ISSCC 2014 Tutorial Transcription Filtering in RF Transceivers Instructor: Borivoje Nikolic

1. Introduction So what I plan to do today is to walk you through some of the concepts in filtering in RF transceivers. My basic goal is to go from some very basic concepts of analog circuit design and signal processing, and walk you through particular issues that are essential for RF transceivers. Filtering is the fundamental operation that we are performing in transceivers, and you are going to see how that is done in a classical way, and what are the new trends that are happening in that field that are represented in this conference. My goal of this tutorial is that as you go through the material and when you finish this material, if you do a little bit of homework, optional homework towards the end, you should be able to probe further by listening to five or six papers that are presented in this conference, and build up on that material. This tutorial will take you from relatively basics to the state-of-the-art in the design of RF transceivers. 2. Outline Here is how I’m planning to do that. First I’ll start with a basic overview of what we have. What are the basic standards, what are the requirements for RF transceivers? I’ll focus primarily on the RF receiver and the filtering chain inside the receiver. Then, I’ll cover baseband circuits – baseband filtering techniques based on active RC filters, Gm-C filters and switched-capacitor filters that are employed in RF receivers today. I’ll also then try to discuss, what does it take to move these filtering techniques up in the filtering chain closer to the antenna? What are the tradeoffs, what is possible, what is not and what are the research trends, illustrated through the design of an N-path filter? And finally I’ll conclude the talk with a little assignment to go follow particular papers in the conference. 3. Outline –RF Filtering Chain I’ll be talking about the RF filtering chain as the first topic of my presentation.

4. Today’s System: LTE Before I get into that, I’ll give you a quick, very rough overview of probably the most prevalent RF system today, which is LTE. LTE stands for Long Term Evolution, and that’s kind of a strange name, which the 3G community gave that as a name towards expansion/extension/ enhancements/evolution towards the 4th-generation cellular communication, and that name stuck. Some people will jokingly call it Long Term Employment as well. The system is based on OFDM modulation – that stands for orthogonal frequency-division multiplexing. This orthogonal frequency-division multiplexing is built on a concept of these subcarriers. Each subcarrier is approximately 15kHz wide, and can modulate subcarriers from QPSK to 64 QAM. We can take these subcarriers and build one resource block. What is called a resource block in LTE is a pack of 12 subcarriers. And then by putting together these resource blocks from 6 to exactly 100 of them, we can build channels. These channels can be wide from 1.4MHz to 20MHz. When we look into this system, there are 43 bands that are either presently allowed for its use in the world, or will be very soon released for use worldwide. There is a particular thing that comes with these bands that is associated with duplexing. Duplexing corresponds to the operation of the transmitter and the receiver, and this duplexing is implemented either through frequency-division duplexing or time-division duplexing. Frequency division duplexing means that the transmitter and the receiver are on at the same time, or that the transmission is happening in both the uplink and the downlink simultaneously at different frequencies. Uplink is the link from the terminal up to the base station; downlink is from the base station down to the terminal. In time-division duplexing, both transmitters are not on at the same time; however they share the same frequency band. So when one of them is transmitting, the other one is listening and vice versa. In LTE, we have both frequency-division duplexing and time-division duplexing, and that depends on the band and the geographical region where we are operating. There are slight differences in how these systems have to implemented, and how the filtering in the entire system has to be implemented for that to work. Of the other things to be mentioned, relevant to LTE systems, one of them is that they typically implement nowadays diversity, which means that there are always two radios listening on the handset side at the same time, in order to provide better coverage. So if one of them is in a radio fade, the other one hopefully is not, and it allows us to preserve the link. It does have to do something later with the budgeting, as I’ll mention briefly. Also, we are expected to support 2x2 MIMO, which means there’ll be 2 transmitters and 2 receivers operating on both sides, operating at the same time, in order to enhance the throughput. Furthermore, the things that we are starting to see deployed now and I am not going to cover to a great detail, are the concepts of carrier aggregation. We have different types of carrier aggregation deployed in these systems, going forward, and they can be quite complex. The

simplest way of that is channel bonding, where we will take two channels perhaps, two 10-MHz, or two 5-MHz channels, and bond them together to build a 10-MHz or 20-MHz channel. We can take two adjacent 20-MHz channels and build a 40-MHz channel to enhance the throughput. But, in the case of inter-band aggregation, we can take two non-adjacent channels and send stuff through them together. This is generally going to happen first on the downlink side, but is expected to happen on the uplink side. A particularly interesting case is the case of so-called inter-band aggregation, where we are supposed to be aggregating channels from two different bands that can be from several 10ths of MHz away, to several 100s of MHz away. Now those are interesting things that we are going to see addressed in the filtering requirements in the baseband, and also up in the receiver chain as we go forward. Again, I’m not going to cover that, I’m just going to hint to it. 5. Transceiver Design Problem When we talk about the basics of the receiver and transmitter design, it can be illustrated in this simple diagram that I have on this slide. The idea is somewhat reflecting a well-known near-far problem, where we are generally trying to demodulate and receive this desired signal that may be just slightly stronger than the noise, and the noise is this yellow stuff that is wideband on the bottom. Now, that relatively weak signal may be buried next to two fairly strong blockers that may be coming from nearby transmitters. In a particular case, these blockers by the standards that are nearby are relatively limited in power, but those that are farther away, may be stronger than that. The goal is to separate the signal from the blockers and the noise. 6. Practical Example: LTE A particular example of an LTE – I can try to put some numbers to illustrate how this looks like. Starting from the noise; the noise is set to be (and this is all referred to 1-MHz of channels, you know how to do the multiplication if you would like to see for example, a 10-MHz channel). If it is thermal noise, then the level of thermal noise at room temperature is -174dBm/Hz, where dBm is (if people are not familiar with) is power referenced to 1mW of power, and is expressed as 10log10 of power divided 1mW. If you are looking at how much noise falls in 1MHz of bandwidth, we have to take that -174dBm, add 60dB to it, and the noise level in a 1MHz noise channel is -114dBm. By the specification of the standard, the receiver is supposed to be able to receive some kind of a signal at 1MHz of bandwidth that is at a power level of -100dBm. At the same time, these blockers can be as strong as -15dBm, that are relatively close to it, so these blockers can be much, much, stronger – 85dB – 8 orders of magnitude stronger than what we have and what we would like to receive. At the same time, the biggest blocker may be our own transmitter in frequency-division duplexing system. Our own transmitter will be putting out something that is

