1 GSW Noise and IP3 in Receivers

Transcription

1 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers GSW Noise and 3 Receivers In many cases, the designers of dividual receiver components (mostly amplifiers, mixers and filters) don t know how any of their customers are gog to put them together to make a receiver. However, the designer of a radio receiver is gog to want to calculate the likely performance of a receiver built from these dividual components. This gives us two problems: how should the dividual components be specified, and how do we combe the specifications of these dividual components to produce a specification for the whole receiver?. The Parameters: Noise and 3 In many cases, the two most critical parameters for a receiver are its noise performance (which determes the sensitivity: the smallest put signal that can be successfully receiver) and its overall third-order ter-modulation products (which determes the largest put signal that can be successfully received). These parameters tend to be dependent: noise is a problem for very small put signals where the ter-modulation products are so small they can be neglected; ter-modulation products are a problem for very large put signals where the signal to noise ratio is so large that noise can be neglected. Second-order ter-modulation products are easier to prevent, sce at least one of the unwanted put signals required to produce some power at the carrier frequency f c must be at least f c / 2 away from the carrier, and is therefore easy to filter out; and higher-order termodulation products tend to produce significant problems only at higher put powers that the third-order effects. In other words, the third-order effects tend to start first: get them under control and you won t have to worry too much about any higher-order effects. There are many other important considerations when designg receivers (perhaps most notably the power consumption), but these are the two I ll consider this chapter... Noise and Noise Power Spectral Density Havg said that I ll talk about noise, I should mention that s it s often more useful to talk about the noise power spectral density. Noise (N) is measured Watts; noise power spectral density (N ) is measured Watts per Hertz. N is a spectral density, so by defition the amount of noise with a range of frequencies from f to f 2 is given by: f2 f N N f df (.) All the noise considered this chapter is white, which means that the noise power spectral density is not a function of frequency, and we can simply write: Third order ter-modulation products are the result of non-lear mixg of two frequencies f c f and f c 2f (where f c is the carrier frequency and f is a small offset frequency, often the channel spacg of the multi-channel radio system beg used). This non-lear mixg produces some power at f c itself, which terferes with the signal the receiver is tryg to receive, and sce it s at the same frequency, it can t be removed by filterg. For more details, see the chapter on Non-Lear Effects. 29 Dave Pearce Page 6/3/29

2 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers N N f f N B (.2) 2 where B is the bandwidth of terest (the frequencies from f to f 2 ). From the pot of view of a receiver, the noise power spectral density is actually more terestg than the total noise power, sce the performance of a receiver with a matched (optimum) filter is a function of E s /N, where E s is the energy one symbol, and N is the noise power spectral density..2 Specifyg Noise: Noise Temperature and Noise Figure There are two common ways to specify the amount of noise that a component will troduce to the signal cha: noise temperature and noise figure. I fd noise temperature more tuitive and easier to understand, but noise figure seems to be more common. They both use the same model of noise: that of treatg the effect of any component as the combation of addg some noise to the put signal, followed by a perfect (noise-free) ga or loss. For example, consider a component that when provided with an put signal S with an put noise power N produces an output signal S out and output noise power N out. Let the power ga of this component be G, so that: S S out G (.3) For those who like pictures, it looks like this: Signal S Noise N Ga G Noise Figure F Signal S out Noise N out Figure - One Noisy Component with Ga We then consider that this component behaves identically to a component that adds a noise power N e = N out / G N to the put signal, and follows this with a perfect, noise-free ga of G, like this: Equivalent Input Noise N e = N out / G N Signal S Noise N Ga G No Added Noise Signal S out Noise N out Figure -2 Equivalent Circuit for One Noisy Component with Ga The amount of noise added to the put of the component this model is called the equivalent put noise of the component. Sce by defition: 29 Dave Pearce Page 2 6/3/29

