1 00:00:00,012 --> 00:00:04,942 We're going to talk about a real communication system today, a so called 2 00:00:04,942 --> 00:00:10,156 modulated communications systems. The idea is of course, instead of using 3 00:00:10,156 --> 00:00:15,070 base band signalling we're now going to shift the message signal up to a higher 4 00:00:15,070 --> 00:00:18,460 band of frequencies. This allows it to get through wire line 5 00:00:18,460 --> 00:00:23,336 and wireless channels more easily. We'll talk about the transmitter and 6 00:00:23,336 --> 00:00:26,465 receiver. Turns out we already understand how these 7 00:00:26,465 --> 00:00:29,555 work. Once I show you how they function, you 8 00:00:29,555 --> 00:00:35,297 know how to think about them because we'lve already talked about it in previous 9 00:00:35,297 --> 00:00:38,070 videos. And finally we'll compute the SNR that 10 00:00:38,070 --> 00:00:40,705 results when you use modulated communication. 11 00:00:40,706 --> 00:00:45,540 So, what does the transmitter look like for modulated communication? 12 00:00:45,540 --> 00:00:50,426 It's very straightforward. So here's our message. 13 00:00:50,426 --> 00:00:55,537 And we get this somewhat curious thing in which we add one. 14 00:00:55,538 --> 00:01:02,105 And the assumption here is, is that the amplitude of the message has been scales 15 00:01:02,105 --> 00:01:07,821 so it's less than or equal to 1, which makes this greater than equal to 0. 16 00:01:07,821 --> 00:01:14,779 Now the reason for this has to do with your sum receivers that require that this 17 00:01:14,779 --> 00:01:21,413 be a positive quantity, but not all, but in order to satisfy everybody most 18 00:01:21,413 --> 00:01:28,475 amplitude modulation systems do this, adding this offset and then scaling the 19 00:01:28,475 --> 00:01:34,384 message accordingly. The transmitter provides a gain, usually a 20 00:01:34,384 --> 00:01:40,115 as a big number. And multiplies by a cosine at what's 21 00:01:40,115 --> 00:01:47,757 called the carrier frequency. And so the idea is that the cosine here is 22 00:01:47,757 --> 00:01:53,198 carrying the message. That's the idea and so fc defines the 23 00:01:53,198 --> 00:01:59,726 characteristics of the transmitters so There's how much, essentially, power is 24 00:01:59,726 --> 00:02:06,350 radiated is determined by a, and what frequency, range is used is determined by 25 00:02:06,350 --> 00:02:10,694 fc. So, it's easier to understand what's going 26 00:02:10,694 --> 00:02:14,766 on here by looking in the frequency domain. 27 00:02:14,767 --> 00:02:20,328 So here's my typical message spectrum, assume it has this rooftop look, and as we 28 00:02:20,328 --> 00:02:26,304 know, from previous videos when we talked about the Fourier transform, we've already 29 00:02:26,304 --> 00:02:29,899 figured out the spectrum of such signals like this. 30 00:02:29,900 --> 00:02:38,162 So, you get a line spectrum corresponding to the fact that, that periodic par, 31 00:02:38,162 --> 00:02:44,670 portion comes out by itself. Added to this, when you multiply a, low 32 00:02:44,670 --> 00:02:49,950 pass message by a cosine, shifts it up and down in frequency. 33 00:02:49,950 --> 00:02:55,438 So you get these two replicas of the message itself. 34 00:02:55,439 --> 00:03:00,918 Now, I want to point out that the message bandwidth here is W. 35 00:03:00,919 --> 00:03:11,983 Don't forget, bandwidth is defined to be the part of the positive frequency axis 36 00:03:11,983 --> 00:03:17,940 devoted to the message, where it has power. 