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

2. Outline
Thermal noise
Noise in circuits
Noise figure

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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 = ( ) 250; g1=
Te2 = 150; g2 = 102
Te3 = ( )x290; g3 =
Te4 = 700; g4 = 103
Te = Te1 + Te2/g1 + Te3/g1g2 + Te4/g1g2g3
Pno = k (Tg + Te) B g
T0 = 290 : Standard temperature

12. 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

13. Reading
Carlson Ch. 9.3, 9.4
Proakis & Salehi 5.5