,915 channel in a box. Has x of t coming from the transmitter n. 20 00:01:29,915 --> 00:01:36,267 And what we call r of t, the received signal coming out of the your channel. 21 00:01:36,267 --> 00:01:41,154 It is delayed. The transmitted signal is delayed by some 22 00:01:41,154 --> 00:01:47,580 amount, and in both situations. It may be attenuated, alpha's supposed to 23 00:01:47,580 --> 00:01:54,790 represent a gain but usually alpha is less than one and certainly hopefully greater 24 00:01:54,790 --> 00:01:59,526 than zero. For the wireline channel, basically alpha 25 00:01:59,526 --> 00:02:05,862 is 1 but for the wireless channel, it can be significantly small, a very small 26 00:02:05,862 --> 00:02:10,343 number indeed. Then to that are added the interference 27 00:02:10,343 --> 00:02:16,384 which usually come from man-made sources and noise which is ubiquitous. 28 00:02:16,384 --> 00:02:22,898 So a simple relationship between input and output is given by this expression. 29 00:02:22,898 --> 00:02:29,403 And we're going to use this in our considerations of communication channels 30 00:02:29,403 --> 00:02:34,052 and systems. So, for a wireline channel, to summarize. 31 00:02:34,053 --> 00:02:40,694 The delay TAU is given by the distance between the transmitter and the receiver 32 00:02:40,694 --> 00:02:46,700 divided by the speed of propagation which is I said is a fraction of a speed of 33 00:02:46,700 --> 00:02:51,768 light in free space. Alpha's basically one and it interference 34 00:02:51,768 --> 00:02:58,004 in the laser base is basically zero. So it makes it kind of uninteresting for 35 00:02:58,004 --> 00:03:01,647 us. But in wireless channels, situations get 36 00:03:01,647 --> 00:03:07,131 much more challenging. There is a delay which in many situations 37 00:03:07,131 --> 00:03:11,332 can be small. But if you're talking about satellite 38 00:03:11,332 --> 00:03:15,808 communications Or even ionospheric, communications. 39 00:03:15,808 --> 00:03:20,556 That delay can become signifigant. What's much more important is the fact 40 00:03:20,556 --> 00:03:25,876 that the attenuation is inversely related to the distance between transmitter and 41 00:03:25,876 --> 00:03:29,336 receiver. The further away you are, the further away 42 00:03:29,336 --> 00:03:35,082 that a receiver is from the transmitter. The smaller the signal you're going to 43 00:03:35,082 --> 00:03:39,035 receive. And who knows what kind of interference 44 00:03:39,035 --> 00:03:44,036 and noise are going to inhibit the reception of the signal. 45 00:03:44,036 --> 00:03:49,964 Now we're going to basically assume that we can control the interference by 46 00:03:49,964 --> 00:03:53,682 filtering. Basically is we're going to see as we go 47 00:03:53,682 --> 00:03:59,490 through that in well designed communication systems the interference 48 00:03:59,490 --> 00:04:03,979 from other communication signals is going to be out of band. 49 00:04:03,979 --> 00:04:09,859 What that means is in a different bandwidth range of frequencies than x will 50 00:04:09,859 --> 00:04:13,034 be. So we can just by floater and get rid of 51 00:04:13,034 --> 00:04:15,305 it. Noise is another problem. 52 00:04:15,305 --> 00:04:21,207 Noise we're going to consider as being wideband, technical term, which means it's 53 00:04:21,207 --> 00:04:24,473 everywhere. And it's going to be out of band. 54 00:04:24,473 --> 00:04:29,053 It's going to be in band. It's going to be in the same transmission 55 00:04:29,053 --> 00:04:34,845 bandwidth, as we have, or as we're using and so we have to worry about it a lot. 56 00:04:34,846 --> 00:04:40,114 But two big effects are going to be the attenuation and the noise. 57 00:04:40,114 --> 00:04:46,620 So we need a way of describing noise in particular because it's going to be a new 58 00:04:46,620 --> 00:04:50,823 signal for us. It has very interesting and special 59 00:04:50,823 --> 00:04:55,357 properties. We're going to use a model called white 60 00:04:55,357 --> 00:04:59,012 noise. The reason it's called white is because it 61 00:04:59,012 --> 00:05:03,857 has powered all frequencies. Just like white, white has equal power 62 00:05:03,857 --> 00:05:09,627 more or less in all the range of colors. Put all the colors together you get white. 