close to an average power of 24dBm in a nearby band, and this spacing can be wide from a few hundred MHz to as low as 40MHz. Now these receivers and transmitters are expected to operate in a variety of bands that are sitting from about 450MHz to about 3.7GHz. Generally, this gross problem of frequency-division duplexing is alleviated and addressed through the use of very good mechanical filters that are built on SAW or FBAR technology. SAW stands for surface acoustic wave and FBAR are thin-bulk acoustic resonators. I’m going to mention those in a bit. 7. Detailed Practical Example: LTE 20 Just to look at an illustration – this is not something that needs to be memorized. This is an example of how the blocking profile looks in a 20MHz-wide LTE system. Generally, now, I’ve had this number at my sensitivity of the receiver at -100dBm. For 1MHz, that translates to -85dBm for a 20MHz-wide signal. The blocking profile is given here. The adjacent channel may be stronger, but not dramatically stronger by about 30dB. And then, it goes up by another 1dB two channels after that. Those are all systems that are sitting in the same band. Outside of the band, these blockers from another band can be much stronger, as strong as -15dBm at 87.5MHz of spacing, and as I mentioned before, our biggest blocker may be our own transmitter transmitting at 23dBm or 24dBm. Generally, we’ll be using a band-select filter to help us out with that. These band-select filters are going to at least be able to attenuate this transmitted signal to -25dBm or so. 8. RF Filters: FDD Speaking of that first filter that is there, on both the transmit and the receive path, which is generally implemented either as a SAW filter or as an FBAR. It’s a mechanical filter with very nice properties. It can sustain a very large signal without being fried, and it provides fairly good attenuation in the stopband, while not adding too much insertion loss in the desired band. So when we look at that, that duplexer is essentially implemented as two filters next to each other, whose response looks like what is shown in this graph. This is shown as insertion loss versus the frequency for the two bands, where we can place perhaps a transmitter and a receiver. What you will see here, is that the attenuation in the stopband is sitting somewhere between -45dB and -55dB, which is very good – it allows these transceivers to operate in a frequency-division mode. There are particularities about these filters that differentiate them by performance and price. They’re generally somewhat set by what we have in the stopband, but more importantly often is the insertion loss that we see here. So now notice that this is shown on a dB scale, and this looks very close to zero, but it is not a zero insertion loss – it adds a loss of about 1dB or 2dB there, and there is also some rounding – we have some more of a loss at the edges of the band.

9. Typical Cellular Transceiver (2013) When we look at a transceiver that is now in production (a cellphone of a 2013 vintage) that supports 2G/3G/LTE operation, it will generally cover about 5 to 10 bands. These bands can be spanning multiples of GHz, and there will be one dedicated duplexer per each band. So when we walk through the transmit chain, essentially the transmitter will be modulating the signal, and putting it through a PA and through the filter out to the antenna through a switch, and this single-pole 10-throw switch is there to select the appropriate band that we are operating in. This one is good enough for selecting 10 bands. Notice that this filter also helps the transmitter achieve the spectral mask, because we do want the transmitter to put very small amount of noise in the receive band, because the receiver is supposed to be operating at the same time and that noise has to be, (people will typically say) 6dB below the thermal noise in that band. On the receive side, the signal will be going from the antenna through the band-select switch, and then through the duplexer and through the receiver. And in general, we would have another set of these receivers, typically another 5 of them for diversity. They are there to help us switch. They are only there on the receive side and will pick that signal if the main signal is in the fade. 10. Software Radio? Now these filters are the main source of inflexibility. It is very hard to design a truly agile cellphone nowadays. When you buy a cellphone nowadays, it is region specific. So if you go and buy the latest and greatest cellphone, that cellphone will be very different if you buy it in the United States, or in Europe, or in Asia, and may not work well in other regions. The reason for that is these front-end filters are set to be region-specific, they are mechanical, and they are not flexible. But at least, you know lots of people are interested in these software radios, and assuming that we can somehow solve the duplexing issue, can we make this radio to be agile enough to operate in any band of the 43 and meet the specifications over there? One thing that people are discussing is a possibility of just on the receive side, duplexer being followed by the low noise amplifier, and then going straight into an analog-to-digital converter. On the transmit side, we would have a digital-to-analog converter going through an additional power amplification and then back through the duplexer. If you look at an LTE system, what we’ll find out is that this A/D converter would have to be able to sample all 3.7 GHz of bandwidth to satisfy the Nyquist rate, therefore the sampling rate would have to be somewhere about 7.4GS/s or greater, and the dynamic range should be larger than 80dB (perhaps even more than that), assuming that we would have about 20dB low noise amplifier gain and the blockers would not be greater than -10dBm. That is challenging, although you know it is perhaps possible to build that kind of an A/D converter, but that would burn way too much power. Luckily, we don’t have to do that.

At the same time, what we have to do on the transmit side, transmit DAC is also a very difficult - remember the requirement there is that outside the band, it should not be putting noise into the receive band, which is operating simultaneously at the same time, and that noise that is coming out of the transmitter has to be below the level of thermal noise there. But assuming that we could build some radio like this, that we can sample the entire band, we can just select the channel digitally. 11. Nyquist ADC is an Overdesign Now we don’t really have to do that, but if we build a Nyquist type of an A/D converter, that would mean that we would be converting the entire band. We would then be demodulating our desired signal, all of the blockers, and we’ll be receiving also all the other radio systems that exist from here, up to 3.7GHz. We don’t need that. We would like to filter these things out as we go along through the receive chain. 12. Direct Conversion Receiver So what we would like when we are trying to get this signal, generally what we do, it is very hard to build a filter that is operating at the RF frequencies, and we are going to discuss why is that hard, and then we are going to take a look at what are the other options? The other options are generally to translate this signal down to the baseband (close to the DC level), and then build an easier filter. This is a concept that is typically associated with direct conversion receivers, where we employ frequency translation to move the signal into the baseband, and then ease out the filter specification. 13. Direct Conversion Receiver So direct conversion receivers are, I’m assuming people are at least familiar with the concept of them, they will consist of a SAW filter on the receive side, followed by an LNA, and then there will be an LO that is either at the carrier frequency of the receive signal or very close to it. And then it will be going through a baseband filter and an A/D converter. Generally baseband filters and A/D converters are designed together because the requirements of the filtering and the dynamic range in these two systems are interchangeable. So depending on how much of a filtering we would like to employ, we would come up with residual requirements for the A/D converter itself. There is another concept that you will see in the papers that are presented at ISSCC where people instead of trying to build a Nyquist A/D converter, they will try to build an oversampled A/D converter. However, where its passband is made to be at the RF frequency, and essentially the A/D converter is only converting stuff that is sitting at around a particular channel. That involves a translation in the frequency of the loop filter inside that ADC and you will see an example of that in this ISSCC. It’s an alternate way of essentially building a direct conversion receiver because this conversion is happening inside the ADC.

