# 實習一 實習一 實習一

(1)

1

2

1

2

(2)





































0

0

0

0

0

(3)





d

) 2 2

cos(

) ) ( 2 cos(

) ( ) (

0 0

0

d d d

d

t f t

f A

t t f A

t t x t x

− +

=

+

=

=

) 2 cos(

)

(t =A πf0t

x td

td

f0

0

f td

(4)



π

w0 f0 w0

f0

0

0

w







0

t

A f0t

A

x

A∈ℜ

f0

θ

f0

A

(5)



) 2 sin 2

(cos

) sin (cos

)

( ( )

ft j ft e

wt j wt e

e e t x

t

t t jw st

π

σ π

σ σ

+

=

+

=

=

= +

) sin(

)

cos(wt j wt

ejwt = + eσt

wt

wt

eσtsin σ >0 σ <0



A：振幅

jwt

0 0

0 0

0

0 0

0 1/f 2 /w

T = = π

0 0

0 0

) ( 0

Q I

t w j c

+

θ

(6)



) (w0t+θ

j



0

1

t

jw

(2 /6)

2

0+π

Aej wt t

x 3 (3 0 /2)

3 ) 2

( = Aej wt+π t

x

(7)





0

j(2πf0t+θ)

j(2πf0t+θ)







T0

0

0

t

t x T t

x( + ) = (), 所有

0

(8)



i

i

n

fi

f0

=

1

0

n

i i

sum

(9)



i

i

=

1

0

n

i i i

sum

i

i

0

0

i



1

1

2

2 / 6 .

0

2 / 3 . 0

2 / 3

(10)



1



2

2

2 / 7 . 0

(11)



(12)



0















T |x(t)|dt<

0

(13)

## 複指數傅利葉級數 複指數傅利葉級數 複指數傅利葉級數 複指數傅利葉級數







0



0

−∞

=

n

t nf j ne c t

x

### ( )

2π 0 cn T T x te j nf0tdt

0

2

0

)

1 ( π

=

T0

T0

T0

T0

dt t T x

c

T 0 0

0

* n

n



0

=

=

1

0 1

0

0

n n n

n

n

T

0

0 0

n

T

0

0 0

(14)



0

0

n n n

n n

n

n

n

n

n*

n n

n n n

n n

n n





=

1

0

0

n

n

(15)



C0

1

0

1

x

0 n

n

Cn

n

=

1

0

0

n

n

n



 C0

0

0



0

0

0

0





(16)

## 週期訊號的功率分析 週期訊號的功率分析 週期訊號的功率分析 週期訊號的功率分析





0 2

0

T x t dt

P T

### T

0

2

*

2

0

* 2

*

0

* 2

0

*

0 2

0

|

|

) 1 (

) 1 (

) 1 (

) ( ) 1 (

) 1 (

0

0 0

0

0

0 0

0

n n n n

n

T

t nf j

n T n

n

t nf j n

T n

t nf j T n

T

c c

c

dt e

t T x c dt e

c t T x

dt e

c t T x dt t x t T x dt t T x

P

−∞

=

−∞

=

−∞

=

−∞

=

−∞

=

=

=

=

=



 

= 

=

=

π π

π





2 2 0

0

n

T

n

−∞

=

=

−∞

=

1

2 2 2

2 0 2

0

0 n

n n n

T n

(17)



0



0



−∞

=

+

−∞

=

−∞

=

n

t nf j n n

t nf j j n n

t nf j n

n

n

2π 0

θ 2π 0

(2π 0 θ )

|

|cn

n

* n

n

n n

n

n



0



n

n

=

1

0

0

n

n

n

(18)





+ + =

=

=

=

− −

=

=

=

1 2 ) ,

1 2 ( ) 1 1 (

0 2 ,

) 0 2 / ] sin(

2 [ 1

) 1 1 (

2 / 2 /

4 /

4 / -

2 0

2 2 /

2 / - 0

0 0

0 0 0

0

k k n

k n n

e n n e

j

dt T e

dt e t T x

c

k jn

jn

T T

t nf j t

nf T j

n T

π π π π

π π

π π

T0 f0

2 1 1 ) 1

1 ( /4

4 / 0 2

/

2 / 0 0

0

0 0

0

=

=

=T

x t dt T

dt

c T

T T

T

## 傅利葉級數範例 傅利葉級數範例 傅利葉級數範例 傅利葉級數範例(續 續 續 續)







−∞

=

−∞

=

+

n k

t f k j k t

nf j

n e

e k c t

x 2 0 2 (2 1) 0

π π

### π

] ) 14 1cos(

) 10 1cos(

) 6 1cos(

) 2 [cos(

2 1

] ) 1 2 ( 2 ) cos[

1 2 ( ) 2 1 2 (

) 1 (

0 0

0 0

0

0

+L

− +

− +

=

+ +

− +

=

=

t f t

f t

f t

f

t f k k

t x

k k

)

1 2 ( ) 1 1 (

; 0 , 0 2;

1

1 2 2

0= = ≠ + = − +

c k k

c

c k k k

(19)

(20)





/4

4 0 / 2

/ 2 /

2

0 2

0

0

0 0

0 0

T

T T

T T

−∞

=

k

2 2

2 2 2

2

−∞

=

−∞

= k

k

(21)



n

n

n

n



0 0

0 0 7

7f − = f

(22)











(23)

## 弦波訊號振幅設定 弦波訊號振幅設定 弦波訊號振幅設定 弦波訊號振幅設定

Sine Wave 之振幅(Amplitude)為1v Sine Wave 1 之振幅(Amplitude)為2v



## 弦波訊號頻率設定 弦波訊號頻率設定 弦波訊號頻率設定 弦波訊號頻率設定

Sine Wave2 之頻率(Frequency)為1000 Hz Sine Wave3 之頻率(Frequency)為2000 Hz



(24)

## 弦波訊號相位設定 弦波訊號相位設定 弦波訊號相位設定 弦波訊號相位設定

Sine Wave 4 之相位(Phase)為 0 (rad) Sine Wave 5 之相位(Phase)為π/2 (rad)













(25)





(26)

2 3 1



-3





(v)

1=1 A



2

A

(27)

Hz

1=1000 f

Hz

2=2000 f



1=0 ϕ

0

2

π π ϕ

ftd

3 0

0

d



(28)

1

A



(v) A2

Hz

1=1000 f

Hz

2=2000 f



(29)

1=0 ϕ

0

2

π π ϕ

ftd

3 0 0

d





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## References

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