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# 第三章常見的基本訊號(一)弦波訊號弦波訊號

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## (一) 弦波訊號

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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

c

a

b

1 2 2

θ

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### x(t) = ω + φ

A：訊號之振幅大小；

f：f= ω /2π，為訊號之頻率，單位是赫茲(Hz)。

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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### n F =f/f s ；ω=2πF=2πf/f s

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𝑥𝑥[𝑛𝑛 + 𝑁𝑁] = 𝑥𝑥[𝑛𝑛]

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

= 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴(𝛺𝛺𝑛𝑛 + 𝛺𝛺𝑁𝑁 + 𝜑𝜑)

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

𝑁𝑁 �

𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑛𝑛𝐴𝐴

𝐴𝐴𝑐𝑐𝐴𝐴𝑐𝑐𝑐𝑐 � 𝑚𝑚 ∈ 𝑍𝑍, 𝑁𝑁 ∈ 𝑍𝑍

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2 2

1 2

2 2 2

2 2

2 2

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## z = +

2 2

1 2

2

θ θ

θ

θ θ

i i

i

i i

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+

− +

+

− +

+

− +

+

) ( )

(

) ( )

( )

( ) (

) (

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

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### ★ 1. Exponential Signalsx t ( ) = Be

a t

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

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/( )

( ) 0 t RC

=

### RC = Time constant

Discrete-time case:

n

where

### discrete-time signal.

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Signals_and_Systems_Simon Haykin & Barry Van Veen

13

/( )

( ) 0 t RC

=

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where

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### Nm N Ω = π

Periodic condition:

or

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

= 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

### [ ] [ ] [ ] 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

.

### <Sol.>

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

### Ω

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

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### φ = − π / 6

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

### 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:

jθ

### = cos θ + j sin θ

Complex exponential signal:

jφ

j t

ω

( )

+

j t

ω

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### −

2. Discrete-time case:

j n

### A Ω + n φ = Be

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)

j n

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t

α

### ω φ t + α >

Example for A = 60,

= 6, and

= 0.

−at

n

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• 題目：指數訊號

• 目標：說明指數訊號之基本定義

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## 習作3-3

• 目標：說明並複習複數之表示法與性質

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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

& 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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ex 3-4 exponential signal (multi).vi

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## End and Question?

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 無線射頻識別 (Radio Frequency Identification, RFID) 系統近年來越來越普及，應用範圍如供

(approximation)依次的進行分解，因此能夠將一個原始輸入訊號分 解成許多較低解析(lower resolution)的成分，這個過程如 Figure 3.4.1 所示，在小波轉換中此過程被稱為

Hanning Window 可用來緩和輸入訊號兩端之振幅，以便使得訊號呈現 週期函數的形式。Hanning Window