Spectral Measurements Case A: Bandwidth exceeds that of available amplifiers TA(f) S(f) channels f1 fN receivers and antennas detector noise, detect directly or split frequencies and then detect before amplification or detection f1 fN f1 fN f Receivers-G1 1) Extreme bandwidth: use multiple 2) If signal large compared to 3) Use passive frequency splittersSpectral Measurements Case B: Bandwidth permits amplificationf1 fN further frequency splitting Case C: Bandwidth permits digital spectral analysis ( fResolution -~)f(V 2M 2 N )f()( NN Φ↔τ φ (Permits ~×100 more B per cm2 silicon) (Reference: Van Vleck and Middleton, Proc. IEEE, 54, (1966) ) G2 1) Amplify before either detection or 1) If computer resources permit, compute N 2B : transform point N per multiplys N log N ≥ ∆ 2) Or 1-bit (or n-bit) samples) (Nspectra M averageExamples of Passive Multichannel Filters IN f1 fnf2 Zo 2. Waveguides f1 resonant cavities at ff virtual short λ/4 fN RCVRRCVR f1 f3f2 filters passive channel-dropping filters Zo at f1 ≠ f G3 1. CircuitsExamples of Passive Multichannel Filters 3. Prism -bound electron(s) red blue fo fprism 5. Cascaded Dichroics plane wave <f1 <f2 fn >f1 >f2 >fn-1(f1>f2…>fn) 4. Diffraction grating ε(f) G4Digital spectral analysis example: autocorrelation 0 0τ f φ(τ) Φ(f) [W Hz-1] analog signals Possible analog implementation: 1) max lag = τmax = NT 2) sample lag, T sec 3) finite integration time τ >> τmax )(ˆ v τφ BRF 0 ffRF × 0 ffIF delay line × ×× ∫ ∫ ∫ LO “local oscillator” v(t) )T(ˆ vφ )(ˆ vφ )NT(ˆ vφ NT = τmax B≤BRF τ τ τ G5 : on based is T2Resolution of autocorrelation analysis )(W)()(ˆ yv τ•τφ=τφ1) W(τ ) 0 τ τ M = NT-τ M 0 f 0 f B Φ v(f) ∗ W(f) ~1/2 τ M Hz Thus Φ v(f) )f(W)f()f(ˆ vv M ∗Φ=Φ∴ 7777 W(f) f0 G6 ½ τ m τ < τAliasing in autocorrelation spectrometers )f(I)f( )t(i)( )f(ˆ )(ˆ v v v v ∗Φ •τφ =Φ =τφ 7777 2) i(t) T0 t I(f) 0 f(Hz)-1/T “Aliasing” is spectral overlap -1/T 0 1/T 2/T B)f(ˆ vΦ 3) Finite averaging time τ adds noise to )f(ˆ,)(ˆ vv Φτφ G7 1/T 2/TAutocorrelation of hard-clipped signals × A/D ( )2 ∫ c o u n t e r delay line × × × LO x(t) vo(t)±1 v(t) )(ˆ v τφ “hard clipping” A/D ⇒ ±1 +1 -1 0 t +1 if v(t) > 0 Receivers-I1Analysis of 1-bit autocorrelation () () 0-0 xt212211 ⎩ ⎨ ⎧ < ≥+∆∆∆ [ ] ==τφ 21x)( ( ) ( ) 21 xx 21 e 12 12 2 2 2 1 ∫∫ ∞ +ρ−− ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ π 12v tt),()(where −=ττ∆ () ( )[ ] ( )∫∫∫∫ ∞− ∞∞ −=τφ 0 0 2121 0 2121x 22 ( ) ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ =+−= ∫∫∫∫ ∞ ∞− ∞∞ 0 0 00 2121 1214 I2 JGRVZM are x , x where x 1 x 1 x sgn , x , x t x Let x sgn x sgn E 1 2 x x 2 2 1 dx dx x sgn x sgn 2 1 ∞ − ρ − ρ − 2 1 x x φ ≡ τ ρ dx dx x , x p dx dx x , x p ∫ ∫ 2 : Note dx dx x , x pPower spectrum for 