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Feb 25, 2024 232 likes •411 views

s tt rtt Prt Prs st s tr P

SLIDE 1

❯s❛❣❡ ♦❢ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❙♦♠❡ Pr❛❝t✐❝❛❧ Pr♦❜❧❡♠s ♦❢ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s

❉♠✐tr② P❛✈❧♦✈

▲❛❜♦r❛t♦r② ♦❢ ❊♣❤❡♠❡r✐s ❆str♦♥♦♠② ■♥st✐t✉t❡ ♦❢ ❆♣♣❧✐❡❞ ❆str♦♥♦♠② ♦❢ t❤❡ ❘✉ss✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ❙t✳ P❡t❡rs❜✉r❣✱ ❘✉ss✐❛

❆♣r✐❧ ✷✶✱ ✷✵✶✽

SLIDE 2

Pr♦❜❧❡♠ ❢♦r♠✉❧❛t✐♦♥✿ ♣r♦♣❛❣❛t✐♦♥ ♦❢ ✉♥❝❡rt❛✐♥t②

❙♦❧❛r s②st❡♠ ❜♦❞✐❡s ❛r❡ ♠♦❞❡❧❡❞ ✇✐t❤ ❛ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿ ˙ x(t) = f(x(t)) ✭❆❝t✉❛❧❧② ✐t ✐s ♠♦r❡ ❝♦♠♣❧❡①✱ ❜✉t ❧❡t ✉s ❧❡❛✈❡ ✐t ❛t t❤❛t ❢♦r ♥♦✇✳✮ ❚❤❡r❡ ❛r❡✿

◮ ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s✿ x(t0) = x0 ◮ P❛r❛♠❡t❡rs {pi} ✭♠❛ss❡s✱ ❣r❛✈✐t② ✜❡❧❞ ❝♦❡✣❝✐❡♥ts✱ ❡t❝✮✳

■❈ ❛♥❞ ♣❛r❛♠❡t❡rs ❛r❡ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ ❛str♦♥♦♠✐❝❛❧ ♦❜s❡r✈❛t✐♦♥s ❛♥❞ ❤❛✈❡ t❤❡✐r ✉♥❝❡rt❛✐♥t✐❡s ❛♥❞ ❝♦rr❡❧❛t✐♦♥s✳ ▲❡t P = (x(1)

0 , . . . , x(n) 0 , p1, . . . , pm)✳

❈♦✈❛r✐❛♥❝❡ ♠❛tr✐① cov(P) ✐s ❡st✐♠❛t❡❞ ❢r♦♠ ♦❜s❡r✈❛t✐♦♥s✳ ❍♦✇ ❞♦ ✇❡ ❡st✐♠❛t❡ cov(x(t))❄

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✷ ✴ ✶✻

SLIDE 3

▲✐♥❡❛r✐③❛t✐♦♥

dx dP =         dx(1) dx(1) · · · dx(1) dx(n) dx(1) dp1 · · · dx(1) dpm ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ dx(n) dx(1) · · · dx(n) dx(n) dx(n) dp1 · · · dx(n) dpm         cov(x) ≈ dx dP cov(P) dx dP

T

dx dP ✭✐s♦❝❤r♦♥♦✉s ❞❡r✐✈❛t✐✈❡✮ ✐s ♥♦✇ ♣❛rt ♦❢ t❤❡ ❞②♥❛♠✐❝ ❡q✉❛t✐♦♥✿ dx dP · = df dP = ∂f ∂x dx dP

dx dP(t0) =    1 · · · · · · ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ · · · 1 · · ·   

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✸ ✴ ✶✻

SLIDE 4

❉✐✛❡r❡♥t✐❛t✐♦♥

❍♦✇ t♦ ❝❛❧❝✉❧❛t❡ ∂f ∂x❄

◮ ◆✉♠❡r✐❝❛❧❛❧❧② ✭❝❛♥ ❛s ✇❡❧❧ ❝❛❧❝✉❧❛t❡ dx dP ❞✐r❡❝t❧②✮

▼♦st s✐♠♣❧❡✱ ❜✉t s✉✛❡rs ❢r♦♠ ♥✉♠❡r✐❝❛❧ ♥♦✐s❡ ❛♥❞ r❡q✉✐r❡s ❡①tr❛ ❝♦♠♣✉t✐♥❣ t✐♠❡✳ ✭❚❤♦✉❣❤ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ dx

