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❋❖❙❉❊▼ ✷✵✶✺

❙♦❢t✇❛r❡ ❉❡✜♥❡❞ ❘❛❞✐♦ ❞❡✈r♦♦♠

❆r✐t❤♠❡t✐❝ ❜❛s❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ❛ q✉❛❞r❛t✉r❡ ❋▼ ❉❡♠♦❞✉❧❛t♦r ❙❉❘ ✐♥ ●♥✉❘❛❞✐♦ ❉❡♥✐s ❇❡❞❡r♦✈ ✭❉▲✸❖❈❑✮

❖✈❡r✈✐❡✇

❖✈❡r✇✐❡✇✿ ✶✳ ❇❛❝❦❣r♦✉♥❞ ♦❢ ❛♥❣❧❡ ♠♦❞✉❧❛t✐♦♥ ✷✳ ❚r❛❞✐t✐♦♥❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r ✸✳ ❆r✐t❤♠❡t✐❝❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r ✹✳ Pr❡s❡♥t❛t✐♦♥ ♦❢ r❡s✉❧ts ✺✳ ❇✉❣ ✜①✐♥❣ ✐♥ ❢❛st ❛t❛♥

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✶

✶✳ ❇❛❝❦❣r♦✉♥❞ ♦❢ ❛♥❣❧❡ ♠♦❞✉❧❛t✐♦♥

• ❡♥❡r❛❧❧ ❛♥❣❧❡ ♠♦❞✉❧❛t✐♦♥ ✐♥ r❡❛❧ ✇♦r❧❞ ✭❘❋✮

ωc ❢r❡q✉❡♥❝② ♦❢ ❝❛rr✐❡r✱ ♦♥ ✇❤✐❝❤ t❤❡ ♠♦❞✉❧❛t❡❞ s✐❣♥❛❧ ✐s tr❛♥s♠✐t❡❞ ✐♥ t❤❡ r❡❛❧ ✇♦r❧❞ f(t) ∈ R ❧♦✇ ❢r❡q✉❡♥❝② ♠♦❞✉❧❛t✐♥❣ s✐❣♥❛❧ y(t) ∈ R r❡❛❧ s✐❣♥❛❧ tr❛♥s♠✐t❡❞ ❜② ωc

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✷

✶✳ ❇❛❝❦❣r♦✉♥❞ ♦❢ ❛♥❣❧❡ ♠♦❞✉❧❛t✐♦♥

❘❡❝❡✐✈✐♥❣ ♦❢ ❛♥❣❧❡ ♠♦❞✉❧❛t✐♦♥ yT(t) ∈ C tr❛♥s♠✐t❡❞ s✐❣♥❛❧ ✐♥ ❧♦✇ ♣❛ss ❛r❡❛ yT(t) = yTR(t)

• I

+j · yTI(t)

• Q

✭✶✮

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✸

✶✳ ❇❛❝❦❣r♦✉♥❞ ♦❢ ❛♥❣❧❡ ♠♦❞✉❧❛t✐♦♥

❋♦r♠❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❆♥❣❧❡ ♠♦❞✉❧❛t✐♦♥ y(t) = A · cos

 Φ(t)  

✭✷✮ PM : Φ(t) = ϕ0 + α · f(t) ✭✸✮ FM : Ω(t) = dΦ(t) dt = ˙ Φ = ω0 + α · f(t) ✭✹✮ y(t) = A · cos

 ω0t + α ·

t

• −∞

f(τ) dτ + ϕ0

 

✭✺✮ yT(t) = A · e

j

• ∆ωt+α·

t

• −∞

f(τ) dτ+ϕ0

• ✭✻✮

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✹

✷✳ ❚r❛❞✐t✐♦♥❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r

▼❛t❤❡♠❛t✐❝❛❧ ❇❛❝❦❣r♦✉♥❞ ❢♦r ❛ tr❛❞✐t✐♦♥❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r ✭■✮ ❇❛s✐❝ ✐❞❡❛ ✐s t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❝✉rr❡♥t s❛♠♣❧❡ ✇✐t❤ ❝♦♥❥✉❣❛t❡ ❝♦♠✲ ♣❧❡① ✈❡rs✐♦♥ ♦❢ ♣r❡✈✐❡✇ s❛♠♣❧❡✳ yT(nts) · y∗