14. Direct Conversion Receiver When we look at how does the direct conversion receiver in a classical form of a direct conversion receiver help ease the design of these filters, it can be generally appreciated through our diagram that I’m showing here in this slide. What we have here is a profile that we would generally see on the receive side. We’ll have our blocker that is close to 0dBm - it is slightly negative, for example. Then the desired signal that is sitting pretty low, and in the worst case, is sitting close to the sensitivity level. However, this desired signal maybe varying quite a bit in its receive power, depending on how far or how close we are to the transmitter. And then there is a noise floor. As we move that signal through the SAW filter, through the first front-end filter, it will provide quite a bit of attenuation out of band. It will knock these out-of-band blockers along with our unwanted transmit signal by about 45dB, 50dB, 55dB. At the same time, due to the insertion loss, it will attenuate the desired signal a bit. Then we’ll go through an LNA. LNA is generally of a wide enough bandwidth that we’ll be amplifying both the blockers and the desired signal and the noise. In addition to amplifying the noise, it will add noise on its own that is generally quantified through the LNA noise figure. So it will add a little bit more noise - it will have a little bit worse signal-to-noise ratio than when we started off, but that’s the feature of the low noise amplifier. And although it is a low noise amplifier, it adds a low amount of noise, but still, a finite amount of noise. Then we’ll go through a frequency translation that will generally be done with very little loss, but will add a little bit more noise, and then we’ll end up going through the baseband filtering and A/D conversion. The role of baseband filters, because they are operating at lower frequency, they can significantly attenuate the blockers while amplifying the desired signal and adding a little bit of noise, such that we present a desired signal and a blocker together to the ADC, and the ADC can separate the signal with high enough signal to noise ratio from the blockers and the noise. 15. A/D Converters Specifications When we try to specify an A/D converter – that is something that is very important for the design of the filters and the entire filtering chain. There are two things that are essential for the A/D specification. One of them is the bandwidth here - how much of a bandwidth we would like to sample with this A/D converter, and then how much of a dynamic range - how much of a signal-to-noise ratio we would like to get? So what we’ll see here is: generally there will be some amount of noise and we would set our A/D quantization to be somewhere over there comparable to it. And then, there is some minimum amount of signal-to-noise that is needed to demodulate the signal. For a particular modulation, the higher the constellation of the signal going from QPSK to 64 QAM, we’ll need more and more of a signal-to-noise ratio to successfully demodulate this signal with low enough bit error rate. We need to add that and then we need to start adding various kinds of margins to this design.

What you will see, is that these margins correspond to the peak-to-average power ratio, which is something that is very important in orthogonal frequency-division multiplexing (OFDM ) signals, like LTE or wireless LAN systems, and then various kinds of other margin of how much of left over blockers are in there, and if these blockers turn out to be stronger than our signal, we have to be able to process them, otherwise the ADC will be saturated by the blocker and not by our signal. 16. A/D Converter for LTE So if we are looking at what people are now specifying what is needed for LTE, they will be typically kicking around this range of about 60-70dB, and depending on who you talk to, you will get a little bit different numbers. These numbers are varying because of the fact that filters are implemented in different ways in different specifications. So if you implement a looser filter, you may need a higher dynamic range A/D converter. If you have a better filter, you can get away with a lower resolution A/D converter. These two things can be traded off for each other, and the key thing is, each one of them costs something - they cost power. Depending on the implementation, you may choose to burn your power budget in one or the other place. What you will see is that they will say that you need 36dB to demodulate a 64QAM signal with low bit error rate. That is much more - that is at least 10dB more than what the theory says, and this accounts for in-band fading and diversity, and the ability to switch between two diversity receivers at least. General margin for peak-to-average power ratio in these systems is about 10dB. It needs to be added there. And then ADC has to have some operating margin of 6-10dB. And then to account for example, the fading and automatic gain control margins, we need to add, depending on what you have and what you need, 5-15dB. When you total all of that, you end up with 60-70dB. One thing that I would like to highlight here is 5-15dB automatic gain control margin. It is one of the very interesting systems that exists in typical receivers. When we find out what we are trying to demodulate, we generally estimate the signal level that we are seeing in the previous frame. For what we need to do, we set up our receiver for the next frame, and the automatic gain control is set to accommodate that, based on the previous frame measurements generally, or some kind of training sequence. Now things in the channel may change between the previous frame and the current frame, or between the preamble and the current design, and the ADC should not saturate because of that. That’s why we have to have that margin. 17. Filtering Requirements Now when you look at that, what kind of a filter do we need with a 70dB ADC? Well, we can have a little bit looser ADC, and if you try to fit these filters over here to accommodate in-band blockers, you’ll find out that you would need to attenuate about 10dB per octave if you have that big (about 70dB) of an ADC. That should tell you that it is good enough to have about a 2nd or 3rd order filter.

