第三章常見的基本訊號(一)弦波訊號弦波訊號

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授課老師：

林俊宏老師資料來源：

蕭子健老師

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第三章常見的基本訊號

(一) 弦波訊號

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• 前言

– 弦波訊號為許多應用的基礎，本章將從弦波之產生、特性、到應用範疇等作說明。

• 目標

– 瞭解不同函數所產生的弦波訊號 – 瞭解弦波訊號之週期特性

• 關鍵名詞

– 正弦訊號 (Sinusoidal signal)

– 指數訊號 (Exponential signal)

– 週期性 (Periodic)

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What does “sinusoidal” mean?

• The curve whose ordinates (縱座標) are proportional to sines of the abscissas (橫坐標) with the equation y=a*sin(x) (Webster’s Third New

International Dictionary)

• 數學上指三角函數之一。直角三角形中，一銳角的對邊除以斜邊所得的值，稱為此角的「正弦」。

(

中華民國教育部重編國語辭典修訂本

)

• 三角函數中的正弦函數(sin)及餘弦函數(cos)只有相位上的差別(兩者相位相差) ， 其餘性質並無不同，因此在討論訊號的時候，正弦函數及餘弦函數所表示的訊號一律都稱為「正弦訊號」。

c

a

b

 



=

= +

 



=

−

( / ) tan

1 ) ( cos )

( sin

/ ) cos(

/ ) sin(

1 2 2

a b where

c a

c b

θ

θ θ

θ

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) t

Acos(

x(t) = ω + φ

A：訊號之振幅大小；

ω ：角頻率，單位是每秒多少弳度(radians/s)；

Φ：初始相位，亦即 t=0 之相角，單位為弳度(radians)；

f：f= ω /2π，為訊號之頻率，單位是赫茲(Hz)。

首先，假設正弦訊號 x(t) 是一個週期性訊號，且 x(t) 的週期為 T，

依據週期的定義可知，T = 1/f = 2π /ω ，因此代入右式：

符合週期性訊號的要件，故可稱「連續時間下之弦波訊號為一週期性訊號」

𝑥𝑥(𝑡𝑡 + 𝑇𝑇) = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔(𝑡𝑡 + 𝑇𝑇) + 𝜑𝜑)

= 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔𝑡𝑡 + 𝜔𝜔𝑇𝑇 + 𝜑𝜑)

= 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔𝑡𝑡 + 𝜔𝜔2𝜋𝜋 𝜔𝜔 + 𝜑𝜑)

= 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔𝑡𝑡 + 2𝜋𝜋 + 𝜑𝜑)

= 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔𝑡𝑡 + 𝜑𝜑)

= 𝑥𝑥(𝑡𝑡)

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) cos(

) ( )

( t = x t = A ω t + φ

y if

時域 (Time Domain) 頻域 (Frequency Domain)

取樣 (Sampling) 取樣週期 (Sampling Period)

一個樣本為多少秒 (second / Sample, s/S)

取樣速率 (Sampling Rate)

每秒多少樣本(Sample/second, S/s)

類比 (Analog) 秒 (second, sec) 頻率 (cycle/sec)；角頻率 (rad/sec)

數位 (Digital) 次 (Time, time) 數位頻率 (cycle/S)；數位角頻率 (rad/S)

時域 (Time Domain) 頻域 (Frequency Domain) 取樣

T _s f _s = 1/T _s

類比

T f；ω = 2πf

數位

n F =f/f _s ；ω=2πF=2πf/f _s

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假設此離散時間之正弦訊號是一週期性訊號的話，就會有一個週期N，滿足下面 情況：(由於為離散時間之狀況，因此週期N與n相同必須為整數)

𝑥𝑥[𝑛𝑛 + 𝑁𝑁] = 𝑥𝑥[𝑛𝑛]

𝑥𝑥[𝑛𝑛 + 𝑁𝑁] = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴(𝛺𝛺(𝑛𝑛 + 𝑁𝑁) + 𝜑𝜑)

= 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴(𝛺𝛺𝑛𝑛 + 𝛺𝛺𝑁𝑁 + 𝜑𝜑)

由式子中可以得知，只有在 ΩN 等於2π 的整數倍時，才有可能使得成立，亦即

𝛺𝛺𝑁𝑁 = 2𝜋𝜋𝑚𝑚 (𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑛𝑛𝐴𝐴) 𝑚𝑚 ∈ 𝑍𝑍, 𝑁𝑁 ∈ 𝑍𝑍 𝛺𝛺 =2𝜋𝜋𝑚𝑚

