C. Noise in Microwave Systems

Transcription

1 10/18/2007 Noise Microwave Systems 1/2 C. Noise Microwave Systems Bad News: Eve if we completely reject the image ad all spurious sigals, there will still be a uwated sigal that will always appear at the detector/demodulator. NOISE Q: What is oise, ad where does it come from? A: HO: Receiver Noise Q: So how do we quatify oise? A: HO: he Statistics of Noise Q: How much exteral oise do we typically see? A: HO: Atea Noise emperature Q: What ab teral oise; how much oise is geerated by a microwave compoet our receiver? A: HO: Equivalet Noise emperature Aother way to specify the oise performace of a microwave compoet is by its Noise Figure. HO: Noise Figure ad SNR

2 10/18/2007 Noise Microwave Systems 2/2 Q: What ab passive devices; do they geerate oise? What is their oise figure? A: HO: Noise Figure of Passive Devices A microwave system (e.g., a receiver) is made of may compoets. We ca (ad must!) determe the overall system oise figure ad/or equivalet oise temperature for a etire system. HO: System Equivalet Noise emperature HO: System Noise Figure

3 10/18/2007 Receiver Noise 1/7 Receiver Noise Q: Say we tue our receiver to a frequecy at which o sigal is preset. Does this mea that the put of the detector/demodulator will be zero? A: Nope! Ufortuately, eve if we completely reject all spurious sigals, there will always be oe uwated sigal that reaches the demodulator/ detector. his uwated sigal is called oise. Noise is a completely radom sigal, ad whe it reaches the demodulator the result is a completely radom demodulator put. his provides the familiar hiss you might here if you tue your radio to a frequecy where o statio exists, or the sow you see o your televisio if you similarly select a chael where o statio exists! Q: Big deal! I d ever tue to a frequecy where there is o sigal. Is this oise really a problem? A: A big problem! Note that we said that oise will always be preset at the detector/demodulator there is o way to completely get rid of it.

4 10/18/2007 Receiver Noise 2/7 As a result, the best we ca hope for (if we completely suppress all spurious sigals) is that oly the desired sigal s ( t ) ad oise ( t ) will reach the demodulator/detector. s ( t) + ( t) î ( t) = i ( t) + ε ( t ) his oise will the cause a error ε ( t ) the demodulated sigal î ( t )! Q: Yikes! How large will this error be? A: It depeds may thgs (e.g., modulatio type, sigal badwidth, sigal power), but most fudametally it depeds o the Sigal-to-Noise Ratio (SNR) at the demodulator put. his ratio is simply the power associated with the sigal ( P s ), to the power associated with the oise ( P ): P = = ( ) ( ) ( ) s SNR SNR db Ps dbm P dbm P hus, if there is a lot of sigal power, ad just a little oise power, the SNR will be large. If the coverse is true, the SNR will be small. he as you might expect the demodulator error ε ( t ) dimishes as SNR creases. hus:

5 10/18/2007 Receiver Noise 3/7 he SNR at the demodulator put must be sufficietly large order for demodulator error to be acceptably small. Q: No problem! If we place eough amplifiers frot the demodulator/detector, the we ca always make P s really large right? P s P s > P s A: Yes but there s oe big catch. Although amplifiers of course crease sigal power P s, they crease the oise power P eve more! hus, we will fd that amplifiers actually decrease SNR! P s SNR P s SNR > P s < SNR his is a tremedous challege for radio egeers ad receiver desigers. We must:

6 10/18/2007 Receiver Noise 4/7 1. Icrease the sigal power (by amplificatio), such that the sigal is large eough to be detected/demodulated. 2. But make sure that the SNR is ot degraded to the extet that the demodulatio error is uacceptable. Q: From where does oise origate? A: wo sources: oe exteral ad oe teral! Exteral Noise * Exteral oise is coupled to the receiver through the receiver atea. It turs that the etire electromagetic spectrum is awash radom eergy (i.e., oise). * his radom eergy has either a specific frequecy, or directio, but stead is spread across all directios ad all frequecies! * As a result, we ca pot our atea ay directio, ad we ca tue our receiver to ay frequecy, but we will always receive a portio of the electromagetic oise! Q: What is the source of this exteral oise?

