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■◆✸✶✼✵✴✹✶✼✵✱ ❙♣r✐♥❣ ✷✵✷✵ P❤✐❧✐♣♣ ❍ä✢✐❣❡r ❤❛❢❧✐❣❡r❅✐❢✐✳✉✐♦✳♥♦ ❊①❝❡r♣t ♦❢ ❙❡❞r❛✴❙♠✐t❤ ❈❤❛♣t❡r ✾✿ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ ♦❢ ❇❛s✐❝ ❈▼❖❙ ❆♠♣❧✐✜❡rs

❈♦♥t❡♥t ■♥t❡r♠❡③③♦✿ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❛♥❞ ❇♦❞❡ P❧♦t ❍✐❣❤ ❋r❡q✉❡♥❝② ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧ ♦❢ ▼❖❙❋❊❚s ✭❜♦♦❦ ✾✳✷✮ ❍✐❣❤ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ ♦❢ ❈❙ ❛♥❞ ❈❊ ❆♠♣❧✐✜❡rs ✭❜♦♦❦ ✾✳✸✮ ❚♦♦❧s❡t ❢♦r ❋r❡q✉❡♥❝② ❆♥❛❧②s✐s ❛♥❞ ❈♦♠♣❧❡t❡ ❈❙ ❆♥❛❧②s✐s ✭✾✳✹✮ ❈● ❛♥❞ ❈❛s❝♦❞❡ ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✺✮ ●❡♥❡r❛❧ ❋❡❡❞❜❛❝❦ ❛♥❞ ❙t❛❜✐❧✐t② ❙tr✉❝t✉r❡ ✭❜♦♦❦ ✶✵✳✶✱ ✶✵✳✽✱ ✶✵✳✾✮ ❙♦✉r❝❡ ❢♦❧❧♦✇❡r ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✻✮ ❉✐✛❡r❡♥t✐❛❧ ❆♠♣ ❍❋ ❆♥❛❧②s✐s ✭❜♦♦❦ ✾✳✼✮

❈♦♥t❡♥t ■♥t❡r♠❡③③♦✿ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❛♥❞ ❇♦❞❡ P❧♦t ❍✐❣❤ ❋r❡q✉❡♥❝② ❙♠❛❧❧ ❙✐❣♥❛❧ ▼♦❞❡❧ ♦❢ ▼❖❙❋❊❚s ✭❜♦♦❦ ✾✳✷✮ ❍✐❣❤ ❋r❡q✉❡♥❝② ❘❡s♣♦♥s❡ ♦❢ ❈❙ ❛♥❞ ❈❊ ❆♠♣❧✐✜❡rs ✭❜♦♦❦ ✾✳✸✮ ❚♦♦❧s❡t ❢♦r ❋r❡q✉❡♥❝② ❆♥❛❧②s✐s ❛♥❞ ❈♦♠♣❧❡t❡ ❈❙ ❆♥❛❧②s✐s ✭✾✳✹✮ ❈● ❛♥❞ ❈❛s❝♦❞❡ ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✺✮ ●❡♥❡r❛❧ ❋❡❡❞❜❛❝❦ ❛♥❞ ❙t❛❜✐❧✐t② ❙tr✉❝t✉r❡ ✭❜♦♦❦ ✶✵✳✶✱ ✶✵✳✽✱ ✶✵✳✾✮ ❙♦✉r❝❡ ❢♦❧❧♦✇❡r ❍❋ r❡s♣♦♥s❡ ✭❇♦♦❦ ✾✳✻✮ ❉✐✛❡r❡♥t✐❛❧ ❆♠♣ ❍❋ ❆♥❛❧②s✐s ✭❜♦♦❦ ✾✳✼✮

■♥t❡rr✉♣t✿ ❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❛♥❞ ❇♦❞❡ P❧♦t

❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ❚❤❡ tr❛♥s❢❡r ❢✉♥❝t✐♦♥ H ( s ) ♦❢ ❛ ❧✐♥❡❛r ✜❧t❡r ✐s ◮ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ♦❢ ✐ts ✐♠♣✉❧s❡ r❡♣♦♥s❡ h ( t ) ✳ ◮ t❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❞❡s❝r✐❜✐♥❣ t❤❡ ■✴❖ r❡❛❧t✐♦♥s❤✐♣ t❤❛t ✐s t❤❡♥ s♦❧✈❡❞ ❢♦r V out ( s ) V in ( s ) ◮ ✭t❤✐s ❧❡❝t✉r❡✦✦✦✮ t❤❡ ❝✐r❝✉✐t ❞✐❛❣r❛♠ s♦❧✈❡❞ q✉✐t❡ ♥♦r♠❛❧❧② ❢♦r V out ( s ) V in ( s ) ❜② ♣✉tt✐♥❣ ✐♥ ✐♠♣❡❞❛♥❝❡s Z ( s ) ❢♦r ❛❧❧ ❧✐♥❡❛r ❡❧❡♠❡♥ts ❛❝❝♦r❞✐♥❣ t♦ s♦♠❡ s✐♠♣❧❡ r✉❧❡s✳

