introduction to spectral analysis (chapter 11) · 2008-03-24 · arthur berg introduction to...

Introduction to Spectral Analysis (Chapter 11)

§11.2: Orthogonal Functions §11.6: Fourier Representation of Continuous-Time Functions Periodogram

Outline

1 §11.2: Orthogonal Functions

2 §11.6: Fourier Representation of Continuous-Time Functions

3 Periodogram

Arthur Berg Introduction to Spectral Analysis (Chapter 11) 2/ 19


Leonhard Euler (1707 — 1783)



Euler’s Formula

Euler’s formula:eiω = cosω + i sinω

gives rise to the Euler identity

eiπ + 1 = 0

In combination with the identities

sinω =e−ω − e−iω

2i

and

cosω =e−ω + e−iω

2iOne can conclude the following set of functions are orthogonal{

sin(

2πktn

), cos

(2πkt

n

): k = 0, 1, . . . ,

[n2

]}Arthur Berg Introduction to Spectral Analysis (Chapter 11) 4/ 19


Outline



3 Periodogram



Jean Baptiste Joseph Fourier (1768 — 1830)



Fourier Representation



Fourier Analysis

A “well behaved” function1, f (t), defined on (−P,P) has the following Fourierseries representation:

f (t) =a0

2+∞∑

j=1

(aj cos(jt) + bj sin(jt))

where

a0 =1P

∫ P

−Pf (t) dt

aj =1P

∫ P

−Pf (t) cos

(jtπP

)dt

bj =1P

∫ P

−Pf (t) sin

(jtπP

)dt

1satisfies the Dirichlet conditions...finite number of extrema in a given interval, finite numberof discontinuities in a given interval, and absolutely integrable over a period



Fourier Analysis

Or when defined on (0, 2P)...

f (t) =a0

2+∞∑

j=1


where

a0 =1P

∫ 2P

0f (t) dt

aj =1P

∫ 2P

0f (t) cos

(jtπP

)dt

bj =1P

∫ 2P

0f (t) sin

(jtπP

)dt



Fourier Approximation

The function f (t) can be approximated by truncating the infinite summation as:

f (t) ≈ a0

2+

k∑j=1




Nyquist Rate

The Nyquist rate is the minimum sampling rate required to avoid aliasing.



Some Examples



Some More Examples

AR(1) with φ = .9



Outline



3 Periodogram



Deterministic Example> t = 1:100> x1 = 2*cos(2*pi*t*6/100) + 3*sin(2*pi*t*6/100)> x2 = 4*cos(2*pi*t*10/100) + 5*sin(2*pi*t*10/100)> x3 = 6*cos(2*pi*t*40/100) + 7*sin(2*pi*t*40/100)> x = x1 + x2 + x3> par(mfrow=c(2,2))> plot.ts(x1, ylim=c(-10,10),+ main = expression(omega==6/100~~~A^2==13))> plot.ts(x2, ylim=c(-10,10),+ main = expression(omega==10/100~~~A^2==41))> plot.ts(x3, ylim=c(-10,10),+ main = expression(omega==40/100~~~A^2==85))> plot.ts(x, ylim=c(-16,16),main="sum")



Periodogram Example> P = abs(2*fft(x)/100)^2> f = 0:50/100> plot(f, P[1:51], type="o", xlab="frequency", ylab="periodogram")



Scaled Periodogram Equation

P(j/n) =

(2n

n∑t=1

Zt cos(2πtj/n)

)2

+

(2n

n∑t=1

Zt sin(2πtj/n)

)2

A time series Z1, . . . , xn with n odd has the exact representation

Zt = a0 +(n−1)/2∑

j=0

[aj cos(2πtj/n) + bj sin(2πtj/n)] .

From this representation of Zt, we have

P(j/n) = a2j + b2

j



Spectral Representation Theorem

Theorem (Spectral Representation)Any stationary process Zt has the representation

Zt =∫ π

0cos(ωt) du(ω) +

∫ π

0sin(ωt) dv(ω)

where u(ω) and v(ω) are uncorrelated continuous process with orthogonalincrements.

This theorem suggests the time series Zt can be approximated as

Zt =q∑

k=1

Uk cos(2πωkt) + Vk sin(2πωkt)

where Uk and Vk are independent zero-mean random variables.



The Link

The periodogram can be obtained from the ACF via the finite Fourier transformof the ACF.

Therefore we will be interested in the Fourier transform of γ(k)...the spectraldensity.


introduction to spectral analysis (chapter 11) · 2008-03-24 · arthur berg introduction to...

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