Welcome, dear students. Welcome to this lecture in the series RF and millimeter-wave circuit design. Today, we're going to talk about synthesizers. Synthesizer is something that you will need a lot. You might wonder why because we already discussed how to design oscillators and you are now capable of designing perfect oscillators. But in reality, of course, when you implement them and you measure them, they are not quite as ideal as you hoped. What's wrong with oscillators and why are synthesizers so popular? It didn't use to be that way. In the old days, when I was young, many radios didn't have a synthesizer at all. They had, in fact, just an oscillator and it was fairly carefully designed and it was tunable, and that allowed you to, for example, provide your local oscillator signal for your mixer. That worked quite well. In the day's that frequencies were low, channels were still far enough apart because allocation was not such a big issue yet. We could also spend a lot of money on radios because people didn't buy so many and they were prepared to pay a lot of money. That allowed us to make radios which had very good inductors, very good variable capacitors, and were carefully tuned manually, in many cases, to their optimal performance. But when you try to make radio today, then very often you don't have these boundary conditions anymore, and so you need to make an oscillator signal that's better. Let's look at the things that we really want from an oscillator signal nowadays. What we need from an oscillator signal is accuracy because we know that the radio spectrum is fully allocated and over-allocated. We don't want to leave a lot of empty spectrum between two transmitters because that's wasted and we could use that for very good purposes and applications. We really want the channels to be close. That means, besides having very good filtering, we also need to have the frequency very accurate. That accuracy nowadays is down into the parts-per-million range and sometimes even better than that. That's definitely a requirement that's hard to meet if you want to fully integrate an oscillator because like you know, all the components that we have on an IC are not very accurate. We are already quite happy if we get components that are, for example, 10 percent accurate. If things would be off by 10 percent, then we definitely would not meet these parts-per-million requirements. Accuracy basically requires us to make sure that we have some reference that allows us to achieve the accuracy that we need and another means to transfer that accuracy from the reference to our oscillator. That means can be a synthesizer. Another problem that we have is phase noise. Because oscillators, as we know, have phase noise, and that phase noise depends on the quality factor of a resonator, and we know that the resonators that we have available at high frequencies on our ICs have fairly limited quality factors. We usually have significant amounts of phase noise. The phase noise has all kinds of complications in our system. For example, in the transmitter, it results in a reduction of the accuracy of constellation points, so an increase of the error factor magnitude of our modulation. That means that we can transfer less information per symbol than we otherwise could, so we would lose data rate or bandwidth efficiency. The receiver, it can do the same thing, but it can also result in reciprocal mixing, meaning that you receive strong interference that mix with the phase noise of your oscillator. Again, that is not desired if you want to pack the spectrum full with many users. Also, for the same application reason, we want to minimize the phase noise. This is, again, something that we can solve. If we can link the oscillator that we designed to a reference that has fairly low phase noise. We've now looked at two of these requirements that really encourage us to go to some kinds of reference length oscillator. How are we going to do that? What is a synthesizer, really? Well, you all might remember from your earlier lectures, the phase-locked loop. In a phase-locked loop, what you do is you have one signal, a reference signal, and you have an oscillator. Hence you compare the output of the oscillator with your reference signal, and you compare your phase. Then you low-pass filter that phase arrow and you use that to control the frequency of your oscillator. If you do it this way and you do that carefully and you design it well, as you might remember from your bachelor courses, then you can make an oscillator that is locked in phase to the phase of your input signal. It's like any feedback loop, a loop that should have enough gain, that should have a dominant pole to ensure stability and to ensure accurate control of your frequency and your phase, but for a radio, this is not completely a solution. Why? Because this gives us an oscillation signal that is the same frequency as a reference signal and, well, if we have reference signal anyway, that is accurate and has a low phase noise, what's the point? I'd might as well just use the reference for all local oscillator signal. We want to have an oscillator that is capable of generating much higher frequencies and much different frequencies and tunable frequencies from a fixed reference, and that's what the synthesizer does. The synthesizer is fairly much like a phase lock loop, except you open up that loop over there, and in the feedback path of the loop, you insert a divider. For now, let's assume that we divide by [inaudible] N. The control loop still does its work. The control loop, if you've designed correctly, will still try to ensure that this error, this Delta Phi goes to zero, so that means that the phase 2, here, becomes equal to the phase 1 or as close as possible. If that happens, we also know, of course, that if the phase 2 is phase 1, then