𝑁𝑁 �

𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑛𝑛𝐴𝐴

𝐴𝐴𝑐𝑐𝐴𝐴𝑐𝑐𝑐𝑐 � 𝑚𝑚 ∈ 𝑍𝑍, 𝑁𝑁 ∈ 𝑍𝑍

Ex.. 請參考 page 3-37, 問題與討論

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複數 (Complex number)

iy x

z = +

實部 (Real part)

虛部 (Imaginary part)

) sin(

) ˆ cos(

ˆ ,

) ( tan )),

sin(

) (cos(

) (

2 2

1 2

2 2 2

2 2

θ θ

θ

i z

and y

x z

where z

z z

x and y

y x

r where i

r

y x

i y y

x y x

x z

iy x z

+

= +

=

= +

=

+ + + +

= +

=

−

複數表示法

三角函數表示法

向量空間表示法

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尤拉公式 (Euler formula)

iy x

z = +

實部 (Real part)

虛部 (Imaginary part)

) sin(

) cos(

,

} Im{

} Re{

) sin(

) cos(

) ( tan ),

sin(

) cos(

, )) sin(

) (cos(

2 2

1 2

2

θ θ

θ

θ θ

θ

θ θ

i e

and y

x z

where e

z z

z i

e

x and y

i e

y x

r where

re i

r z

iy x z

i i

i

i i

+

= +

=

+

= +

=

= +

=

= +

=

−

複數表示法

三角函數表示法尤拉表示法

向量空間表示法

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 

 



−

= +

+

= +

 



+

− +

=

+ +

+

=

≈ +

=

+

− +

+

− +

+

− +

+

) 2 (

) 1 sin(

) 2 (

) 1 cos(

) sin(

) cos(

) sin(

) cos(

) (

) ( )

(

) ( )

( )

( ) (

) (

ϕ ω ϕ

ω

ϕ ω ϕ

ω ϕ

ω ϕ ω

ϕ ω

ϕ ω ϕ

ω

ϕ ω ϕ

ω ϕ ω

t i t

i

t i t

i t

i t i

t i

e i e

t

e e

t

t i

t e

t i

t e

Ae t

A t

x

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Elementary Signals

★ 1. Exponential Signals x t ( ) = Be

^{a t}

1. Decaying exponential, for which a < 0 2. Growing exponential, for which a > 0

B and a are real parameters

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Ex. Lossy capacitor:

Figure

Lossy capacitor, with the loss represented by shunt resistance R.

( ) ( ) 0 RC d v t v t

dt + =

KVL Eq.:

/( )

( ) 0 ^t ^RC

v t

=

V e

⁻

RC = Time constant

Discrete-time case:

[ ]

ⁿ

x n = Br

r = e ^α

where

Figure

(a) Decaying

exponential form of discrete-time signal.

(b) Growing

exponential form of

discrete-time signal.

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Signals_and_Systems_Simon Haykin & Barry Van Veen

13

( ) ( ) 0 RC d v t v t

dt + =

/( )

( ) 0 ^t ^RC

v t

=

V e

⁻

0 )

) ( ( )

( )

(

then circuit,

RC a

s it' if

) ) (

) ( ) (

) ( (

) ) (

( )

(

= +

=>

=

=>

=

=>

=

∫

t dt v

t RC dv

dt t C dv

R t v

dt t C dv

t C i

dt t i C

t t Q

v

R t t v

i R

t i t

v

R c

c R

c c

R

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★ 2. Sinusoidal Signals ( ) cos( ) x t = A ω φ t +

T 2 π

= ω

where

( ) co ( ( s ) )

cos( )

cos( 2 ) cos( )

( )

x t T A t T

A t T

A t

x t

ω φ

ω ω φ ω π φ ω φ

+ = + +

= + +

= +

=

periodicity

◆ Continuous-time case:

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◆ Discrete-time case :

[ ] cos( )

x n = A Ω + n φ

[ ] cos( )

x n + N = A Ω + Ω + n N φ

2 N π m

Ω = ² m radians/cycle, integer ,

N m N Ω = π

Periodic condition:

or

Ex. A discrete-time sinusoidal signal: A = 1,

φ

= 0, and N = 12.