dP ❝❛♥ ❜❡ r✉♥

✐♥ ♣❛r❛❧❧❡❧ ❛♥❞ t❤❡ r❡s✉❧ts ❝❛♥ ❜❡ r❡✉s❡❞✮✳

◮ ❙②♠❜♦❧✐❝❛❧❧②

▼♦st ♣r❡❝✐s❡✱ ❜✉t r❡q✉✐r❡s ♠✉❝❤ t✐♠❡ ❛♥❞ ♠❡♠♦r② t♦ ❡①♣❛♥❞ ❡①♣r❡ss✐♦♥s ❛♥❞ ❝❛♥ ❜❡ ✐♥❡✣❝❡♥t✳ ❘❡q✉✐r❡s ❈❆❙ s♦❢t✇❛r❡✳ ❍❛r❞❧② ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ✇❛② ♣r♦❣r❛♠s ❛r❡ ✇r✐tt❡♥ ✭❝♦♥❞✐t✐♦♥s✱ ❧♦♦♣s ❡t❝✮✳

◮ ❆✉t♦♠❛t✐❝❛❧❧②

▼♦st ❝♦♥✈❡♥✐❡♥t ❢♦r ❝♦♠♣❧❡① ♣r♦❣r❛♠s✳ ❯s❡s ❛ ♠✐♥✐♠❛❧ s❡t ♦❢ ✏♣r✐♠✐t✐✈❡✑ ❞❡r✐✈❛t✐✈❡s ♣❧✉s t❤❡ ❝❤❛✐♥ r✉❧❡✳ ❘❡q✉✐r❡s ❝♦❞❡ ❛♥❛❧②s✐s t♦♦❧s ♦r ♣r♦❣r❛♠♠✐♥❣ ❧❛♥❣✉❛❣❡ s✉♣♣♦rt✳

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✹ ✴ ✶✻

SLIDE 5

❆✉t♦♠❛t✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✇✐t❤ ❞✉❛❧ ♥✉♠❜❡rs

❚❤❡ ✐❞❡❛✿ t♦ ❡❛❝❤ ✈❛❧✉❡ x ✐♥ t❤❡ ♣r♦❣r❛♠✱ ✇❡ ❛tt❛❝❤ ❛♥ x′✳ ■♥st❡❛❞ ♦❢ ❛r✐t❤♠❡t✐❝s ♦♥ x ✇❡ ❤❛✈❡ ❛r✐t❤♠❡t✐❝s ♦♥ (x, x + x′ε)✳ ε2 = 0✳ ❙♦♠❡ ❡①❛♠♣❧❡s ❜♦rr♦✇❡❞ ♦♥❧✐♥❡✿

u, u′

+

v, v′

=

u + v, u′ + v′
u, u′

∗

v, v′

=

uv, u′v + uv′
u, u′

/

v, v′

= u v , u′v − uv′ v2

(u = 0)

sin

u, u′

=

sin(u), u′ cos(u)
u, u′k =
uk, kuk−1u′

(u = 0) ❉✉❛❧ ♥✉♠❜❡r ❛r❡ ❡❛s✐❧② ❡①t❡♥❞❛❜❧❡ t♦ ♠✉❧t✐♣❧❡ ❞❡r✐✈❛t✐✈❡s✿ ❢r♦♠ ❞✉❛❧ ♥✉♠❜❡rs t♦ tr✐♣❧❡ ♥✉♠❜❡rs t♦ (1 + m + n)✲♥✉♠❜❡rs✳

