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2 信号処理の基礎サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.

4 i AI

5 ii z / z 8 8

6 iii

7 iv

8 v LPF HPF BPF BEF z

10 t x(t) t R t t R x(t) x(t) R x(t) x(t) R, t R 1.1

11 mg mg mg x(t) g 1 (t),g 2 (t),...,g N (t) c 1,c 2,...,c N x(t) x(t) =c 1 g 1 (t)+c 2 g 2 (t)+ + c N g N (t) a analysis 1.3 b synthesis x(t) 1.1

12 filter 1.3 c g 1 (t),g 2 (t),...,g N (t)

13 4 2 2 sin cos T 0 T 0 f 0 =1/T 0 1 [s] f 0 [s 1 ] [Hz] x(t) f 0 0,f 0, 2f 0, 3f 0,...

14 2.2 5 n=0 [ ] a n cos(2πnf 0 t)+b n sin(2πnf 0 t) a n b n f e(t) n =0 cos(2π 0f 0 t)=1 sin(2π 0f 0 t)=0 x(t) x(t) =a n=1 [ ] a n cos(2πnf 0 t)+b n sin(2πnf 0 t) + e(t) x(t) cos(2πnf 0 t), sin(2πnf 0 t), n=0, 1, 2,... a n, b n, n =0, 1, 2, f 2π ω =2πf f [s 1 ] [Hz] [rad/s] 2πf 0 2.1

15 6 2 ω 0 =2πf 0 ω [ ] x(t) =a 0 1+ a n cos(nω 0 t)+b n sin(nω 0 t) + e(t) 2.2 n=1 a n b n e(t) x(t) =c 1 g 1 (t)+c 2 g 2 (t)+ + c N g N (t)+e(t) 2.3 x(t) e(t) 2 J = e 2 (t)dt 2 J c 1,...,c N 2 2 J 2.3 [ 2 N J = x(t) c n g n (t)] dt 2.4 n=1 2 J c k 2 J c 1,c 2,...,c N 2 J c k J N =2 x(t)g k (t)dt 2 c n g n (t)g k (t)dt 2.5 c k n=1 x(t)g k (t)dt = N c n g n (t)g k (t)dt, k =1, 2,...,N 2.6 n=1 x(t) y(t) x, y = x(t)y(t)dt 2.6 x, g k = 2.7 N c k g n,g k, k =1, 2,...,N 2.8 n=1

16 2.4 7 R g 1,g 1 g 2,g 1 g N,g 1 g 1,g 2 g 2,g 2 g N,g g 1,g N g 2,g N g N,g N, c c 1 c 2... c N, r x, g 1 x, g 2. x, g N Rc = r 2.10 R c c = R 1 r 2.11 c 1,c 2,...,c N g n,g k =0, n k 2.12 R 1 c k = g k,g k x, g k, k =1, 2,...,N a n b n x(t) e(t) [, ] 1 [0,T 0 ] T0 0 T0 0 T0 0 T0 0 cos(nω 0 t)dt =0, T0 cos(kω 0 t)cos(nω 0 t)dt = sin(kω 0 t)sin(nω 0 t)dt = cos(kω 0 t)sin(nω 0 t)dt =0 n, k =1, 2,... 0 sin(nω 0 t)dt =0 T 0 2, k = n 0, k n T 0 2, k = n 0, k n 2.14