Now for a 60dB ADC, you will need at least 20dB per octave, and you would need a 4th or 5th order filter that we precedes that ADC. These are just illustrations of what we need to do. Again, these filters are not enormous and very complex, but there is a tradeoff between their order and dynamic range of an A/D converter, and we know how to calculate how much power it costs us to add dynamic range to the A/D converter, and we can use that to judge which kind of a filter we would like. 18. Outline A large chunk of this presentation will be on some of these baseband filtering techniques. 19. Baseband Filtering First, I’m going to mention the different types that we have. In general, these filters can be built as active RC: they’re built on passive resistors, capacitors, and opamps; Gm-C: where you really do not need a full blown opamp - we can get away with just building a transconductor and a capacitance at the output; and then switched-capacitor filters that are built out of switches and capacitances. These have different intrinsic properties, and all of them are good enough to build 1st order filters, 2nd order filters, and higher order filters that are needed for these kinds of systems. 20. Filter Specification So, I’ll dive into them now, but before I dive into them, I’ll talk a little bit about the requirements for the filters, and I believe that many people in the audience are familiar with the ways of how we specify these filters, but in general, there is a number of specifications that are given in the filter design - some of them are relevant to the receivers, some of them may not be that relevant. Generally, they will be specified by the width of the passband - up to which frequency we should have relatively flat response and where our signal would be that we would like to receive. What would be the stopband, and how much of attenuation we would like to have in the stopband? At the same time, we would like to specify how much of a loss we would like to have in the passband. And then, there may be specifications about the amount of ripple in both the passband and the stopband, and how quickly can we transition from the passband to the stopband? There are a whole bunch of filter types that are taught in school, which you’ll find out. There will be a Butterworth filter that is generally the filter response that has the flattest possible response in the passband, or there is a Chebyshev filter which has this equi-ripple property, but a much more interesting property for RF receivers is that it has maximum steepness in the attenuation region, and that’s why you’re going to see it very often implemented as 2nd-5th order. Chebyshev filter is what you’re going to see people mention in their papers at ISSCC. There are others: Elliptic or Bessel filters that can be used.

What I am mentioning here is that we cannot have a brick wall filter. The brick wall filter will cost this infinite amount of power - the same amount of power as an infinite SNR A/D converter, and there are tradeoffs here in what can be done with these filters. Generally, as a result, we will be receiving some of the blocker power from the adjacent and other channels, and we would like to specify how much of that blocker power does leak into our A/D and what does A/D need to deal with? 21. Outline Getting into filter topologies. 22. Active RC Filters The first one are active RC filters that you will see sometimes, and I’ll introduce how they work. They are relatively simple to understand for anybody who has basic background in IC design. Generally, they’re implemented using a resistor and then a capacitor in the feedback loop around an opamp. We can calculate the output voltage as a function of the input voltage for each one of them, and if you like to do that, you can do that by using simple superposition. You’ll find the Vout is equal to (-1/SR1C)*Vin1, and then if you turn off Vin1 and turn on Vin2, you’ll find out the response there is also a (-1/SR2C)*Vin2. This is an integrator response. What you got with this circuit is an integrator function and a summing function, which is very convenient for building filters, and these things are generally implemented fully differential. These resistors can be passive resistors, but sometimes can be implemented as transistors in the triode region. One thing to be kept in mind is that these filters need to drive a resistance. These opamps need to drive a resistive load, in general, because there will be a cascade of multiples of them and therefore we need a full-blown opamp, with relatively low output resistance. A great discussion about these filters is for example in Gregorian and Temes’ book, from 1986 that I put in the references. 23. Active RC Filters If we are trying to build a higher order filter, generally, what the textbooks will tell us is, they’ll teach us a theory of biquads, and biquads are these 2nd order building blocks that have a transfer function that is reasonably well defined. You can derive it into a canonical form, and I’ll come back to that in a second. I’m trying to add things in layers here. If you try to look at this filter, and it’s relatively easy to analyze this quadratic function that we have here, we can easily find out what is the Vout as a function of Vin, and we’ll find out that it is of the form that is shown here.

It has two poles and no zeros, so it is a 2nd order quadratic function, which is a typical thing for a biquad to be a 2nd order thing by definition. In the denominator, we will see this canonical form of s2, plus s times something, plus 1 over something in the end here without an s term. That something is generally, when brought in this canonical form, is ω0

2, which is the corner frequency. In this case, it is set by 1 over the square root of R1R2C1C2. By dividing the appropriate terms here, in the middle term we’ll find out what is the Q factor of this filter, and it is set here. What we’ll find out is a characteristic of all of these filters is that while the Q is set by the ratio of various resistors and capacitors, ω is set as a product. In integrated circuit technology, we can control the ratios of capacitors and resistors extremely well - easily to 1% with a little bit of an effort to 0.1%, therefore the Q can be set very precisely, but the bandwidth cannot, and generally what we will see here is in order to set the accurate corner frequency, we need to have a method to either trim or tune these filters. And as I mentioned before, opamps have to be able to drive resistive loads. 24. Bandpass Σ∆ ADC It’s a matter of preference if you would like to implement this kind of a filter - you have to be able to tune it and calibrate it. You’ll find out there are papers in ISSCC that will use this kind of a filter often in a bandpass Σ∆ ADC. Those are complex designs often used for example in base station applications, where this A/D converter that follows the LNA is of a bandpass type, and essentially, what you will have there is some kind of a bandpass filter put into a Σ∆ loop. There is a quantizer and a DAC that is subtracting the error signal from the input, and this bandpass filter has to have an appropriate response to take out our desired signal. An example of that is the Eglund paper from ISSCC this year. 25. Bandpass ∆Σ ADC The way that these filters can be implemented - and I’m not going to try to describe this in gory detail - but generally, what you’ll find out in a paper from a couple of years ago at ISSCC, this filter is implemented as an active RC filter in there, and in order to make sure that they can calibrate and tune it properly, they’ve used in every single filter section, for every opamp and every element, one little DAC. That’s what it takes to do that. Now the power budget for this design was fairly large, and intended operation was for base stations and they can tolerate it a little bit more. 26. Outline Moving on to the next very popular type of baseband filters: Gm-C filters. They are also known as transconductor-capacitance filters.

