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Butterworth IIR LPF Design

Chebyshev IIR LPF Design

Filter Shape Transformation

Homework

             p431: 10.8,  p432: 10.16

Butterworth IIR LPF Design

   Given Features: { f_pass , f_stop , d_p, d_s, f_s }

     Impulse Invariance Method

           a.   ω_p = 2π f_pass ;    ω_s = 2π f_stop ;

           b.   Solve Equation Set:

                 for N and ω_c

            c.  Compute Poles:

            d.  Transfer Function:

  ?/span>        ?/span> ?/span>

            e.  Compare  |H(Ω)| to | H(ω)|

     Bilinear Transform Method

        a.   Ω_p = 2π f_pass / f_s ;   Ω_s = 2π f_stop /f_s;

        b.   ω_p = 2f_s tan(Ω_p) ;  ω_s = 2f_s tan(Ω_s) ;

        c.   Determine Order N and ω_c by:

        d.   Use Bilinear Transform to get h[n]

H(s)  →

→

H(z)  → |H(Ω)|

       e.  Compare |H(Ω)| to | H(ω)|

Butterworth Filter Shape vs Order

Chebyshev IIR LPF Design

   Given Features: { f_pass , f_stop , d_p, d_s, f_s }

    Impulse Invariance Method

          a.   ω_p = 2π f_pass ;    ω_s = 2π f_stop ;

           b.  

           c.   Determine Order N by:

           d.  Compute Poles From

           e.  Build H_a(s)

           f.  Transfer Function:

?/span>

           g.  Compare |H(Ω)| to | H(ω)|

   Bilinear Transform Method

a.   Ω_p = 2π f_pass / f_s ;    Ω_s = 2π f_stop /f_s;

      b.   ω_p = 2f_s tan(Ω_p) ;  ω_s = 2f_s tan(Ω_s) ;

      c.   ε² = (1-d_p)^-2  - 1

      d.   Determine Order N by:

      e.   Use Bilinear Transform get h[n]

→

H(z)  → |H(Ω)|

       f.  Compare  |H(Ω)| to | H(ω)|

Chebyshev Type I

              Filter Shape vs Order

Filter Shape Transformation

S Domain

     Low Pass   →   Low Pass

     Low Pass   →   High Pass

     Low Pass   →   Band Pass

Low Pass   →   Band Stop

z Domain

     Low Pass   →   Low Pass

     Low Pass   →   High Pass

 Low Pass   →   Band Pass

Low Pass   →   Band Pass