[PDF] DIGITAL FILTERING 1 Applications 2 Digital and analog filters PDF filt.pdf

FIR filter: A filter with a Finite Impulse Response The transfer function H(z) of a causal FIR filter is a polynomial in z −1

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DIGITAL FILTERING

Applications

Digital and analog lters: Pros and cons

I IRdigital lters

FIR digital lters

FIR and I IRlters: Pr osand cons

Ideal lters

The app roximationp roblem

8. The realization p roblemI. Selesnick EL 713 Lecture Notes1

APPLICATIONS

Noise supp ression

(a) imaging devices (medical, etc) (b) biosignals (hea rt,b rain) (c) signals sto redon analog media (tap es) 2.

Enhancement of selected frequency ranges

(a) equalizers fo raudio systems (increasing the bass) (b) edge enhancement in images 3.

Removal o ra ttenuationof selected frequenc ies

(a) removing the DC comp onentof a signal (b) removing interfer encesat a sp ecicfrequency ,fo rexample those caused by power supplies 4.

Bandwidth limiting

(a) anti-aliasing lters fo rsampling (b) ensuring t hata transmitted signal o ccupiesonly its alloted frequency band. 5.

Sp ecialop erations

(a) dierentiation (b) integration (c)

Hilb erttrans form

Simulation/Mo deling

(a) simulating c ommunicationchannels (b) mo delinghuman audito rysystem I. Selesnick EL 713 Lecture Notes2

DIGITAL AND ANALOG FILTERS: PROS AND CONS

It was said of early digital lters that they

Cost to omuch

W ereto ola rge

Used to omuch p ower.

But these considerations have become less important with advances in hardware. Digital lters have the following advantages 1. Programmable (lter cha racteristicseasily changed) 2.

Reliable and rep eatable

F reefrom comp onentdrift

No tuning required

5. Sup eriorp erformancein some cases I. Selesnick EL 713 Lecture Notes3

IIR DIGITAL FILTERS

IIR lter:A lter with an Innite Impulse Response.

The transfer functionH(z)of a realizable IIR lter must be a ra- tional transfer function inz1:

H(z) =b0+b1z1++bMzM1 +a1z1+aNzN:

IIR digital lters are implemented using ARMA (autoregressive mov- ing average) dierence equation: y(n) =b0x(n) ++bMx(nM) a1y(n1) aNy(nN) FIR DIGITAL FILTERSFIR lter:A lter with a Finite Impulse Response. The transfer functionH(z)of a causal FIR lter is a polynomial in z 1:

H(z) =b0+b1z1++bMzM:

FIR digital lters are usually implemented using MA (moving aver- age) dierence equation: y(n) =b0x(n) +b1x(n1) ++bMx(nM):I. Selesnick EL 713 Lecture Notes4

FIR AND IIR FILTERS: PROS AND CONS

FIR digital lters have several desirable properties in relation to IIR lters. 1.

FIR lters can have exactly linea rphase.

FIR lters a reautomatically stabl e.

There a reseveral very

exible metho dsfo rdesigning FIR digital lters. 4.

FIR lters a reconvenient to implement.

On the other hand,

1. Linea r-phaseFIR lters can have long dela yb etweeninput and output. 2. If the phase need not b elinea r,then I IRlters can b emuch more ecient to implement.I. Selesnick EL 713 Lecture Notes5

IDEAL FILTERS

The frequency responses of the commonideallters:

Lo w-pass0

!o!o2.Hi-pass 0 !o!o3.Band-pass 0 !a!a!b!b4.Band-stop 0 !a!a!b!b5.Notch 0 !o!oI. Selesnick EL 713 Lecture Notes6

DIGITAL FILTER DESIGN

The lter design process consists of two parts: the approximation problem and the realization problem. The approximation problem deals with the choice of parameters or coecients in the lter's transfer function. The realization part of the design problem deals with choosing a structure to implement the transfer function. The approximation stage can be divided into 4 steps: 1. A desire do ri dealresp onseis chosen (usually in the frequency domain). 2. A class of lters is chosen (fo rexa mple,FIR vs I IR). 3. A design criteria is chosen (least squa reo rminimax). 4. An algo rithmis selected to design the transfer function. The realization stage can also be divided into 4 steps: 1.

A set of structures is chosen.

2. A criteria fo rcompa ringdierent implementations is chosen. 3. The b eststructure is chosen, and its pa rametersa recalculated from the transfer function. 4. The structure is implemented in ha rdwareo rsoft ware.I. Selesnick EL 713 Lecture Notes7

THE APPROXIMATION PROBLEM

The impulse response of ideal lter

d(n) =IDTFTDf(!); whereDf(!)is an ideal low-pass response, for example, is unreal- izable because it is noncausal and of innite duration. The actual frequency response of the lter will be denoted byHf(!) H f(!) =DTFTfh(n)g=1X n=1h(n)ejn!: The DTFT is theZ-transform evaluated on the unit circlez=ej!.

H(z) =Zfh(n)g=1X

n=1h(n)zn H f(!) =H(ej!) The desired (or target) response of the lter will be denotedD(!).

Common approximation criteria:

The square errorcriterion is written as

E 2=Z 2 0

W(!) jHf(!)D(!)j2d!

2. The Minimax (or Chebyshev)error criterion is written as Equotesdbs_dbs2.pdfusesText_2

[PDF] [PDF] DIGITAL FILTERING 1 Applications 2 Digital and analog filters