7 10/18/2007 Receiver Noise 5/7 A: here are three sources: terrestrial, extra terrestrial ad huma-made. errestrial Noise - Every warm object radiates electromagetic eergy (its oe method of heat trasfer)! Defitio of warm Aythg with a temperature above absolute zero (i.e., > 0 K ). * he frequecy spectrum of this emitted electromagetic oise depeds o the temperature of the object. * For objects o the Earth (i.e., terrestrial objects), the temperature is such that the emitted eergy peaks the frared regio. * However, terrestrial objects emit radom eergy across the etire electromagetic spectrum cludg the RF ad microwave regios! Extra-errestrial here are also very warm objects er space! Amog these objects are of course stars ad plaets, but the most sigificat of these extraterrestrial objects is our very ow star the Su!

8 10/18/2007 Receiver Noise 6/7 he Su as you may already kow is really hot. As a result it radiates electromagetic oise at a astoishg rate! Some of this oise is ufortuately radiated the RF/microwave ed of the e.m. spectrum (a great aoyace to us radio egeers), but the oise power radiated by the Su peaks the visible regio of the electromagetic spectrum. Q: Wait a secod! Our eyes detect electromagetic eergy the visible regio of the e.m. spectrum. Why have t I ever oticed this oise? A: You have! What our eyes see is this oise sulight is fact extra-terrestrial electromagetic oise produced by a very hot Su. Jim Stiles he Uiv. of Kasas Dept. of EECS

9 10/18/2007 Receiver Noise 7/7 Huma-Made Noise We humas geerate a heck of a lot of radom oise (both electromagetic ad otherwise)! We have built literally millios of trasmitters, ad each of these radiate oise that was terally geerated! Iteral Noise Ay warm object that efficietly absorbs electromagetic eergy must likewise emit electromagetic eergy ( the form of oise). hese objects clude resistors ad semicoductors! Q: May of the compoets our receiver have resistors ad semicoductors; does this mea that they produce oise? A: Absolutely! his is a major headache for radio egeers. Not oly do we ed up amplifyg the exteral oise coupled to the receiver through the atea, but the receiver itself adds to this radom sigal by the oise it terally geerates!

10 10/18/2007 he Statistics of Noise 1/8 he Statistics of hermal Noise Noise is a completely radom sigal. It caot be described determistically, but it ca be described statistically. For example, cosider the frequecy spectrum of a oise process ( t ) v : 1 j 2πft S ( f ) = v ( t) e dt 2π From this we ca determe the spectral power desity of this oise: N ( f ) = S ( ) 2 W f Hz his fuctio describes how the eergy of a sigal is distributed across the frequecy spectrum. Sce oise is a

11 10/18/2007 he Statistics of Noise 2/8 radom fuctio (i.e., a radom process), its spectral power desity is likewise radom. hus, N ( f ) likewise caot be described determistically, but it ca be described statistically. I other words, we caot state specifically how the oise eergy is distributed across the frequecy spectrum, but we ca describe how it is distributed o average! he fuctio N ( f ) therefore is defed as the average spectral power desity of oise. Now, let s cosider the average spectral power desity of a resistor R at temperature. Q: How could a resistor produce a oise sigal v ( t )? Is t a resistor a passive device that produces o power? A: hat s ot quite true! Sce the resistor is a warm object, the free electros with the device will be movg (due to thermal eergy) a radom I way. his creates a ty electric field g with the device, which tur creates a ty voltage across this resistor. R + v (t) - his voltage is the resistor oise voltage v ( t ). We call this pheomeo thermal oise.

