Filters and NoiseOptional Assessment of Practical Importance

Rubin H Landau

Sally Haerer, Producer-Director

Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu

with Support from the National Science Foundation

Course: Computational Physics II

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Problem: Cleaning Up Noisy a Signal

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Problem: What is pure signal?

Two Simple Approaches1 Autocorrelation functions

2 Digital Filters

Both wide applications

More filters in Wavelet Analysis & Data Compression

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Noise Reduction via Autocorrelation (Theory)

Assumption: Noise just adds to signal

Measure = Signal + Noise

y(t) = s(t) + n(t)

s(t) = ?

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Science: assume simplest (+)

Science: noise ' random (∞ o F)

Recall “random” sequence; ri ; ri+1

⇒ n(t) not correlated with s(t), n(t)3 / 1

Correlation Function c(t)

How measure correlation?

y(t) = sinω, x(t) = sin(nωt + φ) correlated

c(τ) =

∫ +∞

−∞dt y∗(t) x(t + τ) (Correlation Function)

Correlated (τ = lag time = variable):Integrand > 0 for some τ⇒ Constructive interference ⇒ c(τ)→∞

Not correlated:2 functions oscillate independently+, - equally likely⇒ Destructive interference ⇒ c(τ) ' 0

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More Correlation Function

PropertiesExpress c, y∗, x via FT & substitute:

(FT) c(τ) =

∫ +∞

−∞dω′ C(ω′)

eiω′τ√

2π(1)

(Def) c(τ)def=

∫ +∞

−∞dt y∗(t) x(t + τ) (2)

⇒ C(ω) =√

2π Y ∗(ω)X (ω) (3)

Requires convergence to rearrangeRelated to convolution theorem (soon)

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Special Correlation Function: Autocorrelation

Measure Correlation with Itself: a(τ)

a(τ)def=

∫ +∞

−∞dt y∗(t) y(t + τ)

To compute: fold or convolute with self:

y(t) = measured signal

Average over time for “all” τ values

a(0) = “large”

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Averaging Removes Random Noise from a(t)Proof by substitution

y(t) =s(t) + n(t) (Noisy Signal) (4)

ay (τ)def=

∫ +∞

−∞dt y∗(t) y(t + τ) (Def a(t)) (5)

=

∫ +∞

−∞dt [s(t)s(t + τ) + s(t)n(t + τ) + n(t)n(t + τ)]

⇒ ay (τ) '∫ +∞

−∞dt s(t) s(t + τ) = as(τ) QED Magic (6)

So As(ω) '√

2π |S(ω)|2 ∝ Power Spectrum (7)

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How Apply to Data?

Start with Noisy signal

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Power Spectrum (with Noise)

Frequency

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Function y(t) + Noise After Low pass Filter

t (s)

y

tau (s)

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Autocorrelation Function A(tau)

A

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AA B

C D

1 Compute autocorrelationfunction

2 DFT: a(t) ⇒ A(ω)

3 ⇒ power spectrum w/orandom noise

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Autocorrelation Function Exercises (Noise.java)

Example1 Signal: s(t) = 1

1−0.9 sin t ' 1 + 0.9 sin t + (0.9 sin t)2 · · ·

2 DFT ⇒ S(ω), Plot |S(ω)|2.

3 Autocorrelation function a(t) of s(t)?

4 Power spectrum a(t) vs |S(ω)|2?

5 Add noise y(ti) = s(ti) + α(2ri − 1), 0fuss ≤ α ≤ hide

6 Plot y(t), Y (ω), Power spectrum.

7 a(t) → A(ω).

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Filtering with Transforms (Theory)

Action of Filter

g(t) =

∫ +∞

−∞dτ f (τ) h(t − τ) (8)

def= f (t) ∗ h(t) (analog filter) (9)

h(t) def= unit impulse

responseh(t) =

∫ +∞−∞ dτ δ(τ) h(t − τ)

Greens functionh(0) = max, h(< 0) = 0∗ = Convolution

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Convolution Theorem

Filter as Convolution

g(t) =

∫ +∞

−∞dτ f (τ) h(t − τ) (10)

G(ω) =√

2π F (ω) H(ω) (11)

Proof: FT,δ,∫

Simpler in ω than tDigital: response (ωn)

Lowpass: ↓ high ωHighpass: ↓ low ω

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Digital Filters

Filter Coefficients cn = Complete Description

Filter Def: g(t) =

∫ +∞

−∞dτ f (τ) h(t − τ) (12)

Digital Transfer: h(t) =N∑

n=0

cn δ(t − nτ) (13)

⇒ g(t) =N∑

n=0

cn f (t − nτ) (14)

cn: integration wts N point DFT + response

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Exploration: Windowed Sinc Filters (Filter.java)�

Lowpass Filter to Reduce Noise, Aliasing

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Ideal frequency response

Frequency

Amplitude

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Windowed-sinc filter kernel

Time

Amplitude

Ideal lowpassY (ω) = rectangular pulse ⇒ y(t) = sinc function∫ +∞

−∞dω e−iωt rect(ω) = sinc

(t2

)def=

sin(πt/2)

πt/2

⇒ filter out high ω: convolute with sin(ωc t)/(ωc t)Exercise: Repeat random noise addition using sinc filter

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