37 00:03:17,941 --> 00:03:24,368 Well, for the modulated message, looks like it has half the bandwidth of the 38 00:03:24,368 --> 00:03:31,406 original message, and this is always true for simple amplitude modulation is that 39 00:03:31,406 --> 00:03:37,307 the transmission bandwidth doubles and that's just a fact of life. 40 00:03:37,307 --> 00:03:41,443 And this kind of transmitter is really easy to deal with. 41 00:03:41,444 --> 00:03:47,784 This fact that it doubles the transmission bandwidth won't be that much of a factor. 42 00:03:47,784 --> 00:03:55,880 So, let's look at more detail, well, some of the consequqnces of using this kind of 43 00:03:55,880 --> 00:03:59,350 scheme. Suppose we have two different modulation 44 00:03:59,350 --> 00:04:02,678 communication schemes going on at the tsame time. 45 00:04:02,678 --> 00:04:07,809 Because one of them is using a carrier frequency 1, the other one is using 46 00:04:07,809 --> 00:04:12,325 carrier frequency 2. As we know from the wireless situation, if 47 00:04:12,325 --> 00:04:16,444 both signals come in my receiving antenna, so I get 'em both. 48 00:04:16,444 --> 00:04:23,875 Well, as long as these two spectra do not overlap, I can band pass filter. 49 00:04:23,876 --> 00:04:33,617 And remove the second unwanted signal. I don't have to even know it's there. 50 00:04:33,618 --> 00:04:40,798 This is called frequency division multiplexing, where multiplexing means 51 00:04:40,798 --> 00:04:47,262 using more than once. And what we're doing Is using separating 52 00:04:47,262 --> 00:04:53,600 signals in frequency. So all radio stations are all transmiting 53 00:04:53,600 --> 00:04:58,681 on their own carrier frequencies at the same time. 54 00:04:58,681 --> 00:05:02,607 So if you look in the time domain you'll see this mess. 55 00:05:02,608 --> 00:05:09,223 All cellphone signals, all radio signals, all those signals are all occurring at the 56 00:05:09,223 --> 00:05:12,805 same time, but in different frequency bands. 57 00:05:12,806 --> 00:05:19,246 In, which means they're, they're being separated in frequency, which means simple 58 00:05:19,246 --> 00:05:24,799 band pass filtering will allow you to focus on the message that you want. 59 00:05:25,890 --> 00:05:32,490 And there is a little problem here. Suppose the second carrier is not 60 00:05:32,490 --> 00:05:41,670 well-designed, and suppose it overlaps in frequency with somebody else. 61 00:05:41,670 --> 00:05:49,586 Well now, neither, neither of these signals Can be demodulated without error. 62 00:05:49,586 --> 00:05:56,778 This overlap here would cause interference, and it's something you can't 63 00:05:56,778 --> 00:06:01,903 get rid of by linear filtering at all. So this is bad. 64 00:06:01,903 --> 00:06:06,920 So it turns out you have to have some mechanism for regulating. 65 00:06:06,920 --> 00:06:10,780 These carrier frequencies and these bandwidths. 66 00:06:10,780 --> 00:06:15,146 So that there's no overlap among all the various uses that were used in wireless 67 00:06:15,146 --> 00:06:18,368 communication and that's where the government comes in. 68 00:06:18,368 --> 00:06:25,962 In the United States the regulation for what each frequency band can be used for 69 00:06:25,962 --> 00:06:32,618 is very tightly regulated. And this is a rather beautiful depiction 70 00:06:32,618 --> 00:06:39,304 of what can occur in every frequency band. And so let me, let's go through this a 71 00:06:39,304 --> 00:06:44,063 little bit. This is a logarithmic frequency scale, 72 00:06:44,063 --> 00:06:48,160 first of all. Frequency increases going down the page. 73 00:06:48,160 --> 00:06:55,271 It starts at 3 kHz and this first one goes up to 300 kHz. 74 00:06:55,271 --> 00:07:05,470 The rest of these are one decade, 300 kHz, 2-3 MHz, 3 MHz, 30 MHz, etc. 75 00:07:06,690 --> 00:07:11,490 Down here, at very low frequencies, you probably can't read that, it says not 76 