12 10/18/2007 he Statistics of Noise 3/8 Q: What is the average spectral power desity N ( f ) of this thermal oise? A: Usg a buch quatum physics, we fd that the thermal oise produced by a resistor is: ( ) = 0 N f k N W Hz where: is the temperature of the resistor degrees Kelv. ad k = Boltzma's Costat 23 = 138x10. - J K Sce k is a costat, egeers ofte specify temperature stead ofn 0, they call this temperature the oise temperature: = Q: Wait! his fuctio N ( f ) seems to be depedet of frequecy f!?! A: hat s correct! he average spectral power desity of thermal oise is theoretically a costat with respect to frequecy. N 0 k

13 10/18/2007 he Statistics of Noise 4/8 N ( f ) N 0 f I other words, the oise power is (o average) distributed uiformly across the frequecy spectrum o frequecy will have ay more or less (o average) tha ay other frequecy. Noise of this type is called white oise. Q: Ad this fuctio N ( f ) is also depedet of the value of resistace R!?! A: hat is aga correct! he oise that a resistor produces does ot deped o its resistace, it depeds oly o its temperature. However, this resistor caot have the values R = 0 or R = it must be able to absorb power. Q: So N ( f ) N0 = is the average spectral power desity of the thermal oise. What simply is the total power P of the thermal oise?

14 10/18/2007 he Statistics of Noise 5/8 A: We ca determe the total power from the average spectral power desity by tegratg the power desity over all frequecy: P = N ( f ) df Q: Yikes! If we tegrate N ( f ) N0 get fite power! 0 = over all frequecy, we P = N df =??? 0 0 (Is the eergy crisis solved?) A: he reality is, as frequecy gets extremely large, we fd that the average spectral power desity will dimish to zero. lim N f f ( ) = 0 I other words, the result: N ( f ) = k is a approximatio that is valid the RF/microwave regio of the electromagetic spectrum. herefore: P = N f df < 0 ( ) Q: Still, would t the resultg value of P still be quite large?

15 10/18/2007 he Statistics of Noise 6/8 A: Mathematically speakg, yes. But remember, resistors reside circuits with reactive elemets. As a result, every microwave device has a fite badwidth. his fite badwidth will limit the amout of oise power that passes through the receiver to the demodulator. For example, cosider the case where a badpass filter is coected betwee two resistors: R s N ( f ) Badpass Filter ( f ) P R L We ll cosider the resistor o the left to be source of thermal oise, with average spectral power desity: ( ) = 0 N f N While the resistor o the right is a load that absorbs oise power P. he badpass filter has a power trasmissio coefficiet, where: ( f )

16 10/18/2007 he Statistics of Noise 7/8 ( f ) ( f ) 1 0. for f filter passbad 00. for f filter stopbad f hus, the average spectral power desity of the oise at the load is: N f = N f f = N f Ad we coclude: ( ) ( ) ( ) ( ) 0 N ( f ) N ( f ) ( f ) N0 for f filter passbad 00. for f filter stopbad N 0 f

17 10/18/2007 he Statistics of Noise 8/8 Now, if the filter has a passbad that exteds from frequecy f 1 to frequecy f 2 (i.e., badwidth B = f2 f1 ), the total power of the oise at the load is: f 0 f 1 ( ) P = N f df = N ( ) f df N 10. df = NB hus, we ca coclude that the thermal oise produced by some resistor R at temperature, whe costraed to some fite badwidth B (ad it always is costraed this way!), has total power: P = k B = N B 0

18 10/18/2007 Atea Noise emperature 1/4 Atea Noise emperature Q: So, exteral oise will be coupled to our receiver through the receiver atea. What will the average spectral power desity of this oise be? N ( f ) Rx A: Geerally speakg, it will be white oise! I other words, the average spectral power desity of the exterally geerated oise will be costat with respect to frequecy (or at least, costat across the atea badwidth). hus, as far as oise (ad oly oise) is cocered, the receiver appears to have a warm resistor attached to its put! N A Rx

19 10/18/2007 Atea Noise emperature 2/4 We ca specify this put oise terms of average spectral power desity N A, but we typically defe it terms of its oise temperature: A NA k K We call A the atea oise temperature it s the apparet temperature of the warm resistor attached to the receiver put. hus, we ca write: N A = k A Q: Is there some typical value of atea temperature? A: It depeds o which directio the atea is potg! A < 10K

20 10/18/2007 Atea Noise emperature 3/4 * If the atea is poted toward the sky (e.g., satellite commuicatios), the atea oise temperature could be very low, o the order of 10 K or less. he oe big exceptio to this occurs whe you pot your atea at the Su. A 290 K * If the atea is ot poted at the sky, the most of the exteral oise will be geerated by terrestrial sources. It turs that the atea oise temperature this case is simply equal to the physical temperature at the Earth s surface! Of course, this temperature chages somewhat, but expressed degrees Kelv (i.e., with respect to absolute zero) this chage is small. hus, radio egeers typically assume a atea oise temperature of a stadard value of o = 290 K.