63 00:05:09,627 --> 00:05:14,376 So it's called white noise because it has power to all frequencies. 64 00:05:14,376 --> 00:05:19,218 Random phase in amplitude. And here is an example of white noise that 65 00:05:19,218 --> 00:05:23,795 I simulated on the computer. And it's very erratic, very wiggly. 66 00:05:23,795 --> 00:05:27,761 Sometimes if you look very carefully it is not so wiggly. 67 00:05:27,761 --> 00:05:33,231 And sometimes it has these really big values, which really is a random signal. 68 00:05:33,231 --> 00:05:37,041 There one special property that noise signals have. 69 00:05:37,041 --> 00:05:43,152 Is that when you add two noise signals together, where powers add nor their 70 00:05:43,152 --> 00:05:47,639 amplitudes. So we're going to see this in more detail 71 00:05:47,640 --> 00:05:54,846 just a second but this is very important property of noise signals, not so much for 72 00:05:54,847 --> 00:05:57,789 regular, if you will. Non noise signals. 73 00:05:57,789 --> 00:06:03,049 Suppose you add together two sign waves of the same frequency, well their amplitudes 74 00:06:03,049 --> 00:06:07,747 more or less add, depends on the phase. But if you're in phase and at the same 75 00:06:07,747 --> 00:06:12,354 frequency their amplitudes had. But their powers are, there's a greater 76 00:06:12,354 --> 00:06:15,979 power than just the sum of their powers individually. 77 00:06:15,979 --> 00:06:19,712 Not true for noise, for noise the powers add. 78 00:06:19,712 --> 00:06:26,186 So we need to talk about, what's called the power spectrum because we're going to, 79 00:06:26,186 --> 00:06:32,874 communication analyses relies a lot on understanding where signals have power. 80 00:06:32,874 --> 00:06:38,424 And to make it convenient we like to talk about power in the frequency domain 81 00:06:38,424 --> 00:06:43,296 through something called the power spectrum and very simply it is the 82 00:06:43,296 --> 00:06:47,421 magnitude squared of the Fourier transform of the signal. 83 00:06:47,421 --> 00:06:52,725 And that's what a power spectrum is. So the power spectrum for signal S is just 84 00:06:52,725 --> 00:06:58,736 this magnitude squared. Well,as we know magnitude squared is S 85 00:06:58,736 --> 00:07:07,391 times s conjugate of f, but this because of our properties, is just s of minus f. 86 00:07:07,391 --> 00:07:15,307 And so when you multiply these to things together mathematically what you discover 87 00:07:15,307 --> 00:07:20,046 is that power spectra are always even functions. 88 00:07:20,046 --> 00:07:24,561 So, this is going to be important a little bit later. 89 00:07:24,561 --> 00:07:29,387 So, it's an even function. It's just a mirror image about the, 90 00:07:29,388 --> 00:07:33,931 vertical axis. For all power spectra have this property. 91 00:07:33,931 --> 00:07:40,070 There are no exceptions. Okay, now next thing to talk about is this 92 00:07:40,070 --> 00:07:48,488 power spectrum at the output of a filter, so we know that if we have a filter here, 93 00:07:48,488 --> 00:07:53,921 which has x going in a transfer function of capital h. 94 00:07:53,921 --> 00:08:00,417 And y coming out that the Fourier transform of the output is equal to the 95 00:08:00,417 --> 00:08:06,129 transfer function times the Fourier transform of the input. 96 00:08:06,129 --> 00:08:12,174 Well to find a power spectrum you just need the magnitude squared. 97 00:08:12,174 --> 00:08:19,980 And the magnitude squared of a product is the product of the magnitude squared 98 00:08:19,980 --> 00:08:27,329 that's Py, that's Px, and we get that relationship, pretty easy to see. 99 00:08:27,329 --> 00:08:35,375 Not very difficult to derive at all. Now, because of Parseval's Theorem. 