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LINEAR SYSTEMS

Corso di Laurea in Fisica - UNITS

ISTITUZIONI DI FISICA PER IL SISTEMA TERRA

FABIO ROMANELLI

Department of Mathematics & Geosciences University of Trieste

romanel@units.it

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Fabio Romanelli Linear Systems

G(x,s)

Green's function (GF) is a basic solution to a linear

differential equation, a building block that can be used to construct many useful solutions.

If one considers a linear differential equation written as:

L(x)u(x)=f(x)

where L(x) is a linear, self-adjoint differential operator, u(x) is the unknown function, and f(x) is a known

non-homogeneous term, the GF is a solution of:

L(x)u(x,s)=δ(x-s)

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Fabio Romanelli Linear Systems

Why GF is important?

If such a function G can be found for the operator L, then if we multiply the

second equation for the Green's function by f(s), and then perform an integration in the s variable, we obtain:

u(x) = G

(x,s)f(s)ds

Thus, we can obtain the function u(x) through knowledge of the Green's

function, and the source term. This process has resulted from the linearity of the operator L. € L

(x)G(x,s)f(s)ds = δ

(x − s)f(s)ds = f(x) = Lu(x) L G

(x,s)f(s)ds = Lu(x)

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Fabio Romanelli Linear Systems

Linear Systems

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Fabio Romanelli Linear Systems

Convolution

Definition:

f(t) ∗ h(t) = f(τ)h(t − τ)dτ

−∞ ∞

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Fabio Romanelli Linear Systems

Convolution

Pictorially

f(t)

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Fabio Romanelli Linear Systems

Convolution

f(τ)

τ

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Fabio Romanelli Linear Systems

Convolution

Consider the function (box filter):

h(t) = 0 t < − 1 2 1 1 2 ≤ t ≤ 12 0 t > 1 2 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪

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Fabio Romanelli Linear Systems

Convolution

This function windows our function f(t)

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Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

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Fabio Romanelli Linear Systems

Convolution

This function windows our function f(t).

(12)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(13)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(14)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(15)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(16)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(17)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(18)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(19)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(20)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(21)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(22)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(23)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(24)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(25)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(26)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(27)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(28)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(29)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

(30)

Fabio Romanelli Linear Systems

f(τ)

Convolution

This function windows our function f(t)

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Fabio Romanelli Linear Systems

Convolution

This particular convolution smooths out some of the high frequencies in f(t).

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Fabio Romanelli Linear Systems

Various spaces and transforms

Signal

type Continuous time Discrete time Transform Domain

Finite duration Laplace z Continuous complex frequency (s-plane) Finite duration Fourier Discrete-time Fourier (DTFT) Continuous real frequency Periodic Fourier Series Discrete Fourier

Series (DFS)

Discrete real frequency

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Fabio Romanelli Linear Systems

(34)

Fabio Romanelli Linear Systems

Sampling Function

A Sampling Function or Impulse Train is defined by:

S

T

(t) =

k =−∞

δ(t − kT)

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Fabio Romanelli Linear Systems

Sampling Function

The Fourier Transform of the

Sampling Function is itself a sampling function.

The sample spacing is the inverse.

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Fabio Romanelli Linear Systems

Convolution Theorem

The convolution theorem states that convolution in the spatial domain is

equivalent to multiplication in the frequency domain, and viceversa.

f(t) ∗ g(t) ⇔ F(ω)G(ω)

f(t)g(t) ⇔ F(ω) ∗ G(ω)

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Fabio Romanelli Linear Systems

Convolution Theorem

This powerful theorem can

illustrate the problems with our point sampling and provide guidance on

avoiding aliasing.

Consider: f(t) ST(t)

f(t)

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Fabio Romanelli Linear Systems

Convolution Theorem

What does this look like in the Fourier domain?

F(ω)

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Fabio Romanelli Linear Systems

Convolution Theorem

In Fourier domain we would convolve

F(ω)

S(ω)

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Fabio Romanelli Linear Systems

Aliasing

What this says, is that any

frequencies greater than a certain amount will appear intermixed with other frequencies.

In particular, the higher

frequencies for the copy at 1/T

intermix with the low frequencies centered at the origin.

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Fabio Romanelli Linear Systems

Aliasing and Sampling

Note, that the sampling process

introduces frequencies out to infinity. We have also lost the function f(t), and now have only the discrete

samples.

This brings us to our next powerful theory.

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Fabio Romanelli Linear Systems

Sampling Theorem

The Shannon Sampling Theorem

A band-limited signal f(t), with a cutoff frequency of λ, that is sampled with a

sampling spacing of T may be perfectly

reconstructed from the discrete values f[nT] by convolution with the sinc(t) function,

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Fabio Romanelli Linear Systems

Sampling Theory

Why is this?

The Nyquist limit will ensure that the copies of F(ω) do not overlap in the frequency domain.

I can completely reconstruct or determine f(t) from F(ω) using the Inverse Fourier Transform.

(44)

Fabio Romanelli Linear Systems

Sampling Theory

In order to do this, I need to

remove all of the shifted copies of F(ω) first.

This is done by simply multiplying F(ω) by a box function of width 2λ.

F(ω)

S(ω)

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Fabio Romanelli Linear Systems

Sampling Theory

In order to do this, I need to

remove all of the shifted copies of F(ω) first.

This is done by simply multiplying F(ω) by a box function of width 2λ.

F(ω)

S(ω)

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Fabio Romanelli Linear Systems

A Cosine Example

Consider the function f(t)=cos(2πt).

x

ω

F(ω)

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Fabio Romanelli Linear Systems

Sampling Theory

So, given f[nT] and an assumption

that f(t) does not have frequencies

greater than 1/2T, we can write the

formula:

f[nT] = f(t) ST(t) ó F(ω)* ST(ω)

F(ω) = (F(ω)* ST(ω)) Box1/2T(ω)

therefore,

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Fabio Romanelli Linear Systems

A Cosine Example

Now sample it at T=1/2 x ω f(t) 1 F(ω)

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Fabio Romanelli Linear Systems

A Cosine Example

Problem:

The amplitude is now wrong or undefined.

Note however, that there is one

and only one cosine with a frequency less than or equal to 1 that goes

through the sample pts.

ω

1

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Fabio Romanelli Linear Systems

A Cosine Example

What if we sample at T=2/3? x ω f(t) 1 F(ω)

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Fabio Romanelli Linear Systems

Supersampling

Supersampling increases the

sampling rate, and then integrates or convolves with a box filter, which is finally followed by the output

sampling function.

t

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Fabio Romanelli Linear Systems

Sampling and Anti-aliasing

The problem:

The signal is not band-limited.

Uniform sampling can pick-up higher

frequency patterns and represent them as low-frequency patterns.

F(ω)

References

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