17 R 2.13 a 0 = 1 T0 x(t)dt T 0 0 a n = 2 T0 x(t)cos(nω 0 t)dt T 0 0 b n = 2 T0 x(t)sin(nω 0 t)dt T a n b n x(t) T 0 f 0 ω T 0 T 0 =2[s] f 0 f 0 =1/T 0 f 0 =0.5 [Hz] ω 0 ω 0 =2πf 0 ω 0 = π[rad/s] a 0 = 1 Z T0 x(t)dt = 1»Z 1 Z 2 1dt + 0dt = 1 T a 1 = 2 Z T0 x(t)cos(ω 0 t)dt = 2»Z 1 1 cos(πt)dt + T b 1 = 2 Z T0 x(t)sin(ω 0 t)dt = 2»Z 1 Z 2 1 sin(πt)dt + T a 2 = 2 Z T0 x(t)cos(2ω 0 t)dt = 2»Z 1 1 cos(2πt)dt + T b 2 = 2 Z T0 x(t)sin(2ω 0 t)dt = 2»Z 1 1 sin(2πt)dt + T a 3 =0, b 3 = 2 3π a 4 =0, b 4 =0 a 5 =0, b 5 = 2 5π.. 8 >< a n =0, b n = >: 0, n 2 nπ, n Z 2 0 cos(πt)dt =0 0 sin(πt)dt = 2 π Z 2 1 Z cos(2πt)dt =0 0 sin(2πt)dt =0

18 T [Hz] 20 [khz] 405 [THz] 790 [THz] 2.6 i = 1 x y z = x + iy x y z 2.3 a 1 2

19 z x = Re[z], y =Im[z] z = x + iy x iy z z z A = z = x 2 + y b zz =(x + iy)(x iy) =x 2 + y 2 z = zz z ( φ =arg z =tan 1 y ) x

20 (A, φ) x = A cos φ y = A sin φ z e z e z =e x+iy =e x e iy e x e iy φ e iφ 1 φ 2.3 c e iφ =cosφ + i sin φ φ φ cos e iφ =cosφ i sin φ cos φ = eiφ +e iφ, sin φ = eiφ e iφ i z = x + iy = Ae iφ 2.3 d z 1 = A 1 e iφ 1 z 2 = A 2 e iφ 2 z 1 z 2 = A 1 A 2 e i(φ 1+φ 2 ) e iφ φ t φ =2πt e i2πt O e t =0 1+i0 t =1, 2, 3,... t =0 1[Hz] 2.16 e i2πt =cos(2πt)+i sin(2πt) 2.17 cos sin A Ae i2πft f O A 0 1 f f c ce i2πft c = c e i arg c ce i2πft = c e i arg c e i2πft i(2πft+arg c) = c e 2.18

21 12 2 ce i2πft f O c 1 f t =0 arg c 2.3 f τ =(argc)/(2πf) ce i2πft = c e i2πf(t+τ) 2.19 t = τ cos sin cos sin x(t) =a 0 + = a 0 + n=1 n=1 c n c 0 = a 0 c n c n ( an 2 einω 0t + b n 2i einω 0t + a n 2 e inω 0t b n 2i e inω 0t ) + e(t) ( an ib n e inω0t + a ) n + ib n e inω 0t + e(t) = a n ib n, n =1, 2,... 2 = a n + ib n, n =1, 2, x(t) = c n e inω0t + e(t) n= c n c n c n = 1 T0 x(t)e inω0t dt T c n c n n nω 0 nω 0 c n e inω 0t + c n e inω 0t 2.24

22 T 0 T 0 T 0 2 x(t) c n e inω 0t 3.1 n= c n = 1 T0 /2 x(t)e inω0t dt 3.2 T 0 T 0 /2 e(t) 3.2 T 0 [0,T 0 ] [ T 0 /2,T 0 /2] t τ T 0 =2π/ω 0 ω 0 =2πf 0 =2π/T

23 x(t) { T0 /2 ω0 2π n= T 0 /2 } x(τ)e inω0τ dτ e inω 0t T 0 T 0 ω 0 = 2π 0 ω 0 Δω x(t) 1 2π T 0 T0 /2 T 0 /2 f(x)dx = { x(τ) lim n= Δx 0 n= } e i(t τ)nδω Δω dτ f(nδx)δx 3.3 T 0 Δω { } lim e i(t τ)nδω Δω = e i(t τ)ω dω 3.4 Δω 0 n= 3.3 Δω x(t) 1 { } x(τ)e iωτ dτ e iωt dω 3.5 2π Fourier integral x(t) x(t) dt < (, ) 3.1 (, ) x(t) x(t) t x(t) t {x(t 0) + x(t +0)}/2