3 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers G N N N (.4) e out it s straightforward to derive that: N e N G out N (.).2. Noise Temperature The noise power available from a resistor at a certa temperature T is given by: N k T B (.6) where k is the Boltzmann constant (.38 x -23 J/K), T is the temperature of the resistor ( Kelv), and B is the bandwidth 2 of the system ( Hz). This allows a noise power N to be associated with a temperature T. The equivalent noise temperature T e is the temperature of a resistor that would provide a power equal to the equivalent put noise power for a component: N e k T B (.7) e Knowg the noise temperature and the ga of the component (as well as the bandwidth of the signal) together with the put noise is then enough formation to calculate the total output noise power: N G N N G N k T B (.8) out e e That s it. It s quite simple. The only slight problem is if you re just designg the radiofrequency stages of a receiver, you might not know what the bandwidth of the signal is likely to be. In that case we can t calculate the absolute noise powers, the best we can do is calculate the noise power spectral densities N ( W/Hz), where: NB N (.9) In this case if we know the put noise power spectral density N, and the equivalent put noise power spectral density N e, we can calculate the output noise power spectral density N out usg: N G N N G N k T (.) out e e and pass this formation onto the system designer. Often this is what he really wants to know anyway 3. 2 Strictly speakg it s the noise bandwidth of the system, not the 3-dB bandwidth or any other defition of bandwidth. The noise bandwidth of a system is the bandwidth of a perfect brick-wall filter that would let through the same amount of noise. 29 Dave Pearce Page 3 6/3/29

4 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers.2.2 Noise Figure The defition of the noise figure is a little less obvious. It s the ratio of the actual output noise power to the output noise power due to the put noise alone (i.e. assumg the device itself does not add any noise), provided the put noise is due to a resistor at 29 K. To unpick that a bit: suppose a component has a power ga of G, and the output noise power is N out. Some of this output noise power will be G times the put noise power, and some will be due to the additional noise troduced by the component itself. In terms of the equivalent put noise, we can write: out e e N G N N GN GN (.) and note that the noise figure, by defition, is then equal to: Nout GN GNe Ne F (.2) GN GN GN N and sce for the correct value of noise figure, the put noise has to be that due to a resistor at 29 degrees Kelv: Ne Ne F N 29 k B (.3) Also, the amount of equivalent put noise N e you have to add to a noise-free version of a component with a noise figure F to get the same output noise can be determed from rearrangg equation (.2) to get: N N F (.4) So thgs now look like this (for the equivalent circuit for the noisy component): e Equivalent Input Noise N e = N / (F ) Signal S Noise N Ga G No Added Noise Signal S out Noise N out Figure -3 Equivalent Input Noise Terms of the Noise Figure 3 As noted, it s the noise power spectral density that determes the mimum bit error rate for a lk, not the noise power with the signal bandwidth (which is hard to defe, sce many signals have power spectra that gradually roll-off away from the ma lobe, and it s not always obvious how best to defe the bandwidth). 29 Dave Pearce Page 4 6/3/29

5 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers Another very useful result follows from this defition of the noise figure ( fact, this is why the noise figure is used so frequently): the noise figure is the ratio of the put signal to noise ratio to the output signal to noise ratio, aga provided the put noise is equal to that due to a resistor at 29 K, sce: SNR S Nout S G N Ne N Ne Ne F (.) SNR N S GS N N N out out.2.3 The Noise Figure of Passive Components In most cases, for passive components, the noise figure is the verse of the ga. Or perhaps I should say the noise figure is equal to the loss 4, sce passive components can t have a ga greater than one, the output is always smaller than the put. This isn t obvious, but the result is fairly easy to derive if you thk about thgs terms of equivalent circuits. Consider the followg filter: Ga G Noise Figure F Noise N out Figure -4 A Passive with Input Tied to Ground Clearly, the put noise gog to this filter is just that due a resistor at 29 K. However, consider another component followg the filter. As far as it is concerned, all it has on its put is a network of dividual resistors, capacitor and ductors all at the same temperature (29 K). This network could be replaced by an equivalent network composed of an ideal resistor series with an ideal capacitor or ductor. Sce ideal capacitors and ductors are noise-free, this means from the noise pot of view, the output of the passive component just looks like a resistor at 29 K. This is exactly what the put noise looks like to this filter; so the put noise must be equal to the output noise: But the defition of the noise figure is: so here: N N k 29 B (.6) out Nout Nout F (.7) G N G k 29 B 4 Where loss is defed as the put divided by the output ( contrast to the ga beg the output divided by the put). 29 Dave Pearce Page 6/3/29