■♠♣❡❞❛♥❝❡s ♦❢ ▲✐♥❡❛r ❈✐r❝✉✐t ❊❧❡♠❡♥ts r❡s✐st♦r✿ R ✶ ❝❛♣❛❝✐t♦r✿ sC ✐♥❞✉❝t♦r✿ sL ■❞❡❛❧ s♦✉r❝❡s ✭❡✳❣✳ t❤❡ i d = g m v gs s♦✉r❝❡s ✐♥ s♠❛❧❧ s✐❣♥❛❧ ♠♦❞❡❧s ♦❢ ❋❊❚s✮ ❛r❡ ❧❡❢t ❛s t❤❡② ❛r❡✳

❚r❛♥s❢❡r ❋✉♥❝t✐♦♥ ✐♥ ❘♦♦t ❋♦r♠ ❚r❛♥s❢❡r ❢✉♥❝t✐♦♥s H ( s ) ❢♦r ❧✐♥❡❛r ❡❧❡❝tr♦♥✐❝ ❝✐r❝✉✐ts ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❞✐✈✐❞✐♥❣ t✇♦ ♣♦❧②♥♦♠✐❛❧s ♦❢ s ✳ H ( s ) = a ✵ + a ✶ s + ... + a m s m ✶ + b ✶ s + ... + b n s n H ( s ) ✐s ♦❢t❡♥ ✇r✐tt❡♥ ❛s ♣r♦❞✉❝ts ♦❢ ✜rst ♦r❞❡r t❡r♠s ✐♥ ❜♦t❤ ♥♦♠✐♥❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r♦♦t ❢♦r♠ ✱ ✇❤✐❝❤ ✐s ❝♦♥✈❡♥✐❡♥t❧② s❤♦✇✐♥❣ s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❇♦❞❡✲♣❧♦ts✳ ▼♦r❡ ♦❢ t❤❛t ❧❛t❡r✳ ( ✶ + s z ✶ )( ✶ + s z ✷ ) ... ( ✶ + s z m ) H ( s ) = a ✵ ( ✶ + s ω ✶ )( ✶ + s ω ✷ ) ... ( ✶ + s ω n )

❇♦❞❡ P❧♦ts P❧♦ts ♦❢ ♠❛❣♥✐t✉❞❡ ✭❡✳❣✳ | H ( s ) | ✮ ✐♥ ❞❇ ❛♥❞ ♣❤❛s❡ ❡✳❣✳ ∠ H ( s ) ♦r φ ✮ ✈s✳ log( ω ) ✳ ■♥ ❣❡♥❡r❛❧ ❢♦r tr❛♥s❢❡r ❢✉♥❝t✐♦♥s ✇✐t❤ ♦♥❧② r❡❛❧ ♣♦❧❡s ❛♥❞ ♥♦ ③❡r♦s ✭♣✉r❡ ❧♦✇✲♣❛ss✮✿ ❛✮ ω → ✵ + | H ( s ) | ✐s ❝♦♥st❛♥t ❛t t❤❡ ❧♦✇ ❢r❡q✉❡♥❝② ❣❛✐♥ ❛♥❞ ∠ H ( s ) = ✵ o ❜✮ ❢♦r ❡❛❝❤ ♣♦❧❡ ❛s ω ✐♥❝r❡❛s❡s t❤❡ s❧♦♣❡ ♦❢ | H ( s ) | ✐♥❝r❡s❡s ❜② ❞❡❝❛❞❡ ❝✮ ❡❛❝❤ ♣♦❧❡ ❝♦♥tr✐❜✉t❡s ✲✾✵ o t♦ t❤❡ − ✷✵ dB ♣❤❛s❡✱ ❜✉t ✐♥ ❛ s♠♦♦t❤ tr❛♥s✐st✐♦♥ s♦ t❤❛t ❛t ❛ ❢r❡q✉❡♥❝② ❡①❛❝t❧② ❛t t❤❡ ♣♦❧❡ ✐t ✐s ❡①❛❝t❧② ✲✹✺ o

●❡♥❡r❛❧ r✉❧❡s ♦❢ t❤✉♠❜ t♦ ✉s❡ r❡❛❧ ③❡r♦s ❛♥❞ ♣♦❧❡s ❢♦r ❇♦❞❡ ♣❧♦ts ✭✶✴✸✮ ■♥ ❣❡♥❡r❛❧ ❢♦r tr❛♥s❢❡r ❢✉♥❝t✐♦♥s ✇✐t❤ ♦♥❧② r❡❛❧ ♣♦❧❡s ❛♥❞ ♥♦ ③❡r♦s ✭♣✉r❡ ❧♦✇✲♣❛ss✮✿ ❛✮ ω → ✵ + | H ( s ) | ✐s ❝♦♥st❛♥t ❛t t❤❡ ❧♦✇ ❢r❡q✉❡♥❝② ❣❛✐♥ ❛♥❞ ∠ H ( s ) = ✵ o ❜✮ ❢♦r ❡❛❝❤ ♣♦❧❡ ❛s ω ✐♥❝r❡❛s❡s t❤❡ s❧♦♣❡ ♦❢ | H ( s ) | ✐♥❝r❡s❡s ❜② ❞❡❝❛❞❡ ❝✮ ❡❛❝❤ ♣♦❧❡ ❝♦♥tr✐❜✉t❡s ✲✾✵ o t♦ t❤❡ − ✷✵ dB ♣❤❛s❡✱ ❜✉t ✐♥ ❛ s♠♦♦t❤ tr❛♥s✐st✐♦♥ s♦ t❤❛t ❛t ❛ ❢r❡q✉❡♥❝② ❡①❛❝t❧② ❛t t❤❡ ♣♦❧❡ ✐t ✐s ❡①❛❝t❧② ✲✹✺ o

