簡約 1/((1-0.16e^{-iw))(1-0.37e^{-iw})}

解

簡約

\frac{1}{( 1 - 0.16 e ^{- i w} ) ( 1 - 0.37 e ^{- i w} )}

\frac{1 + 0.0592 cos ^{2} ( w ) - 0.0592 sin ^{2} ( w ) - 0.53 cos ( w )}{1 + 0.00350464 cos ^{4} ( w ) + 0.00350464 sin ^{4} ( w ) - 0.062752 cos ^{3} ( w ) + 0.1625 sin ^{2} ( w ) + 0.3993 cos ^{2} ( w ) - 1.06 cos ( w ) + 0.00700928 cos ^{2} ( w ) sin ^{2} ( w ) - 0.062752 sin ^{2} ( w ) cos ( w )} + i \frac{- 0.53 sin ( w ) + 0.1184 cos ( w ) sin ( w )}{1 + 0.00350464 cos ^{4} ( w ) + 0.00350464 sin ^{4} ( w ) - 0.062752 cos ^{3} ( w ) + 0.1625 sin ^{2} ( w ) + 0.3993 cos ^{2} ( w ) - 1.06 cos ( w ) + 0.00700928 cos ^{2} ( w ) sin ^{2} ( w ) - 0.062752 sin ^{2} ( w ) cos ( w )}

解答ステップ

\frac{1}{( 1 - 0.16 e ^{- i w} ) ( 1 - 0.37 e ^{- i w} )}

(1 - 0.16 e^{- i w}) (1 - 0.37 e^{- i w}) = (1 - 0.16 (cos (- w) + i sin (- w))) (1 - 0.37 (cos (- w) + i sin (- w)))

= \frac{1}{( - 0.16 ( cos ( - w ) + i sin ( - w ) ) + 1 ) ( - 0.37 ( cos ( - w ) + i sin ( - w ) ) + 1 )}

(1 - 0.16 (cos (- w) + sin (- w) i)) (1 - 0.37 (cos (- w) + sin (- w) i)) : - 0.0592 sin^{2} (w) + (- 0.16 cos (w) + 1) (- 0.37 cos (w) + 1) + i (0.53 sin (w) - 0.1184 sin (w) cos (w))

= \frac{1}{- 0.0592 sin ^{2} ( w ) + ( - 0.16 cos ( w ) + 1 ) ( - 0.37 cos ( w ) + 1 ) + i ( 0.53 sin ( w ) - 0.1184 sin ( w ) cos ( w ) )}

有理化する

\frac{1}{- 0.0592 sin ^{2} ( w ) + ( - 0.16 cos ( w ) + 1 ) ( - 0.37 cos ( w ) + 1 ) + i ( 0.53 sin ( w ) - 0.1184 sin ( w ) cos ( w ) )} : \frac{0.0592 cos ^{2} ( w ) + i ( 0.1184 sin ( w ) cos ( w ) - 0.53 sin ( w ) ) - 0.0592 sin ^{2} ( w ) - 0.53 cos ( w ) + 1}{0.00350464 sin ^{4} ( w ) + 0.00350464 cos ^{4} ( w ) + 0.1625 sin ^{2} ( w ) + 0.00700928 sin ^{2} ( w ) cos ^{2} ( w ) + 0.3993 cos ^{2} ( w ) - 0.062752 cos ^{3} ( w ) - 0.062752 sin ^{2} ( w ) cos ( w ) - 1.06 cos ( w ) + 1}

= \frac{- 0.0592 sin ^{2} ( w ) + 1 - 0.53 cos ( w ) + 0.0592 cos ^{2} ( w ) + i ( 0.1184 sin ( w ) cos ( w ) - 0.53 sin ( w ) )}{0.00350464 sin ^{4} ( w ) + 0.1625 sin ^{2} ( w ) - 0.062752 sin ^{2} ( w ) cos ( w ) + 0.00700928 sin ^{2} ( w ) cos ^{2} ( w ) - 1.06 cos ( w ) + 0.3993 cos ^{2} ( w ) - 0.062752 cos ^{3} ( w ) + 0.00350464 cos ^{4} ( w ) + 1}

標準的な複素数形式で

\frac{- 0.0592 sin ^{2} ( w ) + 1 - 0.53 cos ( w ) + 0.0592 cos ^{2} ( w ) + i ( - 0.53 sin ( w ) + 0.1184 sin ( w ) cos ( w ) )}{0.00350464 sin ^{4} ( w ) + 0.1625 sin ^{2} ( w ) - 0.062752 sin ^{2} ( w ) cos ( w ) + 0.00700928 sin ^{2} ( w ) cos ^{2} ( w ) - 1.06 cos ( w ) + 0.3993 cos ^{2} ( w ) - 0.062752 cos ^{3} ( w ) + 0.00350464 cos ^{4} ( w ) + 1}

を書き換える:

\frac{- 0.0592 sin ^{2} ( w ) + 1 - 0.53 cos ( w ) + 0.0592 cos ^{2} ( w )}{0.00350464 sin ^{4} ( w ) + 0.00350464 cos ^{4} ( w ) + 0.1625 sin ^{2} ( w ) + 0.00700928 sin ^{2} ( w ) cos ^{2} ( w ) + 0.3993 cos ^{2} ( w ) - 0.062752 cos ^{3} ( w ) - 0.062752 sin ^{2} ( w ) cos ( w ) - 1.06 cos ( w ) + 1} + \frac{- 0.53 sin ( w ) + 0.1184 sin ( w ) cos ( w )}{0.00350464 sin ^{4} ( w ) + 0.00350464 cos ^{4} ( w ) + 0.1625 sin ^{2} ( w ) + 0.00700928 sin ^{2} ( w ) cos ^{2} ( w ) + 0.3993 cos ^{2} ( w ) - 0.062752 cos ^{3} ( w ) - 0.062752 sin ^{2} ( w ) cos ( w ) - 1.06 cos ( w ) + 1} i

= \frac{- 0.0592 sin ^{2} ( w ) + 1 - 0.53 cos ( w ) + 0.0592 cos ^{2} ( w )}{0.00350464 sin ^{4} ( w ) + 0.00350464 cos ^{4} ( w ) + 0.1625 sin ^{2} ( w ) + 0.00700928 sin ^{2} ( w ) cos ^{2} ( w ) + 0.3993 cos ^{2} ( w ) - 0.062752 cos ^{3} ( w ) - 0.062752 sin ^{2} ( w ) cos ( w ) - 1.06 cos ( w ) + 1} + \frac{- 0.53 sin ( w ) + 0.1184 sin ( w ) cos ( w )}{0.00350464 sin ^{4} ( w ) + 0.00350464 cos ^{4} ( w ) + 0.1625 sin ^{2} ( w ) + 0.00700928 sin ^{2} ( w ) cos ^{2} ( w ) + 0.3993 cos ^{2} ( w ) - 0.062752 cos ^{3} ( w ) - 0.062752 sin ^{2} ( w ) cos ( w ) - 1.06 cos ( w ) + 1} i