6 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers k 29 B F (.8) G k 29 B G So, for example, a passive filter with a ga of. would have a noise figure of 2; or decibels, a passive filter with a ga of 3 db has a noise figure of 3 db..2.4 Noise Figures and Noise Temperatures Sce they are measurg the same thg, knowg the noise temperature of a component allows you to calculate the noise figure of the same component very easily (and vice versa). This is often useful: sometimes some components a receiver are specified terms of noise temperatures and others terms of noise figures. The relationship between them is simple enough to work out: from equation (.7) and equation (.3), we get: or for gog the other way from F to T e : Ne k Te B Te F N 29 k B 29 (.9) Te 29 F (.2) So, for example, the noise temperature of a passive component (like a filter, or a length of cable with some loss) is related to its ga by: T e 29 G (.2) For the case of passive components (cludg cables, which troduce a loss between the antenna and the first stage of the receiver) this is sometimes expressed terms of a transmission factor, defed as: S S out G (.22) which gives for passive components an equivalent put noise temperature of: T e 29 (.23).3 Addg Noise Powers You might have noticed that the previous sections, I ve been addg powers, not amplitudes. That s because the noise sources are uncorrelated: the equivalent put noise waveform added by the component is entirely dependent of the comg noise waveform. They both have a mean power (N and N e respectively), but the amplitude of these noise contributions at any time is an uncorrelated random variable. I ll call these noise amplitudes n and n e respectively, with the lower-case n representg the fact that these are amplitudes not powers. 29 Dave Pearce Page 6 6/3/29

7 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers The total put noise amplitude is then: n n n (.24) e so the mean value of the power this total put noise is: 2 2 e 2 2 En 2nne ne 2 2 En Ene 2Enne E n E n n (.2) where E{} represents the expectation (mean) value. However, sce the two noise sources are completely dependent and have a mean value of zero, the product of them is equally likely to be positive (when both have the same sign) or negative (when they have different signs). So the expectation value of the product n e n is zero. That just leaves: e E n E n E n (.26) or other words: the mean power the total put noise is the sum of the mean power the put noise with the mean power the equivalent put noise. n e (t) n e2 (t) - n (t) mean power = n 2 (t) - n e (t) n (t) mean power = (n e (t) n (t)) 2 mean power = 2 - Figure - Addg Noise Contributions This is a general rule when addg uncorrelated noise signals: always add the powers, not the amplitudes..4 The Noise Performance of Receivers To calculate the noise performance of a whole receiver volves combg the noise added by all the stages the receiver: the various filters, mixers and amplifiers than make up the 29 Dave Pearce Page 7 6/3/29

8 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers receiver signal cha. This turns out to be quite simple to do, and the resultg formula has some very important consequences. The trick is to replace all the noise sources throughout the signal path with equivalent noise sources at the begng of the signal path. As we ve seen, each component can be considered, terms of the noise it adds, as if it had a perfect noise-free ga, but an additional noise component added to the put of (F )N i. We can extend this idea to more than one component: N i (F ) N i (F 2 ) N i (F 3 ) Signal S Noise N i Ga G No Added Noise Ga G 2 No Added Noise Ga G 3 No Added Noise Figure -6 Three Noisy Components Series The trick now is to move the additional equivalent noise sources back to the start of the series of components. This is quite simple to do, all you have to do is divide the equivalent put noise contributions of each stage by the total ga of the components up to the start of the component. For example, we can move the second equivalent put noise contribution back to the start of the series by dividg it by G, and add an equivalent put noise of N i (F )/G to the put. For the third component, we ll need to divide the equivalent noise contribution by G G 2. The equivalent circuit for the series of components now looks like this: N i (F ) N i (F 2 )/G N i (F 3 )/G G 2 Signal S Noise N i Ga G G 2 G 3 No Added Noise Figure -7 Movg Noise Contributions to the Input Note that both of the above figures, the output noise power is: N G G G N G G G N F G G N F G N F (.27) 2 3 i 2 3 i 2 3 i 2 3 i 3 and the overall ga is G G 2 G 3. Takg this one stage further, we can add up all these additional noise contributions, and we ll get: 29 Dave Pearce Page 8 6/3/29

