Digital Sampling, Signal Spectra and Bandwidth - A Level Physics

hello today we're going to look at digital sampling signal spectra and bandwidth for a level physics revision we're going to start with digital recording and we really need to go back to basics and say what actually happens when you take a microphone and you have sound coming in and you have an electrical signal coming out what's actually happening inside the microphone well of course the sound is coming to you in sound waves sound waves are longitudinal waves so in other words instead of vibrating as electromagnetic waves do as a sine wave in this way they actually vibrate in and out so at any moment of time the air for example through which the sound is traveling will be either very compressed or it will be rarefied typically if you look at the sound waves you will find that it gets rarefied and then it spreads out and then it becomes rarefied again and then it spreads out and it's rarefied again but that's constantly changing so that the the air is vibrating but in a longitudinal way as the sound waves cause it to vibrate inside the microphone in essence is what's called a diaphragm and you can think of a diaphragm is really stretched piece of balloon skin and there it is and in the middle is a small magnet and around the balloons skin is a coil of wire now sometimes it's the other way around sometimes they put the coil of wire on the balloon skin and they have a magnet outside but here's the way it works when the sound waves which are causing the air to oscillate in this fashion here when they hit the diaphragm I'll call it a diaphragm now and they cause that to vibrate and as that vibrates the magnet also vibrates and now you've got a moving magnet in a coil or alternatively a moving coil in a magnet depending on which way it's been done and wherever you have a moving magnet or in a coil you will get a current flowing and so in this coil a current will flow and the size of that current will reflect the degree of movement of the magnet we've done this in electromagnetism so you've got a current that is essentially analogous to the sound wave that caused it and that's why it's called an analog signal now just think about the sound you might be recording so you're recording an orchestra then although the waves are not sine waves as I've said in terms of the fact that they're not transverse they're longitudinal nonetheless we can represent them as sine waves so if you took something like a flute which is playing a very high pitch high pitch means high frequency so a flutes frequency would look something like this where this of course is the amplitude of the frequency and this is time going by and so you've got a very short wavelength high frequency wave whereas a double bass over time would look like that because and this again is the amplitude because the bass the double bass is a very low note and that means it's frequency is much smaller and that means its wavelength is much longer and all these frequencies this is just two instruments from the orchestra but waves have sound waves have the capability of what's called superposition as all those sound waves approach the microphone they are effectively all joining together and making an an amalgam of a wave so if if we just simply add the double bass and the flute together then you're probably going to get something that looks like this you know you've got the double bass which is the the longer wavelength superimposed on which will be the flute signal on this basis the flute is much softer than the double bass if the flute were about the same volume then you would get a kind of pattern that looks like this but you've got all the other instruments of the orchestra and those waves are hitting the diaphragm and they are creating a pattern which will look like this in terms not now of sound waves but once you've got the current coming out of here you've got current and obviously voltage so if you were to measure this on a cathode ray oscilloscope CRO then this would essentially be the voltage or the current and this would be time so this now becomes the electrical analog signal of the sound waves that caused it and in the old days you would send that electronic signal that you've got from your microphone to a reel-to-reel tape recorder which had a a reel-to-reel would have a tape the tape would have an iron deposit on it and you would go past what were called the heads and the electric current from the microphone would be fed into the heads and they would cause the iron deposits on the tape to form a particular shape and that therefore you will have stored in a sense the sound an analog signal on the tape and then when you play it back the tape will cause the heads to recreate the current that caused it and you can take that current off to a loud speaker and play back the sound that you started with in the first place initially of course this process was fraught all sorts of noise and irregularity was introduced and so the designers of these systems were always trying to get the best and the most faithful reproduction of the sound they were looking for the fidelity if you like the faithfulness of the sound and the higher the fidelity they could get the better consequently they were looking for high fidelity or as we shortened it HiFi so as I say if you were to look at the signal that comes out of the microphone which is an electrical signal analogous to the sound signal that