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

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IIT-Bombay Lecture 38 M. Shojaei Baghini

Noise – Part I

Chapter 07 from Razavi’s book

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IIT-Bombay Lecture 38 M. Shojaei Baghini

Slides Figures

• Unless it’s mentioned figures and contents of

slides are taken from: ‘Design of Analog CMOS Integrated Circuits’ by Behzad Razavi

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IIT-Bombay Lecture 38 M. Shojaei Baghini

• Instantaneous value of a random signal is not

predictable.

• In many cases average power of noise is

predictable.

∞ → × =

∫

−

T dt t v R T P

T T avg 2 2 2 )

( 1 lim

For stationary process

Noise

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IIT-Bombay Lecture 38 M. Shojaei Baghini

Power spectral density (PSD)

• f the noise
• Concept of white noise

Noise Power in Frequency Domain

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IIT-Bombay Lecture 38 M. Shojaei Baghini

2

| ) ( | ) ( ) ( f H f S f S

X Y

=

Theorem: For a linear time-invariant system with transfer function H(s)

A Very Useful Theorem Related to PSD

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IIT-Bombay Lecture 38 M. Shojaei Baghini

Central limit theorem: If many independent random variables with arbitrary PDFs are added PDF of the sum (here x) approaches a Gaussian distribution.

( )

2 2

2

2 1 ) (

σ

π σ

avg

x x X

e x P

− −

= Amplitude distribution and Central Limit Theorem

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IIT-Bombay Lecture 38 M. Shojaei Baghini

Consider two noise sources v1n(t) and v2n(t). Assume vn(t) = v1n(t) + v2n(t) Express average power of vn(t). For stationary independent random signals: v1n(t) × v2n(t) ≈ v1n(t) × v2n(t)

( )

∞ → × + + = ∞ → = + =

∫ ∫

− −

T dt t v t v T P P T dt t v t v T P

T T n n avg avg T T n n avg 2 2 2 1 2 1 2 2 2 2 1

) ( ) ( 2 lim ) ( ) ( 1 lim

Equivalent Power of Multiple Random Signals

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IIT-Bombay Lecture 38 M. Shojaei Baghini

One sided noise power spectral density

and 1 for 4 4 ) (

2

≥ = ∆ = ⇒ = f Hz f kTR v kTR f S

n v

K=Boltzmann constant=1.38 × 10-23 J/°K T↑⇒vn

2 ↑

Example: R=50 Ω ⇒ Sv(f)=0.91 nV/√Hz In other words RMS value of noise power in 1 Hz bandwidth = 0.91 nV

Thermal Noise of Resistor

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IIT-Bombay Lecture 38 M. Shojaei Baghini

1 4 1 4 | ) ( | ) ( ) (

2 2 2 2 2

+ = = f C R kTR j H f S f S

R

• ut

π ω

Example: Passing White Noise from LPF

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IIT-Bombay Lecture 38 M. Shojaei Baghini

C kT RMS v x x dx C kT df f C R kTR f P

• ut

n

• ut

= ⇒       = + = + =

− ∞

∫ ∫

) ( tan 1 1 4 1 4 ) (

, 1 2 2 2 2 2

π Noise shaping For 1pF capacitor, vn,out=64.3µV (RMS) What about HPF?

Independent of R Example: Passing White Noise from LPF (cont’d)

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IIT-Bombay Lecture 38 M. Shojaei Baghini

f R kT R v i

n n

∆ = = 4

2 2 2

Equivalent Current Noise Source

• One sided PSD
• Example of two parallel resistors