the frequency 2 has to be frequency 1 as well. It's then going to be the same as the reference frequency. But if this frequency here at the output is going to be the same as the reference frequency, then this frequency has to be n times f_2, and that is n times the reference frequency. It has to be generated by this oscillator. The good thing is that we can now use the control loop to tune an oscillator to a multiple of a reference frequency, and by changing the division factor, we can now tune the oscillator to any integer multiple of the reference frequency. This is the basis of a synthesizer. It's a control loop, like a phase lock loop. Within feedback, we now introduce a frequency divider, and that means in the forward path, we get frequency multiplication because that's what the control loop achieves for us. Now, there are some variations on this because nothing is ever perfect, but one of the things that is not perfect is that you might not want to have integer multiplication. If you do integer multiplication, then you can make all frequencies that are a multiple of the reference frequency. But if you want to have many channels very close to each other, that means that the distance between these channels has to be very small, but the smallest distance between local oscillators that you can make has to be the reference frequency because that's the difference in frequency that you get between n times the reference frequency and n minus 1 to reference frequency. The reference frequency sets the pitch of the channels in your wireless system. If you want them close, you have to use a very low reference frequency and a very high division number. You still might wonder why there's a problem. Well, because, in principle, divide was not a big deal, I [inaudible] transistors, transistors are small, we like designing transistors anyway, more transistors makes us happier, so what's the point? Now, the point is the second thing, the phase noise. When you look at the same way that you get from your oscillator, if this is frequency and this is the power that you have, then, of course, this is the n times the f ref, that's what we want. What we get for free is phase noise. If this is the natural phase noise of the oscillator, if it's not in the loop, then we would like to improve on that. We would like to use the reference to clean up the oscillator. But you can only clean up the oscillator within the bandwidth of the loop, because loop can only relate the phase of the oscillator to the phase of the input signal within the frequency band where the loop has gained. That way loop can affect some control over the oscillator. That means that if we have this f ref here, we can at most clean up the phase noise inside of here. I'll do that in a different color. We might be able to make something like this that's outside of the f ref bandwidth. You will get the original phase noise of the oscillator again. That's the reason that you don't want your f ref to be very low because then you're only going to clean up your oscillator it cause a very small frequency band. In reality, of course, this is even slightly worse because you will not be able to clean up across the whole bandwidths of the f ref because you also have some requirements like stability, and so you probably can only clean it up for smaller part and you will get some bumping head in many cases. These are details. The basic argument was if you want to clean up your oscillator in terms of phase noise across a wide bandwidth, then you need to have high f ref. If you want your oscillators to be accurate, then you need to have low f ref. These are contradictory requirements. In good destination, you could go for a compromise, and that's what people have done for many years. But it's not, of course, what we as engineers want. We don't want necessarily compromise, we want to have the best of both worlds. How do we do that? How can we really make the best of both worlds and still have a high f ref and clean up that way the phase noise of the oscillator because of wide bandwidth and still have tuning in very small steps? Well, we could do that conceptually if we would go for the same circuit. We still have reference frequency. We still have a phase comparator. We have a low pass filter. We have [inaudible]. Nothing new there yet. But here comes the trick, what if instead of dividing by n, we could divide by x with x naught being an integer with a rational number or even a real number? But let's start with the rational number, that's the ratio between two integers. Wouldn't that be great? Because if we now would get with the same reasoning, x times f reference, but x would not have to be an integer. We could have frequencies fairly close to each other and we could still have a large bandwidth of the system because the bandwidth could still be related to a very high f ref, hence we would have the best of all worlds. Now the only thing we have to figure out is how to divide by a non-integer. One way to do that is to have conceptually two dividers. If we cannot divide by x, but let's say x is between n and n plus 1. What we can do is we can make two dividers. We can make one divider that divides by n, like so, and we can have one divider that divides by n plus 1. Of course we cannot connect these outputs like this, so we need a switch here to either choose one or the other. We have a switch here that allows us to connect to either this divider or to that divider. Now if you put the switch in this position, then we get n times f ref. If we would put the switch in the other position, then we get n plus 1 times f ref. One time our frequency is too low, one time our frequency is too high. But what if we would switch the switch very, very quickly, faster than this loop filter could follow? Then you would get an average frequency out of the