caused it here's a cathode ray oscilloscope the vertical axis is going to be depending on what you look at the voltage or the current and the so this is voltage or current going upwards and you just have a time base going across so you're actually seeing how the wave changes or how the current changes with time then you get what essentially just looks like bubbly goop on the cathode ray oscilloscope because that is the amalgamation or the superposition of all the sound waves that have contributed to the sound that the microphone has picked up and this pattern of course is constantly changing with time so you will just see it constantly moving now one way of overcoming the fact that to this digit to this analog signal all sorts of noise can be added and when you record it on the tape you lose some quality and when you play it back you lose some more quality what you can do is to digitize it and this is the way you digitize you as it were freeze-frame this shape so just essentially stop it for a moment and let's suppose that you get something like this what you do is you superimpose on that a grid computers are good at doing this sort of thing and let's suppose we have I've got here 1 2 3 4 5 6 7 7 lines and let's separate the grid up like this and it's a bit like the pixels in the photograph what you're going to say is that for each square we are going to look and say what is the as it were average current in that square so for example here we would say it's this one here we would say it's maybe this here we would say it's maybe this here we would say it's maybe this here we say it's this don't criticize my kind of assumptions you know computers do this much better than I can here we'd say it's this here we'd say it's this so we've now got the average or not the average value the sort of the value per square and this is one two three four five six seven and so the digital value of that signal would be represented as four six one five two three one one okay that is the digital value of that signal now clearly you can see that that means we would get a signal that was four six one five two three one what's 1 1 and that doesn't look very much like this but I think you'll agree that if you make the divisions much narrower in both directions then you will get to a point where the digitized signal is very close to the analyze of the analog signal that you were measuring so we've got a problem here it's a trade-off issue if you have too few gradations too few of these as it were pick cell equivalence then you're not going to reproduce the sound in the digital signal that was the original analog but if you have too many then of course remember every single one of these is going to be a depending on how many gradations you have you're going to need a number of bits to retain each of these numbers then that is going to overload your computer you're going to have a massive amount of data to process and a massive amount of data to store and the way this is resolved is that for CDs the vertical component in other words the number of levels that you split your sing signal up into the voltage signal is 65,536 and that is equivalent to 16 bits or 2 bytes so what you essentially do is to divide this up into lots of different levels and you give each of these in each column you give one value which represents the level of the signal at that point and that can be anything from nought if there's no signal at all to 65536 if it's added as it were it's maximum volume and each of those numbers therefore needs 2 bytes in order for 65,536 gradations to be achieved just to remind you this is as it were the voltage or the current and this along here is time so these divisions represent the amount of time that we're going to elapse between each sampling of the wave and this way is called the sampling rate this way is called the digital resolution so the digital resolution is 65,536 different alternatives levels if you like this rate is called the sampling rate and the sampling rate for most CDs these days is 44,000 100 Hertz in other words you measure the level of the current or the voltage 44100 times per second now how did they determine to use this number for the digital resolution in other words the number of gradations this way and this number for the sampling rate in other words the number of gradations per second that way well let's take the bit resolution or the digital resolution first you can have too many gradations ironically let's suppose that you had a signal that look like this and you want to set up the number of gradations by which you're going to measure so you're going to define the level of the voltage at each point to that level of resolution well it's quite possible that noise will be added to the signal and you might actually get a signal it looks like and if you have these too close together you'll start to identify the noise resolution and the last thing you want is to reproduce the noise in the digital signal and so what you do is you look at the total voltage difference which we call VT in other words the complete range of the voltage or the current and you look at the range of the noise which we'll call the noise and there is a formula which have been it's been developed by the digital recording experts which is that the maximum number of bits that you need that is to record this number of great gradations it turns out it comes out to about 16 so you can get the 65,000 different alternatives