Receivers, Antennas, and Signals. Professor David H. Staelin Fall 2001 Slide 1

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1 Receivers, Anennas, and Signals Professor David H. Saelin 6.66 Fall 00 Slide A

2 Subjec Conen A Human Processor Transducer Radio Opical, Infrared Acousic, oher Elecromagneic Environmen B C Human Processor Communicaions: A C (radio, opical) Acive Sensing: A C (radar, lidar, sonar) Transducer Passive Sensing: B C (sysems and devices: environmenal, medical, indusrial, consumer, and radio asronomy) 6.66 Fall 00 Slide A

3 Subjec Offers Physical conceps Mahemaical mehods, sysem analysis and design Applicaions examples Moivaion and inegraion of prior learning 6.66 Fall 00 Slide 3 A3

4 Subjec Ouline Review of signals and probabiliy Noise in deecors and sysems; physics of deecors Receivers and specromeers; radio, opical, infrared Radiaion, propagaion, and anennas Signal modulaion, coding, processing and deecion Communicaions, radar, radio asronomy, and remoe sensing Parameer esimaion 6.66 Fall 00 Slide 4 A4

5 Review of Signals Signal Types o be Reviewed: Pulses (finie energy) Periodic signals (finie energy per period) Random signals (finie power, infinie energy) 6.66 Fall 00 Slide 5 A5

6 Have Finie Energy : Define noaion " " for e.g. v() V (f) Pulses v() Define Fourier Transform: V (f) v () = v()e = V(f)e j N πf ω v () + jπf d df d < [ vols/hz = vol sec ] [ vols] Fourier Transform : Dimensions mus only be self consisen ; e.g. v() can be dimensionl ess, vols, meers, newons, ec Fall 00 Slide 6 B

7 S (f) V (f) Energy Specral Densiy S(f) V (f) = v()e j πf ω S(f) can have dimensions of : sec if is ime and v is dimensionless m (vols/hz) Joules/Hz E ceera if is disance and v is dimensionless if is ime and v() is vols where (v/hz) = v /sec Hz = v / sec ( ) ( ) ( ) if S(f) is dissipaed in a -ohm resisor by v() vols, where Joules = vols sec/ohm 6.66 Fall 00 Slide 7 B

8 R( τ ) S (f) S (f)? v()v Claim: R ( τ) S(f) Auocorrelaion Funcion R( τ )e ( τ ) d j π f τ = V(f) V (f) = dτ = [ ] v sec or [] Reverses V(f) Q.E.D. { } jf τ v()v ( τ ) d e dτ { j π f } { } v()e d v ( + j π f )e d = R ( τ ) R (0) = = Therefore: + j π f τ S(f)e v () d = df S(f) df J, ec. sign of d for = Parseval' s Theorem d consan 6.66 Fall 00 Slide 8 B3

9 Compac Transform Noaion v () R ( τ ) [ v ] [ v /Hz] V(f) V(f) S(f) [ ] v sec [ v / Hz ] [ Joules ] [ J/Hz] If power is dissipaed in a -ohm resisor 6.66 Fall 00 Slide 9 B4

10 Define Uni Impulse ε ε 0 ε u o () δ () where lim uo() d =, u o() = 0 for > 0 u n () u o () u Impulse 0 n () d Sep u () u () 0 0 Ramp Slope = Fall 00 Slide 0 B5

11 Define Convoluion 0 a () b() a() b( τ ) d = c( τ) a() b() a () b() = c( τ) * τ Fall 00 Slide B6

12 a ( Useful Transformaion Pairs for Pulses o ) ( f ) a () A A (f) a() e d u o A(f) e j ω o j πf u o () = δ () Have energy (f) (reaed as special pulses) ( f ) j πf a () jωa a () = A(f)e df u n n () (jω) a () u e j ω o α () e / ( f ) A f o ( jω + α ) ωo πf o ω πf 6.66 Fall 00 Slide C

13 Transforms: Even/Odd Funcions a e () where: R e a a { A( f )} e o () () [ a() + a( ) ]/ = a ( ) [ a() a( ) ]/ = a ( ) e o EVEN ODD so: a() = a e () + a o () e.g. a () a e () = + a o () () a o j { } Im A(f) 6.66 Fall 00 Slide 3 C