00:07:11,490 --> 00:07:14,254 allocated. You can do anything you want up to about 77 00:07:14,254 --> 00:07:18,396 10 kilohertz carrier frequency. And like I said for wireless it doesn't 78 00:07:18,396 --> 00:07:22,816 propagate very well at all, and no one's going to really care if you build your own 79 00:07:22,816 --> 00:07:26,930 transmitter, frequency range. It won't go very far so it doesn't 80 00:07:26,930 --> 00:07:31,546 interfere with anybody else. But above that it's very tightly 81 00:07:31,546 --> 00:07:35,927 regulated. The client's frequency here is 300 82 00:07:35,927 --> 00:07:43,396 gigahertz, that's very, very high. And everything in between, you can see is 83 00:07:43,396 --> 00:07:47,863 modulated. The AM radio band is here. 84 00:07:47,863 --> 00:07:56,896 The FM band is somewhere here, and it may look like the FM band is narrower then the 85 00:07:56,896 --> 00:08:03,570 AM band, but that's only because this is a logarithmic scale. 86 00:08:03,570 --> 00:08:06,930 This is going from 300 kilohertz to three megahertz. 87 00:08:06,930 --> 00:08:14,535 This scale goes from 50 megahertz up to five, I'm sorry, 30 megahertz, up to 300 88 00:08:14,535 --> 00:08:19,415 megahertz. So this bandwidth is actually much wider 89 00:08:19,416 --> 00:08:23,612 than it appears. And things become busier, if you will, 90 00:08:23,612 --> 00:08:29,525 more things are allocated because it's the bandwidth that matters in a logarithmic 91 00:08:29,525 --> 00:08:32,803 scale. Things look much more compressed than they 92 00:08:32,803 --> 00:08:37,058 would be in the linear scale. The we point out that in some frequencies 93 00:08:37,058 --> 00:08:42,164 bands you can use them for more than one thing as long as you don't interfere with 94 00:08:42,164 --> 00:08:46,972 something else. But we there is so much demand for carrier 95 00:08:46,972 --> 00:08:53,803 frequencies that some of them are, the government regulates them to be multi-use, 96 00:08:53,803 --> 00:08:57,380 we have to do that. And actually, some are reserved for radio 97 00:08:57,380 --> 00:08:59,927 astronomy. It's kind of interesting, I think that's 98 00:08:59,927 --> 00:09:03,017 one of the bands reserved for radioastronomy, interesting. 99 00:09:03,017 --> 00:09:08,325 Okay, well, how are we going to receive this thing, how are we going to receive a 100 00:09:08,325 --> 00:09:14,196 amplitude modulated message. So, this is what we get what the, what 101 00:09:14,196 --> 00:09:20,017 comes out of the channel and out of the, in the frequency domain. 102 00:09:20,018 --> 00:09:26,410 We get the broadband noise, white noise and there's our, our message signals. 103 00:09:26,410 --> 00:09:29,009 And pretty clearly, we want to get rid of all this. 104 00:09:30,280 --> 00:09:36,597 Stuff that out-of-band noise and we do that by band pass filtering. 105 00:09:36,597 --> 00:09:43,397 So this is a, stands for band pass filter as a center frequency equal to the carrier 106 00:09:43,397 --> 00:09:49,013 frequency and has a badwidth of 2w because that's frequency range. 107 00:09:49,013 --> 00:09:52,990 So once you go through that band pass filter, this is what you get. 108 00:09:52,990 --> 00:10:00,664 You're only left with in-band noise and the message that we want. 109 00:10:00,664 --> 00:10:05,837 Now I point out that the message that we want is not at the right frequency. 110 00:10:05,837 --> 00:10:10,792 The original message started down here. How do we get the message down in 111 00:10:10,792 --> 00:10:14,394 frequency, back to where it originally started? 