the number of bits you need is log to the base 2 of the total variation of the voltage divided by the variation according to the noise and that actually comes out to 16 bits approximately 16 bits if you do any more than that and you think you're getting an even greater quality what you will actually do is to start sampling or to start measuring the degree of noise and you can do without that now what about the sampling rate which is the number of times per second that we measure the voltage or the current well let's suppose that the actual signal this is the analog signal looks something like this so we've got actually quite a high-frequency signal coming in but let's suppose we sample it here this is a time signal this is the voltage so this is how the cut the voltage is varying with time so we'll sample it at this time here then we'll sample it at this time here then we'll sample it at this time here then we'll sample it at this time here and then we'll sample it at this time here okay so we're doing regular sampling that's supposed to be reasonably regular so we take a sample snapshot at that point at that point at that point at that point at that point what will that digital picture look like it will look as though we've got a wave that looks like that that's a very long wave a very low frequency so ironically with a very long sampling period you can convert high frequency sounds into a low frequency wave so you have to be able to take these samples sufficiently frequently that you don't get that kind of mistake happening and the general rule is that you need to take the the frequency of sampling has got to be twice that of the highest frequency sound that you're likely to record now the highest frequency that the human ear can usually hear is about 20 kilohertz and that tends to be young people as you get older the top range reduces and for older people generally 16 kilohertz is the maximum they're likely to be over here but since we're going to be recording for young people we must take that into account so if 20 kilohertz is the highest frequency you're recording then the sampling frequency has got to be double that which is 40 kilohertz and as I've said what they've actually used for various technical reasons is forty four point one kilohertz or forty four thousand one hundred times per second different systems use different rates some go as high as 96 kilohertz but of course what you've got to think about is that the more times you do it per second the more data you're going to receive so how much data is there going to be when we digitize a three-minute song so let's take a three-minute song well we've already said that we're going to sample 44,000 100 times per second so every second you're going to do 44100 measurements and each of those measurements is going to need 16 bits because we're going to have 65,000 alternative values for the level of the voltage so we're gonna need 44,100 samples per second times 16 bits for each sample we're going to record in stereo so you need 2 times that and of course this is just per second and we're doing it for 3 minutes which is 180 seconds that will tell us the total number of bits we need but of course there are 8 bits per byte so so we can convert this now 2 bytes and if you work that out it should come to 32 megabytes so you need 32 megabytes of storage to store the information that is coming from a three-minute song so what's the advantage of digitizing a signal well digital signals can be sent and received and reproduced more easily because once you've digitized the signal you can never corrupt it unless you of course do something drastic but you can't add noise to a digital signal it's either a 1 or a 0 it can't be anything else you can use digital signals to represent different types of information we've already shown how digital signals can be used for images in another video and here we're using digital signals for sound and of course because we're talking about digits we can process those easily on computers because that's what computers do best they process binary digits but of course we have to recognize that since your digitizing you can never exactly reproduce the original signal but in fact you can do it sufficiently well that it achieves what you want to achieve now we're going to look at how we transmit information well let's consider a signal here of course we're looking as usual at the voltage and here we're looking at the frequency this is of course again it's time and so we will have a high frequency and a low frequency and what's called the bandwidth is the gap between the lowest frequency or the highest frequency minus the lowest frequency so if you're talking about sound the highest frequencies are likely to be about 20 kilohertz and the lowest frequency well that might be 100 Hertz or maybe maybe 50 Hertz so you might as well say that the bandwidth is 20 kilohertz because that is the difference between the highest and lowest frequency when it comes to television signals that's for sound that is here if you're talking about television signals then you need to obviously contain a lot more information and that the bandwidth for that tends to be 8 megahertz and the way that signals are transmitted over the air is to use a carrier wave so what will typically happen is that a sine wave carry a wave electromagnetic wave usually in the radio wave of the spectrum VHF or UHF and sometimes shortwave of course sometimes