agc dsp agc dsp professor a g constantinides 1 digital filter specifications only the magnitude...

Professor A G Constantinides

Digital Filter SpecificationsDigital Filter Specifications Only the magnitude approximation problem Four basic types of ideal filters with magnitude

responses as shown below (Piecewise flat)

0 c –c

HLP(e j)

0 c –c

HHP (e j)

–c1 c1 –c2 c2

HBP (e j)

–c1 c1 –c2 c2

HBS(e j)

Digital Filter SpecificationsDigital Filter Specifications These filters are unealisable because

(one of the following is sufficient) their impulse responses infinitely long

non-causal Their amplitude responses cannot be

equal to a constant over a band of frequencies

Another perspective that provides some understanding can be obtained by looking at the ideal amplitude squared.

Digital Filter SpecificationsDigital Filter Specifications Consider the ideal LP response squared

(same as actual LP response)

0 c –c

HLP(e j)

0 c – c

Digital Filter SpecificationsDigital Filter Specifications The realisable squared amplitude

response transfer function (and its differential) is continuous in

Such functions if IIR can be infinite at point but

around that point cannot be zero. if FIR cannot be infinite anywhere.

Hence previous defferential of ideal response is unrealisable

Digital Filter SpecificationsDigital Filter Specifications A realisable response would

effectively need to have an approximation of the delta functions in the differential

This is a necessary condition

Digital Filter SpecificationsDigital Filter Specifications For example the magnitude response

of a digital lowpass filter may be given as indicated below

Digital Filter SpecificationsDigital Filter Specifications In the passband we

require that with a deviation

In the stopband we require that with a deviation

1)( jeG

0)( jeG s

p eG ,1)(1

ssjeG ,)(

Digital Filter SpecificationsDigital Filter SpecificationsFilter specification parameters

- passband edge frequency

- stopband edge frequency

- peak ripple value in the passband

- peak ripple value in the stopband

Digital Filter SpecificationsDigital Filter Specifications

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

Peak passband ripple

dB Minimum stopband attenuation dB

)(log20)( 10 jeGG

)1(log20 10 pp

)(log20 10 ss

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

Example - Let kHz, kHz, and kHz

7pF 3sF25TF

56.01025

)107(23

24.01025

)103(23

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 or M for reduced computational complexity

Selection of Filter TypeSelection of Filter Type

zdzdzddzpzpzpp

Selection of Filter TypeSelection of Filter Type

FIR real digital filter transfer function is a polynomial in (order N) with real coefficients

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

If a linear phase is desired then we must have:

(More on this later)

nznhzH0

][][ nNhnh

Selection of Filter TypeSelection 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 is considerably higher than that of an equivalent IIR filter meeting the same specifications; this leads to higher computational complexity for FIR

FIR DesignFIR DesignFIR Digital Filter DesignFIR 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

Finite Impulse Response Finite Impulse Response FiltersFilters

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

0).()(

nznhzH

FIR: Linear phaseFIR: Linear phase

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

FIR: Linear phaseFIR: Linear phase To develop expression for phase

response set transfer function (order n)

In factored form

Where , is real & zeros occur in conjugates

nnzhzhzhhzH ...)( 2

)1().1()( 12

zzKzH i

1,1 ii K

Let where

)()()( 21 zNzKNzH

)1ln()1ln()ln())(ln(2

ii zzKzH

)1()( 11

i )1()( 12

Expand in a Laurent Series convergent within the unit circle

To do so modify the second sum as )

11ln()ln()1ln(

So that

1ln()1ln()ln()ln())(ln(2

ii zzznKzH

znKzH2

2 )ln()ln())(ln(

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 haveand

)()()( jj eAeH

Fundamental RelationshipsFundamental Relationships

hence (note Fourier form)

jmNmj e

jnKeH2

2)ln())(ln(

)())(ln())(ln())(ln( )( jAeAeH jj

Nm cos)()ln())(ln(

Nm sin)()(

FIR: Linear phaseFIR: 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 (normalised) equal to the number of zeros outside the unit circle

FIR: Linear phaseFIR: Linear phase It follows that zeros of linear phase

FIR trasfer functions not on the circumference of the unit circle occur in the form 1 ij

