EEM478-DSPHARDWARE WEEK12:FIR & IIR Filter Designeem.eskisehir.edu.tr/mfidan/EEM 478/icerik/eem478_dsphw_week12… · Analog IIR Filter Design Commonly used analog filters : •Lowpass

EEM478-DSPHARDWARE

WEEK12:FIR & IIR Filter Design

PART-I : Filter Design/Realization

• Step-1 : define filter specs (pass-band, stop-band, optimization criterion,…)

• Step-2 : derive optimal transfer functionFIR or IIR design

• Step-3 : filter realization (block scheme/flow graph)direct form realizations, lattice realizations,…

• Step-4 : filter implementation (software/hardware)accuracy issues, …question: implemented filter = designed filter ?

Lecture-2 : FIR & IIR Filter Design • FIR filters

• Linear-phase FIR filters• FIR design by optimization

Weighted least-squares design, Minimax design• FIR design in practice

`Windows’, Equiripple design, Software (Matlab,…)

• IIR filters• Poles and Zeros• IIR design by optimization

Weighted least-squares design, Minimax design• IIR design in practice

Analog IIR design : Butterworth/Chebyshev/ellipticAnalog->digital : impulse invariant, bilinear transform,…Software (Matlab)

FIR Filters

FIR filter = finite impulse response filter

• Also known as `moving average filters’ (MA)• N poles at the origin z=0 (hence guaranteed stability) • N zeros (zeros of B(z)), àll zero’ filters• corresponds to difference equation

• impulse response

Linear Phase FIR Filters

• Non-causal zero-phase filters :example: symmetric impulse response

h[-L],….h[-1],h[0],h[1],...,h[L]h[k]=h[-k], k=1..L

frequency response is

i.e. real-valued (=zero-phase) transfer function

kL

Linear Phase FIR Filters• Causal linear-phase filters = non-causal zero-phase + delayexample: symmetric impulse response & N even

h[0],h[1],….,h[N]N=2L (even)h[k]=h[N-k], k=0..L

frequency response is

= i.e. causal implementation of zero-phase filter, by introducing (group) delay

kN0

Linear Phase FIR Filters

Type-1 Type-2 Type-3 Type-4N=2L=even N=2L+1=odd N=2L=even N=2L+1=oddsymmetric symmetric anti-symmetric anti-symmetrich[k]=h[N-k] h[k]=h[N-k] h[k]=-h[N-k] h[k]=-h[N-k]

zero at zero at zero atLP/HP/BP LP/BP BP HP

PS: `modulating’ Type-2 with 1,-1,1,-1,.. gives Type-4 (LP->HP)PS: `modulating’ Type-4 with 1,-1,1,-1,.. gives Type-2 (HP->LP)PS: `modulating’ Type-1 with 1,-1,1,-1,.. gives Type-1 (LP<->HP)PS: `modulating’ Type-3 with 1,-1,1,-1,.. gives Type-3 (BP<->BP)PS: IIR filters can NEVER have linear-phase property ! (proof see literature)

Filter Specification

Ex: Low-pass

FIR Filter Design by Optimization

(I) Weighted Least Squares Design :• select one of the basic forms that yield linear phase

e.g. Type-1

• specify desired frequency response (LP,HP,BP,…)

• optimization criterion is

where is a weighting function


• …this is equivalent to

= `Quadratic Optimization’ problem


• Example: Low-pass design

optimization function is

i.e.

FIR Filter Design by Optimization• a simpler problem is obtained by replacing the F(..) by…

where the wi’s are a set of (n) selected sample frequenciesThis leads to an equivalent (`discretized’) quadratic optimization problem:

+++ : simple- - - : unpredictable behavior in between sample freqs.

Compare to p.27


• …then all this is often supplemented with additional constraints

Example: Low-pass (LP) design (continued)pass-band ripple control :

stop-band ripple control :


Example: Low-pass (LP) design (continued) a realistic way to implement these constraints, is to impose the

constraints (only) on a set of sample frequencies , e.g. in the pass-band

and in the stop-bandThe resulting optimization problem is :

minimize :

subject to (pass-band constraints)(stop-band constraints)

= `Quadratic Programming’ problem (see HG94)


(II) `Minimax’ Design :• select one of the basic forms that yield linear phase

e.g. Type-1



where is a weighting function


• this is equivalent to

• the constraint is equivalent to a so-called `semi-definiteness’ constraint

where D>=0 denotes that the matrix is positive semi-definite

Filter Design by Optimization

• a realistic way to implement these constraints, is to impose the constraints (only) on a set of sample frequencies :

i.e. a `Semi-Definite Programming’ (SDP) problem, for which efficient interior-point algorithms and software are available.