T((n − 1)ts)

✭✼✮ = A·e

j

• ∆ωtsn+α·

nts

• −∞

f(τ) dτ+ϕ0

• ·

A·e

−j

• ∆ωts(n−1)+α·

(n−1)ts

• −∞

f(τ) dτ+ϕ0

• = A2 · e

j

• ∆ωts+α·

nts

• (n−1)ts

f(t) dt

• ✭✽✮

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✺

✷✳ ❚r❛❞✐t✐♦♥❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r

▼❛t❤❡♠❛t✐❝❛❧ ❇❛❝❦❣r♦✉♥❞ ❢♦r ❛ tr❛❞✐t✐♦♥❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r ✭■■✮ ❚❤❡ s♣❡❝✐✜❝ ✐♥t❡❣r❛❧ ❝❛♥ ❜❡ ❛♣♣r♦①✐♠❛t❡❞ ❜② s♠❛❧❧ ✈❛❧✉❡ ♦❢ ts ✐♥ ❢♦❧❧♦✇✐♥❣ ✇❛②✿

nts

• (n−1)ts

f(t) dt ≈ tsf(nts) ◆♦✇ ✇❡ ❝❛♥ s✐♠♣❧✐❢② ✭✽✮ (8) = A2 · e

j

• ∆ωts+α·ts·f(nts)
• = A2 · e

j

• 2π∆fts+2πdev·ts·f(nts)
• ❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺

♣❛❣❡ ✻

✷✳ ❚r❛❞✐t✐♦♥❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r

◗✉❛❞r❛t✉r❡ ❋▼✲❉❡♠♦❞ ✐♥ ●♥✉❘❛❞✐♦

• ut(n) = Gain

1 2πts

·arc

• (yT(n) · y∗

T(n − 1)

• = ∆f + dev · f(nts)

✭✾✮ ✧❛r❝✧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❝❛♥ ❜❡ ❜❛s❡❞ ♦♥ ❛r❝t❛♥✱ ❛r❝❝♦s✱ ❛r❝s✐♥ ❛♥❞ ♥❡❡❞s ♦❢t❡♥ ❤✐❣❤ ❝♦♠♣✉t❛t✐♦♥ ❡✛♦rt✳

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✼

✸✳ ❆r✐t❤♠❡t✐❝❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r

▼❛t❤❡♠❛t✐❝❛❧ ❇❛❝❦❣r♦✉♥❞ ❢♦r ❛♥ ❛r✐t❤♠❡t✐❝❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r ✭■✮ ❇❛s✐❝ ✐❞❡❛ ✐s t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦❢ ❞❡r✐✈❛t✐♦♥ s✐❣♥❛❧ ✇✐t❤ ✇✐t❤ ❝♦♥❥✉❣❛t❡ ❝♦♠♣❧❡① s✐❣♥❛❧✳ ˙ yT(t) · y∗

T(t)

✭✶✵✮ ˙ yT(t) = j · A ·

• ∆ω + α · f(t)
• · e

j

• ∆ωt+α·

t

• −∞

f(τ) dτ+ϕ0

• y∗

T(t) = A · e −j

• ∆ωt+α·

t

• −∞

f(τ) dτ+ϕ0

• ⇒

˙ yT(t) · y∗

T(t) = j · A2

• ∆ω + α · f(t)
• ✭✶✶✮

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✽

✸✳ ❆r✐t❤♠❡t✐❝❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r

▼❛t❤❡♠❛t✐❝❛❧ ❇❛❝❦❣r♦✉♥❞ ❢♦r ❛♥ ❛r✐t❤♠❡t✐❝❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r ✭■■✮ ˙ yT(t) · y∗