medium wave sometimes long wave and then what you do is you add to that the actual signal that you're trying to send so the signal will be superimposed on the band wave on the carrier wave rather some set looks something like this the signal is embedded within it when you get to your television or your radio that will use essentially the LRC circuit that I described in another video and that will be able to tune itself to the wavelength that this carrier wave has and once it has received you get what's called resonance with an LRC circuit it picks up this wave are the more than any other that might be floating around in the atmosphere at the time and then it strips off the carrier wave and leaves just the basic signal that you had in the first place but there is of course a problem which we know about waves and that is that they are capable of interfering so if we have a transmitter here and a transmitter here and they are both transmitting signals at the same wavelength and you're standing here with your radio your radio will pick up both signals but there will be an interference between them and so it's likely that you will get a combination of the two programs and that will not be making very comfortable listening so within this area here you're going to have very bad interference consequently you have to make sure that the carrier wave coming from this transmitter is significantly different from the carrier wave from that transmitter and in fact the idea is that you try to make sure that they are at least one band width apart in practice you should probably find that the carrier waves are significantly different in wavelength to ensure that there is no and that there is no interference in terms of television in the UK the bandwidth the or rather the spectrum that has been allocated radio frequencies from is a very scarce resource there's only a limited amount of it so it has to be used carefully and sparingly but in the UK the electromagnetic spectrum which has been set aside for television broadcasting off air that is where you've got a transmitter on the ground as opposed to a satellite is 417 megahertz up to 860 megahertz and I said before that the band width was 8 megahertz that is the frequency bandwidth to contain the entire signal and what that means is that this bandwidth of or other frequency spectrum of the electromagnetic spectrum contains 48 channels so you can only send 48 different as it were wavelengths of carrier waves and those are numbered 29 sorry 21 to 68 so BBC ITV channel 4 or get a channel and they are allowed to broadcast on one of these channels which will be 8 megahertz wide now in practice you can't you might think you can only therefore have took 48 transmitters each transmitting one of these channels but actually you can do better than that because if you think about if this is if this is England you might have a transmitter a Newcastle and a transmitter in Exeter well they might transmit on the same channel because by the time the signal from Newcastle has got all the way down to Exeter it's going to be so small it's going to be almost indeed undetectable so the interference is going to be trivial but you couldn't have two transmitters almost side-by-side transmitting on the same channel because then you get interference so you can reuse the channels as long as you're far enough away as far as radio is concerned then the typical bandwidth that has been all the typical spectrum that's been allocated is 30 megahertz to 300 megahertz for example BBC Radio 2 is 88 to 91 fm as they call it which is 88 megahertz to 91 megahertz the bandwidth required for each channel is naught point 2 megahertz or 200 kilohertz and so you can see that you've got a total spectrum of 300 - 300 minutes each channel is naught point 2 megahertz wide so that gives you a total of 1 5 3 oh sorry 1 3 5 Oh stations I'll just write that here 1 3 5 Oh separate channels in this spectrum here so again you might think that you're limited to only 1,350 1,350 radio broadcast channels but actually again that's not entirely true because you can repeat some of these if you are far enough away the same frequency can be used for transmission because they're so far away that the intensity of the signal from one will be trivially small by the time it reaches the other so actually you can have more than 1,350 radio stations but if you want to transmit in digital let's just think for a moment about what that means how many bits are you gonna have to transmit in one second so what were your computer have to process if you digitized all this sound well we said that there are 44,100 samples per second each of those requires 16 bits we're going to broadcast in stereo and so that is the number of bits per second that you're going to need to be able to transmit your computer is going to have to be able to process that amount and of course as we said before you can divide that by 8 and that now gets you the number of bytes per second that has to be transmitted so you need computer with a fair degree of processing power to be able to just push that out at that rate across the airwaves

Channel: DrPhysicsA

Views: 56,562

Keywords: Physics, A level, Digital, Sampling, Signal Spectra, Bandwidth, microphone, sound, digitise, analogue, sampling rate, digital resolution, noise, frequency, transmission, BBC, radio, TV, spectrum, bits, bytes

Id: MQFElk0lllA

Length: 28min 58sec (1738 seconds)

Published: Mon May 13 2013