112 00:10:14,394 --> 00:10:19,554 We put it up here originally so we can get through the channel, this is the 113 00:10:19,554 --> 00:10:25,316 transmitter matching the channels characteristics, antennas with much, much 114 00:10:25,316 --> 00:10:28,597 better at high frequencies than at base band. 115 00:10:28,598 --> 00:10:32,020 So we did that we got it through now we have to move it back. 116 00:10:32,020 --> 00:10:35,928 How do we do that? Well, little surprise here. 117 00:10:35,928 --> 00:10:41,112 So, let's look at this little mathematical property. 118 00:10:41,112 --> 00:10:45,972 Suppose we take x, and multiply it again by a cosine. 119 00:10:45,972 --> 00:10:52,335 So the original transmitted message was, a times 1 plus n times a cosine, and 120 00:10:52,335 --> 00:10:57,187 multiplying it again by cosine gives me cosine squared. 121 00:10:57,187 --> 00:11:05,266 What's cosine squared of theta? Well, cosine squared of theta is one half 122 00:11:05,266 --> 00:11:12,011 times one plus cosine of 2 theta. So we wind up with something at twice the 123 00:11:12,011 --> 00:11:19,030 carrier frequency that we started with. Let's expand this. 124 00:11:19,030 --> 00:11:31,162 So there's the first part and you get the same thing times this cosine. 125 00:11:31,163 --> 00:11:41,499 So, believe it or not, remodulating by the cosine gives us back a signal that's now 126 00:11:41,499 --> 00:11:48,762 in it's original frequency band. We also though, have a signal they're 127 00:11:48,762 --> 00:11:54,484 sitting up at twice the carrier frequency. I think I know how I'm going to get rid of 128 00:11:54,484 --> 00:11:57,783 that. I'm still, I'm simply going to filter it 129 00:11:57,783 --> 00:12:03,647 out to get rid of that, and when I do that I'm left with my original message. 130 00:12:03,648 --> 00:12:13,150 So this is the where we started coming out of the front end, and after multiplication 131 00:12:13,150 --> 00:12:21,470 by the cosine and low pass filtering, we wind up with our message down where we 132 00:12:21,470 --> 00:12:26,778 want it. We're going to low pass it to get rid of 133 00:12:26,778 --> 00:12:31,543 these things. And we're just going to have to live with 134 00:12:31,543 --> 00:12:36,300 the noise that wound up in the same band and that's where we'll have to calculate a 135 00:12:36,300 --> 00:12:41,327 signal to noise ratio. So, here's the demodulator in all of it's 136 00:12:41,327 --> 00:12:45,030 glory. Consists of what we call the front-end. 137 00:12:45,030 --> 00:12:51,200 Which just is a simple band pass filter to get rid of Extraneous communications, 138 00:12:51,200 --> 00:12:56,257 interference we don't care about and to get rid of out of hand noise. 139 00:12:56,258 --> 00:13:01,146 And as we know that gives us a spectrum that looks like this. 140 00:13:01,146 --> 00:13:08,502 Then after we multiply by a cosign and low pass filter we're left now with our. 141 00:13:08,502 --> 00:13:15,396 Demodulated message consisting of the original message, that's what we want, 142 00:13:15,396 --> 00:13:19,050 plus noise. So we need to calculate some signal to 143 00:13:19,050 --> 00:13:23,309 noise ratios. Well, for all kinds of reasons, I want to 144 00:13:23,309 --> 00:13:27,407 compute the signals and rates ratio here and here. 145 00:13:27,408 --> 00:13:32,504 So the front end output is going to be of interest because it's a little bit simpler 146 00:13:32,504 --> 00:13:36,215 to compute. And also it has it, it's interesting to 147 00:13:36,215 --> 00:13:43,478 compare these two s, s and r's. So, for the front end the message part is 148 00:13:43,478 --> 00:13:47,934 easy. So this is just x of t, times the 149 00:13:47,934 --> 00:13:56,778 amplitude that we used and this is the attenuation introduced by the channel and 150 00:13:56,778 --> 00:14:02,794 this is the power in this. And it turns out