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✺ ✴ ✶✻

SLIDE 6

❈❛s❡ ❢♦r ❛ ❞♦♠❛✐♥✲s♣❡❝✐✜❝ ❧❛♥❣✉❛❣❡

■♥ ♦r❞❡r t♦ ❜❡ ✉s❛❜❧❡ ❢♦r t❤❡ t❛s❦✱ ♣r♦❣r❛♠ ❝♦❞❡ ♠✉st

◮ ❇❡ r❡str✐❝t✐✈❡ ❡♥♦✉❣❤ ✭♥♦ ❢❛♥❝② ♠❡♠♦r② ❛❝❝❡ss✱ ♥♦ ❝♦♥❞✐t✐♦♥s

✐♥✈♦❧✈✐♥❣ st❛t❡ ✈❛r✐❛❜❧❡s

◮ ◆❡✈❡rt❤❡❧❡ss ❜❡ ❡①♣r❡ss✐✈❡ ❡♥♦✉❣❤ t♦ ❞❡✜♥❡ ❝♦♠♣❧❡① s②st❡♠s

✭❧♦♦♣s✱ ❝♦♥❞✐t✐♦♥s ✐♥✈♦❧✈✐♥❣ ♥♦♥✲st❛t❡ ✈❛r✐❛❜❧❡s✮

◮ ◆♦t ❤❛✈❡ ❡①t❡r♥❛❧ ❝❛❧❧s ✭♦r ❤❛✈❡ t❤❡♠ ♣r♦♣❡r❧② ❛♥♥♦t❛t❡❞✮ ◮ ❍❛✈❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ✇❤❛t st❛t❡ ✈❛r✐❛❜❧❡s ❛r❡ ✐s♦❝❤r♦♥♦✉s

❞❡r✐✈❛t✐✈❡s

◮ Pr♦✈✐❞❡ ✐♥t❡r❢❛❝❡ t♦ ❜❡ ❝❛❧❧❡❞ ❢r♦♠ ❛ ❣❡♥❡r✐❝ ♥✉♠❡r✐❝❛❧

✐♥t❡❣r❛t♦r ❛s ❛ ✏❜❧❛❝❦ ❜♦①✑ ❚❤❡ ♠♦st ❝♦♥✈❡♥✐❡♥t ❝❤♦✐❝❡ ✐s t♦ ❝r❡❛t❡ ❛ ❉❙▲✳

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✻ ✴ ✶✻

SLIDE 7

LANDAU✿ ▲❆◆❣✉❛❣❡ ❢♦r ❉②♥❛♠✐❝❛❧ s②st❡♠s ✇✐t❤ ❆❯t♦♠❛t✐❝

❞✐✛❡r❡♥t✐❛t✐♦♥

◮ ❊①♣❧✐❝✐t❧② t②♣❡❞❀ ❤❛s r❡❛❧s✱ ✐♥t❡❣❡rs✱ ❛♥❞ ❛rr❛②s ♦❢ ✐♥t❡❣❡rs ♦r

r❡❛❧s

◮ ❋✉♥❝t✐♦♥s✱ ❧♦♦♣s✱ ❝♦♥❞✐t✐♦♥s ◮ ●❧♦❜❛❧ ❝♦♥st❛♥ts✱ ❢✉♥❝t✐♦♥ ❛r❣✉♠❡♥ts ✭✐♠♠✉t❛❜❧❡✮✱ ❛♥❞

t❡♠♣♦r❛r② ✈❛r✐❛❜❧❡s ✭♠✉t❛❜❧❡✮

◮ ✏P❛r❛♠❡t❡rs✑✿ ❢✉♥❝t✐♦♥s ❞♦ ♥♦t ❤❛✈❡ t❤❡♠ ❞✐r❡❝t❧② ❛s

❛r❣✉♠❡♥ts✱ ❜✉t ❤❛✈❡ ❞❡r✐✈❛t✐✈❡s ✇✳r✳t t❤❡♠ ✐♥ t❤❡ st❛t❡ ✈❡❝t♦r

◮ ❆♥♥♦t❛t✐♦♥s ❢♦r ❞❡r✐✈❛t✐✈❡s ✐♥ t❤❡ st❛t❡ ✈❡❝t♦r ✭✐♥♣✉t✮ ❛♥❞

r❡t✉r♥❡❞ ✈❡❝t♦r ✭♦✉t♣✉t✮

◮ ❖♣t✐♦♥ t♦ ♠❛♥✉❛❧❧② ❞✐s❝❛r❞ ❛✉t♦♠❛t✐❝ ❞❡r✐✈❛t✐✈❡s ✇❤❡r❡ t❤❡②

❛r❡ ♥♦t s✐❣♥✐✜❝❛♥t ■♠♣❧❡♠❡♥t❡❞ ✐♥ ❘❛❝❦❡t ♣❧❛t❢♦r♠ ✭❋❡❧❧❡✐s❡♥ ❡t ❛❧✳ ✷✵✶✽✮✳ ❈✉rr❡♥t❧② LANDAU ❝♦♠♣✐❧❡s t♦ ❘❛❝❦❡t✱ ❈ ❝♦❞❡ ❣❡♥❡r❛t✐♦♥ ✐s ♣❧❛♥♥❡❞✳