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Example 1.7 Discrete-Time Sinusoidal Signal

A pair of sinusoidal signals with a common angular frequency is defined by

1 [ ] sin[5 ]

x n = π n x n ₂ [ ] = 3 cos[5 π n ]

1 2

[ ] [ ] [ ] y n = x n + x n

and

(a) Both x₁[n] and x₂[n] are periodic. Find their common fundamental period.

(b) Express the composite sinusoidal signal

In the form y[n] = Acos(Ωn +

φ

), and evaluate the amplitude A and phase

φ

.

<Sol.>

(a) Angular frequency of both x₁[n] and x₂[n]:

5 radians/cycle π

Ω = 2 2 2

5 5

m m m

N π π

= = π =

Ω

This can be only for m = 5, 10, 15, …, which results in N = 2, 4, 6, …

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sin( ) 1 and cos( ) 3

A φ = − A φ =

1 2

sin( ) amplitude of [ ] 1 tan( )

cos( ) amplitude of [ ] 3 x n

x n φ φ

φ

= = = −

sin( ) 1 A φ = −

( ¹ ) ²

sin / 6

A π

= − =

− [ ] 2cos 5

y n = ^   π n − π 6 ^  

φ = − π / 6

Accordingly, we may express y[n] as (b) Trigonometric identity:

cos( ) cos( ) cos( ) sin( )sin( ) A Ω + n φ = A Ω n φ − A Ω n φ

x

₁[n] + x₂[n] with the above equation to obtain that Let Ω = 5π, then compare

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★ 3. Relation Between Sinusoidal and Complex Exponential Signals

1. Euler’s identity:

e

^j^θ

= cos θ + j sin θ

Complex exponential signal:

B = Ae

^j^φ

cos( ) Re{

^{j t}

} A ω φ t + = Be

^ω

( )

cos( ) sin( )

j t

j j t

j t

Be

Ae e Ae

A t jA t

ω

φ ω φ ω

ω φ ω φ

+

=

= + + +

( ) cos( ) x t = A ω φ t +

◇ Continuous-time signal in terms of sine function:

( ) sin( ) x t = A ω φ t +

sin( ) Im{

^{j t}

}

A ω φ t + = Be

^ω

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π / 4 π / 4

−

2. Discrete-time case:

cos( ) Re{

^{j n}

}

A Ω + n φ = Be

^Ω and

3. Two-dimensional representation of the complex exponential e ^{j Ω n} for Ω = π/4 and n = 0, 1, 2, …, 7.

:

Projection on real axis: cos(Ωn);

Projection on imaginary axis: sin(Ωn)

Figure

Complex plane, showing eight points uniformly distributed on the unit circle.

sin( ) Im{

^{j n}

}

A Ω + n φ = Be

^Ω

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★ 4. Exponential Damped Sinusoidal Signals

( )

^t

sin( ), 0

x t = Ae

⁻^α

ω φ t + α >

Example for A = 60,

α

= 6, and

φ

= 0.

Figure

Exponentially damped sinusoidal signal Ae

^−at

sin( ω t), with A = 60 and α

= 6.

◆ Discrete-time case:

[ ]

ⁿ

sin[ ]

x n = Br Ω + n φ

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習作 3-1

• 題目：正弦訊號 (Sinusoidal signal)

• 目標：何謂正弦訊號，及如何模擬與產生?

• 程式：ex 3-1 Sinusoidal signal.vi

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習作3-2

• 題目：指數訊號

• 目標：說明指數訊號之基本定義

• 程式： ex 3-2 Exponential signal.vi

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習作3-3

• 題目：複數(Complex number)及正交(Orthogonal)

• 目標：說明並複習複數之表示法與性質

• 程式：ex 2-3 Addition continuous.vi & ex2-3 Addition discrete.vi

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習作3-4

• 題目：實指數訊號

• 目標：說明實指數訊號之定義與意涵

• 程式：ex 3-4 exponential signal (continuous).vi

& 3-4 exponential signal (discrete).vi Case 1：當a為正數時(a>0)，隨 t 的增加成指數遞增。

Case 2：當a為負數時(a<0)，隨 t 的增加成指數遞減。

Case 3：當a為零時(a=0)，，亦即是任何時間 t 下訊號值皆為常數B。

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有何不同?

ex 3-4 exponential signal (multi).vi

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End and Question?

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第三章常見的基本訊號(一)弦波訊號弦波訊號

授課老師：

林俊宏 老師 資料來源：

蕭子健 老師