EE354 : Communications System I Lecture 8,9: Thermal noise Aliazam Abbasfar

Outline Thermal noise Noise in circuits Noise figure

Gaussian process If {X(t1), …, X(tN)} are jointly Gaussian Properties : mX(t) and RX(t1,t2) gives complete model If X(t) passed through an LTI system, the output signal is Gaussian process as well If WSS, then strictly stationary and ergodic

White process If X(t) has flat power spectral density Total power =  GX(f) = c RX(t) = c d(t) X(t1) and X(t2) are uncorrelated if t1 ≠ t2 Total power =  Limited power when goes through a filter

Thermal noise Random movements of free electrons in a resistor Gaussian process Zero mean and finite power T: temperature [K] k: Boltzmann constant (1.38×10-23 joule/K) h: Planck constant (6.625×10-34 joule-sec) Almost flat PSD ( up to 1012 Hz) Practically a white process Gn(f) = 2kTR Noise power in band-limited systems : 4kTB R Channel bandwidth

AWGN Additive white Gaussian noise Gn(f) = N0/2 and Rn(t) = N0/2 d(t) SNR = SD/ND Noise can be modeled as voltage source Best power delivery when matched Gn(f) = kT/2 W/Hz Independent of R N0 = kT (T = 290 K) is -174 dBm T0 = 290 : Standard temperature

Filtered noise Reduce noise power by filtering Gno(f) = Gn(f) |H(f)|2 Ideal filter with bandwidth B |H(f)| = rect( f/2B) Gno(f) = N0/2 rect( f/2B) Rno(t) = N0 B sinc (2Bt) Pno = N0 B Colored noise Uncorrelated for t1-t2 = n/2B

Noise equivalent bandwidth Noise power in practical filters Assume an ideal filter with gain : g= |H(f)|2max Pno = g N0 BN (noise equivalent bandwidth ) BN is usually greater than 3dB bandwidth RC filter g = 1 BN = 1/4RC Pno = kT/C g and Bn specifies output noise power – given by factory

Effective noise temperature Output power of any white noise source Pno = k Teff B Noise in two-port networks Pno = kTgBNg + Pni = k(Tg+Te)BNg Pni : internal noise Te : Effective noise temperature Pno /Pni = (1 + Te/Tg) Te = Input referred noise temperature A passive network with loss = L Te = (L-1)T Circuit design

Noise figure Noise enhancement factor (T0 = 290 K) nf = Pno /Pni = (1 + Te/T0) nf = (Si/Ni)/ (So/No) A passive network with loss = L (at T0) nf = L Cascade of amplifiers Te = Te1 + Te2/g1 + Te3/g1 g2 + … nf = nf1 + (nf2-1)/g1 + (nf3-1)/g1 g2 + … First stage should have high gain T0 = 290 : Standard temperature

Noise figure - example Receiver working at T=250 K Antenna : Tg = 50 K Cable : L = 1 dB Amplifier : Te = 150 K, g = 20 dB Mixer : L =3 dB, Nf = 3 dB Amp : Te = 700 K , g = 30 dB Te1 = (L-1)T = (100.1 -1) 250; g1= 10-0.1 Te2 = 150; g2 = 102 Te3 = (100.3-1)x290; g3 = 10-0.3 Te4 = 700; g4 = 103 Te = Te1 + Te2/g1 + Te3/g1g2 + Te4/g1g2g3 Pno = k (Tg + Te) B g T0 = 290 : Standard temperature

Sample circuits Circuit Te (K) Nf (dB) Ideal LNA (VG) 10 0.2 LNA (G) LNA (VG) 10 0.2 LNA (G) 100 1.3 LNA (M) 300 3 Amplifier 500 4.5 Circuit design

Reading Carlson Ch. 9.3, 9.4 Proakis & Salehi 5.5