第四章常見的基本訊號(二)非弦波訊號非弦波訊號

授課老師：

林俊宏 老師 資料來源：

蕭子健 老師

第四章 常見的基本訊號

(二) 非弦波訊號

• 前言

– 介紹奇異訊號、方波、三角波、鋸齒波等訊號。

• 目標

– 瞭解奇異訊號基本的定義

– 瞭解利用 LabVIEW 產生非弦波訊號

• 關鍵名詞

– 步階訊號 (Step Signal)、脈衝訊號 (Impulse Signal)、斜坡訊號 (Ramp Signal)

– 方波、三角波、鋸齒波

★ Step Function

◆ Discrete-time case:

{ ^{1, 0, 0 0}

[ ] _n ⁿ u n = _< ^≥

x[n]

n 1 2 3 4

−1 0

−2

−3

1

★ Impulse Function

◆ Discrete-time case:

{ ^{1, 0}

[ ] 0, 0

n n

δ = n ⁼ ≠

{ ^{1, 0}

( ) 0, 0

u t t

t

= < >

◆ Continuous-time case:

注意：u(0) 不存在

注意：u[0] 存在

Example: Rectangular Pulse

Consider the rectangular pulse x(t) shown in following figure (a). This pulse has an amplitude A and duration of 1 second. Express x(t) as a weighted sum of two step functions.

Figure

(a) Rectangular pulse x(t) of amplitude A and duration of 1 s, symmetric about the origin. (b) Representation of x(t) as the difference of two step functions of amplitude A, with one step function shifted to the left by ½ and the other shifted to the right by ½; the two shifted signals are denoted by x₁(t) and x₂(t), respectively. Note that x(t) = x₁(t) – x₂(t).

<Sol.>

, 0 0.5

( ) 0, 0.5

A t

x t t

 ≤ <

=   >

1. Rectangular pulse x(t):

1 1

( ) 2 2

x t = Au t    +    − Au t    −   

 



<

= >

0 ,

0

0 ,

) 1

( t

t t u

 



−

<

−

= >

+ t T

T T t

t

u 0 ,

, ) 1

(

 



<

= >

0 ,

0

0 ,

) ) (

( )

( t

t t

t x u t x

步階訊號 (Step signal)

習作 4-1

• 題目：連續時間的步階訊號 (Step signal)

• 目標：何謂連續時間的步階訊號，及如何模擬與產生?

• 程式：ex 4-1 Step function (continuous).vi





−

<

−

= >

+





<

= >

T t

T T t

t u

t t t

u

, 0

, ) 1

(

0 , 0

0 , ) 1

(

★ Impulse Function

◆ Discrete-time case:

{ ^{1, 0}

[ ] 0, 0

n n

δ = n ⁼ ≠

◆ Continuous-time case:

( ) t 0 for t 0

δ = ≠

( ) t dt 1

∞

δ

−∞

=

∫

應用：非破壞性檢測 ( ) t lim

x t ( )

δ

=

◆ Properties of impulse function:

1. Even function: δ ( − = t ) δ ( ) t 2. Sifting property:

0 0

( ) ( ) ( )

x t δ t t d t x t

∞

−∞

− =

∫

3. Time-scaling property:

( ) at 1 ( ), t a 0 δ = a δ >

1. Rectangular pulse approximation:

( ) at lim

x at ( )

δ

=

2. Unit area pulse Time scaling

Area = 1/a

Restoring unit area ax

(at)

Figure

Steps involved in proving the time-scaling property of the unit impulse. (a) Rectangular pulse xΔ(t) of amplitude 1/Δ and duration Δ, symmetric about the origin. (b) Pulse xΔ(t) compressed by factor a. (c) Amplitude scaling of the compressed pulse, restoring it to unit area.

<p.f.>

★ Ramp Function

1. Continuous-time case:

, 0

( ) 0, 0

t t

r t t

 ≥

=   <

( ) ( ) r t = tu t

or

2. Discrete-time case:

, 0

[ ] 0, 0

n n

r n n

 ≥

=   <

Figure

Ramp function of unit slope.

or

[ ] [ ] r n = nu n

Figure

Discrete-time version of the ramp function.

x[n]

n 1 2 3 4

−1 0

−2

−3

4

注意：n 為整數