●❡♥❡r❛❧ r✉❧❡s ♦❢ t❤✉♠❜ t♦ ✉s❡ r❡❛❧ ③❡r♦s ❛♥❞ ♣♦❧❡s ❢♦r ❇♦❞❡ ♣❧♦ts ✭✶✴✸✮ ❛✮ ✜♥❞ ❛ ❢r❡q✉❡♥❝② ω mid ✇✐t❤ ❡q✉❛❧ ♥✉♠❜❡r k ♦❢ ③❡r♦s ❛♥❞ ♣♦❧❡s ✇❤❡r❡ z ✶ , ..., z k , ω ✶ , ..., ω k < ω mid ⇒ | H ( s ) | ≈ K | z ( k + ✶ ) ... | z m | | ω ( k + ✶ ) | ... | ω n | ∠ H ( s ) ≈ ✵ o ❛♥❞ t❤❡ ❣r❛❞✐❡♥t ♦❢ ❜♦t❤ | H ( s ) | ❛♥❞ ∠ H ( s ) ✐s ③❡r♦

●❡♥❡r❛❧ r✉❧❡s ♦❢ t❤✉♠❜ t♦ ✉s❡ r❡❛❧ ③❡r♦s ❛♥❞ ♣♦❧❡s ❢♦r ❇♦❞❡ ♣❧♦ts ✭✷✴✸✮ ❜✮ ♠♦✈✐♥❣ ❢r♦♠ ω mid ✐♥ t❤❡ ♠❛❣♥✐t✉❞❡ ♣❧♦t ❛t ❡❛❝❤ | ω i | ❛❞❞ ✲✷✵❞❇✴❞❡❝❛❞❡ t♦ t❤❡ ♠❛❣♥✐t✉❞❡ ❣r❛❞✐❡♥t ❛♥❞ ❢♦r ❡❛❝❤ | z i | ❛❞❞ ✰✷✵❞❇✴❞❡❝❛❞❡ ❝✮ ♠♦✈✐♥❣ ❢r♦♠ ω mid ✐♥ t❤❡ ♣❤❛s❡ ♣❧♦t t♦ ❤✐❣❤❡r ❢r❡q✉❡♥❝✐❡s ❛t ❡❛❝❤ ω i ❛❞❞ ✲✾✵ o t♦ t❤❡ ♣❤❛s❡ ✐♥ ❛ s♠♦♦t❤ tr❛♥s✐t✐♦♥ ✭r❡s♣❡❝t✐✈❡❧② − ✹✺ o r✐❣❤t ❛t t❤❡ ♣♦❧❡s✮ ❛♥❞ ✈✐❝❡ ✈❡rs❛ t♦✇❛r❞s ❧♦✇❡r ❢r❡q✉❡♥❝✐❡s✳

●❡♥❡r❛❧ r✉❧❡s ♦❢ t❤✉♠❜ t♦ ✉s❡ r❡❛❧ ③❡r♦s ❛♥❞ ♣♦❧❡s ❢♦r ❇♦❞❡ ♣❧♦ts ✭✸✴✸✮ ❞✮ ❋♦r t❤❡ ③❡r♦s t♦✇❛r❞s ❤✐❣❤❡r ❢r❡q✉❡♥❝✐❡s ✐❢ t❤❡ ♥♦♠✐♥❛t❡r ✐s ♦❢ t❤❡ ❢♦r♠ ( ✶ + s z i ) ❛❞❞ ✰✾✵ o ❛♥❞ ✐❢ ✐ts ♦❢ t❤❡ ❢♦r♠ ( ✶ − s z i ) ✭r❡❢❡r❡❞ t♦ ❛s r✐❣❤t ❤❛❧❢ ♣❧❛✐♥ ③❡r♦ ❛s t❤❡ s♦❧✉t✐♦♥ ❢♦r s ♦❢ z i ) ✐s ♣♦s✐t✐✈❡✮ ❛❞❞ ✲✾✵ o t♦ t❤❡ ♣❤❛s❡ ✵ = ( ✶ − s ✐♥ ❛ s♠♦♦t❤ tr❛♥s✐t✐♦♥ ✭✐✳❡✳ r❡s♣❡❝t✐✈❡❧② ± ✹✺ o r✐❣❤t ❛t t❤❡ ③❡r♦s✮ ❛♥❞ ✈✐❝❡ ✈❡rs❛ t♦✇❛r❞s ❧♦✇❡r ❢r❡q✉❡♥❝✐❡s✳