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Definition & Properties 

Diff. Eq., Transfer Function & DTFT

Sinusoid Explanation

Magnitude & Phase Response

Poles, Zeros & Filter’s Shape

Homework:

p274: 7.1,   p275: 7.5,   p278: 7.20

Frequency Response频率响应

   The discrete time Fourier transform (DTFT) of

        System Impulse Response.

F{ h[n] } = H (Ω),
F-1{H (Ω) } = h[n],

Properties

       Given H(Ω) of h[n]:

        P1. Periodicity

             H(Ω) = H(Ω+2π)

        P2. Time Delay

             F{ h[n-M] } = exp(-jMΩ) H (Ω)

        P3. Frequency Shift #

             F{ exp(-jΩ₀n) h[n] } =  H (Ω + Ω₀)

        P4. Linearity

        F{ a h₁[n] + b h₂[n] } = aH₁ (Ω) + b H₂ (Ω)

        where  H₁(Ω)=F{h₁[n] }, H₂(Ω)=F{ h₂[n] }

        P5. Conjugacy  #

             F{ h^* [n] } = H^* (-Ω)

              When h[n] is real, H^* (-Ω) = H (Ω)

        P6. Difference

            F{ h[n] - h[n-1] } = [1-exp(-jΩ)] H (Ω)

        P7. Flipping  #

             F{ h[-n] } = H (-Ω)

        P8. Upsampling  #

             F{ h[n/k] } = H (kΩ)

        P9. Differential  #

F{ n h[n] } =

        P10. Convolution   #

           F{ x[n] * h[n] } = X(Ω) H(Ω)

           where  F{ h[n] } = H(Ω) , F{ x[n] } = X(Ω)

        P11. Product   #

F{ h1[n] h2[n] } =

           where  F{ h₁[n] } = H₁(Ω) , F{ h₂[n] } = H₂(Ω)

        P12. Parseval  Theorem:   #

             |H (Ω)|²  : Power Spectrum Density

Difference Equation,

Transfer Function & DTFT

   Difference Equation of ARMA Model

   自回归滑动平均模型的差分方程

                 a₀y[n] + a₁y[n-1] + L + a_Ny[n-N]

              = b₀x[n] + b₁x[n-1] + L + b_Mx[n-M]

   After DTFT using P2,

                 Y(Ω) [a₀ + a₁e^-jΩ + L + a_Ne^-jNΩ ]

              = X(Ω) [b₀ + b₁e^-jΩ + L + b_Me^-jMΩ ]

   So,

              H(Ω) = Y(Ω) / X(Ω)

   Transfer Function

                        z    =>    exp(jΩ)

Sinusoid Explanation 

       When  the input x[n] is sinusoidal,

            x[n] = A cos(nΩ₀ +Φ_x)

       Short form notation of DTFT

            X(Ω₀) = A Ð Φ_x   ,  H(Ω₀) = G Ð Φ

       According to

            Y(Ω) = X(Ω)H(Ω)

       We have

            Y(Ω₀) = X(Ω₀)H(Ω₀) = AG Ð (Φ_x+Φ)

       Thus

                  y[n] = AG cos(nΩ₀ +Φ_x+Φ)

       Frequency Response is a collection

        Magnitude Gain & Phase Delay

        for all the frequencies.

                    H(Ω) = G(Ω) Ð Φ(Ω)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Magnitude  & Phase Response

     The Shape of the filter is the shape of the

     Magnitude Response.

     Rewrite H(Ω) into:

            H(Ω) = |H(Ω)| Ð Φ(Ω)

            H(Ω + 2π) = H(Ω)

            |H(-Ω)| = |H(Ω)|

             Φ(-Ω) = - Φ(Ω)

 

     Magnitude Response         Phase Response

    

    

                        

     Filter  Response

Responses	Unit
|H(Ω)|	None
20 log|H(Ω)|	dB
Φ(Ω)	Degree   or  Radian
Φ(f )	Degree   or  Radian

 

     Filters Examples: Comb Filter 梳状滤波器

        Comb Filter Input            Comb Filter Output

               Input (“Hello?                   Output (“Hello hello?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Poles, Zeros & Filter’s Shape

       Single Pole Filter:

    ， ，

            

 

       Three 2-Pole Filters

          

 

         Any Filter:

                  

                 

 

Distance	Filter
Poles Closer to    or Zeros Farther from ejΩ	Larger Gain     More Selective
Poles Farther from or Zeros Closer to  ejΩ	Smaller Gain     Less Selective

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