• Conclusion:(I) weighted least squares design(II) minimax designprovide general `framework’, procedures to translate filter

design problems into standard optimization problems • In practice (and in textbooks):

emphasis on specific (ad-hoc) procedures : - filter design based on `windows’- equiripple design

FIR Filter Design using `Windows’

Example : Low-pass filter design• ideal low-pass filter is

• hence ideal time-domain impulse response is

• truncate hd[k] to N+1 samples :

• add (group) delay to turn into causal filter


Example : Low-pass filter design (continued)• ps : it can be shown that time-domain truncation corresponds to solving a

weighted least-squares optimization problem with the given Hd, and weighting function

• truncation corresponds to applying a `rectangular window’ :

• +++: simple procedure (also for HP,BP,…)• - - - : truncation in the time-domain results in `Gibbs effect’ in the frequency

domain, i.e. large ripple in pass-band and stop-band (at band edge (=discontinuity)), which cannot be reduced by increasing the filter order N.


Remedy : apply windows other than rectangular window:• time-domain multiplication with a window function w[k] corresponds to

frequency domain convolution with W(z) :

• candidate windows : Han, Hamming, Blackman, Kaiser,…. • window choice/design = trade-off between side-lobe levels (define peak

pass-/stop-band ripple) and width main-lobe (defines transition bandwidth)

FIR Equiripple Design

• Starting point is minimax criterion, e.g.

• Based on theory of Chebyshev approximation and the àlternation theorem’, which (roughly) states that the optimal d’s are such that the `max’ (maximum weighted approximation error) is obtained at L+2 extremal frequencies…

…that hence will exhibit the same maximum ripple (èquiripple’)• Iterative procedure for computing extremal frequencies, etc. (Remez

exchange algorithm, Parks-McClellan algorithm) • Very flexible, etc., available in many software packages• Details omitted here (see textbooks)

FIR Filter Design Software

• FIR Filter design abundantly available in commercial software• Matlab:

b=fir1(n,Wn,type,window), windowed linear-phase FIR design, n is filter order, Wn defines band-edges, type is `high’,`stop’,…b=fir2(n,f,m,window), windowed FIR design based on inverse Fourier transform with frequency points f and corresponding magnitude response mb=remez(n,f,m), equiripple linear-phase FIR design with Parks-McClellan (Remez exchange) algorithm

IIR filters

Rational transfer function :•

N poles (zeros of A(z)) , N zeros (zeros of B(z))• infinitely long impulse response• stable iff poles lie inside the unit circle• corresponds to difference equation

= also known as ÀRMA’ (autoregressive-moving average)

IIR filtersFrequency response versus pole-zero location :(cfr. frequency response is z-transform evaluated on the unit circle)

Example-1 : Low-pass filter

poles at

DC (z=1)

Nyquist freq (z=-1)

pole pole

ReIm

IIR filtersFrequency response versus pole-zero location :

Example-2 : Low-pass filter

poles at

zeros at

DC

Nyquist freq

zero

pole pole


Example-3 : Band-pass filter

poles at

zeros at

DCzero

zero

pole


* pole near unit-circle introduces `peak’ in frequency responsehence pass-band can be set by pole placement

* zero near unit-circle introduces `dip’ in frequency responsezero on unit-circle introduces `transmission zero’ in frequency

responsehence stop-band can be emphasized by zero placement

IIR Filter Design

+++• low-order filters can produce sharp frequency response• low computational cost

- - -• design more difficult• stability should be checked/guaranteed• phase response not easily controlled

(e.g. no linear-phase IIR filters)• coefficient sensitivity, quantization noise, etc. can be a

problem

IIR Filter Design by Optimization

(I) Weighted Least Squares Design :• IIR filter transfer function is



where is a weighting function• stability constraint :


(II) `Minimax’ Design :• IIR filter transfer function is



where is a weighting function• stability constraint :


These optimization problems are significantly more complex than those for the FIR design case… :• Problem-1: presence of denominator polynomial leads to rational

approximation, and hence to highly non-linear optimization (instead of -for instance- quadratic optimization in the FIR design case)• A possible procedure (`Steiglitz-McBride’) consists in iteratively (k=1,2,…)

minimizing

…which leads to iterative Quadratic Programming, etc.. (similar for minimax)


• Problem-2: stability constraint (zeros of a high-order polynomial are related to the polynomial’s coefficients in a highly non-linear manner)

• Solutions based on alternative stability constraints, that e.g. are affine functions of the filter coefficients, etc…