24 x(t) 3.5 (, ) x(t) t 0 x(t 0 0) x(t 0 +0) {x(t 0 0) + x(t 0 +0)}/2 x(t 0 ) 3.5 { } X(ω) =F[x(t)] = x(t)e iωt dt x(t) =F 1 [X(ω)] = 1 2π X(ω)e iωt dω x(t) 3.7 X(ω) X(ω) x(t) X(ω) X(ω) 2 arg X(ω) 3.1 x(t) (, ) x(t) t x(t) t {x(t 0) + x(t +0)}/2 x(t) 3.3 T 0 x(t) 3.1 c 3.6 X(ω) = T0 /2 T 0 /2 x(t)e iωt dt ω ω 0 =2π/T 0 ω nω 0 X(nω 0 )= T0 /2 T 0 /2 x(t)e inω 0t dt c n = 1 T 0 X(nω 0 ) 3.9 T 0 x(t) c n x(t) X(ω) ω 0 =2π/T 0 1/T x(t) 1 T 0 n= X(nω 0 )e inω 0t 3.10 x(t) X(ω) ω 0 =2π/T 0 X(nω 0 )

25 T 0 x(t) X(ω) ω = nω 0 0 2π T 0 X(nω 0 ) X(ω) = 2π T 0 X(ω), ω = nω 0, n Z 3.1 x n (t) X n (ω) 3.3 a >0 ( e at, t 0 x 1 (t) = a 0, t < 0 ( te at, t 0 x 2 (t) = b 0, t < 0 x 3 (t) =e a t c 3.3

26 a a 3.6 X(ω) = Z 0 = 1 a + iω e at e iωt dt = Z 0 he (a+iω)ti b b 3.6 Z 0 0 e (a+iω)t dt = 1 a + iω Z X(ω) = te at e iωt dt = te (a+iω)t dt 0 0» t Z 1 = a + iω e (a+iω)t 0 0 a + iω e (a+iω)t dt =0+ 1 Z e (a+iω)t 1 dt = a + iω (a + iω) 2 8 >< t n 1 x(t) = (n 1)! e at, t 0 >: 0, t < 0 1 X(ω) = (a + iω) n c c 3.6 X(ω) = = Z Z 0 e a t e iωt dt e at e iωt dt + Z 0 e at e iωt dt X(ω) = 1 a iω + 1 a + iω = 2a a 2 + ω x(t) ax(t)+by(t) F[ax(t)+by(t)] = af[x(t)] + bf[y(t)] ax(t)+by(t) a b 3.6 x(t) x( t) t τ

27 28 3 F[x( t)] = = = x( t)e iωt dt x(τ)e iω( τ) ( 1)dτ x(τ)e i( ω)τ dτ = X( ω) x( t) X(ω) 3.6 x(t) x(t/a) a >0 τ = t/a dτ =(1/a)dt dt = adτ F[x(t/a)] = = a x(t/a)e iωt dt = x(τ)e iωaτ a dτ x(τ)e iωaτ dτ = ax(ωa), a > 0 F[x( t/a)] = ax( ωa), a > 0 a F[x(t/a)] = a X(aω), a 0 a 0 x(t) X(ω) 1/a a 3.4 a x(t) X(ω) 2 b 2 a =2 1/ x(t) x(t τ) F[x(t τ)] = x(t τ)e iωt dt t τ = a F[x(t τ)] = = x(a)e iωa e iωτ da x(a)e iωa da e iωτ = X(ω)e iωτ τ x(t) X(ω) e iωτ 3.4 a

