CHAPTER 7 Digital Filter Design

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CHAPTER 7Digital Filter Design

• Wang Weilian
• wlwang_at_ynu.edu.cn
• School of Information Science and Technology
• Yunnan University

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Outline

• About Digital Filter Design
• Bilinear Transformation Method of IIR Filter
  Design
• Design of Lowpass IIR Digital Filters
• Design of Hignpass, Bandpass, and Bandstop IIR
  Digital Filter
• FIR Filter Design Based on Windowed Fourier
  Series
• Computer-Aided Design of Digital Filters
• Digital Filter Design Using MATLAB

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About Digital Filter Design

• The most important step in the development of a
  digital filter Determine a realizable transfer
  function G(z)
• Digital Filter Specifications
• (1) magnitude response specifications in the
  passband and the stopband are given with some
  acceptable tolerances
• (2) A transition band is specified between
  the passband and the stopband to permit the
  magnitude to drop off smoothly

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About Digital Filter Design
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About Digital Filter Design

• Passband edge frequency
• Stopband edge frequency
• Peak ripple value of passband
• Peak ripple value of stopband
• Peak passband ripple
• Minimum stopband attenuation
• Sample frequency FT

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About Digital Filter Design
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About Digital Filter Design

• Selection of the Filter Type
• (1)The objective of digital filter design is
  to develop a causal transfer function H(z)
  meeting the frequncy specifications.
• (2)FIR and IIR Digital Filter
• FIR Digital Filter
• IIR Digital Filter

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About Digital Filter Design
FIR IIR
Impulse Response finite infinite
System Function H(z)P(z) H(z)P(z)/D(z)
Structure diagram Have feedback No feedback
Phase response Exact linear phase hn hn-N ________________
Zero-poles Only have zeros Both zeros and poles
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About Digital Filter Design

• The order NFIR of an FIR filter is higher than
  the
• order NIIR of an equivalent IIR filter meeting
  the
• same magnitude specifications
• The ratio NFIR/ NIIRis typically of the order of
  10 or more (the IIR filter usually is
  computationally more efficient)

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About Digital Filter Design

• Basic Approaches to Digital Filter Design
• Step1convert the digital filter specifications
  into analog lowpass prototype filter
  specifications
• Step2determine the analog lowpass filter
  transfer function Ha(s)
• Step3transform Ha(s) into the desired digital
  filter transfer function G(z)

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About Digital Filter Design

• Why analog?
• (1)Analog approximation techniques are highly
  advanced
• (2)They usually yield closed-form solutions
• (3)Extensive tables are available for analog
  filter design
• (4)Many applications require the digital
  simulation of analog filters

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About Digital Filter Design

• How to convert an analog prototype transfer
  function Ha(s) into a digital IIR transfer
  function G(z)?
• (1)the imaginary(j ) axisin the s-plane be
  mapped onto the unit circle of the z-plane
• (2)A stable analog transfer functon be
  transformed into a stable digital transfer
  function

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About Digital Filter Design

• Estimation of the Filter Order
• IIR The order of G(z) is determined from the
  transformation being used to convert Ha(s) into
  G(z)(The determination of Ha(s) is refered to
  Eq.(5.33),(5.41),or(5.51)
• FIR(lowpass digital filter)
• For narrowband filter
• For wideband filter

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Bilinear Transformation Method of IIR Filter
Design

• Bilinear transformation is more commonly used to
  design IIR digital filters based on the
  conversion of analog prototype filters
• The Bilinear Transformation
• S-plane to z-plane
• G(z) Ha(s)
• The transformation is a one-to-one mapping.
  It maps a single point in the s-plane to a unique
  point in the z-plane

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Bilinear Transformation Method of IIR Filter
Design
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Bilinear Transformation Method of IIR Filter
Design
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Bilinear Transformation Method of IIR Filter
Design