FIR: Linear phaseFIR: Linear phase For Linear Phase t.f. (order N-1) so that for N even:

)1()( nNhnh

0).().()(

n znhznhzH

0).1().(

n znNhznh

mn zznh nNm 1

FIR: Linear phaseFIR: Linear phase for N odd:

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

NhzznhzH

cos).(2.)(N

Tj NnTnheeH

FIR: Linear phaseFIR: Linear phase II) While for –ve sign

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

III) For N odd with +ve sign

sin).(2.)(N

Tj NnTnhjeeH

cos).(2

FIR: Linear phaseFIR: 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]

sin).(.2)(

Tj NnTnhjeeH

FIR: Linear phaseFIR: 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

Design of FIR filters: Design of FIR filters: WindowsWindows

(i) Start with ideal infinite duration (ii) Truncate to finite length. (This

produces unwanted ripples increasing in height near discontinuity.)

(iii) Modify toWeight w(n) is the window

)().()(~

nwnhnh

WindowsWindowsCommonly used windows

Rectangular Bartlett Hann Hamming Blackman

Kaiser

21 NnN

cos46.054.0

cos08.02

cos5.042.0

Kaiser windowKaiser window Kaiser window

β Transition width (Hz)

Min. stop attn dB

2.12 1.5/N 30

4.54 2.9/N 50

6.76 4.3/N 70

8.96 5.7/N 90

ExampleExample• Lowpass filter of length 51 and 2/ c

0 0.2 0.4 0.6 0.8 1

Lowpass Filter Designed Using Hann window

0 0.2 0.4 0.6 0.8 1

Lowpass Filter Designed Using Hamming window

0 0.2 0.4 0.6 0.8 1

Lowpass Filter Designed Using Blackman window

Frequency Sampling MethodFrequency 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

)(kH)(zH

Linear-Phase FIR Filter Linear-Phase FIR Filter Design by OptimisationDesign by Optimisation

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

)()()( AQH

4Typefor),2/sin(

3Typefor),sin(

2Typefor/2),cos(

1Typefor,1

Modified form of weighted error function

)]()()()[()( DAQW E

])()[()()()(

)](~)()[(~ DAW

)()()(~ QWW

)(/)()(~ QDD

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]

)](~)cos(][~)[(~)(0

DkkaWL

][~ ka

)( jeA][~ ka

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.

FIR Digital Filter Order FIR Digital Filter Order EstimationEstimation

Kaiser’s Formula:

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

2/)(6.14

)(log20 10

Hermann-Rabiner-Chan’s Formula:

]2/))[(,(),( 2

psspsp FDN

sppsp aaaD 1031022

101 log])(log)(log[),(

])(log)(log[ 61052

104 aaa pp ]log[log),( 101021 spsp bbF

4761.0,07114.0,005309.0 321 aaa

4278.0,5941.0,00266.0 654 aaa

51244.0,01217.11 21 bb

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

Fred Harris’ guide:

where A is the attenuation in dB Then add about 10% to it

2/)(20 ps

agc dsp agc dsp professor a g constantinides 1 digital filter specifications only the magnitude...

agc dsp agc dsp professor

g constantinides

digital filter design

digital lowpass filter

iir real digital filter

stopband slide

deviation slide

unrealisable slide

Documents

agc dsp agc dsp professor a g constantinides©1 fourier...

agc networks

an analytical construction of constantinides’ …an...

unstanding agc

receiver fm & agc - yuyunsitirohmah's...

agc application

incoming agc

sdffdsgu agc

professor a g constantinides 1 digital signal processing &...

agc dsp agc dsp professor a g constantinides©1 adaptive...

agc 35 agc 25 - de dietrich 2020-05-09 · 6.1 general...

magnet faqs - constantinides - magnetics - 2013€¦ ·...

agc meditations

agc dsp agc dsp professor a g constantinides©1 hilbert...

an analytical construction of constantinides’ social...

constantinides bakeries

steve constantinides, director of technology arnold ... ·...

agc plibrico

documentación agc

an analytical construction of constantinides’ social...