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✼ ✴ ✶✻

SLIDE 8

❈♦❞❡ ❡①❛♠♣❧❡ ✭♣❛rt ✶✮

const int N = 3 # Sun , Earth -Moon barycenter , Moon (geocentric) # Annotated parameters: geocentric Moon initial position and velocity parameter initial [6] real[N * 6 + 36 + N * 6] xdot ( real[N * 6 + 36 + N * 6] x, # N-body state + derivatives real[N] GM) # masses { # annotating derivatives in the state vector x[12:18] ’ initial [0 : 6] = x[N * 6 : N * 6 + 36] x[12:18] ’ GM[2] = x[N * 6 + 36 : N * 6 + 36 + N * 6] real[N * 6] xb , xdotb # like x and xdot , but barycentric # Neglect the effect of the Moon ’s orbit and mass to the Sun discard xdotb [3 : 6] ’ initial [0 : 6] discard xdotb [3 : 6] ’ GM[2] xb[0 : 6] = x[0 : 6] # Sun is already barycentric # Transfer time derivatives from x to their xdot counterparts for j = [0 : N] xdot[j * 6 : j * 6 + 3] = x[j * 6 + 3: j * 6 + 3 + 6]

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✽ ✴ ✶✻

SLIDE 9

❈♦❞❡ ❡①❛♠♣❧❡ ✭♣❛rt ✷✮

for k = [0 : 6] { # Calculate barycentric Earth and Moon xb[6 + k] = x[6 + k] - x[12 + k] * 1 / (1 + GM[1] / GM[2]) xb[12 + k] = xb[6 + k] + x[12 + k] } for i = [0 : N] # Apply Newtonian laws for j = [0 : N] if i != j { real dist2 = sqr(xb[6 * i] - xb[6 * j]) + sqr(xb[6 * i + 1] - xb[6 * j + 1]) + sqr(xb[6 * i + 2] - xb[6 * j + 2]) real dist3inv = 1 / (dist2 * sqrt(dist2 )) for k = [0 : 3] xdotb [6*i + 3 + k] += GM[j] * (xb[6*j + k] - xb[6*i + k]) * dist3inv } xdot [0:6] = xdotb [0:6] # Back to EMB + geocentric Moon coordinates for xdot for k = [0 : 6] { xdot [12+k] = xdotb [12+k] - xdotb [6+k] xdot [6+k] = (xdotb [6+k] * GM[1] / GM[2] + xdotb [12+k]) / (1 + GM[1] / GM[2]) } # Annotating the parameter derivatives in xdot xdot[N * 6 : N * 6 + 36] = xdot [12:18] ’ initial [0 : 6] xdot[N * 6 + 36 : N * 6 + 36 + N * 6] = xdot [12:18] ’ GM[2] }

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✾ ✴ ✶✻

SLIDE 10

❙✉♠♠❛r② ♦❢ t❤❡ ♣r♦♣♦s❡❞ s♦❧✉t✐♦♥

◮ ❙✐♥❣❧❡ r✉♥ ♦❢ ❆❞❛♠s✕❇❛s❤❢♦rt❤✕▼♦✉❧t♦♥ ♥✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t♦r

❢♦r t❤❡ ♦r❜✐ts ❛♥❞ ✐s♦❝❤r♦♥♦✉s ❞❡r✐✈❛t✐✈❡s

◮ ❊q✉❛t✐♦♥s ♦❢ ♠♦t✐♦♥ ✭r✐❣❤t✲❤❛♥❞ s✐❞❡ ❢♦r t❤❡ ✐♥t❡❣r❛t♦r✮ ❝♦❞❡❞

✐♥ ▲❆◆❉❆❯ ❛♥❞ ❝♦♠♣✐❧❡❞ ❜② ❘❛❝❦❡t ✇✐t❤ ❛✉t♦♠❛t✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✈✐❛ ❞✉❛❧ ♥✉♠❜❡rs