T(t) = (i′ + j · q′) · (i − j · q)

= (i′ · i + q′ · q)

• =0

+j · (i · q′ − i′ · q) ! = j · A2

• ∆ω + α · f(t)
• ⇒

(i · q′ − i′ · q) = A2

• ∆ω + α · f(t)
• ✭✶✷✮

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✾

✸✳ ❆r✐t❤♠❡t✐❝❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r

▼❛t❤❡♠❛t✐❝❛❧ ❇❛❝❦❣r♦✉♥❞ ❢♦r ❛♥ ❛r✐t❤♠❡t✐❝❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r ✭■■■✮ ◆♦✇ ❧❡t ✉s ❛ss✉♠❡ t❤❛t in := i(nts) in−1 := i((n − 1)ts) qn := q(nts) qn−1 := q((n − 1)ts) ◆♦✇ ✇❡ ❝❛♥ ✇r✐t❡ t❤❡ ❧❡❢t s✐t❡ ♦❢ ✭✶✷✮ ❛s ❢♦❧❧♦✇ i · q′ − i′ · q = in qn − qn−1 ts − in − in−1 ts qn = 1 ts (inqn − inqn−1 − inqn + in−1qn) = 1 ts (in−1qn − inqn−1) ✭✶✸✮

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✶✵

✸✳ ❆r✐t❤♠❡t✐❝❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r

▼❛t❤❡♠❛t✐❝❛❧ ❇❛❝❦❣r♦✉♥❞ ❢♦r ❛♥ ❛r✐t❤♠❡t✐❝❛❧ ❋▼✲❉❡♠♦❞✉❧❛t♦r ✭■❱✮ ◆♦✇ ✇❡ ♣✉t t❤❡ ✭✶✸✮ ✐♥t♦ ✭✶✷✮ ❛♥❞ ♦❜t❛✐♥✿ 1 ts (in−1qn − inqn−1) = A2

• ∆ω + α · f(nts)
• 1

2πts · (i2

n + q2 n)

• A2

(in−1qn − inqn−1) = ∆f + dev · f(nts) ✭✶✹✮ ◆♦✇ ✇❡ ❝❛♥ r❡✇r✐t❡ t❤✐s r❡s✉❧t s✐♠✐❧❛r ❧✐❦❡ ✭✾✮

• ut(n) = Gain

1 2πts

·in−1qn − inqn−1 i2

n + q2 n

= ∆f + dev · f(nts) ✭✶✺✮

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✶✶

✹✳ Pr❡s❡♥t❛t✐♦♥ ♦❢ r❡s✉❧ts

◆♦✇ ❜♦t❤ ❉❡♠♦❞✉❧❛t♦rs ✐♥ ❝❛rt❡s✐❛♥ ❢♦r♠ ✧❛r❝✧ ❜❛s❡❞ ❋▼✲❉❡♠♦❞✉❧❛t♦r✿

• ut(n)