21 10/18/2007 Atea Noise emperature 4/4 o 290 K his is approximately room temperature o Earth. hus, the average spectral power desity of the oise eterg a receiver is typically (if A = o!) assumed to be: N A = k = o -23 (. ) 138x = 400x10. = 400x W Hz mw Hz Ofte, this value is expressed terms of dbm/hz (yuck!) as: A ( ) = N dbm/ Hz.

22 10/18/2007 Equivalet Noise emperature 1/7 Equivalet Noise emperature I additio to the exteral oise coupled to the receiver through the atea, each compoet of a receiver geerates its ow teral oise! For example, cosider a amplifier with ga G ad badwidth B: N G P = BN Here there is o put sigal at the amplifier put, other tha some white (i.e., uiform across the RF ad microwave spectrum) oise with average spectral power desity N. At the put of the amplifier is likewise oise, with a average spectral power desity of N. his put average spectral power desity N is typically ot widebad, but stead is uiform oly over the badwidth of the amplifier: N ( f ) N for f badwidth B N for f side badwidth B

23 10/18/2007 Equivalet Noise emperature 2/7 W/Hz N N ( f ) N ( f ) f hus, the oise power at the put is: B 0 f 2 f 1 ( ) P = N f df N = BN Q: he amplifier has ga G. So is t N = G N, ad thus P = G BN?? df A: NO!! his is NO correct! We will fd that the put oise is typically far greater tha that provided by the amplifier ga: N G N

24 10/18/2007 Equivalet Noise emperature 3/7 Q: Yikes! Does a amplifier somehow amplify oise more tha it amplifies other put sigals? A: Actually, the amplifier caot tell the differece betwee put oise ad ay other put sigal. It does amplify the put oise, creasg its magitude by ga G. Q: But you just said that N G N!?! A: his is true! he reaso that N G N is because the amplifier additioally geerates ad puts its ow oise sigal! his terally geerated amplifier oise has a average spectral power desity (at the put) of N. hus, the put oise N cosists of two parts: the first is the oise at the put that is amplified by a factor G (i.e., GN ), ad the secod is the oise geerated terally by the amplifier (i.e., N ). Sce these two oise sources are depedet, the average spectral power desity at the put is simply the sum of each of the two compoets: N = G N + N Q: So does this oise geerated terally the amplifier actually get amplified (with a ga G) or ot?

25 10/18/2007 Equivalet Noise emperature 4/7 A: he teral amplifier oise is geerated by every resistor ad semicoductor elemet through the amplifier. Some of the oise udoubtedly is geerated ear the put ad thus amplified, other oise is udoubtedly geerated ear the put ad thus is ot amplified at all, while still more oise might be geerated somewhere the middle ad thus oly partially amplified (e.g., by 0.35 G). However, it does ot matter, as the value N does ot specify the value of the oise power geerated at ay pot with the amplifier. Rather it specifies the total value of the oise geerated through the amplifier, as this total oise exits the amplifier put. As a result, we ca model a oisy amplifier (ad they re all oisy!) as a oiseless amplifier, followed by a put oise source producg a average spectral power desity N : Amplifier N G + N = G N + N Ideal Noiseless Amplifier w/ ga G N * Noise Source w/ average SPD N

26 10/18/2007 Equivalet Noise emperature 5/7 Note however that this is ot the oly way we ca model terally geerated oise. We could alteratively assume that all the terally geerated oise occurs ear the amplifier put ad thus all this oise is amplified with ga G! Amplifier N Noise Source w/ average SPD N G + G N * G ( ) N = G N + N G = GN + N Ideal Noiseless Amplifier w/ ga G Note here that the oise source ear the put of the amplifier has a average spectral power desity of N G. It is fact this model (where the teral oise is assumed to be created by the put) that we more typically use whe cosiderg the teral oise of a amplifier! o see why, recall that we ca alteratively express the average SPD of oise terms of a oise temperature ( degrees Kelv): N = k