100 00:08:35,376 --> 00:08:43,115 If you recall Parseval's theorem, lets say that the interval in time which we call 101 00:08:43,115 --> 00:08:50,573 the power is equal to the interval of the 3 magnitude square root by tens form but 102 00:08:50,573 --> 00:08:57,872 that's for us in other power spectrum. And because we now know that power spectra 103 00:08:57,872 --> 00:09:04,736 are even we, the interval from minus infinity to infinity, is just twice the 104 00:09:04,736 --> 00:09:10,626 interval from 0 to infinity. Because it's even, this interval is going 105 00:09:10,626 --> 00:09:15,424 to be the same as that interval. So it's a little bit simpler for 106 00:09:15,424 --> 00:09:19,888 calculational of purposes just to integrate it with a positive frequency 107 00:09:19,888 --> 00:09:24,765 axis and multiply the answer by two. And finally, I want to point out this 108 00:09:24,765 --> 00:09:30,975 little interesting result, that you can think about the power of the signal within 109 00:09:30,975 --> 00:09:36,735 a given frequency range as being the interval of the power spectrum over that 110 00:09:36,735 --> 00:09:42,931 frequency range. The reason that comes about is because of 111 00:09:42,931 --> 00:09:48,802 that result. So, suppose capital H is a band pass 112 00:09:48,802 --> 00:09:56,218 filter between f1 and f2. And so we know what that does is that the 113 00:09:56,218 --> 00:10:05,262 output y only provide, only thing in the output is the frequency components of the 114 00:10:05,262 --> 00:10:12,563 input in that frequency range. Well the power in the output is just given 115 00:10:12,563 --> 00:10:17,270 by the interval. And so, when it is a band pass filter on 116 00:10:17,270 --> 00:10:23,055 like this, magnitude at h of s squared is just 1 in that frequency range and that 117 00:10:23,055 --> 00:10:27,863 interval simplifies to that. So you can interpret this band pass 118 00:10:27,863 --> 00:10:33,431 filtering to be, I'm only measuring the power in the input signal over a given 119 00:10:33,431 --> 00:10:37,304 frequency range. So, once you have the power spectrum for a 120 00:10:37,304 --> 00:10:42,195 signal you can figure our how much power has in different ranges of frequencies, 121 00:10:42,195 --> 00:10:45,365 simply by integrating over that frequency range. 122 00:10:45,365 --> 00:10:50,615 It's very useful for us as you are going to see here is a second, especially as you 123 00:10:50,615 --> 00:10:55,380 talk about noise. So going back to white noise, as we said 124 00:10:55,380 --> 00:11:01,318 by definition, white noise signals have a constant power spectrum. 125 00:11:01,318 --> 00:11:07,573 So your power spectrum using some sense the easiest, is just a constant. 126 00:11:07,573 --> 00:11:14,388 And that amplitude of the constant it turns out as a not over 2 And we'll see 127 00:11:14,388 --> 00:11:18,856 why there's this over tuned factor in just a second. 128 00:11:18,856 --> 00:11:24,884 Now, what's the power in white noise? It's supposed to be in the inner row of 129 00:11:24,884 --> 00:11:30,031 the power spectrum, what do you get for the power in white noise? 130 00:11:30,031 --> 00:11:35,338 Of course, the answer is it's infinite. So white noise is a fiction. 131 00:11:35,338 --> 00:11:40,691 It doesn't exist in the real world. But it's a very convenient fiction, very 132 00:11:40,691 --> 00:11:44,574 useful for us, in analyzing communications problems. 133 00:11:44,574 --> 00:11:49,680 And believe it or not we're going to assume in any channels the channel adds 134 00:11:49,680 --> 00:11:53,346 white noise, infinite power noise, to our signal. 135 00:11:53,347 --> 00:12:00,073 It's going to turn out it's easy to remove that noise because of filtering. 136 00:12:00,073 --> 00:12:06,527 So again the power and noise signal we going to give in frequency range is just 137 00:12:06,527 --> 00:12:12,767 going to be given that the spectral, so called spectral height times the 138 00:12:12,767 --> 00:12:17,572 bandwidth. The range of frequencies of interest, and 139 00:12:17,572 --> 00:12:22,778 that's called bandwidths. So the bandwidth is f 2 minus f 1. 140 00:12:22,778 --> 00:12:30,776 So, again, using our result that we shown on the previous slide, the power spectrum 141 00:12:30,776 --> 00:12:37,769 of the output of a linear filter, when white noise serves as the input Is n mod 142 00:12:37,769 --> 00:12:44,396 over 2 the power spectrum of the input times magnitude of h of f squared. 