28 τ c e iωτ ω ωτ [rad] X (ω) = ω ω X ( ω) = 3.6 x (t)e iωt dt x (t)e iωt dt = F[x (t)] 3.11

29 30 3 x(t) x (t) =x(t) X ( ω) =X(ω) Re[X(ω)] = Re[X( ω)] Im[X(ω)] = Im[X( ω)] x(t) ω X(ω) ω X(ω) 3.4 x(t) e iωt =cos(ωt) i sin(ωt) Im[X(ω)] = x(t)sin(ωt) dt =0 x(t) Re[X(ω)] = x(t)cos(ωt) dt =0 3.4 a b d c a e x n (t) X n (ω) a (3.6) 3.5

30 b e a d b a x 1 (t) =1, 1/2 t 1/2 3.6 X 1 (ω) = Z 1/2 1/2 e iωt dt = 1 ˆe iωt 1/2 iω = 1 e iω/2 e iω/2 1/2 iω = 2 e iω/2 e iω/2 = 2 ω ω 2i ω sin 2 x 1 (t) b x 2 (t) x 1 (t) τ =1/2 X 2 (ω) X 1 (ω) e iτω X 2 (ω) =X 1 (ω)e iω/2 = 1 iω = i ω ( 1+e iω ) e iω/2 e iω/2 e iω/2 e iω =cos(ω) i sin(ω) Re[X 1 (ω)] = sin ω ω Im[X 1 (ω)] = 1 (cos ω 1) ω c x 3 (t) x 2 (t) X 3 (ω) =X 2 ( ω) X 3 (ω) =X 2 ( ω) = = i ω (1 eiω ) i ω ( 1+eiω ) d x 4 (t) =x 2 (t)+( 1)x 3 (t) X 4 (ω) =X 2 (ω) X 3 (ω) = i ω ( 1+e iω ) i ω (1 eiω ) = 2i (cos(ω) 1) ω x 4 (t) e x 5 (t) x 2 (t) 4 x 5 (t) =x 1 (t/4) a =1/4 X 5 (ω) = 1 ω a X 2 a = 4i ω ( 1+e 4iω )

31 t ω ω ω x( ω) = 1 2π X(t)e iωt dt 3.6 F[X(t)] = 2πx( ω) x(t) X(ω) X(t) 2πx( ω) 3.7 X(ω) X(ω a) F 1 [X(ω a)] = 1 2π X(ω a)e iωt dω ω a = b F 1 [X(ω a)] = 1 2π = x(t)e iat X(b)e iat e ibt db F[x(t)e iat ]=X(ω a) t ω a d dt x(t) = 1 2π iax(a)e iat da [ ] d F dt x(t) = 1 iax(a) F[e iat ]da 2π 3.3 e iat 2πδ(ω a) 3.25 [ ] d F dt x(t) = 1 iax(a) 2πδ(ω a) da 2π = iωx(ω) 3.15 x(t) x(t) X(ω) iω 3.7 x(t) 2 = 1 (2π) 2 X(ω)X (ω )e iωt e iω t dωdω

32 t x(t) 2 dt = 1 (2π) 2 { } X(ω)X (ω ) e iωt e iω t dt dωdω { } e iωt 2πδ(ω ω) x(t) 2 dt = 1 X(ω) 2 dω 2π x(t) y(t) z(t) X(ω) Y (ω) Z(ω) Z(ω) =X(ω)Y (ω) x(t) y(t) z(t) x(t) y(t) Z(ω) = x(t)e iωt dt y(t)e iωt dt z(t) = 1 { x(t 1 )e iωt 1 dt 1 y(t 2 )e iωt 2 dt 2 }e iωt dω 2π { 1 } = x(t 1 )y(t 2 ) e iω(t (t 1+t 2 )) dω dt 1 dt 2 2π 3.16 δ(t) F[δ(t (t 1 + t 2 ))] = 1 e iω(t 1+t 2 ) δ(t (t 1 + t 2 )) = 1 2π e iω(t 1+t 2 ) e iωt dω 3.16 { } z(t) = x(t 1 )y(t 2 )δ(t (t 1 + t 2 )) dt 1 dt 2 t 1 + t 2 = τ z(t) = x(t 1 )y(τ t 1 )δ(t τ)dt 1 dτ τ = t z(t) = x(t 1 )y(t t 1 )dt 1