= Gain

1 2πts

·arc

• (inin−1 + qnqn−1) + j · (qnin−1 − inqn−1)
• =

∆f + dev · f(nts) ✭✶✻✮ ✧❛r✐t❤♠❡t✐❝✧ ❜❛s❡❞ ❋▼✲❉❡♠♦❞✉❧❛t♦r✿

• ut(n) = Gain

1 2πts

·in−1qn − inqn−1 i2

n + q2 n

= ∆f + dev · f(nts) ✭✶✼✮

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✶✷

✹✳ Pr❡s❡♥t❛t✐♦♥ ♦❢ r❡s✉❧ts

❈♦♠♣❛r✐s♦♥ ♦♥ ❝♦♠♣✉t❛t✐♦♥ ♣❡r❢♦r♠❛♥❝❡ ✧❛r❝✧ ■♠♣❧❡♠❡♥t❛✐♦♥

• ❊◆❊❘■❈

❙❙❊✸ ✶✶✱✷✽s ✾✱✾✵s ✧❛r✐t❤♠❡t✐❝✧ ■♠♣❧❡♠❡♥t❛t✐♦♥

• ❊◆❊❘■❈

❙❙❊✸ ✶ t❤r❡❛❞ ✸✱✶✵s ✶✱✽✵s ✷ t❤r❡❛❞s ✶✱✻✺s ✶✱✵✹s ✸ t❤r❡❛❞s ✶✱✷✸s ✵✱✽✶s ✹ t❤r❡❛❞s ✵✱✾✽s ✵✱✻✺s ❚✐♠❡s ❛r❡ ♠❡❛s✉r❡❞ ❢♦r ❝♦♠♣✉t❛t✐♦♥ ♦❢ s♦♠❡ ♠✐❧❧✐♦♥s ♦❢ s❛♠♣❧❡s ♦♥ ❛ ✹ ❝♦r❡ ✐✼✲❈P❯✳

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✶✸

✹✳ Pr❡s❡♥t❛t✐♦♥ ♦❢ r❡s✉❧ts

❱✐s✉❛❧✐③❛t✐♦♥ ♦❢ ❞❡♠♦❞✉❧❛t✐♦♥ ❡rr♦r ❉❡✈✐❛t✐♦♥ ✐♥ ❍③ fmod ✐♥ ❍③ fs ✐♥ ❍③ εarc ✐♥ ❞❇ εarith ✐♥ ❞❇ ✶✵✵✵ ✶✵ ✽✵✵✵ ✲✶✵✼ ✲✷✷ ✶✵✵✵ ✶✵ ✶✹✵✵✵ ✲✶✶✶ ✲✸✷ ✶✵✵✵ ✶✵ ✷✺✵✵✵ ✲✶✶✶ ✲✹✷ ✶✵✵✵ ✶✵ ✹✺✵✵✵ ✲✶✶✶ ✲✺✷ ✶✵✵ ✶✵ ✽✵✵✵ ✲✶✵✺ ✲✻✷ ✶✵✵ ✶✵✵ ✽✵✵✵ ✲✼✷ ✲✻✵ ✶✵✵ ✹✵✵ ✽✵✵✵ ✲✹✽ ✲✹✻ ✶✵✵✵ ✶✵✵✵ ✽✵✵✵ ✲✸✷ ✲✷✵ ✶✷✵✵ ✷✹✵✵ ✷✽✽✵✵ ✲✸✾ ✲✸✹

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✶✹

✺✳ ❇✉❣ ✜①✐♥❣ ✐♥ ❢❛st ❛t❛♥

❇✉❣ ✜①✐♥❣ ✐♥ t❤❡ ❢❛st ❛t❛♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ■ error(x) := fast atan(x) − arctan(x)

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✶✺

✺✳ ❇✉❣ ✜①✐♥❣ ✐♥ ❢❛st ❛t❛♥

❇✉❣ ✜①✐♥❣ ✐♥ t❤❡ ❢❛st ❛t❛♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ■■ ■♥ t❤❡ ♦r✐❣✐♥❛❧ ❱❡rs✐♦♥ t❤❡ ♠❛①✐♠❛❧ ❡rr♦r ✇❛s ❜❡❧♦✇ 0.111◦✱ ✐♥ t❤❡ ✜①❡❞ ✈❡rs✐♦♥ t❤❡ ❡rr♦r ✐s ❜❡❧♦✇ 8.2 · 10−5◦

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✶✻

❊♥❞

◗✉❡st✐♦♥s❄

❋♦r ❧❛t❡r q✉❡st✐♦♥s✿ ❉❡♥✐s✳❇❡❞❡r♦✈❅❣♠①✳❞❡

❋❖❙❉❊▼ ✷✵✶✺✱ ✵✶✲❢❡❜✲✷✵✶✺ ♣❛❣❡ ✶✼