• Topic of ongoing research, details omitted here


• Conclusion:(I) weighted least squares design(II) minimax designprovide general `framework’, procedures to translate filter

design problems into ``standard’’ optimization problems • In practice (and in textbooks):

emphasis on specific (ad-hoc) procedures : - IIR filter design based analog filter design (s-domain design) and analog->digital conversion

- IIR filter design by modeling = direct z-domain design(Pade approximation, Prony, etc., not addressed here)

Analog IIR Filter Design

Commonly used analog filters :• Lowpass Butterworth filters

all-pole filters characterized by magnitude response…. (N=filter order)

Poles of H(s)H(-s) are equally spaced points on a circle of radius in s-plane

poles of H(s)N=4

poles of H(-s)


Commonly used analog filters :• Lowpass Butterworth filters

monotonic in pass-band & stop-band

`maximum flat response’: (2N-1) derivatives are zero atand

N


Commonly used analog filters :• Lowpass Chebyshev filters (type-I)

all-pole filters characterized by magnitude response (N=filter order)

is related to passband ripple are Chebyshev polynomials:


Commonly used analog filters :• Lowpass Chebyshev filters (type-I)

All-pole filters, poles of H(s)H(-s) are on ellipse in s-planeEquiripple in the pass-bandMonotone in the stop-band

• Lowpass Chebyshev filters (type-II)Pole-zero filters based on Chebyshev polynomialsMonotone in the pass-bandEquiripple in the stop-band

• Lowpass Elliptic (Cauer) filtersPole-zero filters based on Jacobian elliptic functionsEquiripple in the pass-band and stop-band

(hence) yield smallest-order for given set of specs


Frequency Transformations :• Principle : prototype low-pass filter (e.g. cut-off frequency = 1 rad/sec)

is transformed to properly scaled low-pass, high-pass, band-pass, band-stop,… filter• example: replacing s by moves cut-off frequency to

• example: replacing s by turns LP into HP, with cut-off frequency

• example: replacing s by turns LP into BP

• etc...

Analog -> Digital

• Principle :design analog filter (LP/HP/BP/…), and then convert it to a digital filter.

• Conversion methods: - convert differential equation into difference equation- convert continuous-time impulse response into discrete-time impulse response

- convert transfer function H(s) into transfer function H(z)• Requirement: the left-half plane of the s-plane should map into the

inside of the unit circle in the z-plane, so that a stable analog filter is converted into a stable digital filter.

Analog -> Digital

• Conversion methods: (I) convert differential equation into difference equation : -in a difference equation, a derivative dy/dt is replaced by a `backward difference’ (y(kT)-y(kT-T))/T=(y[k]-y[k-1])/T, where T=sampling interval.-similarly, a second derivative, etc…-eventually (details omitted), this corresponds to replacing s by (1-1/z)/T in Ha(s) (=analog transfer function) :

-stable analog filters are mapped into stable digital filters, but pole location for digital filter confined to only a small region (o.k. only for LP or BP)

jws-plane z-plane

j

1

Analog -> Digital• Conversion methods: (II) convert continuous-time impulse response into discrete-time impulse response : -given continuous-time impulse response ha(t), discrete-time impulse response is where T=sampling interval.

-eventually (details omitted) this corresponds to a (many-to-one) mapping

-aliasing (!) if continuous-time response has significant frequency content above the Nyquist frequency

s-planejw

z-planej

1

Analog -> Digital

• Conversion methods: (III) convert continuous-time system transfer function into discrete-time system transfer function : Bilinear Transform-mapping that transforms (whole!) jw-axis of the s-plane into unit circle in the z-plane only once, i.e. that avoids aliasing of the frequency components.

-for low-frequencies, this is an approximation of-for high frequencies : significant frequency compression (`warping’)(sometimes pre-compensated by `pre-warping’)

s-planejw

z-planej

1

IIR Filter Design Software

• IIR filter design considerably more complicated than FIR design (stability, phase response, etc..)

• (Fortunately) IIR Filter design abundantly available in commercial software

• Matlab:[b,a]=butter/cheby1/cheby2/ellip(n,…,Wn),

IIR LP/HP/BP/BS design based on analog prototypes, pre-warping, bilinear transform, … immediately gives H(z) (oef!)

analog prototypes, transforms, … can also be called individuallyfilter order estimation tooletc...

EEM478-DSPHARDWARE WEEK12:FIR & IIR Filter Designeem.eskisehir.edu.tr/mfidan/EEM 478/icerik/eem478_dsphw_week12… · Analog IIR Filter Design Commonly used analog filters : •Lowpass

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