9 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers N i (F ) N i (F 2 )/G N i (F 3 )/G G 2 Signal S Noise N i Ga G G 2 G 3 No Added Noise Figure -8 Combg Noise Contributions This suggests we can treat the entire cha of active stages as havg a ga of G G 2 G 3, and a noise figure of F where: N ( F ) N ( F ) N ( F ) / G N ( F ) / G G i i i 2 i 3 2 F F ( F ) / G ( F ) / G G (.28) Extendg this result to more active stages, we get: F F F F F... (.29) G G G G G G and this is the Friis formula for combg the noise outputs of cascaded stages. Usg this formula, and the gas and noise figures of the stages, we can calculate an overall noise figure for the whole receiver..4. Workg with Noise Temperatures We can do the calculation equally well with noise temperatures. Usg the result from equation (.9), we can replace all the noise figures with noise temperatures sce: T F e (.3) 29 and that turns the Friis formula (equation (.29)) to: e2 e3 e4 Te T e T T T T T T T e2 e3 e4 e Te G GG 2 GG 2G3 G G G G G G (.3) where T e is the overall noise temperature for the receiver, and T e, T e2, T e3 etc are the noise temperatures of the dividual stages. This is perhaps easier to remember (you don t have to remember to subtract one from all the noise figures except the first one). 29 Dave Pearce Page 9 6/3/29

10 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers.4.2 Receiver Sensitivity Sce the noise the origal put N i can be taken to be ktb, where k is the Boltzmann constant (.38 x 23 ), T is the temperature (conventionally taken to be 29 K) and B is the bandwidth, all we need to know is the noise bandwidth of the filters, and we can calculate the total signal to noise ratio at the output of the receiver for any level of put signal. The smallest value of put signal which provides a certa mimum output signal to noise ratio is known as the sensitivity of the receiver. Unfortunately, there isn t a sgle defition of sensitivity, sce the radio receiver designer often doesn t know what level of output signal to noise ratio will be required for the whole system. A common solution is to defe the sensitive of a receiver terms of the mimum detectable signal (MDS). This the is the put signal level that results a signal to noise ratio at the output of db ( other words, the same signal power and noise power). Then a system designer wantg to know what level of put signal would be required to provide a signal to noise ratio of 6 db at the output just has to take the MDS and add 6 db..4.3 Example of Noise Figure Calculation Consider the dual-conversion superhet receiver shown below: Local Oscillator f Local Oscillator f 2 Bandpass LNA Amp Amp To low-frequency circuits Ga (db) NF (db) (dbm) Figure -9 - Example Dual-Conversion Super-Heterodyne Receiver The noise figure, ga, and third-order termodulation tercept pot 3 is given for all the relevant components the design. (It will be assumed that the low-frequency circuits do not add significantly to the noise or non-learity of the receiver.) This receiver has ne component stages, and we know the noise figure and ga of all of them. However, sce they are all given decibels, and the Friis formula is terms of real noise powers and power ratios, the first thg we need to do is take the gas and noise figures out of decibels and back to real powers and ratios. This can be conveniently done usg a table like the followg (spreadsheets are great at this sort of thg): You might notice that the noise figures of the passive components (filters) are all equal to the loss through the filters (and therefore mus one times the ga db). As discussed, this is a general result for all passive components. 29 Dave Pearce Page 6/3/29

11 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers Stage Ga (db) Ga up to this stage (db) Ga up to this stage (lear) Noise Figure (db) Noise Figure (lear) Contribution to Overall Noise Figure Bandpass First (.8 ) /.62 =.4 Amplifier (LNA) First (2. ) / 8.93 =. First Mixer (4.7 ) / = 3.32 Second (.78 ) /.22 =.69 Second (2. ) /.63 =.8 Amplifier Second (.8 ) / 63. =.23 Mixer Third (2. ) / 398 =.2 Third Amplifier ( ) / 99 =. Then add up all the noise contributions to the overall noise figure: F (.32) so this receiver has an overall noise figure of 8.8 (which is 9.4 db). If the channel bandwidth is 2 khz, then the noise power at the put is: ktb = = 8 W 2 dbm (.33) and if we happen to know that the mimum signal-to-noise ratio required to mata an acceptable bit error rate at the output of the receiver (the put to the detector stage) is 6 db (or 3.98 lear terms), then we can work out the sensitivity of this receiver (and it s easier to work db here, so we can subtract rather than havg to divide): S dbm N dbm SNR db 9.4 out F db SNR db SNR db 9.4 S dbm N dbm SNR db 9.4 S out out dbm dBm (.34) If we didn t know what mimum signal-to-noise ratio was required for acceptable performance, all we could do is consider the sensitivity to be the mimum detectable signal MDS), which is the put signal that gives an output signal power equal to the output noise power. Usg this defition, the sensitivity would be 6 db less, at. dbm. 29 Dave Pearce Page 6/3/29