143 00:12:44,396 --> 00:12:52,820 So, in most situations when you have some sort of filter, like this, the power of 144 00:12:52,820 --> 00:12:59,372 the output of that filter is now finite. Even though the power coming in is 145 00:12:59,372 --> 00:13:03,352 infinite. So this is a convenient way of modeling 146 00:13:03,353 --> 00:13:07,386 noise having any kind of spectral properties at all. 147 00:13:07,386 --> 00:13:11,606 Is that you pass it through the appropriate kind of filter. 148 00:13:11,606 --> 00:13:17,453 So, the beginnings that start with white noise because it's very, very easy to do. 149 00:13:17,454 --> 00:13:26,894 Now, in analyzing the communication systems the bottom line measure equality 150 00:13:26,894 --> 00:13:33,627 is the signal to noise ratio. So the idea is that, we're going to wind 151 00:13:33,627 --> 00:13:40,557 up In almost all cases with a signal plus noise, our, our final signal that we 152 00:13:40,557 --> 00:13:48,257 receive after going through the receiver, is going to be a signal relayed component 153 00:13:48,257 --> 00:13:54,540 and a noise related component. We want a measured quality of how big the 154 00:13:54,540 --> 00:14:00,112 signal is relative to the noise. And the signal to noise ratio is the 155 00:14:00,112 --> 00:14:05,042 standard measure. It's the power in the signal part, divided 156 00:14:05,042 --> 00:14:10,774 by the power in the noise part. And clearly, the bigger the SNR, the 157 00:14:10,774 --> 00:14:14,968 better the quality it is, the smaller the noise is. 158 00:14:14,969 --> 00:14:20,898 Well, given by what we've seen, seen in terms of power speculator, it's the ratio 159 00:14:20,898 --> 00:14:25,655 of these 2 quantities. Power and the signal is given by that, and 160 00:14:25,655 --> 00:14:31,325 the power of the noise is given by that. And also as we've seen SNRs frequently 161 00:14:31,325 --> 00:14:38,240 expressed in decibels in DB. So here's the general lay of the land in 162 00:14:38,240 --> 00:14:46,004 terms of our channel model. The channel introduces a delay, it 163 00:14:46,004 --> 00:14:55,012 attenuates, adds interference and noise. We're, like I said, we're basically going 164 00:14:55,012 --> 00:15:01,814 to assume interference is not there. And that gives us the final expression for 165 00:15:01,814 --> 00:15:06,279 the signal to noise ratio coming out of the channel. 166 00:15:06,279 --> 00:15:12,164 As being this quantity over here. So, certainly, the effects of attenuation 167 00:15:12,164 --> 00:15:17,144 in a channel really hurt the numerator that makes can make it small in the 168 00:15:17,144 --> 00:15:22,556 wireless situation as you get further and further away from the transmitter. 169 00:15:22,556 --> 00:15:27,989 This number gets smaller and smaller and smaller, so the s and r goes down like the 170 00:15:27,989 --> 00:15:34,602 square of that attenuation factor. And the noise part depends, of course, on 171 00:15:34,602 --> 00:15:40,005 the spectral height. But also depends on the bandwidth over 172 00:15:40,005 --> 00:15:44,857 which x exists. So I'm assuming that x is some signal in 173 00:15:44,857 --> 00:15:49,780 the frequency domain. Which exists from frequency fl. 174 00:15:49,780 --> 00:15:58,259 A lower frequency to some upper frequency. And the only thing we need to consider is 175 00:15:58,259 --> 00:16:07,315 what's called the in-band noise. I'm about to show you how that works so 176 00:16:07,315 --> 00:16:13,678 next talk about resign the transmitter and receiver. 177 00:16:13,679 --> 00:16:19,512 For a channel, so we have now our channel model, and now we know what the 178 00:16:19,512 --> 00:16:25,778 characteristics of the channel are. What we're going to do for all kinds of 179 00:16:25,778 --> 00:16:29,821 reasons is simply transmit the message as it is. 180 00:16:29,821 --> 00:16:36,998 The transmitter is going to provide some gain usually gain, the gain is much bigger 181 00:16:36,998 --> 00:16:41,082 than 1. To bind some power to the signal, but it's 182 00:16:41,082 --> 00:16:46,545 just going to send it as it is. Most message signals have a low pass 183 00:16:46,545 --> 00:16:51,240 spectrum. They, they're support for their frequency, 184 00:16:51,240 --> 00:16:57,166 transforms usually goes from 0 frequency up to some upper frequency. 