1-bit signal Change variables x2 x1 θ dr rdθ r x1 = r cos θ x2 = r sin θ dx1dx2 = rdr dθ ( ) ( ) ∫ ∫ π ∞ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ θ− − π⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ θ=τφ 2 0 0 1 12r 2 2 x 1e 12 1 2 rdd4)( 2 2 ( )( )∫ −θπθ= π 2 0 2 112 1d4 I3 ρ − ρ − ρ − 2 sin 2 1 ρ − ρ − 2 1 2 sinPower spectrum for 1-bit signal ( )( )∫ −θπθ= π 2 0 2 112 1d4 () ( ) 1sin22 141dsin1 1 4 14 1 0 x − ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎟ ⎠ ⎞⎜ ⎝ ⎛ ρ+π π = φπ =τφ − π ∫ () () ⎟ ⎠ ⎞⎜ ⎝ ⎛ τφπ =φ xv ˆ 2sinˆˆ () ( ) ( ) Tx )ˆWhere τ=τφ 0 b0 p(b)p(a) b ˆρ (see Burns & Yao, Radio Sci., 4(5) p. 431 (1969)) θ∆φ 2Let I4 a ρ − ρ − 2 1 2 sin 2 1 − φ ρ − ρ − ρ ≡ τ -v(t sgn v(t) sgn exact not b if bias has : NoteSpectral response & sensitivity: autocorrelation receiver ;B f1f T)f( eff rms ∆−αβ≅σ ≅β channel bandwidth (S. Weinreb empirical result, MIT EE PhD thesis, 1963) “Apodizing” weighting functions: α ∆ f 1.099 0.87 0.69 N s Nfs Ns first sidelobe -7 dB -16 dB -29 dB Note trade between spectral resolution, sidelobes in Φ (f) and ∆ Trms ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = τ =∆ N;N T 1f M s τ τ τ uniform raised cosine blackman 0 0 0 I5 ∆τ 6. 1 f 60. 0 f 13. 1 taps #Spectral response & sensitivity: autocorrelation receiver If N delay-line taps, how many spectral samples Ns? 1 W(τ) τ0 τM Say uniform weighting of φ(τ): Then B = Ns • ∆f = Ns /2τM) where spectral resolution ∆f ≅ 1/2τm for orthogonal channels from boxcar W(τ) ( ) ( )B2N Ms ===τ=∴ W(f) for adjacent channel Mτ f In practice: raised cosine widens ∆f by 1/0.6 ≅ 1.7, so Ns ≅ N/1.7 W(f) I6 •(1taps # N rate nyquist at 2B 1 T B NT 2 2 1Types of “power” Delivered Available Exchangeable () { } {}jt evt R t sin tω∆ = ω+ ω Vg gZ LZ Rg g + -+ -V { }( ) ∗ ∗ =∆ ∆∆ g ZP PVIR2 1P LDavailable Dedelivered Receivers-K1 Receivers – Gain and Noise Figure Ve Re V cos Im V + j XZ if i.e., , P maxDelivered and Available Power { }( ) ∗ ∗ =∆ ∆∆ g ZP PVIR2 1P LDavailable Dedelivered RL ∞ ge ZR-ge ZR PD 0Zf ge < ( )-finitePP gL ZZD →∆ ∗ = K2 0g = 0 ge ZR-ge ZR PA PD RL 0Zf ge > Z if i.e., , P max R : Ioption power le exchangeab Z Im R : IDefinition of Gain gZ LZG 1 2 Gpower (= Gp) ∆ 12 DD PP Gavailable (= GA) ∆ 12 AA PP Gtransducer (GT) power ∆ 12 AD PP ∆Gexchangeable (=GE) 1 2 E E P P Ginsertion (= GI) ∆ 1 2 D D P P with amplifier without amplifier Note: GA, GE don’t depend on ZL do depend on Zg (via PE2) K3Definition: Signal-to-Noise Ratio (SNR) First define: N1 = N2 = S1 = S2 = 1 zWH− 222111 NSSNR;NS ∆∆ Vg 1 2 gZ LZ F G ( )gE = K4 exchangeable noise power spectrum @ Port 1 same, at 2 exchangeable signal power spectrum @ Port 1 same, at 2 SNR Define Z f G RecallDefinition: Noise Figure F …