12 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers.4.4 Comparative Noise Contributions Lookg at the contributions that each dividual stage makes to the overall noise figure for the receiver, the contribution of each stage (except the first stage) could be written as: Fi G j i j (.3) other words the contribution of each stage is its own noise figure mus one, divided by the total ga up to the put of the stage. After the second amplifier, the combed ga is so high that it doesn t really matter whether subsequent stages are low-noise or not, it won t make much difference to the overall noise performance of the receiver. Of much more concern are the first few stages the receiver cha, where the signal level is low, and there hasn t been much ga. That s where low-noise design is much more important. In this example, the largest sgle contribution to the overall noise figure is from the first mixer, and the second-largest contribution is from the bandpass filter before the front-end lownoise amplifier. To improve the noise performance of the receiver it would be nice to get rid of this filter, but that would allow the entire output of the receive antenna to be fed straight to the put of the sensitive low-noise amplifier: and if there are any strong emitters around (for example someone with a mobile phone walks past) they might block the wanted signal. In general, for low-noise design you need as much low-noise ga as early as possible the receiver cha. Once the signals are large, you can usually forget about noise. Exactly the opposite is true for the other problem with receivers discussed this chapter: the 3 rd -order termodulation products.. Calculatg Intermodulation Distortion The power of the 3 rd -order termodulation product I d the output of a non-lear stage fed by three equal-power puts is given by 6 : I d G 3 S (.36) 2 3 where S is the power the put signal ( Watts), G is the power ga of the stage, and 3 is the third-order termodulation tercept pot (also Watts). If this is the power, then the rms amplitude of this distortion must be: d I g S 3/2 rms d 3 (.37) where g is the amplitude ga of the stage, rather than the power ga G. Sce the power ga is the square of the amplitude ga, this means: 6 For the derivation of this formula, see the chapter on non-lear effects. 29 Dave Pearce Page 2 6/3/29

13 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers 2 g G (.38) A radio receiver with several non-lear elements series will have contributions to the total amount of termodulation distortion from each non-lear stage. Just as before, we can readily calculate the amplitude of the overall termodulation product by usg the technique of replacg every real component with an ideal lear component, with an additional termodulation term added to the put, replacg somethg like this: Signal S Interference I Power Ga G Amplitude Ga g 3 = 3 Signal S out Interference I out Figure - Non-Lear Component Addg Intermodulation Product with somethg like this: Signal S Interference I Effective Input de Interference rms S 3 3/2 Power Ga G Amplitude Ga g Lear ( 3 = fite) Signal S out Interference I out Figure - Equivalent Model of Non-Lear Component Note that the amplitude of the required effective put termodulation product term that s added to the put to the non-lear component is: de rms drms S g 3 3/2 (.39) so that when it is multiplied by the amplitude ga of the non-lear stage, we end up with the result for the output term, given equation (.37)... Addg Intermodulation Product Terms At this pot it might be worth pausg to consider why we always work with powers the case of noise, but I m talkg about envelopes and rms amplitudes the case of the termodulation products. The reason is that the termodulation products are not dependent between the different stages; they all result from the same signals passg through the components. So they all have the same shape, and that means I have to add them coherently. Consider addg two signals that are the same shape, s e (t) and s (t). The result is a signal with a magnitude of s e (t) s (t), and that has a mean power of: e e e E s t s t E s t E s t E s t s t (.4) 29 Dave Pearce Page 3 6/3/29