33 34 3 x(t) y(t) x(τ)y(t τ) dτ 3.17 x(t) y(t) (x y)(t) 3.17 t τ = σ x(t) y(t) = y(σ)x(t σ) dσ = y(σ)x(t σ) dσ x(t) =ax 1 (t)+bx 2 (t) 3.17 z(t) =a { x 1 (t) y(t) } + b { x 2 (t) y(t) } x(t) y(t) x(t)y(t) 2π F[x(t)y(t)] = 1 X(ω) Y (ω) 2π X(ω) 2 X(ω) 2 = X(ω)X (ω) 3.18 x(t) X (ω) =X( ω) X( ω) x( t) X(ω) 2 = X(ω)X( ω) = F[x(t)]F[x( t)] = F[x(t) x( t)] [ ] = F x(t)x( (τ t)) dt [ ] = F x(t)x(t τ) dt 3.19 x(t) R(τ) = x(t)x(t τ) dt 3.20 x(t) R(τ) x(t) X(ω) 2

34 ax(t)+by(t) ax(ω)+by (ω) x( t) X( ω) x(t/a) a X(ωa) x(t τ ) X(ω)e iωτ X(t) 2πx( ω) x(t)e iat X(ω a) d dt x(t) iωx (ω) x(t) y(t) x(t)y(t) X(ω)Y (ω) 1 X(ω) Y (ω) 2π R( τ ) = x(t)x(t τ)dt X( ω) X(ω) =F[x(t)], Y (ω) =F[y(t)] a { 1, t τ P τ (t) = 0, P τ (t) F[P τ (t)] = τ τ = 1 iω 1 e iωt dt [e iωt] τ τ = 1 iω (eiτω e iτω ) =2 sin(τω) 3.21 ω sin(t)/t sinc(t) sinc(t) sinc(t) 3.21 F[P τ (t)] = 2τ sinc(τω)

35 70 5 x(t) H(s) = s +19 (s + 30)(s +9.8) y(t) G(s)I(s) X(s) X(s) = 1 s s +10 x(t) = e 20t +e 10t Y (s) =H(s)X(s) Y (s) = s s s s +30 y(t) = 0.010e 20t 2.250e 10t e 9.8t e 30t x(t) y(t) s = 19 s = 9.8 s = 20 s = 10 Y (s) y(t) 3 4 s = 9.8 s = H(s) = 1 s a x(t) =e iωt y(t) G(s)I(s) x(t) =e iω X(s) X(s) = 1 s iω Y (s) Y (s) =H(s)X(s) = 1 (s a)(s iω) = 1 iω a 1 s a + 1 «s iω y(t) = eat iω a + eiωt iω a 5.4 a

36 H(s) s x(t) y(t) h(t) h(t) dt < 5.26 s s H(s) G(s) G(s) G(s)I(s) H 1 (s) H 2 (s) H(s) H(s) =H 1 (s)h 2 (s) H 1 (s) H 2 (s) H(s) H(s) =H 1 (s)+h 2 (s) H 1 (s) H 2 (s) H(s) x(t) H n (s), n =1, 2,...,N y(t) y(t) Y (s) X(s) H n (s), n =1, 2,...,N 5.4