• Digital filter design procedure
• Step1 the invert bilinear transformation is
  applied
• to the digital filter specifications to
  arrive at the specifications of the analog
  filter function
• Step2 the bilinear transformation is employed to
  obtain the desired digital transfer function G(z)
  from the analog transfer function Ha(s) desired
  to meet the analog filter specifications

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Bilinear Transformation Method of IIR Filter
Design

• When T2(T has no effect on the G(z))
• If then lt1
• If gt0 then gt1

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Bilinear Transformation Method of IIR Filter
Design

• When and

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Bilinear Transformation Method of IIR Filter
Design

• Design of Digital IIR Notch Filters
• Example a second-order IIR notch filter
• Analog transfer function
• Applying a bilinear transformation
• Rewrite it
• Notch frequency
• Notch bandwidth

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Design of Lowpass IIR Digital Filters

• Steps of designing a lowpass IIR digital filter
• Step1 get the digital filter specifications(
  )
• Step2 convert to analog filter specifications
  with bilineat transformation
• Step3 design analog transfer function Ha(s)
• Step4 transfer Ha(s) to H(z) since

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Design of Lowpass IIR Digital Filters

• Example
• Passband edge frequency is 0.25 with a
  passband ripple of 0.5dB
• Stopband edge frequency is 0.55 with a
  stopband attenuation
• of 15dB
• Then

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Design of Lowpass IIR Digital Filters

• From the passband ripple of 0.5dB obtaining
• From the stopband attenuation of 15dB
  obtaining
• Then
• since
• then we get

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Design of Lowpass IIR Digital Filters

• The transfer function of third-order lowpass
  Butterworth is
• Then we can get

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Design of Highpass, Bandpass, and Bandstop IIR
Digital Filters

• To design IIR filters there are two approches can
  be followed
• First approch
• Step1 prewarp the digital frequency
  specifications to arrive at the specifications of
  an analog filter of the same type.

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Design of Highpass, Bandpass, and Bandstop IIR
Digital Filters

• Step2 convert the frequency specifications of
  HD(s) into that of a prototype analog lowpass
  filter HLP(S)
• ( s is the Laplace transform variable of the
  prototype analog lowpass filter HLP(S) and is
  the Laplace transform variable of the desired
  analog filter )

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Design of Highpass, Bandpass, and Bandstop IIR
Digital Filters

• Step3 Design the analog lowpass filter HLP(S)
  using the method described in Section 5.4
• Step4 convert the transfer function HLP(S) into
  HD(S) using the inverse of the frequency
  transformation used in step2
• Transform the transfer function HD(S) using the
  bilinear transformation to arrive at the desired
  digital IIR transfer function GD (Z)

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Design of Highpass, Bandpass, and Bandstop IIR
Digital Filters

• The second approach
• Step1 prewarp the digital frequency
  specifications to arrive at the specifications of
  an analog filter of the same type.

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Design of Highpass, Bandpass, and Bandstop IIR
Digital Filters

• Step2 convert the frequency specifications of
  HD(s) into that of a prototype analog lowpass
  filter HLP(S)
• ( s is the Laplace transform variable of the
  prototype analog lowpass filter HLP(S) and is
  the Laplace transform variable of the desired
  analog filter )

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Design of Highpass, Bandpass, and Bandstop IIR
Digital Filters

• Step3 Design the analog lowpass filter HLP(S)
  using the method described in Section 5.4
• Step4 convert the transfer function HLP(S) into
  the transfer function GLP(Z) of an IIR digital
  filter using the bilinear transformation
• Step5 transform GLP(Z) into the desired digital
  transfer function GD(z) using the appropriate
  spectral transformation discussed in Section 7.5

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Design of Highpass, Bandpass, and Bandstop IIR
Digital Filters

• The functions we usually used in Matlab
• lp2hp transform the lowpass analog filter to
  highpass analog filter
• lp2bp transform the lowpass analog filter
  to bandpass analog filter
• lp2bs transform the lowpass analog filter
  to bandstop analog filter