oscillator, and that would be somewhere between n and n plus 1. If I would keep the switch equal amounts of time in one position as in the other position, then I would get an average frequency which would be halfway between n and n plus 1, so I would get an output frequency that would be n plus 0.5. Times f ref. I could make this way with equal switching time. I could make, for example, N plus 0.5 times f ref. We gain the factor of two. You could imagine by adding more and more dividers, I could go better and better and better. But I can also take a different approach because if I add more and more dividers, then it becomes expensive. What I also can do, I can change the fraction of the time that dispatches in one position or in the other position. You can easily intuitively understand that if you put the switch much longer in the position where it's connected to the divide by N plus 1, and only for a relatively short amount of time where it's connected to the output of divide by N, that your output frequency would still be between N times and N plus 1 times your f ref, but closer to N plus 1 time. Changing this duty cycle will allow you to tune anywhere between these two values, and that allows you to get a [inaudible] continuous tuning. In reality of course, you want to synchronize the switch with the clock frequency of these divisions because otherwise, all kinds of unpredictable situations might occur. There's a limit to how fast you can switch. That means that in reality, there's still a number of discreet steps that you can realize, but much closer than you could get with fully integer PLL. Of course there's a price you pay, and at prices that there's filtering of the loop filter is not perfect. Part of that switching will still be visible in frequency modulation of the output. You will see side-bands occurring in your output spectrum related to the switching between these two dividers. There's going to be some sparse in your synthesizer output. But if you design it carefully, you make a good loop filter, you can in fact, and people do in fact, make synthesizers that are tunable with very small steps and have fairly good phase noise across a wide band. But it's not perfect. I can already see your faces in my mind. You want more. Can't we do even better? Well, yes, you can. This is called a fractional synthesizer. But what if I would not switch this switch with constant duty cycle? At moment, what I do in a fractional synthesizer, for example, is put in 90 percent in one,10 in the other. But if I want to achieve that, I could make some variations. One time I might do 100 percent in one position, 80 percent in the other. Next time I do 90 and 80, then I do 85 and 95. As long as the average is the same, what happens then is I add some randomness to the disturbance of my oscillator. Rather than get discrete sparse, I get more of a noise-like increase of the noise flow, but without any discrete tones or peaks. The ideal way of doing that, the best way that we currently know, is to use something that we also use in data converters, which is essentially a very similar approach but then in the amplitude domain. We have Sigma Delta converters where we generate noise shapes imperfections in the reproduction of an input signal and amplitude domain. By shaping that noise carefully, we not only can make a random echo that is noise-like and therefore doesn't contain any peaks in the frequency spectrum. But we can even move to noise to places where we don't really serve off homemade so much, so usually far out of the way. That's something we can also do. I'm going to do that in green. We can add a Sigma Delta modulator to operate the switch to still get the average duty cycle that we're looking for, but now in a really carefully, noise shaped random way such that we move the noise as far away as possible. We have as small as possible discrete sparse, and we have a synthesizer that allows for very fine tuning with still cleaning of the noise. The phase noise can cause a large frequency range. These are the types of synthesizers the architectural said they would like to introduce for the classical synthesizer. The implementation of each of these synthesizers you also can do in different ways. Traditionally, synthesizers are done, people would say mostly analog. Because we have an analog VCO over there, we have an analog loop filter, the phase comparator. You can argue because as the input it takes two signals, but it really compares the phases and then gives an analog voltage out that represents the difference in phase. The input signals in amplitude, they do not matter so much. It's only the zero-crossings that count. So you could argue that it's not a traditional analog circuit. But then, if you really want to make a difference, we need to have a better characterization of what we call analog and digital. As a sidestep, I would like to offer you my preferred classification. If you talk about digital and analog, then a digital usually we mean discrete time and discrete amplitude, zeros and ones. Analog, we usually mean continuous time, we know the signal at any moment in time, and we also have continuous amplitude, and because we can have amplitudes that are not zeros and ones, but really any value that you can think of. But these are two extremes. You can also think of different combinations. For example, you have time-discrete. For example, what you do in switched capacitors, where the amplitude is continuous, but the time is discrete. You also can have circuits that are, let's call them amplitude-discrete, where you have continuous time, but where you only allowed a certain level of amplitudes, like binary signals that are time continuous. Going back to this, when you look, for example, at a divider, the amplitude is discreet, it's either one or a zero, but the time, the moment of