ECE 145A / 218 C, notes set 13: Very Short Summary of Noise
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1 class otes, M. odwell, copyrighted 009 C 45A / 8 C, otes set 3: Very hort ummary o Noise Mark odwell Uiversity o Calioria, ata Barbara rodwell@ece.ucsb.edu , ax
2 Backgroud / tet class otes, M. odwell, copyrighted 009 Noise is a complex subject Mth Math Physics Device oise models dl Circuit oise aalysis - port oise represetatios ti ystems oise aalysis ystem sesitivity calculatios l We ca oly touch upo a very ew poits here. More dtil details i other classes.
3 class otes, M. odwell, copyrighted 009 Probability Distributio Fuctio: adom Variable Durig a experimet, a radom variable The probability that x lies P{ x < x < x} X ( x dx X ( x is x x betwee X x ad x the probability distributio uctio. X (x takes o a particular value x. is x x x
4 Mea value, Variace, tadard Deviatio class otes, M. odwell, copyrighted 009 Mea Value o X : [ X ] X X x X ( x dx xpected value o : X [ X ] X x X ( x dx The variace σ x o rom its average value σ X σ X ( X x ( X x X ( x The stadard deviatio σ. X is its root - mea - square deviatio [ ] ( x x ( x x o X dx X is simply the square root o the variace
5 The Gaussia Distributio class otes, M. odwell, copyrighted 009 The Gaussia distributio : X ( x πσ The mea ( x x exp ( x x σ ( x x ad the stadard deviatio ( σ are as deied previously. Because o the * cetral limit theorem*, physical radom processes arisig rom the sum o may small eects have probability distributios close to that o the Gaussia. X (x σ x x x
6 Pair o adom Variables class otes, M. odwell, copyrighted 009 a experimet, a pair o radom variables takes o speciic particular values x ad y. X ad Y Their joit behavior is described by the joit distributio P { A < x < B ad C < y < D} ( x, y dxdy D C B A XY ( x, y XY
7 Correlatio ad Covariace class otes, M. odwell, copyrighted 009 The correlatio o XY X ad Y is [ XY ] xy ( x, y dxdy XY The covariace o C XY X ad Y is [ ( X x ( Y y ] XY x y Note that correlatio ad covariace are i either X or Y have zero mea values. the same we are dealig with AC oise sigals (subtractig o the mea values are zero. DC bias,
8 adom Processes class otes, M. odwell, copyrighted 009 A voltagev(t varyig (i some sese radomly with time. Measured at times t ad t, V ( t has valuesv ( t adv ( t. v(t v(t v(t t t t
9 Autocorrelatio Fuctio o A adom Process class otes, M. odwell, copyrighted 009 Write or simplicity V V ( t, V V ( t ( t, t [ V, V ] the process is statioary, the time origi does ot matter, ad [ V ( t, V ( t ] [ V ( t 3, V ( t 3 t t ] [ V ( t, V ( t τ ] The ( τ [ V ( t, V ( t τ ]
10 Power pectral Desity o a adom Process class otes, M. odwell, copyrighted 009 Power spectral desity ( ω ( τ exp( jωτ Fourier trasorm o dτ ( τ ad ( ω are a trasorm pair, so ( τ ( ω exp( jωτ ( τ : π dω The power delivered to a load isv so the average value o But [ V o, ] (0 the oise power is π ( ω itegratig the power spectral gives us the oise power. / dω load, [ V ]/ load desity (& dividig by Load
11 igle-ided Hz-based pectral Desities class otes, M. odwell, copyrighted 009 Double - ided pectral Desities V ( t V ( t τ π ( τ [ ] ( jω exp( jωτ dω ( jω ( τ exp( jωτ dτ igle - ided Hz - based pectral Desities ( τ [ V ( t V ( t τ ] ( j exp( jπ τ ( j ( τ exp( j π τ dτ d
12 class otes, M. odwell, copyrighted 009 igle-ided Hz-based pectral Desities- Why? Why this otatio? The sigal Power power i the badwidth low {, } low high high ( j d ( j d ( j Load high Load ( j / Load is directly the Watts o sigal power per Hz o sigal badwidth at requecies ear to. low Load high low d
13 Thermal Noise: esistaces class otes, M. odwell, copyrighted 009 For h << kt, resistors produce oise voltage. ( j 4 kt By a Thevei - 4kT ( j Norto trasormatio, the oise curret is
14 Available Thermal Noise Power class otes, M. odwell, copyrighted 009 Maximum power traser : load matched to erator. With matched load, voltage across load is With matched hdload, curret through h loadis N N / / Give that ( j 4kT or ( j 4kT load d P d kt P load is the maximum (the available oise power, hece i a badwidth Δ, P, kt Δ available oise All resistors have equal available oise power. Ay compooet uder thermal equilibrium (o bias ollows this law.
15 Noise rom a Atea class otes, M. odwell, copyrighted 009 d P available, oise d kt ( j 4kT e( Z The atea has both Ohmic ad radiatio resistaces. rad The Ohmic resistace has a oise voltage o spectral desity 4 kt ambiet Ohmic, where T ambiet is the physical atea temperature The radiatio resistace has a oise voltage o spectral desity 4kT o ield rad, where T ield is the average temperature the regio rom which the atea receives sigal power,rad loss,loss ter - galactic space is at.7 Kelvi (Big Bag
16 hot oise: PN juctios class otes, M. odwell, copyrighted 009 * *, give a DC curret DC, the arrival idepedet o every other electro, o each electro is statistically * the * the DC curret has a oise luctuatio ti o power spectral desity ( j q DC DC currets i resistors are * ot * a statistically idepedet low o ad *do ot * erate short oise. electros, Currets i PN juctios * DO*exhibit shot oise.