27 10/18/2007 Equivalet Noise emperature 6/7 hus, we ca express the put oise terms of a put oise temperature: N = k N k or the put oise temperature as: N = k N k Similarly, we ca describe the teral amplifier oise, whe modeled as beg geerated ear the amplifier put, as: N ke G = Where oise temperature e is defed as the equivalet (put) oise temperature of the amplifier: N e kg Note this equivalet oise temperature is a device parameter (just like ga!) it tells us how oisy our amplifier is. Of course, the lower the equivalet oise temperature, the better. For example, a amplifier with = 0 K would produce o teral oise at all! e

28 10/18/2007 Equivalet Noise emperature 7/7 Specifyg the teral amplifier oise this way allows us to relate put oise temperature ad put oise temperature a very straightforward maer: Amplifier Noise Source w/ oise temperature e + G * e ( ) = G + Ideal Noiseless Amplifier w/ ga G e ( ) = G + e hus, the oise power at the put of this amplifier is: P N B = k B ( ) = Gk + B e

29 10/18/2007 Noise Figure ad SNR 1/9 Noise Figure ad SNR Of course, additio to oise, the put to a amplifier a receiver will typically clude our desired sigal. Say the power of this put sigal is P s. he put of the amplifier will therefore clude both a sigal with power P s, ad oise with power P : P,N P,P s G s where: ad: P s = G P s P = N + G k B e ( ) = Gk + B e I order to accurately demodulate the sigal, it is importat that sigal power be large compariso to the oise power. hus, a fudametal ad importat measure radio systems is the Sigal-to-Noise Ratio (SNR): SNR Ps P

30 10/18/2007 Noise Figure ad SNR 2/9 he larger the SNR, the better! At the put of the amplifier, the SNR is: SNR P = P s GPs = Gk( + e ) B Ps = k B ( + ) e Moreover, we ca defe a put oise power as the total oise power across the badwidth of the amplifier: P = N B = k B Ad thus the put SNR as: SNR P Ps = = P k B s Now, let s take the ratio of the put SNR to the put SNR: e = 1 + ( + ) SNR P k s e B = SNR kb Ps + e =

31 10/18/2007 Noise Figure ad SNR 3/9 Sce e > 0, it is evidet that: SNR SNR e = 1+ > 1 I other words, the SNR at the put of the amplifier will be less tha the SNR at the put. his is very bad ews! his result meas that the SNR will always be degraded as the sigal passes through ay microwave compoet! As a result, the SNR at the put of a receiver will be the largest value it will ever be with the receiver. As the sigal passes through each compoet of the receiver, the SNR will get steadily worse! Q: Why is that? After all, if we have several amplifiers our receiver, the sigal power will sigificatly crease? A: rue! But remember, this ga will likewise crease the receiver put oise by the same amout. Moreover, each compoet will add eve more oise the teral oise produced by each receiver compoet.

32 10/18/2007 Noise Figure ad SNR 4/9 hus, the power of a sigal travelg through a receiver creases but the oise power creases eve more! Note that the ratiosnr SNR essetially quatifies the degradatio of SNR by a amplifier a ratio of oe is ideal, a large ratio is very bad. So, let s go back ad look aga at ratio SNR SNR : SNR SNR e = 1 + Note what this ratio depeds o, ad what it does ot. his ratio depeds o: 1. e (a device parameter) 2. (ot a device parameter) his ratio does ot deped o: 1. he amplifier ga G. 2. he amplifier badwidth B. We thus might be tempted to use the ratio SNR SNR as aother device parameter for describg the oise performace of a amplifier. After all, SNR SNR depeds

33 10/18/2007 Noise Figure ad SNR 5/9 o e, but does ot deped o other device parameters such as G or B. Moreover, SNR is a value that ca geerally be easily measured! But the problem is the put oise temperature. his ca be ay value it is depedet of the amplifier itself. For example, it is evet that as the put oise creases to fity: SNR e lim = lim 1+ = 1 SNR I other words, if the put oise is large eough, the terally geerated amplifier oise will become sigificat, ad thus will degrade the SNR very little! Q: Degrade the SNR very little! his meassnr = SNR! Is t this desirable? A: Not this stace. Note that if creases to fity, the: P s lim SNR = lim = 0 k B I other words, the SNR does is ot degraded by the amplifier oly because the SNR is already as bad (i.e., SNR = 0 ) as it ca possibly get!