37 72 5 x(t) =e iωt x(t) =e iωt X(s) =1/(s iω) H(s) y(t) Y (s) =H(s)X(s)+G(s)I(s) = H(s) s iω + G(s)I(s) iω H(s) Y (s) = A, t s iω A 4.27 A = H(s) s iω (s iω) s=iω = H(s) s=iω y(t) =H(s) s=iω e iωt, t H(s) s=iω h(t) H(s) s = iω H(ω) H(ω) t ω 2.6 e iωt H(ω) arg H(ω) H(ω) H(ω) arg H(ω) a<0 H(ω) eiωt 5.8 a eat iω a a <0 t iω a e iωt H(ω) =1/(iω a)

38 a t = y(t) = h(τ)x(t τ) dτ 5.31 y(t) h(t) x(t) Y (ω) H(ω) X(ω) 5.31 Y (ω) =H(ω)X(ω) 5.32 X(ω) ω H(ω) Y (ω) H(ω) H(ω) H(ω) arg H(ω) x(t) y(t) =x 2 (t) 5.5 1[rad/s] x(t) =cos(t) y(t) = (cos(2t)+1)/2 2[rad/s] 5.10 H(s) H(ω) H(ω) ( 1, 0 t T h(t) = 0, h(t) H(s) = Z T 0 1 e st dt = 1 s (1 est ) H(s) s = iω H(ω) = 1 iω (1 eiωt )

39 74 5 H(ω) = 1 ω 1 eiωt 1/ ω 1 e iωt 5.10 a b H(ω) ω 0 c 5.10 H(ω) RC 4.4 A v A (t) i(t) B v B (t) v A (t) = 1 C Z t v B (t) =Ri(t) i(τ) dτ + Ri(t) 5.11 CR v A (t) v B (t) H(s) H(ω) R =10 6 [Ω] C =10 6 [F] H(ω) arg H(ω)

40 «1 V A (s) = sc + R I(s), V B (s) =RI(s) H(s) = V B(s) V A (s) = R 1/(sC)+R = 1 1+1/(sRC) s = iω H(ω) = H(ω) = 1 1+1/(iωRC) 1 p 1+1/( ω 2 R 2 C 2 ) ω 1/(RC) H(ω) 1 ω 1/(RC) H(ω) ω RC ω 1 ω ω a b arg H(ω) =arg1 arg 1+ 1 «iωrc =0 arg 1 i «ωrc = tan 1 1 «ωrc «1 =tan 1 ωrc 5.12 H(ω) arg H(ω) 5

41 76 5 t =0 t =0 H(s) s H(s) s s = iω H(ω) ω H(ω) RLC 4.4 A v A (t) i(t) B v B (t) v A (t) = 1 C Z t i(τ) dτ + Ri(t)+Li (t) v B (t) =Li (t) i (t) i(t) v A (t) v B (t) 5 1 H(s) 2 3 L C R ω c 5 L =1[mH] C =1[ F] R 0Ω 70 Ω 6 2 R R 7 R = [Ω] 5.13 LCR

42 H(ω) H(s) H(s) =H (s ) H(ω) H(s) s = iω H(ω) =H ( ω) H(ω) ω arg H(ω) ω 0 H(ω) arg H(ω) ω 0 H(ω) ω H(ω) arg H(ω) gain [db] H(s) =1/s 6.1 H(ω) =1/(iω) = i/ω ω 0

43 H(s) =1/s ω 0 90 [deg] 20 [db/dec] [db] 1/10 H(s) =s 6.2 H(s) 6.2 H(s) =s

44 H(s) =P M (s β k ) k=1 N (s α k ) k=1 6.1 H αk (s) =1/(s α k ) H βk (s) =(s β k ) α k β k 6.1 { M }{ N } H(s) =P H βk (s) H αk (s) 6.2 k=1 k=1 H(s) s = iω H(ω) M N 20 log 10 H(ω) =20log 10 P + 20 log 10 H βk (ω) + 20 log 10 H αk (ω) k=1 k=1 6.3 M N arg H(ω) = arg H βk (ω)+ arg H αk (ω) k=1 k=1 6.4 α k β k α β α H(s) = α /(s α) α 1 H(s) = α /(s α) 6.3 α = 10 2 H(s) = α 2 /(s α) 2 α = H(s) = α 2 /((s α)(s α )) α = 10e i5π/6 α = 10e i5π/ a d 10 a 1 1 b c d ω <10 H(ω) =1 10 [rad/s] 10 [rad/s]