27. Gm-C Filters Their basic operation is described like this. The topology is very simple and similar to what we had before. The input is passed through a transconductor, converted to a current that is proportional to the transconductance times Vin, and then is fed into a capacitor. This Vout is equal to -GmVin/sC. You can see the equivalent circuit over here to help you understand how that thing operates. By looking at it, I can see that again, I have a 1st order transfer function here, and I can implement easily an integrator and a summer by putting two of these together. Again, often implemented fully differentially, you can implement it simply by using diff pairs - the transconductors can be implemented as diff pairs. And what we have here is a multi-input Gm-C integrator, where the output signal is equal to the negative sum of the two transconductances multiplying the two inputs. 28. Gm-C Biquad I can use that again to build a biquad, and here is an example again, of a lowpass biquad, that is shown. You can go through an exercise of analyzing it. What does it take to build this kind of a transfer function? But again you’ll find out two properties: the Q is set as the ratio of Gms and capacitances and ω0 is a function of the products. What you’ll find out is that the other nice property is that DC gain and the Q ratio are going to turn out to be ratio dependent, and ω0 depends on absolute values. Therefore, tuning is often necessary for these filters if we want to have an accurate corner frequency. Now, to talk about tuning, it would require an entire whole tutorial, and I’ll just refer you to various literature towards the end. 29. Gm-C Filters One example of a good literature piece is the Laker and Sansen textbook on analog integrated circuits. 30. Higher-Order Gm-C Filters If I would like to build a higher order filter, there are different ways how we can do that. One of them is just by putting together these biquads, and I can get a higher order function, I can get the 4th order function by putting two biquads together. These functions can be of a different kind, in a canonical form. We can have a lowpass here that would have a transfer function like that, or a bandpass would look like that. Generally, what we will see is that the biquad is essentially a 2nd order transfer function - it has a 12dB per octave attenuation in the stopband. If we need more than 12dB, we would need to add more biquads.

And the theory of that is relatively well-developed, and often this is something that would be done through a MATLAB exercise. MATLAB has pretty good filter synthesis functions that will give you the coefficients of the biquads that you need. 31. Gyrator There are other ways how you can synthesize these filters and they are probably a little bit easier to understand, or visually more appealing to see what is exactly happening there. The way how you do that (this is a technique that I learned from Bram Nauta) is a technique that is based on gyrators. The gyrator is an interesting network element that was I think invented a long time ago - 60 years ago by Tellegen, who was a circuit theoretician, that has a transfer matrix that looks like this: v1 is proportional to i2 and v2 is proportional to i1. And as a result, what you have there is if you put an impedance at the output of that network (that gyrator), what we find out is that the input impedance seen there is the inverse of the impedance that we have at the output. It’s a very easy exercise to run through. 32. Gyrator So if we put a capacitor at the output of a gyrator, what we see from the input port is an inductor. Since inductors are not convenient to be implemented in integrated circuit technology - they take quite a bit of area, it’s much easier to implement capacitors, and this is a very effective way to make inductors out of capacitors. Now how complicated is it to design a gyrator? Well it’s not very complicated either. You can again go through an exercise here and verify that probably the easiest way to build a gyrator is given by putting two capacitors into a feedback form and setting their transconductances to be equal to 1/r, where r is equal to 1/Gm, and then capacitance here is set to be Gm

2L. Then L, the inductance, is going to be equal to r2C. So this is a way to build an inductor. 33. Bandpass Gm-C Biquad Then what we can do (and there are simple rules of doing that), we can design a passive filter of a form that we like, and what I’ve said, in cellular systems or in wireless transceivers in general, we don’t need very complex filters. We can get away with something that is between 3rd order and 5th order, and these 3rd order to 5th order filters can be reasonably well visualized. You can try to design one of these using a passive network, and again you can use MATLAB to help you design a passive filter. For example, if it is just a 2nd order filter, it would look like this: R, C, L. And then what you can do by using an appropriate transformations, we can transform each of the resistances into transconductances, inductances into equivalent gyrators, and capacitances we would like (especially if they are grounded) to leave them as they are.

Here is an example of how this transformation can be done for an RLC network, over here, where the L is the same as what we have had in the previous slide. There is a great example of that in the rules in this Nauta’s book, which still does exist online on Springer’s website. 34. Gm-C Filter at RF? The next thing that I would like to discuss now since we have mastered the design of baseband filters, and we can design them either for now as active RC networks or Gm-C networks. We’ll see that we can design them all as a switched-capacitor networks as well in a bit. But, if we can do that, can we just take this filter and move it up closer to the input of the receiver such that we do not have to downconvert this signal and make a much more flexible receiver? Can we build just an ideal Gm-C filter at the input? 35. Noise in Gm-C Filters In order to do that, to understand what and what cannot be done, we need to add this thing about noise. Noise exists in all the filters, and what I’ve mentioned earlier on, is that this noise adds to the overall noise budget in the receiver. Noise exists in the active RC filters and exists in the Gm-C filters. We just start with a thermal noise. The way this thermal noise is modeled is if you have a noisy transconductor, a noiseless transconductor can be very easily shown that the real transconductor can be modelled as a noiseless transconductor and a noise source. This noise source has a noise current that is equal to 4kTγ, where γ is the noise factor times Gm times the frequency. It has a wideband noise and we only integrate that noise in the frequency of interest ∆f. 36. Noise in Gm-C Biquads So when we would like to find out what is the effect of the noise of each of these transconductors to the output of the filter, we need to find out what is the noise transfer function. We need to find out what is the contribution of each of these transconductors, and find out how do they transfer to the output. What you’ll find out if you go through that exercise is that the output noise power is proportional to 4kT/C times the quality factor of this filter, times the noise factor, and then the rest turns out to be exactly the same as the group delay of this filter. The group delay is essentially a derivative of the frequency response of the filter. Which means that if you would like to have a filter with a high Q factor, which is desirable, that means we’ll have a high noise peak as well - presenting one of the basic tradeoffs that we have there. 37. Noise in Gm-C Filters