34 10/18/2007 Noise Figure ad SNR 6/9 Coversely, as the put oise temperature decreases toward zero, we fd: SNR lim SNR 0 0 e = lim 1 + = Q: Yikes! he amplifier degrades the SNR by a fite percetage! Is t this udesirable? A: Not this stace. Note that if decreases to zero, the: P s lim SNR = lim = 0 0 k B Note this is the perfect SNR, ad thus the ratio SNR SNR will likewise be fity, regardless of the amplifier. Ayway, the pot here is that although the degradatio of SNR by the amplifier does deped o the amplifier oise characteristics (i.e., e ), it also o the oise put to the amplifier (i.e., ). his put oise is a variable that is urelated to amplifier performace Q: So there is o way to use SNR SNR as a device parameter?

35 10/18/2007 Noise Figure ad SNR 7/9 A: Actually there is! I fact, it is the most prevalet parameter for specifyg microwave device oise performace. his measure is called oise figure. he oise figure of a device is simply the measured ratio SNR SNR exhibited by a device, for a specific put oise temperature. I repeat: for a specific put oise temperature. his specific oise temperature is almost always take as the stadard room temperature of = 290 K. Note this was likewise the stadard atea oise temperature assumptio. hus, the Noise Figure (F ) of a device is defed as: o SNR F SNR e = 1 + = 290K = 290K e = K

36 10/18/2007 Noise Figure ad SNR 8/9 It is critically importat that you uderstad the defitio of oise figure. A commo mistake is to assume that: SNR SNR F = his is ot geerally true! Note this would oly be true if = 290K, but this is almost ever the case (i.e., 290K geerally speakg). hus, a correct (but widely repeated) statemet would be: he oise figure specifies the degradatio of SNR. Whereas, a correct statemet is: he oise figure specifies the degradatio of SNR, for the specific coditio whe = 290K, ad for that specific coditio oly he oe exceptio to this is whe a atea is coected to the put of a amplifier. For this case, it is evidet that the put temperature is = = 290K : A = = 290K A G,F SNR = SNR F

37 10/18/2007 Noise Figure ad SNR 9/9 Note that sce the oise figure F of a give device is depedet o its equivalet oise temperature e, we ca determe the equivalet oise temperature e of a device with kowledge F: e F = 1 + e ( F 1) 290K 290K = Oe more pot. Note that oise figure F is a uitless value (just like ga!). As such, we ca easily express it terms of decibels (just like ga!): ( ) = F db log F Like ga, the oise figure of a amplifier is typically expressed db.

38 4/17/2005 Noise Figure of Passive Devices 1/4 Noise Figure of Passive Devices Recall that passive devices are typically lossy. hus, they have a ga that is less tha oe we ca defe this terms of device atteuatio A: 1 A = G where for a lossy, passive device G < 1, therefore A > 1. Q: What is the equivalet oise temperature e or oise figure F of a passive device (i.e., ot a amplifier)? A: he equivalet oise temperature of a passive device ca be show to be approximately (trust me!): e ( 1) = A where is the physical temperature of the passive device. ypically we assume this physical temperature to be 290K, so that: ( ) = A 1 290K e

39 4/17/2005 Noise Figure of Passive Devices 2/4 A, e = G ( + e ) ( ) = + A e hus, we fd that the put oise temperature of a passive device is: ( ) = G + e + e = A ( A 1290 ) K = + A A 290K = + 290K A A his result is very terestg, ad makes sese physically. As atteuatio A approaches the lossless case A = 1, we fd that =. I other words the oise passes through the device uatteuated, ad the device produces o teral oise! Just like a legth of lossless trasmissio le! O the other had, as A gets very large, the put oise is completely absorbed by the device. he oise at the device put is etirely geerated terally, with a oise temperature = 290K equal to its physical temperature.

