Lecture 8: Digital Filter Structures and Filter Toolbox

Mr. Iskandar Yahya

Prof. Dr. Salina A. Samad

Basic Elements

We need 3 elements for designing digital filters:Adder – 2 input and 1 output. Addition of two signals:

x1(n), x2(n) x1(n) + x2(n)

Multiplier – Also called gain. x(n) ax(n)

Delay – Also called shifter or memory. Delays the signal by one or more samples: x(n) x(n -1)

Using these elements, we can describe various structures of IIR and FIR filters

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IIR Filter StructuresSystem function of IIR filter:

an and bn are coefficients and the order is N if aN ≠ 0.

The difference equation representation of IIR filters:

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IIR Filter Structures3 structures to implement an IIR filter:

Direct Form – According to (2) and have 2 parts, namely moving average part and recursive part (the numerator and denominator). There are 2 versions; direct forms 1 and direct forms 2.

Cascade Form – System function as in (1) is factored into biquads and is represented as a product of these biquads. Each biquad is direct form and entire system is implemented as a cascade of biquad sections.

Parallel Form – Similar to cascade form, but use partial fraction to represent the system function.

Review………Discrete-time system:

Discrete-time system has discrete-time input and output signals

Review………Linear time-invariant (LTI) system:

Discrete-time system is LTI if its input-output relationship can be described by the linear constant coefficients difference equation

The output sample y() might depend on all input samplesthat can be represented as

))(()( kxy

Review………MATLAB example 1:

N = 80; k = 0:(N-1);

b0 = 1;

b1 = -1;

b2 = 1;

B = [b0 b1 b2];

f = 1/8;

x = sin(2*pi*f*k+pi/6);

y = filter(B,1,x);

subplot(2,1,1)

systemFIR(0,0,4,5,10,'b')

subplot(2,1,2)

plot(k,x,'go', k,y,'bo',...

k,x,'g-', k,y,'b-')

legend('input','output')

MATLAB filter command

corresponds to the symbol

)2()1()()( 210 nxbnxbnxbny )2()1()()( 210 nxbnxbnxbny

))(()( kxny

Review………Impulse response:

Response of a system to the unit impulse sequence is called the unit impulse response or impulse response for short

))(()( knh ))(()( knh

Review………MATLAB example 2:N = 16; k = 0:(N-1);

b0 = 1;b1 = -1;b2 = 2;

B = [b0 b1 b2];

x = (k==0);

y = filter(B,1,x);

subplot(3,1,1)systemFIR(0,0,4,5,10,'b')

subplot(3,1,2)stem(k,x,'r')ylabel('input')

subplot(3,1,3)stem(k,y,'b')ylabel('output')

)2()1()()( 210 nxbnxbnxbny )2()1()()( 210 nxbnxbnxbny

))(()( knh ))(()( knh

FIR and IIR Filters and SystemWhat are FIR and IIR systems?

A discrete system is said to be an FIR system if its impulse response has zero-valued samples for n > M > 0

Integer number M is called the length of the impulse response

IIR system is a discrete system with an infinite impulse response

FIR = Finite Impulse Response IIR = Infinite Impulse Response

FIR and IIR Filters and SystemMATLAB example 3:N = 80; k = 0:(N-1);

a = 0.97;

B = [0 1];

A = [1 -a];

x = (k==0);

y = filter(B,A,x);

subplot(3,1,1)draw1stIIR(0,0,4,5,10,'b')

subplot(3,1,2)stem(k,x,'r')ylabel('input')

subplot(3,1,3)stem(k,y,'b')ylabel('output')

)1()1()( naynxny )1()1()( naynxny

The impulse response does not vanish after

finite number of samples

Basic FIR structuresDirect form, Transposed form, Cascade form,

Linear-phase, Symmetric

Direct form 2nd order

Basic FIR structuresTransposed direct form 2nd order

Basic FIR structuresCascade direct form

Basic FIR structuresDirect form (Tapped delay line)

Basic FIR structuresTransposed direct form

Basic FIR structuresLinear-phase type 1

Basic FIR structuresLinear-phase type 2

Basic FIR structuresSymmetric FIR type I

Basic FIR structuresSymmetric FIR type II

Basic IIR structuresDirect form, Transposed form

Direct form I 2nd order

Basic IIR structuresDirect form I 2nd order

Basic IIR structuresDirect form II 2nd order

Basic IIR structuresCascade direct form II

Basic IIR structuresTransposed direct form II 1st order

Basic IIR structuresTransposed direct form II 2nd order

Simple FIR design example4th order linear-phase filter

Simple FIR design exampleFile, Export …

Simple FIR design exampleEdit, Convert Structure …

Simple FIR example designs

Classical digital filter design

Overview:SpecificationStep 1: Approximation - Select Chebyshev Type

IStep 2: Realization - Cascade of biquadsStep 3: Study of imperfections - QuantizationsRedo design steps

Step 1: Approximation Step 2: Realization Step 3: Study of imperfections

Step 4: Implementation

Classical digital filter designStart with specification:

Classical digital filter designStart with specification:

Classical digital filter designStep 1: Approximation; Select Chebyshev

Type I

Classical digital filter designExport coefficients: File, Export

Classical digital filter designStep 2: Realization; Direct-form II biquads

Classical digital filter designStep 3: Study of imperfections:

Quantization The spec is violated!

