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Digital Filter Specifications. We discuss in this course only the magnitude approximation problem There are four basic types of ideal filters with magnitude responses as shown below. Digital Filter Specifications.

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Digital filter specifications
Digital Filter Specifications

  • We discuss in this course only the magnitude approximation problem

  • There are four basic types of ideal filters with magnitude responses as shown below

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Digital filter specifications1
Digital Filter Specifications

  • These filters are unealizable becausetheir impulse responses infinitely long non-causal

  • In practice the magnitude response specifications of a digital filter in the passband and in the stopband are given with some acceptable tolerances

  • In addition, a transition band is specified between the passband and stopband

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Digital filter specifications2
Digital Filter Specifications

  • For example the magnitude response of a digital lowpass filter may be given as indicated below

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Digital filter specifications3
Digital Filter Specifications

  • In the passband we require that with a deviation

  • In the stopband we require that with a deviation

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Digital filter specifications4
Digital Filter Specifications

Filter specification parameters

  • - passband edge frequency

  • - stopband edge frequency

  • - peak ripple value in the passband

  • - peak ripple value in the stopband

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Digital filter specifications5
Digital Filter Specifications

  • Practical specifications are often given in terms of loss function (in dB)

  • Peak passband ripple

    dB

  • Minimum stopband attenuation

    dB

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Digital filter specifications6
Digital Filter Specifications

  • In practice, passband edge frequency and stopband edge frequency are specified in Hz

  • For digital filter design, normalized bandedge frequencies need to be computed from specifications in Hz using

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Digital filter specifications7
Digital Filter Specifications

  • Example - Let kHz, kHz, and kHz

  • Then

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Selection of filter type
Selection of Filter Type

  • The transfer function H(z) meeting the specifications must be a causal transfer function

  • For IIR real digital filter the transfer function is a real rational function of

  • H(z) must be stable and of lowest order N for reduced computational complexity

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Selection of filter type1
Selection of Filter Type

  • For FIR real digital filter the transfer function is a polynomial in with real coefficients

  • For reduced computational complexity, degree N of H(z) must be as small as possible

  • If a linear phase is desired, the filter coefficients must satisfy the constraint:

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Selection of filter type2
Selection of Filter Type

  • Advantages in using an FIR filter -

    (1) Can be designed with exact linear phase,

    (2) Filter structure always stable with quantised coefficients

  • Disadvantages in using an FIR filter - Order of an FIR filter, in most cases, is considerably higher than the order of an equivalent IIR filter meeting the same specifications, and FIR filter has thus higher computational complexity

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Fir design
FIR Design

FIR Digital Filter Design

Three commonly used approaches to FIR filter design -

(1)Windowed Fourier series approach

(2)Frequency sampling approach

(3) Computer-based optimization methods

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Finite impulse response filters
Finite Impulse Response Filters

  • The transfer function is given by

  • The length of Impulse Response is N

  • All poles are at .

  • Zeros can be placed anywhere on the z-plane

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Fir linear phase
FIR: Linear phase

  • Linear Phase:  The impulse response is required to be

  • so that for N even:

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Fir linear phase1
FIR: Linear phase

  • for N odd:

  • I) On we have for N even, and +ve sign

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Fir linear phase2
FIR: Linear phase

  • II) While for –ve sign

  • [Note: antisymmetric case adds rads to phase, with discontinuity at ]

  • III) For N odd with +ve sign

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Fir linear phase3
FIR: Linear phase

  • IV) While with a –ve sign

  • [Notice that for the antisymmetric case to have linear phase we require

    The phase discontinuity is as for N even]

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Fir linear phase4
FIR: Linear phase

  • The cases most commonly used in filter design are (I) and (III), for which the amplitude characteristic can be written as a polynomial in

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Fir linear phase5
FIR: Linear phase

For phase linearity the FIR transfer function must have zeros outside the unit circle

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Fir linear phase6
FIR: Linear phase

  • To develop expression for phase response set transfer function

  • In factored form

  • Where , is real & zeros occur in conjugates

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Fir linear phase7
FIR: Linear phase

  • Let

    where

  • Thus

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Fir linear phase8
FIR: Linear phase

  • Expand in a Laurent Series convergent within the unit circle

  • To do so modify the second sum as

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Fir linear phase9
FIR: Linear phase

  • So that

  • Thus

  • where

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Fir linear phase10
FIR: Linear phase

  • are the root moments of the minimum phase component

  • are the inverse root moments of the maximum phase component

  • Now on the unit circle we have

    and

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Fundamental relationships
Fundamental Relationships

  • hence (note Fourier form)

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Fir linear phase11
FIR: Linear phase

  • Thus for linear phase the second term in the fundamental phase relationship must be identically zero for all index values.

  • Hence

  • 1) the maximum phase factor has zeros which are the inverses of the those of the minimum phase factor

  • 2) the phase response is linear with group delay equal to the number of zeros outside the unit circle

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Fir linear phase12
FIR: Linear phase

  • It follows that zeros of linear phase FIR trasfer functions not on the circumference of the unit circle occur in the form

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Design of fir filters windows
Design of FIR filters: Windows

(i) Start with ideal infinite duration

(ii) Truncate to finite length. (This produces unwanted ripples increasing in height near discontinuity.)

(iii) Modify to

Weight w(n) is the window

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Windows
Windows

Commonly used windows

  • Rectangular 1

  • Bartlett

  • Hann

  • Hamming

  • Blackman

  • Kaiser

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Kaiser window
Kaiser window

  • Kaiser window

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Example
Example

  • Lowpass filter of length 51 and

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Frequency sampling method
Frequency Sampling Method

  • In this approach we are given and need to find

  • This is an interpolation problem and the solution is given in the DFT part of the course

  • It has similar problems to the windowing approach

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Linear phase fir filter design by optimisation
Linear-Phase FIR Filter Design by Optimisation

  • Amplitude response for all 4 types of linear-phase FIR filters can be expressed as

    where

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Linear phase fir filter design by optimisation1
Linear-Phase FIR Filter Design by Optimisation

  • Modified form of weighted error function

    where

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Linear phase fir filter design by optimisation2
Linear-Phase FIR Filter Design by Optimisation

  • Optimisation Problem - Determine which minimise the peak absolute value of

    over the specified frequency bands

  • After has been determined, construct the original and hence h[n]

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Linear phase fir filter design by optimisation3
Linear-Phase FIR Filter Design by Optimisation

Solution is obtained via the Alternation Theorem

The optimal solution has equiripple behaviour consistent with the total number of available parameters.

Parks and McClellan used the Remez algorithm to develop a procedure for designing linear FIR digital filters.

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Fir digital filter order estimation
FIR Digital Filter Order Estimation

Kaiser’s Formula:

  • ie N is inversely proportional to transition band width and not on transition band location

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Fir digital filter order estimation1
FIR Digital Filter Order Estimation

  • Hermann-Rabiner-Chan’s Formula:

    where

    with

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Fir digital filter order estimation2
FIR Digital Filter Order Estimation

  • Fred Harris’ guide:

    where A is the attenuation in dB

  • Then add about 10% to it

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Fir digital filter order estimation3
FIR Digital Filter Order Estimation

  • Formula valid for

  • For , formula to be used is obtained by interchanging and

  • Both formulae provide only an estimate of the required filter order N

  • If specifications are not met, increase filter order until they are met

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