185 00:16:57,166 --> 00:17:03,898 For speech it goes up to like 6 or 7 kilohertz, audio music has a somewhat 186 00:17:03,898 --> 00:17:09,136 higher frequency or range. Still, it's a low-pass spectrum. 187 00:17:09,136 --> 00:17:15,088 So given that's the way we're going to do it, we're hopefully going to look at the 188 00:17:15,088 --> 00:17:21,424 channel model and the transmitter design is going to be determined by how we think 189 00:17:21,424 --> 00:17:26,608 we're going to send it and the characteristics of channel in that 190 00:17:26,608 --> 00:17:31,316 frequency range. All as we know baseband communication does 191 00:17:31,316 --> 00:17:36,794 not work well for wireless channels. They just low pass signals do not. 192 00:17:36,795 --> 00:17:43,502 Propagate very well for wireless channels. So it's rarely, if ever used for wireless 193 00:17:43,502 --> 00:17:47,943 channels. For wireline, there is also an attenuation 194 00:17:47,943 --> 00:17:52,524 as we've seen. I mean a high frequency limit that we get 195 00:17:52,524 --> 00:17:57,606 no attenuation. So it is used though for wireline channels 196 00:17:57,606 --> 00:18:03,554 to for communication over limited distances that is definitely a case. 197 00:18:03,554 --> 00:18:08,918 So let's see if we can figure out, the goal is going to be to figure out the 198 00:18:08,918 --> 00:18:15,532 receiver that goes with this transmitter. And to figure out the signal to noise 199 00:18:15,532 --> 00:18:21,792 ratio in the recovered message. There, so here's what's coming out of the 200 00:18:21,792 --> 00:18:27,124 receiver, out of the channel rather, in the frequency domain. 201 00:18:27,124 --> 00:18:34,186 So I'm assuming like I always do, that my message signal has this triangular 202 00:18:34,186 --> 00:18:41,466 spectrum just a habit of mine, just to use a, a triangular looking spectrum. 203 00:18:41,466 --> 00:18:48,605 So it has a highest frequency of w. It has a band width of w, its so called 204 00:18:48,605 --> 00:18:54,900 band width is equal to W. And again bandwidth is defined to be that 205 00:18:54,900 --> 00:19:01,801 portion of the positive frequency axis over which the signal is non zero. 206 00:19:01,801 --> 00:19:07,073 So there's our good old message signal, like that. 207 00:19:07,073 --> 00:19:11,937 It's contained in white noise. Which is very broad band. 208 00:19:11,937 --> 00:19:15,209 I show this as a blue thing. It's everywhere. 209 00:19:15,209 --> 00:19:21,541 Well, clearly, if I was to use that as my demodulated message devise. 210 00:19:21,541 --> 00:19:24,595 If I used that, just sent that to the sync. 211 00:19:24,595 --> 00:19:30,695 The signal to noise ratio would be 0. Because white noise has infinite power. 212 00:19:30,696 --> 00:19:38,290 It's very clear we don't need this portion of the noise. 213 00:19:38,290 --> 00:19:43,523 Why keep it? It's not what we call in band, in the 214 00:19:43,523 --> 00:19:49,050 bandwidth, in the frequency band of the message. 215 00:19:49,050 --> 00:19:56,856 So most receivers, the first thing they do, is remove what's called out of band 216 00:19:56,856 --> 00:20:00,828 noise. It's out of the bandwith of, of the 217 00:20:00,828 --> 00:20:06,691 transmitted signal. So in this case, since we are doing the 218 00:20:06,691 --> 00:20:13,426 baseband with a low pass filter this is called front end filtering. 219 00:20:13,426 --> 00:20:20,047 Virtually every receiver in the communications world, first thing it does 220 00:20:20,047 --> 00:20:25,151 to the signal coming out of the channel is low cast filter. 221 00:20:25,151 --> 00:20:31,037 And the base band communications situation, that is going to be our 222 00:20:31,037 --> 00:20:36,764 message. Signal, it's going to be reconstructed, so 223 00:20:36,764 --> 00:20:45,296 now the message in the frequency domain looks like this bandbase signal. 224 00:20:45,296 --> 00:20:52,406 You have the message, part that transmitted message, comes through without 225 00:20:52,406 --> 00:20:59,621 altering, being altered by the low pass filter, but the noise is certainly cut 226 00:20:59,621 --> 00:21:03,378 down. Now, what is the signal to noise ratio? 