Now we can go through a little bit more of an analysis, taken straight from Bram Nauta’s book. This is the exact same expressions as I’ve had in the previous slide, and I can try to integrate it in the band of interest here. I can integrate it to infinity because there is some attenuation in this filter, and I can find out that this noise essentially is 2kT/C*Q times the noise factor. If I try to find out how does that affect my power, I would like to find out: how do I add the power to this whole system? In general, my power is set by the ratio of the signal power to the noise power, and by substituting the output noise, I’ll find out that the SNR at the output of this filter is given by the ratio of v2max, and the noise power given in the previous expression. The power dissipation is generally proportional to the Gm. The more Gm we would like, the more current we need to pump into a transconductor, and therefore the power is proportional to the sum of the transconductances that we have there. ω0 is set by gm/C. 38. Gm-C Fundamental Limitation This gives us this very nice, interesting, and exact expression, that the power is directly proportional to the quality factor of the filter and the signal-to-noise ratio, and then also proportional to some of the transconductor properties, which means simply that the larger the SNR we would like, the higher the center frequency and the higher the Q factor we have to pay for each of those premium in power. This says that in order to keep the power for given requirements, we need to keep the frequency response of these filters as low-pass in a limited bandwidth. It is possible to make them run at very high frequencies; it’s easy to design this kind of filter to run at 10GHz, but it is going to cost us a lot of power if we would like to maintain the signal-to-noise ratio and the quality factor. So we have to have different solutions for that. 39. Outline The next that I’ll try to address are the switched-capacitor filters, so let’s get into them. 40. Switched-Capacitor Fundamentals A switched-capacitor filter is something that is very popular for implementation of various signal processing and filtering techniques. The way this thing operates is generally there is a capacitor and there are two switches around it. These two switches are operating in a non-overlapping fashion; there is a first phase and a second phase. In the first phase, we put some charge on that capacitor, and then in the second phase, we take some charge from that capacitor and put it at the output. The amount of charge that is transferred from the input to the output in each of these cycles that consist of two phases, is proportional to the capacitance times the voltage difference. And you can calculate what the current here, and that current is going to turn out to be proportional to the sampling frequency, the capacitance, and the voltage difference; and you can see that if you look

at it, it looks like a resistor when looking from the input to the output, but with an equivalent resistance, whose value is 1/fsC. So this looks like a resistor and it’s also an adaptable resistor, whose resistance I can change by changing the sampling frequency or operating frequency of this design. 41. Low-Pass SC Filter I can use that resistor to build a simple 1st-order RC filter. The way how we do that is: we take a resistor, put it together with a capacitor, replace this resistor (in the same way how we have been doing before) with a switched-capacitor network that has this C1 and we will find out that ω3dB is equal to 1/RC - in this case 1/ReqC2, and if you just plug in what I’ve had in the previous slide, you will see that it is equal to fsC1/C2. So this is very nice, this is a very desirable property. We have a corner frequency now set by the ratio of C1 and C2, and the sampling frequency. So I can adjust the filter by changing the sampling frequency, for example. Just as a little side note, you can easily figure out that the quality factor of a filter like this is not going to be capacitance ratio driven. You can’t get everything, but there are ways you can get around that as well. 42. Charge-transfer analysis A quick analysis of how these filters look like - this is what I found out is a very easy way of figuring out how they work. This is a little bit of a complicated slide (probably could have used an animation) but let’s walk through it. This is very simply described here. So what we typically do is we separate the operation of this filter to phases, and what is going to be of interest for me, will be: what happens at the onset of phase one, what happens at the onset of phase two which is a half into the clock cycle, and then what happens when I fully switch back a whole cycle later. So at the onset of phase one at time instant n, what I have is: φ2 is open and φ1 closes, therefore C1 samples the input voltage Vin. I can draw a Gaussian surface around this, but it’s easy to see the amount of charge on the capacitor C1 is equal to Q1, which is C1*Vin at the time instant nTs. I’ll call that (I’ll use a shorthand notation) that is equal to Vin(n). Then there is a little typo in your notes, you may want to change this over here. At half into the clock cycle when φ1 and φ2 change, C1 transfers charge to C2 – essentially, φ1 is open and φ2 is closed, therefore these two capacitances redistribute the charge between them until the potentials are equalized. So, the sum of the charge that is going to be at these two capacitances on this Gaussian surface at time (n+1/2), has to be equal to the sum of the charges that were there before the switches changed their polarities. I can substitute the values that we have had before, where each of the charges are equal to the capacitance times the voltage ratio. At the next time instant, when we open φ2 and close φ1, the charge again has to stay constant on C2, and Q2 at (n+1) has to be equal to Q2 at (n+1/2), and from there I can derive what is the

dependence of Vout at (n+1) as a function of Vin and Vout at n. These are simple capacitor transfer functions. Now what I have here - it’s a linear dependence on the input voltage and the output voltage, but in the previous time instant. There is an inherent memory function added here, and this filter, therefore has memory. 43. Transfer Function The expression is rewritten here, and this memory effect can be easily seen for example in the z-domain, where one cycle delay corresponds to the factor of z-1. You can see that the transfer function here in the Z-domain is equal to (C1/(C1+C2))*z-1/(1-(C2/(C1+C2)z-1). When you look at that transfer function, if you’re familiar with things in the z-domain, even if you’re rusty, you’ll recognize it as a 1st-order transfer function z-domain with 1 unit delay with a pole whose value is C2/(C1+C2), and the DC gain at H(z=1) corresponds with a DC gain of 1. 44. Sampled Data Domain Just a quick note: switched-capacitors operate in a sampled domain - they operate with sampled signals, and if I sample a signal at uniform time instances Ts, 2 Ts, 3Ts, 4Ts, 5Ts, 6Ts, and so on, then in the frequency domain, I’m going to get some filtering function and then I’m going to get images of this spectrum that are repeated at every multiple of fs. The z-transform is essentially a sum of these samples x(n), multiplied to the z-n going from -∞ to +∞. One thing here, if I do not want to have this aliasing of everything that is happening at the multiples of fs and so on, I would need to have some anti-aliasing function in front. I’ll come to that in about a second. 45. Switched-Capacitor Noise But since we talked about the noise in Gm-C filters, and we have found out that Gm-C filter noise is essentially set by the transconductor noise, and the transconductor noise has this nice expression for it. Let’s take a look at what is the switched-capacitor noise? The noise in switched-capacitor filters comes from the switches themselves - they do have some kind of resistance and these capacitor samples that charge them. So what we’ll find out is that each of these switches produces some amount of noise that is sampled onto C at the end of the corresponding phase, and then it is sampled with a noise variance of kT/C. Therefore, I get this noise charge packet that is q = kTC. q2 should be here if I want to look into the power. So also S2 noise is going to be sampled on C2, and since we are assuming a wideband noise, then these two packets will be uncorrelated and the noise charge transferred is equal to 2kTC. We can work through this expression, (what is the noise current and power spectral density?), and what will turn out is this very nice property that the power spectral density of this noise is going to be equal to 4kTCfs, which will in turn be the same as for a physical resistor with an equivalent