that power, that one half 151 00:14:02,794 --> 00:14:08,477 comes from the cosine basically because it, the power in cosine is half. 152 00:14:08,478 --> 00:14:14,764 And if you work it all out you easily decide that this is the power in the 153 00:14:14,764 --> 00:14:19,470 message. The power in the noise is just as easy, in 154 00:14:19,470 --> 00:14:27,246 fact easier, to calculate because it lives in a frequency band that's centered around 155 00:14:27,246 --> 00:14:32,714 the carrier frequency. And so it's been with this 2W and you 156 00:14:32,714 --> 00:14:39,146 multiply by 2 times naught over 2, 2 to the positive, negative frequency, 2 out of 157 00:14:39,146 --> 00:14:45,482 2 is the specter of light, and we wind up with a signal to noise ratio coming out of 158 00:14:45,482 --> 00:14:52,106 the front end filter that's given by this expression, which should look pretty 159 00:14:52,106 --> 00:14:59,124 familiar compared to the base band case. We clearly the transmitters amplitude gain 160 00:14:59,124 --> 00:15:03,634 is to compensate for the attenuation introduced by the channel. 161 00:15:03,634 --> 00:15:09,181 And the bigger the noise is, the smaller the SNR, and the less happier we're going 162 00:15:09,181 --> 00:15:12,966 to be. And the smaller the attenuation the less 163 00:15:12,966 --> 00:15:17,047 happy we're going to be the SNR may not be good enough. 164 00:15:17,048 --> 00:15:24,081 Well now let's calculate the signal to noise ratio in the N hat. 165 00:15:24,082 --> 00:15:29,187 So, to do that we simply look at the message part. 166 00:15:29,188 --> 00:15:34,038 And on a previous slide I showed you. And after you multiply by the cosign which 167 00:15:34,038 --> 00:15:38,889 you get for the message part and it's just going to be this power which I think is 168 00:15:38,889 --> 00:15:44,037 pretty easily seen to be that we square all the constants up front in the power m. 169 00:15:44,038 --> 00:15:50,688 I should point out that the power in n is not very big and since we've done consider 170 00:15:50,688 --> 00:15:56,388 it, because remember we imposed the condition that the amplitude of the 171 00:15:56,388 --> 00:16:01,182 message that we never got bigger then one in absolute value. 172 00:16:01,182 --> 00:16:06,608 So this power is not a big number. Any gain that's provided in, in amplitude 173 00:16:06,608 --> 00:16:10,724 modulation schemes is provided by the transmitter through. 174 00:16:10,724 --> 00:16:16,294 Now the noise is the part where we have some fun. 175 00:16:16,294 --> 00:16:24,980 So after you do the moving up, moving down of the spectrum and, and wide up down 176 00:16:24,980 --> 00:16:29,215 here. The message part is given by the Voice 177 00:16:29,215 --> 00:16:36,920 power spectrum in R tilda, the part of R tilde that's related to the white noise, I 178 00:16:36,920 --> 00:16:41,742 call that N tilde. It, after multiplying by the cosine that 179 00:16:41,742 --> 00:16:47,853 power spectrum got shifted up and down in frequency, and they both wound up here. 180 00:16:47,853 --> 00:16:51,900 There are other parts that wound up in. Frequency ranges we won't care about 181 00:16:51,900 --> 00:16:57,102 because they're going to be filtered out. And the four again comes from the power 182 00:16:57,102 --> 00:17:02,330 considerations in the cosine. So, all we have to do is add up these 183 00:17:02,330 --> 00:17:09,144 components here. So, we get a two for positive and negative 184 00:17:09,144 --> 00:17:14,000 frequency. Each of those had spectral height N naught 185 00:17:14,000 --> 00:17:18,304 over 2. The bandwidth here is W and the 2 comes 186 00:17:18,304 --> 00:17:24,963 from the fact that powers add, that means that power