17 class otes, M. odwell, copyrighted 009 Device Noise Models: Descriptive, No xplaatio Bipolar Trasistor thermal oise shot oise FT
18 Two-Port Noise Descriptio class otes, M. odwell, copyrighted 009 Through the methods o circuit aalysis, the iteral oise erators o a circuit ca be summed ad represeted by two oise erators ad. The spectral desities o ad must be calculated ad speciied. The cross spectral desity * must also be calculated ad speciied. * Cross spectral desity : V ( j V ( τ exp( jπ τ dτ Where V ( τ [ V ( t ( t τ ] is the * cross - correllatio uctio o V ad.
19 Calculatig Total Noise class otes, M. odwell, copyrighted 009 the erator just has 4 kt N 4, thermal oise, epreset the combiatio o ampliier voltage ad curret oise by a sigle source Total N N Z We ca ow calculate the spectraldesity o { } * Z g e Z, total, ampliier g Z e jx g this { ( } total oise :
20 igal / Noise atio o Geerator class otes, M. odwell, copyrighted 009 V sigal,, total ad, are i series ad see the same load impedace. The ratios o powers delivered by these will ot deped upo the load. Thereore cosider the aviable oise powers. The sigal power available rom the erator is Psigal available Vsigal M / 4,, we cosider a arrow badwidth betwee( the the available oise power rom N, is P [ ] ( j Δ / 4 oise, available, erator, sigal Δ / ad ( sigal Δ /, The sigal/oise ratio o the erator is the N P P sigal, available V oise, available, erator, sigal, M ( j Δ / 4 / 4 V sigal, M kt / 4 Δ
21 igal / Noise atio o GeeratorAmpliier class otes, M. odwell, copyrighted 009 igal power available rom the erator : P V / 4 Ni Noise power available rom erator : Poise, av, sigal, available sigal, M ( j Δ / 4 kt Δ Noise power available rom ampliier : P 4, total, ampliier oise, av, Amp ( j Δ / igal/oise ratio icludig ampliier oise: N P P sigal, available P ( j Δ V sigal, M / 4 / 4 oise, avail, oise, avail, amp total, total V sigal, M / 4 ( j Δ / 4 kt Δ ( j Δ / 4
22 class otes, M. odwell, copyrighted 009 Noise Figure: igal / Noise atio Degradatio Noise igure sigal/oise ratio beore addig ampliier sigal/oise ratio beore addig ampliier igal/oise ratio beore addig ampliier : N V sigal, M kt Δ / 4 igal/oise ratio ater addig ampliier : N total ( V sigal, M / 4 j Δ / 4 kt Δ Noise igure ( j Δ / 4 total Δ kt kt Δ total ( j / 4 Noise igure kt ampliier available iput oise kt power
23 Calculatig Noise Figure class otes, M. odwell, copyrighted 009 Noise igure total ( j / 4 kt We also kow that : Z e { Z * },, g total ampliier g We ca calculate l rom this a expressio or oise igure : F * Z e( Z s 4kT s
24 Miimum Noise Figure class otes, M. odwell, copyrighted 009 Noise igure varies as a uctio o Z jx : F Z s ( * e Zs 4kT Ater some calculus, we ca id a mimimum oise igure ad a erator impedace which gives us this miimum : F Z mi opt opt 4kT jx ( [ ] m e[ ] opt m [ ] m[ ] j Pit Poits a optimum to remember are ( F Z varies with Z, hece ( which gives a miimum F(3. there is
25 Noise Figure i Wave Notatio class otes, M. odwell, copyrighted 009 Writte F F mi istead i terms o wave parameters, 4 r Γ Γ [ ] Γ [ Γ ] s s opt opt These describe cotous i the Γs plae o costat oise igure :" oise igure circles", i.e. a descriptio o with source the variatio o relectio coeiciet. oise igure
26 Low-Noise Ampliier Desig Yue & odwell, CC hort Course, Nov. 006 Desig steps are i-bad stabilizatio: this is best doe at output port to avoid degradig oise iput tuig or F mi 3 output tuig (match 4 out-o-bad stabilizatio Note that tuig or miimum oise igure requires a *mismatch* o the ampliier iput; ampliier gai thereore must lie below the trasistor MAG/MG. Note that tuig or miimum oise igure implies that ampliier iput is mismatched: iput relectio coeiciet is thereore ot zero! Discrepacy i iput oise-match & gai-match ca be reduced by addig source iductace Z s Z o Z s oise match V
27 xample LNA Desig: 60 GHz, 30 m ige BJT Yue & odwell, CC hort Course, Nov. 006 gai & oise circles ater iput matchig gai & oise circles ater iput matchig ote compromise betwee gai & oise tuig
28 Noise Figure o Cascaded tages Yue & odwell, CC hort Course, Nov. 006 F cascade F F F3 F4 G G G G G G A A A A A A3... This is the Friis oise igure ormula. G Ai are the available power gais. The relatioship applies whether or ot iter - stages are matched.
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