Noise Part I Chapter 07 from Razavis book IIT-Bombay - PowerPoint PPT Presentation

2 Noise – Part I Chapter 07 from Razavi’s book IIT-Bombay Lecture 38 M. Shojaei Baghini

3 Slides Figures • Unless it’s mentioned figures and contents of slides are taken from: ‘Design of Analog CMOS Integrated Circuits’ by Behzad Razavi IIT-Bombay Lecture 38 M. Shojaei Baghini

4 Noise • Instantaneous value of a random signal is not predictable. • In many cases average power of noise is predictable. T 2 1 2 ) ∫ = lim ( P v t dt × avg T R For stationary process − T 2 → ∞ T IIT-Bombay Lecture 38 M. Shojaei Baghini

5 Noise Power in Frequency Domain Power spectral density (PSD) of the noise • Concept of white noise IIT-Bombay Lecture 38 M. Shojaei Baghini

6 A Very Useful Theorem Related to PSD Theorem: For a linear time-invariant system with = transfer function H(s) 2 ( ) ( ) | ( ) | S f S f H f Y X IIT-Bombay Lecture 38 M. Shojaei Baghini

7 Amplitude distribution and Central Limit Theorem Central limit theorem: If many independent random variables with arbitrary PDFs are added PDF of the sum (here x ) approaches a Gaussian distribution. ( ) 2 − x x − avg 1 = σ 2 ( ) 2 P x e X σ π 2 IIT-Bombay Lecture 38 M. Shojaei Baghini

8 Equivalent Power of Multiple Random Signals Consider two noise sources v 1n (t) and v 2n (t). Assume v n (t) = v 1n (t) + v 2n (t) Express average power of v n (t). T 2 1 ( ) ∫ = + = 2 lim ( ) ( ) P v t v t dt 1 2 avg n n T T For stationary independent − 2 → ∞ random signals: T v 1n (t) × v 2n (t) ≈ v 1n (t) × v 2n (t) T 2 2 ∫ = + + × lim ( ) ( ) P P v t v t dt avg 1 avg 2 1 n 2 n T T − 2 → ∞ T IIT-Bombay Lecture 38 M. Shojaei Baghini

9 Thermal Noise of Resistor One sided noise power spectral density = ⇒ = ∆ = ≥ 2 ( ) 4 4 for 1 and 0 S f kTR v kTR f Hz f v n K=Boltzmann constant=1.38 × 10 -23 J/ ° K 2 ↑ T ↑⇒ v n Example: R=50 Ω ⇒ S v (f)=0.91 nV/ √ Hz In other words RMS value of noise power in 1 Hz bandwidth = 0.91 nV IIT-Bombay Lecture 38 M. Shojaei Baghini

10 Example: Passing White Noise from LPF 1 = ω = 2 ( ) ( ) | ( ) | 4 S f S f H j kTR π + out R 2 2 2 2 4 1 R C f IIT-Bombay Lecture 38 M. Shojaei Baghini

11 Example: Passing White Noise from LPF (cont’d) Noise shaping ∞ 1 kT ∫ = = ( ) 4 P f kTR df π + out 2 2 2 2 4 1 R C f C 0   dx ∫ − =  1  tan x +  2  1 x kT Independent of R ⇒ = ( ) v RMS , n out C For 1pF capacitor, v n,out =64.3 µ V (RMS) What about HPF? IIT-Bombay Lecture 38 M. Shojaei Baghini

12 Equivalent Current Noise Source 2 4 v kT = = ∆ 2 n i f n 2 R R • One sided PSD • Example of two parallel resistors IIT-Bombay Lecture 38 M. Shojaei Baghini