40 4/17/2005 Noise Figure of Passive Devices 3/4 Just like the put of a resistor at physical temperature = 290K Q: So, what is the oise figure F of a passive device? Now, we determed earlier that the oise figure of a twoport device is related to its equivalet oise temperature as: F = 1 + e 290K o herefore, the oise figure of a passive device is: ( A ) 1290K F = K = 1+ ( A 1) = A hus, for a passive device, the oise figure is equal to its atteuatio! F = 1 G = A So, for a active two-port device (e.g., a amplifier), we fd that two importat ad depedet device parameters are ga G ad oise figure F both values must be specified.

41 4/17/2005 Noise Figure of Passive Devices 4/4 However, for passive two-port devices (e.g., a atteuator), we fd that atteuatio A ad oise figure F are ot oly completely depedet they are fact equal! Fally, we should ot that the value A represets the atteuatio (i.e., loss) of ay passive device ot just a atteuator. For example, A would equal the sertio loss for a switch, filter, or coupler. Likewise, it would equal the coversio loss of a mixer. hus, you should ow be able to specify the oise figure ad equivalet oise temperature of each ad every two-port device that we have studied!

42 10/18/2007 System Equivalet Noise emperature 1/6 System Equivalet Noise emperature Say we cascade three microwave devices, each with a differet ga ad equivalet oise temperature: G, 1 e1 G, 2 e 2 G, 3 e 3 G, e hese three devices together ca be thought of as oe ew microwave device. Q: What is the equivalet oise temperature e of this overall device? A: First of all, we must defe this temperature as the value such that: e ( ) = G + e or specifically:

43 10/18/2007 System Equivalet Noise emperature 2/6 G e = Q: Yikes! What is the value of G? A: he value G is the total system ga; other words, the overall ga of the three cascaded devices. his ga is particularly easy to determe, as is it simply the product of the three gas: G = GGG Now for the hard part! o determe the value of, we must use our equivalet oise model that we studied earlier: Amplifier Noise Source w/ oise temperature e + G * e ( ) = G + Ideal Noiseless Amplifier w/ ga G e

44 10/18/2007 System Equivalet Noise emperature 3/6 hus, we cascade three models, oe for each amplifier: G 1 G 2 G = e 1 * e 2 * e 3 * We ca observe our model ad ote three thgs: ( ) = G e1 ( ) = G e1 ( ) = G e1 Combg these three equatios, we fd: ( ) ( ) ( ) = GGG + + GG + G e1 2 3 e 2 3 e 3 a result that is likewise evidet from the model.

45 10/18/2007 System Equivalet Noise emperature 4/6 Now, sce = 3, we ca determe the overall (i.e., system) equivalet oise temperature e : G GGG GG G e = ( + ) + ( ) + ( ) e1 2 3 e 2 3 e 3 = e2 e3 = e G GG GGG Moreover, we will fd if we cascade a N umber of devices, the overall oise equivalet temperature will be: e2 e3 e4 en e = e G GG GGG GGG G N 1 I assume that you ca use the above equatio to get the correct aswer but I wat to kow if you uderstad why your aswer is correct! Make sure you uderstad where this expressio comes from, ad what it meas. Look closely at the above expressio, for it tells us somethg very profoud ab the oise a complex microwave system (like a receiver!).

46 10/18/2007 System Equivalet Noise emperature 5/6 Recall that we wat the equivalet oise temperature to be as small as possible. Now, look at the equatio above, which terms this summatio are likely to be the largest? * Assumg this system has large ga G, we will fd that the first few terms of this summatio will typically domate the aswer. * hus, it is evidet that to make e as small as possible, we should start by makg the first term as small as possible. Our oly optio is to simply make e1 as small as we ca. * o make the secod term small, we could likewise make e 2 small, but we have aother optio! We could likewise make ga G 1 large! Note that makg G 1 large has additioal beefits, as it likewise helps mimize all the other terms the series! hus, good receiver desigers are particularly careful ab placg the proper compoet at the begg of a receiver. hey covet a device that has high ga but low equivalet oise temperature (or oise figure). G Big e Small he ideal first device for a receiver is a low-oise amplifier!

