Definition of Spectrum 谱

Spectra of Non-Periodic Signal

非周期信号的谱

Spectra of Periodic Signal

               周期信号的谱

Frequency Characteristics

              频率特征

Usage of Spectrum 应用举例

Homework:  p309: 8.10,  p309: 8.13,  p313: 8.20

Definition定义

   Merriam-Webster Online Dictionary:

     a: A continuum连续统一体 of color formed

       when a beam of white light is dispersed

       分散as by passing through a prism棱镜,

       so that its component wavelengths are

       arranged in order.   --- 光谱

    b: Any of various continua连续统一体 that

        resemble类似 a spectrum光谱 in

        consisting of an ordered arrangement

        by a particular characteristic, such as

        frequency, energy, etc.

      Examples

         Optics1光谱    Optics2热谱

         Computer Color色谱

    For Signal Processing:

     Spectrum = Fourier Transform of signal.

       Time Domain       Frequency Domain

            x ( t )        =FT=>     X (f )

         Fourier Transform (FT):

          Rewrite X(f ) into:

          X(f ) = |X(f )| exp[ jθ(f )]

          |X(f )| : Magnitude Spectrum幅度谱

           θ(f )  : Phase Spectrum相位谱

       Properties:

               | X(- f )| = |X(f )|

               θ(- f )  = - θ(f )

                Q1: Y?

           For all square and rectangular non-

       periodic signals, magnitude spectra

       have the shape of the absolute value

       of a sinc (sine Cardinal主正弦) function.

       It is also known as:

        Sampling Function or Filtering Function.

       sinc(t) = sin(t) / t , sinc_π(t) = sin(π t) / (π t) 

   sinc(x)                           sinc_π(x)

-10π< x <10π

Spectra of Non-Periodic Series

非周期序列的谱

              x [ n ]   =DTFT=>  X (Ω)

    Discrete Time Fourier Transform:

: Digital Frequency数字频率

       Rewrite X(Ω) into:

       X(Ω) = |X(Ω)| exp[ jθ(Ω)]

       Magnitude Spectrum: |X(Ω)|   幅度谱

       Phase Spectrum:        θ(Ω)     相位谱

       They are Periodic with 2π Period. Q2: Y?

        Non-Periodic Signals

        Magnitude Spectrum

        Phase Spectrum

      |X(Ω)|  &  θ(Ω) are Continuous Spectra.

Spectra of Periodic Series

周期序列的谱              

               x [ n ]   =DFS=>    c_k

       Discrete Fourier Series (DFS):

              离散付氏级数

       Fourier Coefficient:付氏系数

         where N is the Period of x[n]. 

                    c_k=c_k+N   Page 292 8.3 ~

         Rewrite c_k into:  c_k = |c_k | exp[ jΦ_k]

        Magnitude Spectrum: |c_k |    幅度谱

        Phase Spectrum:        Φ_k     相位谱

        They are Periodic with N Period.

        Periodic Signal

        Magnitude Spectrum

         Phase Spectrum

     |c_k |  & Φ_k  are Discrete Spectra 离散谱

        － Line Spectra 线谱

          Sub Summary : Page 294 Fig8.11

Frequency Characteristics

             频率特征

  Signal's Frequency Characteristics

         Spectrum of horse whinny嘶叫           

             (non-periodic)

       Spectrum of square wave (periodic)

  System Frequency Characteristics:

        Frequency Response 频率响应

h(t)    =FT=>   |H(f )|

Filter Gain: |H(f )|/max(|H(f )|)

Usage of Spectrum

       A. Speech Synthesis

Real "eee" Sound

            |X(f)| of Real "eee" Sound

Synthesized合成的 "eee" Sound

            |X(f)| of Synthesized "ee"sound

       B. New Media

                  Water Sound