Classical digital filter designRedo step 1: Approximation, Increase

order

Classical digital filter designRedo step 3: Study of imperfections

(Quantization) The spec is OK

Classical digital filter designStep 4: Implementation as a C header file

Digital filters: FIR filter designOverview

FIR filtersDesign techniquesFIR filter design

using Filter Design Toolbox

Digital filters: FIR filter designFIR filter

FIR filter is characterized by a unit sample response that has a finite duration

An advantage of FIR filters compared to IIR counterparts is that FIR filters can be designed with exactly linear phase

Linear phase is important for applications where phase distortion due to nonlinear phase can degrade performance (e.g., data transmission)

Digital filters: FIR filter designDesign of FIR filters

Windowing techniquesEquiripple FIR filtersLeast squares and related techniquesLinear programming (LP) and mixed integer

linear programming (MILP)Computationally efficient FIR filters using

periodic subfilters as building blocks

Digital filters: FIR filter designWindowing techniques

One of the earliest design techniquesCoefficients can be obtained in

closed formThere is no need for solving complex

optimization problemsMost frequently used window functions are

Bartlett, Hann, Hamming, Blackman, and Kaiser

Windows

Digital filters: FIR filter designFilter Design Toolbox windows

The main role of the window is to damp out the effects of the Gibbs phenomenon that results from truncation of an infinite series.

Kaiser

Hann

Hamming

Blackman

Bartlett

Digital filters: FIR filter designEquiripple FIR filtersDesign of FIR filters can be accomplished through

iterative methods

An iterative method is known as the weighted-Chebyshev method; an error function is formulated in terms of linear combination of cosine functions and is then minimized by using a very efficient multivariable optimization algorithm known as the Remez exchange algorithm

When convergence is achieved, the error function becomes equiripple

Amplitude of the error in different frequency bands is controlled by applying weighting to the error function

Equiripple

remez designs a linear-phase FIR filter using the Parks-McClellan algorithm

Constrained Equiripplefirceqrip

Digital filters: FIR filter design

Least squares and related techniquesBased on approach of minimizing, in some

specified manner, the mean squared error between the transfer function of the filter and the frequency characteristics of a desired filter response

Four approaches exist: unweighted list squares, frequency sampling, weighted list squares, and eigen filter methods

Least square linear-phase

Least square linear-phase FIR filter design (firls)

Least Pth-norm

Maximally flatSymmetric FIR Butterworth filter (maxflat)

Advanced digital filter design

Overview:Drawback of the classic designAdvanced design paradigmAFDesign toolboxExample design

Advanced digital filter design

Drawback of the classic designThe principal drawback of the classical digital

filter design is in returning only one solutionThat solution can be unacceptable for many

practical implementationsAdvanced approach provides

reduced filter complexity for the desired performance

better performances for the required complexity

Advanced digital filter design

Advanced design paradigmAdvanced digital filter design introduces

straightforward procedures to map the filter specification into a design space, that is, a set of ranges for parameters that we use in the filter design

We search this design space for the optimum solution according to given criteria, such as optimized group delay

Advanced digital filter design

Design example 1Design a lowpass filter that satisfies the

following specification:Fp = 1600 HzFs = 1696 HzAp = 0.2 dBAs = 40 dBFsamp = 8 kHzGoal: Optimize group delay

Advanced digital filter design

Classic design

Fp = 1600; Fs = 1696;

Ap = 0.2; As = 40;

Fsamp = 8000; Fnyquist = Fsamp/2;

Wp = Fp/Fnyquist;

Ws = Fs/Fnyquist;

[N, Wn] = ellipord(Wp, Ws, Ap, As)

[num,den] = ellip(N, Ap, As, Wn)

[H,f]=freqz(num, den, 512, Fsamp);

[Gd,fGd]=grpdelay(num,den,512,Fsamp);

plot(fGd,Gd);

figure; plot(f,-20*log10(abs(H)));

Gdmax is over 100

Advanced digital filter designAdvanced design: AFDesign

Gdmax has been reduced to 70

Advanced digital filter designDesign parameters

Design parameters

D = {n, , , fp}

Filter order n

Selectivity factor

Ripple factor

Actual passband edge fp

Advanced digital filter designDesign alternative D1

D1 design has higher attenuation in the stopband than is required by the specification. We choose D1 design when we prefer to achieve as large an attenuation as possible in the stopband.

Advanced digital filter designDesign alternative D2

D2 design has lower attenuation in the passband than is required by the specification. We choose this design when we prefer to achieve as low an attenuation as possible in the passband.

Advanced digital filter designDesign alternative D3a

D3a design has the sharpest magnitude response. Disadvantages of D3a can be very high Q-factors and large variation of the group-delay in the passband. This is MATLAB default design.

Advanced digital filter designDesign alternative D3b

D3b design has the sharpest magnitude response. Disadvantages of D3b can be very high Q-factors. The variation of the group delay can be high, but its maximal value can be moved into the transition region, so the group-delay variation can be acceptable in the passband.

Advanced digital filter designDesign alternative D4a

D4a has a smoother magnitude response, and that is the main reason for lower Q-factors and smaller variation of the group delay in the passband.

Advanced digital filter designDesign alternative D4b

D4b has a smoother magnitude response. D4b usually yields very low Q-factors and small variation of the group delay in the passband. D4b has a very low ripple factor.

Advanced digital filter designGroup delay

The maximum delay is obtained for D3a and D3b, while D4a and D4b have lower group-delay variation.

Advanced digital filter designAFDesign: D5 – minimal Q-factor

Gdmax has been reduced to 60

Advanced digital filter designAFDesign: Bandpass