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Approximate An Ideal Low Pass Filter 理想低通滤波器的近似

Windows for FIR Filter Design

FIR滤波器设计用的窗函数

Filter Features 滤波器指标

Steps of Windowed FIR Design 窗函数FIR滤波器设计步骤

Examples

HomeWork: p379: 9.12, p380: 9.25

Approximate Ideal Low Pass Filter

Ideal Low Pass Filter Shape

   Impulse Response of Ideal LPF

Infinite Length, Non-Casual!

Windowing!

      Rectangular Window

      Its Frequency Response:   

 Multiply h_d [n] with Window w[n]:

h[n] = h_d [n] w[n]

           H(Ω) = H_d(Ω) * W(Ω)

Non-Ideal Low Pass Filter Shape:

   Cut-off frequency:  Ω_c    

20log|H(Ω_c)|/|H(0)| = 3 dB

  Finite Length in Time,

                  Smearing(拖尾) in Frequency.

Practicability 实用性

Quality 品质

Filter Features

Non-Ideal Low Pass Filter Shape:

      Pass band edge frequency: Ω_p

_{通带边缘频率}

      Stop band edge frequency: Ω_s

_{阻带边缘频率}

      Transition width: |Ω_s - Ω_p|

_过度带宽

      Pass band ripple: d_p

_通带波纹

      Stop band ripple: d_s

_阻带波纹

      Stop band attenuation:   -20log(d_s)

_阻带衰减

Windows for FIR Filter Design

Hanning Window:	|n| ≤ N
Hamming Window:	|n| ≤ N
Blackman Window:	|n| ≤ N
Kaiser Window:	Zero Order Modified Bessel Function 零阶修正贝塞尔函数 |n| ≤ N

Steps of Windowed FIR Design

Low Pass Filter

According to the Given Filter Specification

                { f_pass , f_stop , f_s, d_s }

        ----------------------------------------------------------------------

       1. Set  f_d = (f_pass+ f_stop)/2

       2. Compute

Ωd= 2π fd /fs ,

       3. According to 20log(d_s), From Table 9.3,

           Choose  w[n] and the Filter Length (2N+1) .

4. Calculate impulse response

h[n] = h_d[n]w[n].

5. Shift the h[n] to the right by N samples.

       ------------------------------------------------------------------------

P345, Example 9.7

Band Pass & High Pass Filters

Band Pass Filter { fstop,  fpass, f0, fs, ds}	High Pass Filter { fstop, fpass , fs, ds }
1. fd = f0 - (fpass+ fstop)/2 4. h[n]=hd[n]w[n]cos(nΩ0) Ω0 = 2π f0 / fs	1. fd = fs - (fpass+ fstop)/2 4. h[n]=hd[n]w[n]cos(nπ)

P355, Figure 9.45              P359, Example 9.11

Band Stop = Low Pass + High Pass

H_BS(Ω) = H_LP(Ω) + H_HP(Ω)

h_BS[n] = h_LP[n] + h_HP[n]

Ω_p^LP < Ω_p^HP

Band Pass = Low Pass x High Pass

H_BP(Ω) = H_LP(Ω) H_HP(Ω)

h_BP[n] = h_LP[n] * h_HP[n]

Ω_p^LP > Ω_p^HP

.

Examples

Filtering Speech

Original     Low Pass Filtering      High Pass Filtering

Filtering Music

Original     Low Pass Filtering