45 H(s) = α /(s α) α = H(s) = α 2 /(s α) 2 α = ω 20 [db/dec] 2 40 [db/dec] [db] 2 1 ω =0 0[deg] ω

46 H(s) = α 2 /((s α)(s α )) α = 10e i5π/ H(s) = α 2 /((s α)(s α )) α = 10e i5π/ s

47 [deg] [deg] 1 ω iπ/2 0 2 π 0 N Nπ/2[rad] β = 10 H(s) =(s β)/ β 6.8 β = 10e i5π/9 H(s) =((s β)(s β ))/ β β = 10 H(s) =(s β)/ β 6.9 β = 10e i5π/9 H(s) =((s β)(s β ))/ β 2

48 N Nπ/2[rad] H(s) H(s) = H(s) = α 1 α 2 2 (s α 1 )(s α 2 )(s α 2 ) a α 1 = 1, α 2 = 100e i5π/6 α 1 2 α 2 (s β 1 ) β 1 (s α 1 )(s α 1 )(s α 2) α 1 =e i3π/4, α 2 = 100, β 1 = 10 b 6.10 a b 6.10 a H(s) s = [rad/s] 20 [db/dec] 90 [deg] s = 100e ±i5π/ [rad/s] 40 [db/dec] 180 [deg] b H(s) s =e ±i3π/ [rad/s] 40 [db/dec] 180 [deg] s = [rad/s] 20 [db/dec] 90 [deg] s = [rad/s] 20 [db/dec] 90 [deg]

49 a b a b

50 LPF HPF BPF BEF APF h(t) =δ(t τ) δ(t) 1 τ e iτω H(ω) =e iτω

51 H(ω) =1, arg H(ω) =τω ω arg H(ω) =τω ω 6.15 a arg H(ω) = 0.1ω b 0.1 [s] arg H(ω) = 0.1 ω c arg H(ω) ω d(ω) =argh(ω)/ω [s] ω 6.15

52 H(ω) =1 2. H(ω) =0 1 2 H(ω) [db/dec] [db/dec] s ω c n LPF s ω c n n [ ( 2k 1 α k = ω c exp i 2n π + π )], k =1,...,n H(s) = P n (s α k ) k=1 6.6 P H(0) = 1 P = n k=1 α k h(t) 2 h(t) = n A k e αkt,, A k =(s α k )H(s) k=1 s=αk 6.7 ω c = 1 [rad/s] 10 LPF 6.16 a ω c = 1 [rad/s] n =2, 3,...,10 LPF b d

53 LPF 20n [db/dec] α k ( ( )) 1 1 Re(α k ) Re(α k )sinh n sinh Rip/5 1 Rip [db] H(s) h(t) P H(ω c ) =1/ 2 ω c = 1 [rad/s] 10 LPF 6.17 a ω c =1 [rad/s] n =2, 3,...,10 LPF b d

54 LPF 6.18 LPF ω c n LPF s n

55 90 6 α k = R( cos(sin 1 (p k )) + ip k ), k =1, 2,...,n 6.9 p k = 1 2k +1 n H(s) h(t) R P H(0) =1 ω c = 1 [rad/s] 10 LPF 6.18 a ω c S = 1 [rad/s] n =2, 3,...,10 LPF b d 6.4 LPF HPF BPF BEF LPF LPF HPF BPF BEF HPF HPF BPF BEF LPF 6.19 a 20 [db/dec] LPF b c HPF 1 HPF LPF n s n HPF HPF LPF 6.19 LPF HPF

56 2013 Printed in Japan ISBN

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s ... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z