zero-crossing can happen at any moment, it's not synchronized to any clock, so it's time continuous. This I would call an amplitude-discrete circuit. This is an analog circuit, because even though the input signals are represented in the time domain rather than in the voltage domain, that doesn't really matter. We can represent the signal by current, by voltage, by charge, by time, by any parameter we want, it's still an analog signal, or can be an analog signal, or it can be amplitude-discrete, or it can be time-discrete, or it can be digital. In this case, we have a signal coming in on both inputs. This signal it's the time that contains the information, but it's still continuous information in the time domain. Therefore, we have continuous time and we have continuous amplitude because the time can change continuously, not [inaudible]. So this I would consider to be also analog. This would be amplitude-discrete, and this will also be analog, so this is mostly analog. Of course, nowadays the methods for designing digital circuits are much nicer, there's better tools for that, the models are better in the sense that their predictions are more reliable, and that's because there's more margins in there. The fact that in analog, going back here, everything is continuous, that means that there's no redundancy. If your amplitude is a little bit off, then there's an error, if your time is little bit off, it's an error. In digital, since you have discrete time and discrete amplitude, if your amplitude is a little bit off, you can still round to the nearest amplitude and nothing happened. If your time is a little bit off, you can re-clock to an exact synchronized clock and nothing happens. There's redundancy in there, the fact that not all times are allowed, not all amplitudes are allowed, that allows you to do some kind of error correction because of that redundancy, because of that extra information that is not used to represent the signal. Things in between, the time-discrete and amplitude discrete, they have some redundancy, either in the time domain or in the amplitude domain. It would be nice to make synthesizers that are also more robust. You can do that, there's something that's called ADPLL, all-digital PLL. It's essentially the same thing. You have a reference signal then you have something that's comparing the phase, and then we have something that's a low pass filter, then we have something that's the DCO, and then we have a divider, and then we go back to the phase detector. But now we're going to try to implement a lot of things digitally. What we're going to do is we're going to do away with the regular implementation of the phase detector. Instead, what we're going to do is, we're going to use a time-to-digital converter TDC. What this thing does, it's like an HD converter, it looks at a time difference of the zero crossing of the FF and the divided oscillator segments, so the output of the divide by n. That time difference is represented as a digital signal at the output. You can make this, for example, by making change of inverters. You can imagine if you put a signal at the input, then that ripples through over time to the next inverter, next inverter, next inverter. If you then latch all the outputs of the inverters at the moment, at the second zero crossing happens. Then the output of the latch contains the information of the time difference quantize to delay of a signal inverter. That is one way to make a digital implementation of the phase detector. Although, whether it's truly digital, again, going back to our classification, it has analog inputs, and it has a digital output, so it's more like an HD converter, mixed signal. Then we have a digital filter. This can be 2D digital. Digital in, digital out. Just generates low-pass representation of the digital input signal. The oscillator takes the digital information and basically based on the digital information, it generates an analog output, generally is a sine wave. The way you would do that is similar to what we've discussed and the oscillator video. You can use switches and capacitors to tune an oscillator. There we discussed it in the context of allowing a larger tuning range by not just using a varicap, but using switched capacitors that you switch parallel or not to the very varicap to allow for a larger tuning range. But if you leave out the varicap, then you can use these switches and capacitors to make a PCO that can be digitally tuned to any number of discrete frequencies. That's what's happening here. That's the equivalent of D2i converter. Then we have divide by n, but that divide by n has discrete amplitude, but continuous time. So we would call that a discrete amplitudes circuit. As you can see, we have discrete amplitude, we have the equivalent of AD, mixed-singal circuits, the equivalent of DA, again, a mixed-signal circuit, and one truly digital circuit. You might argue that to name all digital PLL is maybe overstating things a little bit, but that doesn't take away that it is more robust. These mixed signal circuits, they still needs the attention that any mixed signal circuit needs if you want to make it high-performance, but at least the filter can be fully digital and automatically synthesized. Also, the divider, usually, with some care, can also be at least partially automatically generated for the frequency dividers at lower frequencies. This implementation form can be used for any of the architectural for an antigen, PLL. As entered your synthesizer for fractional insecticides and for sigma delta, but gives a more robust implementation if you're careful. With that, I would like to conclude this lecture on the basics of synthesizers. In the next lecture, we'll look at two different fundamental types of synthesizers and discuss their properties. Thanks for your attention and see you next time.