Download session 8,9

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
 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
2 f
R
f
exp(
) 1
kT
T: temperature [K]
k: Boltzmann constant (1.38×10-23 joule/K)
h: Planck constant (6.625×10-34 joule-sec)
G n (f) 



2( kT) 2
E[n (t)] 
R
3
 f 
V2

 2kTR 1 
Hz
2kT


2
 Almost flat PSD ( up to 1012 Hz)
 Practically a white process
 Gn(f) = 2kTR
 Noise power in band-limited systems : 4kTB R
V2
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
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 )

1
2
BN   H(f) df
g0
 BN is usually greater than 3dB bandwidth
 RC filter
 g = 1
 BN = 1/4RC
 Pno = kT/C
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
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
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
Te2
Te3
Te4
=
=
=
=
(L-1)T = (100.1 -1) 250; g1= 10-0.1
150; g2 = 102
(100.3-1)x290; g3 = 10-0.3
700; g4 = 103
 Te = Te1 + Te2/g1 + Te3/g1g2 + Te4/g1g2g3
 Pno = k (Tg + Te) B g
Sample circuits
Circuit
Te (K)
Nf (dB)
Ideal
LNA (VG)
LNA (G)
0
10
100
0
0.2
1.3
LNA (M)
Amplifier
300
500
3
4.5
Reading
 Carlson Ch. 9.3, 9.4
 Proakis & Salehi 5.5