14 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers and this time the fal term is not zero (see the figure below). If the signals are exactly the same shape, then we could write: where k is a constant. In this case, we get: e s t k s t (.4) E s t s t E k s t k E s t (.42) e e e You can t just add the powers any more, you have to add the amplitudes. workg with amplitudes here. That s why I m s e (t) n e2 (t) - s (t) mean power = s 2 (t) - s e (t) s (t) mean power = (s e (t) s (t)) 2 mean power = 4 - Figure -2 Addg Intermodulation Product Contributions..2 Intermodulation Products and Multi-stage Receivers We can extend this idea (of replacg real components with ideal lear components with additional equivalent put termodulation product terms) to a series of components, just like we did for noise. For example, suppose we had a cha of components, and two of them had significant non-learities: Signal S Ga G Lear Ga G 2 Ga G Non-Lear 3 Lear 3_2 Ga G 4 Non-Lear 3_4 Figure -3 Sample System with Four Stages, Two Non-Lear Expressg the non-lear stages terms of an ideal lear stage and an additional contribution from the distortion gives: 29 Dave Pearce Page 4 6/3/29

15 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers 3 _ 2 3/2 GS 3 _ 4 3/2 GG 2G3S Signal S Ga G Lear Ga G 2 Lear Ga G 3 Lear Ga G 4 Lear Figure -4 Equivalent System with Two Stages of Non-Learity Note the equivalent distortion put to the second stage is: 3_ 2 3/2 GS (.43) sce the signal put at this stage has a power of G S (it s G times more powerful than the put to the whole cha, sce it s just come through the first stage with a ga of G ). Similarly, the signal put to the last stage has an equivalent put distortion component of: 3_ 4 3/2 GG 2G3S (.44) sce the put to this stage has a power of G G 2 G 3 S. Then we move these additional contributions back to the start of the whole system. They are amplitudes, so movg them back through a component with a power ga of G will reduce the amplitude by a factor G. Hence the contribution of the second stage becomes: G G S S 3/2 3/2 3_ 2 3_ 2 and the contribution of the fourth stage becomes: G G G 2 3 G (.4) 2 3 G G G S S 3/2 3/ _ 4 3_ 4 and that produces an equivalent system that looks like this: G G G (.46) G 3 _ 2 S 3/2 GG 2G3 S 3 _ 4 3/2 Signal S Ga G Lear Ga G 2 Lear Ga G 3 Lear Ga G 4 Lear Figure - Combg Non-Lear Stages Comparg this to a one-stage system with termodulation distortion added to the put: d S 3 3/ 2 (.47) 29 Dave Pearce Page 6/3/29

16 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers suggests that the total amount of termodulation products added by all the stages is equivalent to a perfectly lear system with an termodulation product term added to the put of the whole series of components, of amplitude: G G G G d S S 3/ /2 3 _ 2 3 _ 4 G G G G _ 2 3 _ 4 S 3/2 (.48) This is exactly what would happen if the system were replaced by a sgle non-lear component with a ga of G G 2 G 3 G 4 and an 3 of: 3 system G G G G 2 3 3_ 2 3 _ 4 (.49) In other words, we can derive an equivalent 3 for the whole system, and use this to calculate the total amount of termodulation product troduced by all the components the system. Note that this formula could be expressed the form: 3 system Ga up tostage with Ga up tostage with 3 _ 2 3 _ 4 3 _ 2 3 _ 4 (.) which suggests how it can be expanded for use with any number of non-lear terms: 3system i Ga up tostage with 3 _ 3 _ i i (.)..3 Example of Intermodulation Calculation Consider the superhet receiver from section.4.3: Local Oscillator f Local Oscillator f 2 Bandpass LNA Amp Amp To low-frequency circuits Ga (db) NF (db) (dbm) Figure -6 - Example Dual-Conversion Super-Heterodyne Receiver (Aga) 29 Dave Pearce Page 6 6/3/29