spectra add. 187 00:17:24,964 --> 00:17:29,389 So, they're both a constant power spectrum, so you just get a factor of 2 188 00:17:29,389 --> 00:17:34,564 and over 4 is, Is because we're already in the power domain here, the formatted power 189 00:17:34,564 --> 00:17:39,178 spectrum. So, we get the four, put it all together, 190 00:17:39,178 --> 00:17:45,118 the noise power is in the D over 2. So that gives us a signal to noise ratio 191 00:17:45,118 --> 00:17:50,422 that's perceived and a received message that's given by this. 192 00:17:50,423 --> 00:17:56,410 It turns out you can show that there is going to be no other receiver that gives 193 00:17:56,410 --> 00:18:00,686 you a bigger signal from noise ratio than that quantity. 194 00:18:00,687 --> 00:18:03,832 So this is what we call, is called optimal. 195 00:18:03,832 --> 00:18:11,484 You can't do better than this. And so you cannot improve the signal to 196 00:18:11,484 --> 00:18:19,832 noise ratio over this quantity. And it depends on channel and transmitter 197 00:18:19,832 --> 00:18:30,312 characteristics in an obvious way. Now we point out that the snr here is half 198 00:18:30,312 --> 00:18:37,292 of what it is here. So there's a gain in this receiver part, 199 00:18:37,292 --> 00:18:41,492 by a factor of two over the SNR that's here. 200 00:18:41,492 --> 00:18:44,640 And it's very interesting story about how that comes about. 201 00:18:44,640 --> 00:18:49,304 It's not at all obvious, and have to leave that for another day. 202 00:18:49,305 --> 00:18:55,676 So, now we have a characterization of virtually all modulated communication 203 00:18:55,676 --> 00:18:59,995 systems, at least in terms of amplitude modulation. 204 00:18:59,995 --> 00:19:06,678 So, the transmitter multiples the message by echo sine. 205 00:19:06,678 --> 00:19:12,668 The receiver as a front end filter followed by another. 206 00:19:12,668 --> 00:19:16,437 Essentially modulation scheme followed by a low pass. 207 00:19:16,438 --> 00:19:18,994 These are very easy to build and construct. 208 00:19:18,994 --> 00:19:25,298 So, because we amplitude modulate, we can choose these carrier frequencies and now 209 00:19:25,298 --> 00:19:31,058 send many, many, many signals all at the same time but in different frequency 210 00:19:31,058 --> 00:19:34,105 bands. It's just the advantage of thinking about 211 00:19:34,105 --> 00:19:38,540 signals in the frequency domain. In communication systems, it's very clear. 212 00:19:38,540 --> 00:19:42,595 Frequency domain is very, very important to understand. 213 00:19:42,595 --> 00:19:47,154 We also, you move it up in frequency, so it gets through these wireless channels 214 00:19:47,154 --> 00:19:50,165 more easily. Antennas work much, much more efficiently 215 00:19:50,165 --> 00:19:54,164 when you move them up. In the way that we've seen. 216 00:19:54,165 --> 00:19:58,780 You now have the muliplexing situation. You can do this multiplexing and the 217 00:19:58,780 --> 00:20:02,570 signal to noise ratio we wind up with is fundamental. 218 00:20:02,570 --> 00:20:07,823 Very important to remember this. This is a guildeline for how well 219 00:20:07,823 --> 00:20:11,970 assisting you may be concerned about works in terms of. 220 00:20:11,971 --> 00:20:20,022 The amplitued provided by the transmitter and how much the attenuation is in the 221 00:20:20,022 --> 00:20:24,228 channel. And don't forget, we've assumed that alpha 222 00:20:24,228 --> 00:20:30,140 is some constant divided by the distance between the transmitter and receiver. 223 00:20:30,140 --> 00:20:36,515 So the further away you are from the transmitter, the smaller your SNR is going 224 00:20:36,515 --> 00:20:39,563 to be by the square of the distance.