resistance 1/fsC. So the noise analysis of switched-capacitors filters (as a bottom line of this discussion) is the same as the noise analysis of standard RC networks. 46. SC Integrator We can build integrators out of these filters. Simply, I can build the same integrator as I’ve built with an active RC network, by replacing my resistor that I’ve had before with a switched-capacitor. Then it would be feeding through this capacitor placed into the feedback around an opamp. What is interesting here is, you do not have to have a full-blown opamp - and you can get away with a transconductor because we do not need to drive resistances in general. A transfer function as given in this expression, is (again by looking at what I’ve shown in the previous slide) easy to derive. Some things to keep in mind: these switches are implemented as MOS transistors, and each of these MOS transistors has parasitic capacitances associated with it, and each of these parasitic capacitances is nonlinear. That affects the fidelity of the charge that is being processed here. Generally, it will cause a nonlinear dependence on Vin, and that is undesirable. That is why this topology is called a parasitic-sensitive switched-capacitor integrator. There are ways to build them to be insensitive to this bottom plate capacitance. That’s a topic on its own, and I think that was a tutorial at ISSCC about 10 years ago, so you can look that up. But essentially, it involves using twice as many switches as we want, to additionally be able to reset these capacitances to make sure the bottom plate capacitances are always reset to ground. Again, this is a well-known and developed technique. 47. RF Receiver So how does this switched-capacitor filter fit into a receiver then? They became quite popular in receivers about 10 years ago, and some people have had the preference to use them. One of the interesting things here to wonder about is the anti-aliasing requirements. Do I need to build a separate filter in order to provide anti-aliasing, such that I do not have these images repeated at fs’? 48. Passive Mixer as an AA Filter Well, the good news is this comes for free from the receiver architecture. Anti-aliasing filter can be provided by the mixer itself. What we have here, is an example that I’ve shown, where we have a transconductor driving a mixer. I’m showing a single-balanced version, just shown for simplicity, where we are sampling the input to this capacitor here and the mixer has two phases. The mixer output is a sampled-and-held input signal that also has a very nice property that is prefiltered through this sinc type of anti-aliasing function.

So what is happening here? We take our RF signal, process it through a transconductance, through a Gm, and produce an output current that is shown here. But at the same time, we can look at this in the charge domain, where we are producing out of that, a charge packet that is being switched onto this capacitor CH. And this charge packet is essentially being added onto the charge that is on top of that capacitor in each sampling instant. What I have there is an inherent integrating function, and the capacitor is keeping a running sum of the input that is coming through the mixer. As a result, I get this nice integrating response, which gives us an appropriate anti-aliasing function - not perfect, but good enough to process the signal further. 49. Lossy Discrete-Time Integrator The next thing that we need to introduce in order to understand these switched-capacitor filters used in receivers, is the notion a lossy integrator. Why do I want a lossy integrator? Well there are several reasons, but the one that is the most important is going to become apparent from the next slide. But in this case, the way I can build this lossy integrator - this is what I’ve had: this is my mixer dumping charge on this large capacitor that essentially a function of a loseless discrete-time integrator. Then I can take that charge and transfer onto a smaller capacitor by using this phase 1, and then I reset that capacitor in phase 2, and I lose that charge - that’s why it is lossy. What does this do? Well it sets the DC gain, and you can calculate that the DC gain and set by the ratio of CH and CR. In addition to that, it sets my bandwidth of the integrator. 50. Cascade of Two Integrators But more importantly, what it does is it provides us a way to cascade multiple integrators. One of the challenges of building these switched-capacitor integrators is in cascading them. My information here is stored in a charge. In order to cascade that charge, I have to take the voltage as a measure of that charge, and perhaps pass it through a transconductor or an opamp in order to cascade things. Well, if I would like to move these filters to higher frequencies, I cannot afford to have an active element there. Here is a way to do that by using that revolving (small) capacitor; it allows me to move the charge from this node to the next, and process it through another stage of switched-capacitor filtering. I’ve added loss, and if CR/CH is relatively small, I have a relatively small loss. This is essentially manipulating this through phases 1, 2, and then resetting in phase 3, to allow me to move the charge from here to there, and then onto CH2 and then add a 2nd-order filtering response. Now, I have to reset this capacitor periodically, or after every time I’ve transferred the charge such that I do not add the memory to the whole system. But the transfer function is shown here. I get the 2nd-order responses where this α1 is set by the ratio of CR and CH. 51. Higher-Order Filter

I can build a higher-order filter by doing this. I just need to do that with n consecutive phases and move this charge once stored (sampled) on CH - I can move it to CR, and then CR can feed it to CH2, CH3, CHn and in theory I can build an infinite-order IIR filter out of that. That filter can be programmable, and you can look at that how Tohidian and Staszewski have done that in last year’s ISSCC and this year’s ISSCC - a very effective and elegant way of building these filters. Also, a little homework question, what is a disadvantage of this? You may want to discuss that with the authors as well. 52. SC FIR Filter We can build FIR filters, and FIR filters in switched-capacitors are built in a very easy way; and I don’t have to spend that much time on them. The way that they are built is that we have this transconductance that converts the input RF signal into the voltage, and I can sample that signal in parallel on several banks of capacitors. I can do that with multiple phases of the LO, for example, in this case it’s shown as four phases, and inside of each of these capacitor banks, I can have different capacitance ratios. The way that I can form the FIR transfer function, is then visually obvious from here. I can take any of these values that will be storing a different value - some quantity that is proportional to the input voltage, and then I can sum them together to form the FIR response of my liking. 53. Add a Decimation Filter This is a 4-tap example shown here. In general, when we are doing this switched-capacitance type of processing, I don’t want to present the sample rate that is corresponds to the RF signal to the A/D converter, and there is a need to decimate it. You can do that by using this combination of IIR and FIR filters, where the FIR filter is used as inherent decimator. What we have here, we again sample our signal, our charge packet on CH1, transfer it to CR1; at the same time, we build a second copy of that for the 2nd tap of the FIR on CR2, and then transfer both of those in φ2 to another capacitor CH2. This lowers the sampling rate by a half, as you can see here, φ1A and φ1B are tied to the local oscillator, and then φ2 and φ3 are divided by 2. This essentially gives me a transfer function as shown down here with built-in 2-tap FIR, and intrinsic decimation by 2. I can continue doing that as long as I like. 54. GSM Example An example of that is how TI built a GSM filter. This is a relatively simplified form/drawing of that FIR filter. You have the exact same structure as I’ve shown before. LO samples on this first holding capacitor CH1, and then that is transferred into two phases to CR1 and CR2, to these two revolving/rotating capacitors, and then FIR summation to CH2, and then continuing at a lower rate with another IIR filter.