47 10/18/2007 System Equivalet Noise emperature 6/6 Q: Why do t the devices at the ed of the system make much of a differece whe it comes to oise? A: Recall that each microwave device adds more oise to the system, As a result, oise will geerally steadily crease as it moves through the system. * By the time it reaches the ed, the oise power is typically so large that the additioal oise geerated by the devices there are sigificat ad make little crease the overall oise level. * Coversely, the oise geerated by the first device is amplified by every device the overall system this first device thus typically has the greatest impact o system oise temperature ad system oise figure.

48 10/18/2007 System Noise Figure 1/6 System Noise Figure Say we aga cascade three microwave devices, each with a differet ga ad oise figure: G,F 1 1 G,F 2 2 G,F 3 3 G,F hese three devices together ca be thought of as oe ew microwave device. Q: What is the oise temperature of this overall device? A: Recall that we foud the overall equivalet oise temperature of this system to be: e2 e3 e = e1 + + G GG Likewise, the equivalet oise temperature of each device is related to its oise figure as: ( ) = F 1290K e

49 10/18/2007 System Noise Figure 2/6 Combg these two expressios we fd: ( ) ( ) ( ) ( ) F K F K F 1290K = F1 1290K + + G GG ad thus solvg for F : ( ) ( ) 1 F 1290K F 1290K F = ( F1 ) K K G1 GG 1 2 F2 1 F3 1 = F G GG F F = F1 + + G1 GG 1 2 herefore, the overall oise figure for the device cosistg of three cascade amplifiers ca be determed solely from the kowledge of the ga ad oise figure of each dividual amplifier! F2 1 F3 1 F = F1 + + G GG Moreover, we fd that if we costruct a system of N cascaded microwave compoets, the the overall oise figure of the system ca be determed as: F F 1 F2 1 F3 1 4 FN 1 = F G1 GG 1 2 GGG GGG 1 2 3GN 1

50 10/18/2007 System Noise Figure 3/6 * It is aga evidet from spectio of this equatio that the first device the cascaded cha will likely by the most sigificat device terms of the overall system oise figure. * We come to the same coclusio as for e make the first device oe with low teral oise (small oise figure F 1 ) ad high ga G. * I other words, make the first device your receiver a Low-Noise Amplifier (LNA)! Oe other very importat ote: Although we used oly amplifiers our examples for system equivalet oise temperature ad system oise figure, the results are likewise valid for passive devices! Just remember, the ga G of a passive device is simply the verse of its atteuatio A: 1 G = A Now, let s exame oe importat system made up of cascaded microwave compoets a receiver! At the put of every receiver is a atea. his atea, amog other sigals, delivers oise power to the put, with a temperature that is typically = 290K A.

51 10/18/2007 System Noise Figure 4/6 A = 290K Receiver with: Ga G Noise Figure F IF Badwidth B,N,P Q: What is the put oise (i.e.,n, or P ) of this receiver? A: Recall that the put oise temperature is: ad sce: ( ) ( ) = G + e = G + A e ( 290 e ) = G K + ( ) = F 1290K e we coclude that the put oise temperature is: ( 290 e ) ( 290 ( 1) 290 ) ( 290K ) = G K + = G K + F K = GF herefore, the average spectral power desity at the put is:

52 10/18/2007 System Noise Figure 5/6 N = k = kgf while the put oise power is: P = N B ( 290K ) ( 290 ) = kgf K B Now, compare these values to their respective put values: = = 290K A ( 290 ) N = k K ( 290 ) P = k K B Note for each of the values, the put is a factor G F greater tha the put: N P = = = GF N P However, I aga emphasize, this expressio is oly valid if = 290K!! Of course, the ratio of the sigal put power to the sigal put power is: Ps = G P s

53 10/18/2007 System Noise Figure 6/6 hus, the sigal power is creased by a factor G, while the oise power is creased by a factor GF. his is why there is reductio SNR by a factor F!