14 Transforms: Operaors and Gaussians a () a () A (f) A (f) a () a v() () A (f) A (f) 0 σ v () σ σ A π e e ( / σ) / Ae A ( σω) e / A ( τ / σ ) ( σω) π All Gaussians 6.66 Fall 00 Slide 4 C3

15 x() Characerized by: Linear Sysems h() y() h() = " Impulse Response, " where y() x() h() Tes : If Noe: A (B A B A (B C) x() = = B δ (), hen y() = h() + C) = (A B) + (A C) A = (A B) C x( τ)h( If h() = δ (), hen y() = x() τ )dτ " superposi ion Disribuive Commuaive Associaive inegral" 6.66 Fall 00 Slide 5 C4

16 Alhough v Periodic Signals ()d =, v() T o v ()d m = m = 0 < where Period T Infinie energy, finie power T 0 T T 3T Fourier Series: T / jm(π / T) V m v ()e d where m = 0, ±, ± T T / ω = π v() = V m m = e jm π f o o f o ( f /T) o, Fall 00 Slide 6 E

17 Auocorrelaion, Power Specrum Auocorrelaion Funcion: R ( τ ) T T / T / v ()v ( τ ) d = V m = m e jm π f o τ Power Specrum: Φ m V m = T T / T / R ( τ)e jm πf o τ d 6.66 Fall 00 Slide 7 E

18 Compac Noaion v () R ( τ ) V m V m Φ m Φ (f) Typical dimensions: [ vols ] [ vols] [ ] [ ] vols vols In -ohm resisor: [ was ] [ W] 6.66 Fall 00 Slide 8 E3

19 Transforms of Impulse Trains a v() a/t V m T 0 T 3 / T 0 / T f a / T ( ) R τ ( a / T) V m Φ (f) 6.66 Fall 00 Slide 9 T 0 T e.g. Area = v () ah τ / h τ a is impulse value a rea 3 / T = a / T 0 / T R ( τ ) = R (0) h / h 0 / h Le h so area a / T becomes impulse value a h T f τ = a h / T E4

20 Receiver Noise Processes Receivers, Anennas, and Signals Professor David H. Saelin 6.66 Fall 00 Slide 0

21 Random Signals Random signals generally have finie power, infinie energy, and are unpredicable Example: [ ] WH z Φ(f) 0 f MAX f Since we have infinie informaion for infinie ime and a finie frequency band, hen V(f) is no an analyic funcion and our approach mus be differen. New definiions are required Fall 00 Slide G

22 Expeced Value of x Finie or infinie ensemble of x i () E x { dx [ ()] x i () p x i() } x p ( x ) x i i i ε " ensemble" ( ) p x i [ ( ) ] p x i () x i () x j A random signal is drawn from some ensemble Auocorrel aion Funcion : φ v [ ] (, ) E v ( ) ( ) v 6.66 Fall 00 Slide G

23 v() is wide-sense saionary if: φ v Saionariy (, ) = φ ( +, + ) = φ( τ) where v, v() is sric-sense saionary if: E g v [ { ( ), v ( ),..., v ( )}] = E [ g { v ( + ), v ( + ),..., v ( + )}] n n v() is Ergodic if: v() is wide-sense saionary and T φv ( τ ) = lim v() v ( ) T T τ d, T i.e., ensemble average = ime average ransiions Oherwise v() is occur only Non-saionary e.g.: a clock icks (ime-origin sensiive) Fall 00 Slide 3 τ for any funcion g for all, G3

24 Transform Diagram: Random Signals v() (?) φ v (τ) Φ(f) Typical Ses of Unis [ V ] (?) [ V ] (?) [ V ] [ V /Hz ] [W] [W/Hz] Power o -ohm resisor Power specral densiy 6.66 Fall 00 Slide 4 G4

25 Power Specral Densiy T j π Φ ( f) = lim E v ()e T T T Why use E[ ] if v() is ergodic? Because lim σ ( f) 0! where σ (f) E Φ d [ (f) Φ (f ] T T T ) Specral resoluion increases wih T, becoming infinie as T e.g. Infinie informaion in finie bandwidh unless ensemble is averaged Power Specral Densiy Compuaion: For a single ergodic waveform, ake ensemble average over successive inervals of widh T. Use T adequae o yield desired or meaningful specral resoluion. f 6.66 Fall 00 Slide 5 G5