55. GSM Filter Response In composite, they built a transfer function that looks like that, and a frequency response that looks like this. This is a composite: the black line is a composite of frequency responses of all these. What we have here, the HP is the first decimating filter (the first output of the LO), and then there are two frequency responses overlaid on top of it. What you can see here by setting the sampling frequencies and capacitance ratios, you can place the nulls and attenuation points where you have significant blockers that you would like to get rid of. 56. Bandpass Filtering Another way we can use a bandpass switched-capacitor filter is inside a bandpass or down-converting ∆Σ modulator. This is a very effective way. If you would like to build a bandpass ∆Σ, the way how we do that is by building a band-pass response inside, and we will build a bandpass filter in there by using the same principle: we used in LO to sample on CH1, and then we move over to this rotating capacitor, and then from the rotating capacitor, we add another 2nd order filter. So what we have here inside is a 2nd-order switched-capacitor filter embedded inside of a 2nd-order loop, and there are several instances of how this has been built as a receiver. 57. Outline The next thing that I’m going to talk relatively briefly about is another way of moving the switched-capacitor filters very close to the antenna. 58. N-Path Filtering: Basic Idea These are the N-path filters and here is the basic idea. I can have a large bank of filters that are sampled in frequency with these very simple switched-capacitor networks, just samplers (S11, S12), that are staggered in a very similar way to what we have seen in the previous slide. I can build these filters in there to be low-pass filters, and effectively by eliminating this second switch at the output, I can show that the frequency response of this filter looking in from the RF port, is a band-pass function. It’s an easy transformation that you can do based on the equations that I’ve shown before. The way that you can see that; this is a very old technique that was invented 50-something years ago, and did not have really good use so far, but has become popular in RF receivers as a way to move the filtering very close to the antenna. 59. Lowpass to Bandpass The way how this can be done, is again, through filter transformations. I have a 1st-order lowpass here that I would like to build as a 1st-order N-path bandpass. I set my sum of my capacitors

here - all of them are equal and small to be equal to N*C. As a result, the input transfer function can be shown to be a bandpass around ωLO. 60. Higher-order N-Path BPFs The way how people have used that is to build N-path filters as these parallel LC tanks and synthesize higher-order bandpass filters. Again, it is not good to design a 1st-order filter at a high frequency - you need a higher-order filter. I need perhaps, in order to build this as a real receiver, 7th or 8th order at least. Here is an example of how you can build a 6th-order filter that was shown in last year’s ISSCC. You can start to build an all-pole 6th-order bandpass filter built out of passive elements (Ls and Cs), and then you can use gyrators to convert series LC to parallel using a transformation that exists in Nauta’s book that I’ve shown. What you end up with here are the inductors and capacitors shorted to ground with some gyrators in between. 61. Higher-order N-Path BPFs As a result, by converting these LC bandpass filters into N-path filters by putting a series of capacitors and appropriate switches, gyrators in between, what Darvishi and Nauta have done last year (and it was well described in this JSSC paper - special issue from ISSCC last year), is a tunable bandpass filter that works from 0.1GHz-1.2GHz. And it is worthy, or almost worthy of being a front-end filter of a receiver. 62. Conclusion So what I’ve covered so far is a number of techniques that we can use for filtering inside of RF receivers. What still stands is that the front-end band select filters based on SAW filters or FBAR filters are indispensable. We do not have any other technology known that has that kind of selectivity and immunity to very high signal levels that can be coming out of our transmitter, that can be built electronically to replace these (essentially) mechanical filters. But they are a major cause of inflexibility in the receivers, and there is a lot of demand to replace them with some electronic, fully-tunable filters. What I’ve talked about are Gm-C and active RC filters that are used in the basebands. I’ve shown that they fundamentally cannot be moved to the RF frequency without burning too much power. But the switched-capacitor filters and N-path filters (a particular instance of switched-capacitor filters) can; however, there is still a challenge of their linearity when they’re exposed to very high input signal levels. And what we’re seeing reported at ISSCC for example this year and in previous years, is the march towards building these flexible filters there. What can change this picture in going forward? Can we build these N-path filters or just basic switched-capacitor filters to be immune to the transmitter power? It’s unclear and we need advances over there.

There are things where if we can figure out how to build a flexible transmit filter - maybe they can be used, or maybe tunable MEMS, which we have had as evening panels at ISSCC - maybe they’ll come to maturity and help us build all of this. 63. To Probe Further But the point of this tutorial was not to discuss that. The point of the tutorial was to take you from basic knowledge of signal processing and analog circuit design, and bring you up to the state-of-the art. And if you would like to really get the maximum out of this tutorial, if it was useful for you, I would try to go through these analysis techniques that I’ve shown, get the proceedings and try to analyze a few of these papers. Particularly interesting ones are the first ones that I’m mentioning here for example: Tohidian, Madadi, and Staszewski, you can analyze directly by applying what I’ve shown to you. And there are a few others that you can take a look and see what you can get out of there. That essentially concludes this tutorial. I’ve put a number of references over there and I believe we have some time for questions now.