17 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers This receiver has four stages with significant non-learities. It s usually easier to keep the ga and 3 db and dbm respectively itially sce the division then becomes a subtraction; it s only when combg the contributions from the different stages that thgs have to be taken out of db so that the contributions can be added together. Just like with noise, this sort of calculation can be readily done with a spreadsheet, and this case the results might look somethg like this: Stage Ga up to this stage (db) 3 of this Gupto _ stage (dbm) log 3i i G upto _ i First LNA First Mixer Second Amplifier Second Mixer hence the resultant 3 of the whole system is: 3_ i 3system mw 4.37 dbm (.2) (Unlike the noise analysis earlier the chapter, I ve only considered here the components with a significant non-learity. If you were designg a spreadsheet you might have to consider all the components, and give a value of fity to the 3 of the lear components, but if dog the calculation by hand we can be a bit more telligent and just consider the non-lear components.)..4 Comparative Intermodulation Contributions Lookg at the contributions that each dividual stage makes to the overall 3 tercept pot for this receiver, the sgle largest contribution is from the second mixer, even though this component has by far the largest 3. This isn t surprisg: the power the termodulation product generated at each stage varies with the cube of the signal power, and it s the later stages which tend to have the largest signal powers... 3 and Channel Selectg s There is one potential trap when workg out the overall 3 of a receiver, and that is to remember that order to generate any termodulation product at the signal frequency, you need two other frequencies to be present. If you ve filtered out all of the other frequencies, then further non-lear components the receiver aren t gog to make any difference. If the third amplifier the receiver above actually had an 3 of dbm: 29 Dave Pearce Page 7 6/3/29

18 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers Local Oscillator f Local Oscillator f 2 Bandpass LNA Amp Amp To low-frequency circuits Ga (db) NF (db) (dbm) then you might thk that this would give the entire receiver a very low 3, sce the signal would be so large at this stage. However, if the image filter just before this amplifier gets rid of all the signals except the wanted signal, then there aren t any other signals left to produce any termodulation products, and the 3 of this component can be neglected. It s only the non-lear stages up to the filter than selects an dividual channel or an dividual carrier frequency that need to be considered when calculatg an overall 3 (after this pot there won t be any power left at adjacent carrier frequencies that can be mixed together to create any distortion the fal signal)..6 Receiver Dynamic Range The dynamic range of a receiver is the range of put signals that give acceptable performance. In general, this is impossible to calculate without knowg what the receiver is gog to be used for, but if we defe the lowest signal put power as the mimum detectable signal (the signal that provides an equal amount of signal power and noise power the output), and we defe the highest put signal power to be the put signal power that produces a third-order termodulation product power equal to the output noise power, then we can produce a figureof-merit for the receiver. This is known as the spurious-free dynamic range (SFDR). It turns out that there is a very simple and memorable formula for the spurious-free dynamic range terms of the overall 3 and the mimum detectable signal (MDS). Equation (.36) gives power the termodulation product terms of the put signal powers S and the ga G, and puttg this equal to the output given by the MDS (which is just the MDS times the ga): and a bit of algebra on this then gives: G 3 Id S G MDS (.3) /3 3 S MDS (.4) This is the maximum put power for the spurious free dynamic range. The mimum is just the MDS itself. So the ratio of the two is: Max Power M Power / /3 2/ /3 MDS MDS MDS MDS (.) 29 Dave Pearce Page 8 6/3/29

19 Gettg Started with Communications Engeerg GSW Noise and 3 Receivers which terms of db is: 2 SFDRdB 3 dbm MDS dbm (.6) 3 For the example receiver considered above, this works out to be: db (.7) 3.7 Tutorial Questions ) In the example given the notes, by how much would the noise performance improve if the second amplifier and the second image filter were the other way around (i.e. the amplifier came first)? What would be the disadvantage (if any) of dog this? 2) A passive filter has an sertion loss of 2 db. What can you deduce about its noise figure and third-order tercept pot? 3) What would happen to the spurious-free dynamic range of this receiver if the ga of the second amplifier were reduced to db? 4) Consider the sgle-conversion superhet design shown below: Local Oscillator f Local Oscillator f 2 Bandpass LNA Amp Amp To low-frequency circuits Ga (db)??? 3.??? 3.??? NF (db) (dbm) The receiver noise bandwidth is 2 khz. Calculate: a) The equivalent receiver noise figure; b) The expected output noise power if the receive antenna is potg at the ground; c) The receiver s third-order termodulation tercept pot; d) The spurious-free dynamic range. Assume all the filters are passive (i.e. they conta no active components). Why are the first three components the order they are? Why not combe the bandpass filter with the image filter to save costs? ) Consider the super-heterodyne receive the example above. If you were asked to change any one component order to improve this dynamic range, which component should you choose? 29 Dave Pearce Page 9 6/3/29