◮ Pr♦♣❛❣❛t✐♦♥ ♦❢ ✉♥❝❡rt❛✐♥t② ❜② ❧✐♥❡❛r✐③❡❞ ❡st✐♠❛t✐♦♥ ❢r♦♠ t❤❡

✐♥✐t✐❛❧ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ◆♦ ♣♦❧②♥♦♠✐❛❧ ❛❧❣❡❜r❛ ②❡t✳

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✶✵ ✴ ✶✻

SLIDE 11

❊st✐♠❛t✐♦♥ ♦❢ ▼♦♦♥✬s ♦r❜✐t ❛❝❝✉r❛❝②

◮ ❋r♦♠ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥

σ(xi)2 =

n+m

dxi dPj dxi dPk cov(P)jk

◮ ❋r♦♠ ▼♦♥t❡✲❈❛r❧♦ s✐♠✉❧❛t✐♦♥

❈❤♦❧❡s❦② ❞❡❝♦♠♣♦s✐t✐♦♥✿ cov(P) = LT L✱ ✇❤❡r❡ L(n+m)×(n+m) ✐s ❛ ❧♦✇❡r tr✐❛♥❣✉❧❛r ♠❛tr✐① ❙❛♠♣❧❡ xrandom = LT y✱ ✇❤❡r❡ yi (1 ≤ i ≤ n + m) ❛r❡ ✉♥❝♦rr❡❧❛t❡❞ N(0, 1) r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t❡st✱ P = {x0, y0, z0, ˙ x0, ˙ y0, ˙ z0, ωz0, β, γ, kv/cT , τ, τR1, τR2}

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✶✶ ✴ ✶✻

SLIDE 12

❯♥❝❡rt❛✐♥t② ♣r♦♣❛❣❛t✐♦♥ ❜② ▼♦♥t❡✲❈❛r❧♦

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✶✷ ✴ ✶✻

SLIDE 13

❈♦♠♣❛r✐s♦♥✿ ❳ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ▼♦♦♥

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✶✸ ✴ ✶✻

SLIDE 14

❈♦♠♣❛r✐s♦♥✿ ❨ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ▼♦♦♥

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✶✹ ✴ ✶✻

SLIDE 15

❈♦♠♣❛r✐s♦♥✿ ❩ ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ ▼♦♦♥

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✶✺ ✴ ✶✻

SLIDE 16

❖t❤❡r ❛♣♣r♦❛❝❤❡s

❚❛②❧♦r ♠❡t❤♦❞ ♦❢ ♥✉♠❡r✐❝❛❧ ✐♥t❡❣r❛t✐♦♥ ✭❏♦r❜❛ ❛♥❞ ❩♦✉✱ ✷✵✵✹✮

◮ ❆♣♣r♦①✐♠❛t❡s x(t) ❛s ❛ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ◮ ❍❡❛✈✐❧② r❡❧✐❡s ♦♥ ❛✉t♦♠❛t✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ❝♦❞❡ ❣❡♥❡r❛t✐♦♥ ◮ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ❝❛♥ ❜❡ ✉s❡❞ t♦ ♣r♦♣❛❣❛t❡ ✉♥❝❡rt❛✐♥t② ◮ ❘❡q✉✐r❡s ❤✐❣❤❡r ♦r❞❡r ❞❡r✐✈❛t✐✈❡s✱ s♦ ❞✉❛❧ ♥✉♠❜❡rs ✇✐❧❧ ♥♦t

✇♦r❦ ❏❡t tr❛♥s♣♦rt ✭Pér❡③✲P❛❧❛✉✱ ▼❛s❞❡♠♦♥t✱ ●♦♠❡③✱ ✷✵✶✸✮ ❈❤❡❜②s❤❡✈ ♣♦❧②♥♦♠✐❛❧ ❛❧❣❡❜r❛ ✭❘✐❝❝❛r❞✐✱ ❚❛r❞✐♦❧✐✱ ❱❛s✐❧❡✱ ✷✵✶✺✮

❉♠✐tr② P❛✈❧♦✈ ❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✐♥ ❈❡❧❡st✐❛❧ ▼❡❝❤❛♥✐❝s P❈❆ ✬✷✵✶✽✱ ❊■▼■✱ ❙t✳ P❡t❡rs❜✉r❣ ✶✻ ✴ ✶✻