227 00:21:03,378 --> 00:21:10,008 It's going to be the attenuation squared introduced by the channel, times the gain 228 00:21:10,008 --> 00:21:15,928 squared of our transmitter. Times the power in the message, whatever 229 00:21:15,928 --> 00:21:21,868 that is, divided by n nought w. N nought times the bend with the noise, 230 00:21:21,868 --> 00:21:26,417 which is just w. So you can see that the transmitter is 231 00:21:26,417 --> 00:21:32,435 here as a gain to try to fight the attinuation introduced by the channel 232 00:21:32,435 --> 00:21:36,926 basically. The bigger the noise is in the channel, 233 00:21:36,926 --> 00:21:40,730 the smaller the SMR. That's pretty obvious. 234 00:21:40,730 --> 00:21:47,503 I wouldn't worry too much about band width here, because the power in the message 235 00:21:47,503 --> 00:21:51,844 will probably change if you change its band width. 236 00:21:51,844 --> 00:21:57,861 This part of the SNR, basically, could be a constant, for most situations. 237 00:21:57,861 --> 00:22:03,763 May or may not be, but it, I would not say that the wider the bandwidth, the smaller 238 00:22:03,763 --> 00:22:07,166 the SNR has to be. That's certainly is not true. 239 00:22:07,166 --> 00:22:15,082 Now, another thing I want to point out, is the those of you who are, who know what's 240 00:22:15,082 --> 00:22:22,314 going on, who know that m hat, the, the signal part of the m hat is equal to alpha 241 00:22:22,314 --> 00:22:29,478 times g times the message plus filtered noise, which I'll just call n tilde. 242 00:22:29,478 --> 00:22:37,206 So do, don't we need to worry about alpha times g and in these situations we do not 243 00:22:37,206 --> 00:22:42,874 demand that be 1. This corresponds to the knob on the radio. 244 00:22:42,874 --> 00:22:49,109 Volume control if you will. Will make up for it by amplification a 245 00:22:49,109 --> 00:22:53,898 little bit later. But if you amplify this entire signal 246 00:22:53,898 --> 00:22:58,522 multiply it by a constant. You do not change the SNR. 247 00:22:58,522 --> 00:23:02,655 So, we can study the SNR for this signal as it is. 248 00:23:02,655 --> 00:23:07,439 And that will be a sufficiently characterized quality. 249 00:23:07,439 --> 00:23:13,687 And we will fix the game problem, if you will, by further amplification. 250 00:23:13,688 --> 00:23:21,783 So we now have designed for ourself a communication system. 251 00:23:21,783 --> 00:23:31,069 The model that we've developed, for just the channel, describes the delay the, The 252 00:23:31,069 --> 00:23:36,614 interference and the noise that the channels add. 253 00:23:36,615 --> 00:23:41,716 They said nothing good happens in a channel, saying some get bigger and they 254 00:23:41,716 --> 00:23:47,409 get delayed and sometimes attenuated by a lot especially in the wireless situation. 255 00:23:47,409 --> 00:23:51,463 Interference gets added. Who knows what it is, depends on the 256 00:23:51,463 --> 00:23:55,142 situation. And the noise is always prevalent. 257 00:23:55,142 --> 00:24:01,416 You don't know really how bit it's going to be in any given situation. 258 00:24:01,416 --> 00:24:07,172 And you have to do the best you can. To combat the presense of the noise and 259 00:24:07,172 --> 00:24:13,282 the interference for that matter, most receivers from wireless, consist of a 260 00:24:13,282 --> 00:24:17,478 front end filter. To remove out-of-band noise and 261 00:24:17,478 --> 00:24:23,750 out-of-band interference, again out-of-band means not in the bandwidth of 262 00:24:23,750 --> 00:24:27,591 the transmitter sig, not in the bandwidth of x. 263 00:24:27,591 --> 00:24:33,933 Y keep the noise in any other signals floating around that are out of band when 264 00:24:33,933 --> 00:24:38,276 I band pass. May be a low pass filter for, a base band 265 00:24:38,276 --> 00:24:41,842 situation. A band passed over for modulated 266 00:24:41,842 --> 00:24:45,406 situations can get rid of all other signals. 267 00:24:45,406 --> 00:24:51,501 So, that's why most wireless receivers, right at the front, there's a fold. 268 00:24:51,501 --> 00:24:58,101 And then further processing goes on, communications to yield the reconstructed. 269 00:24:58,101 --> 00:25:02,690 Message. We'll talk about those kind of situations 270 00:25:02,690 --> 00:25:04,580 in succeeding videos.