Eng. Julian S. Bruno - SASE · Imaginary Part 21 Pole/Zero Plot SASE 2012 Eng. Julian S. Bruno 0 50...

Eng. Julian S. Bruno

REAL TIME

DIGITAL

S

www.electron.frba.utn.edu.ar/dplabSASE 2012

SIGNAL

PROCESSING

FIR and IIR.

� Design parameters.

Digital Filters

� Design parameters.

� Implementation types.

� Constraints.

SASE 2012 Eng. Julian S. Bruno

Filters: General classification


Filters: General classification

SW Smith, The Scientist and Engineer’s guide to DSP. California Tech. Pub. 1997.SASE 2012 Eng. Julian S. Bruno

Filters: Frequency domain features

� Fast rollroll--offoff , flat passbandpassband and high attenuation in attenuation in stopbandstopband are stopbandstopband are desirable features.

� They can’t be achieved at the same time, so the design procedure is a trade trade offoff between these between these features.features.


Filters: Time domain features

� Step response evidence the most important time domain features.

� In this case the trade off trade off is between fast step between fast step is between fast step between fast step response and overshootovershoot .

� Overshoot is an undesirable distortion and must be avoided.

� Simetry is a “fingerprint” of a linear phase response system (FIR).


Filters: Phase response

The phase response phase response of a filter gives the radian phase shift radian phase shift experienced by each


experienced by each sinusoidal component of the input signal. Sometimes it is more meaningful to consider phase phase delaydelay

Filters: Group and Phase Delay


Filters: Group and Phase Delay

|H(w)| = 1

140

Phase Delay

200

250

300

350

Normalized Frequency: 0.1000977Group delay: 343.8103

Gro

up

del

ay (

in s

amp

les)

Group Delay


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

40

60

80

100

120Normalized Frequency: 0.1000977Phase Delay: 112.9315

Normalized Frequency (×××× ππππ rad/sample)

Ph

ase

Del

ay (

sam

ple

s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

50

100

150

Normalized Frequency (×××× ππππ rad/sample)

Gro

up

del

ay (

in s

amp

les)

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-1

-0.5

0

0.5

1

Real Part

Imag

inar

y P

art

Pole/Zero Plot

0 200 400 600-1

0

1

0

1

Filters: Group Delay

W = 0.1

W = 0.32

4Input

Gd(0,1)= 343 samples

0 200 400 600-1

0

0 200 400 600-1

0

1

0 200 400 600-1

0

1


W = 0.5

W = 0.7

0 200 400 600 800-4

-2

0

2

4Output

0 200 400 600 800-4

-2

0

1

0 200 400 600-1

0

1

1

Filters: Phase Delay

1

Pd(0,1)= 113 samples

250 300 350 400 450 500-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


250 300 350 400 450 500-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Digital Filters: Design steps

� Requirement specifications.

� Coefficients calculation.

� Filter realization:

� Structure.

� Wordlength effects.

� Software or Hardware implementation.

EC Ifeachor, BW Jervis. Digital Signal Processing. A Practical approach. Second Edition. Prentice Hall.


Digital Filters: Specifications

pA

pδ−1

pδ+1

( )ωH

Ideal filter 1

ppp H ωωδωδ ≤≤+≤≤− 0 ,1)(1

10

120log dB

1p

pp

Aδδ

+= −


Passband

pδ−1

Stopband

0 pω sω

sδ ω π cω

sA

Actual filter

Transition band

πωωδω ≤≤≤ s ,)( sH

1020log dBs sA δ= −

Ap : Ripple in band (dB)As : Band-stop attenuation (dB)π-ωs : Band-stop start/endωp-ωs : Band-pass start/end

FIR

Finite Impulse Response


FIR filters: General characteristics

� FIR digital filters use only currentcurrent and past input past input samplessamples . Require no feedback.

� Their transfer function has zeros around the Z zeros around the Z planeplane and poles at the originpoles at the origin , so FIR filters are planeplane and poles at the originpoles at the origin , so FIR filters are always stablealways stable .

� They can easily be designed to be linear phase linear phase and constant delay and constant delay by making the coefficient sequence symmetric.

� Finite duration of nonzero input values / Finite duration of nonzero output values. Finite Impulse Finite Impulse Response.Response.


FIR filters: General characteristics

Being x(n) and h(n) are arrays of

UTN - FRBA 2011 Eng. Julian S. Bruno

� Being x(n) and h(n) are arrays of numbers. If we want to compute y(n) we have to multiply and sum the last M samples, being M the length of h(n).

� As you can see, any linear system uses multiplications, accumulations (sums), and loops intensively.

FIR filters: Perfect linear phase

( )

[ ]samplesM

delay

fgdpd f

2

1−=

∂Θ∂

−==


� Zero phase filters are a particular case of linear phase with zero delay.

2

gd = 0gd = 25 delay = 25

� Type I: h(n) = h(N-n) with even N .� Type II: h(n) = h(N-n) with odd N.� Type III: h(n) = -h(N-n) with even N.� Type IV: h(n) = -h(N-n) with odd N.

Linear phase FIR filter

� Type IV: h(n) = -h(N-n) with odd N.


Type I Even order - Even symmetric

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Imag

inar

y P

art

10

Pole/Zero Plot

0

0.1

0.2

0.3

0.4

0.5Impulse Response

Am

plit

ude


0 5 10 15 20

-61.1544

-53.4141

-45.6737

-37.9334

-30.1931

-22.4527

-14.7124

-6.9721

0.7683

Frequency (kHz)

Mag

nitude

(dB)

Magnitude (dB) and Phase Responses

-8

-7

-6

-5

-4

-3

-2

-1

0

Pha

se (ra

dia

ns)

-1 -0.5 0 0.5 1 1.5 2

-1

-0.8

-0.6

-0.4

Real Part

0 20 40 60 80 100 120 140 160 180 200-0.2

-0.1

0

Time (useconds)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Imag

inar

y P

art

11

Pole/Zero Plot

Type II Odd order – Even symmetric

0.1

0.2

0.3

0.4

0.5

0.6Impulse Response

Am

plitud

e

-1 -0.5 0 0.5 1 1.5

-1

-0.8

-0.6

Real Part


We can´tdesign highpass filters(HPF)

0 50 100 150 200-0.1

0

Time (useconds)

0 5 10 15 20

-60

-50

-40

-30

-20

-10

0

Frequency (kHz)

Mag

nitude

(dB

)


-8.4029

-7.0167

-5.6306

-4.2444

-2.8583

-1.4721

-0.086

Phas

e (r

adia

ns)

Type III Even order - Odd symmetric

-0.5

0

0.5

1

Imag

inar

y P

art

20

Pole/Zero Plot

-0.1012

-0.0506

0

0.0506

0.1012

0.1519

0.2025

0.2531Impulse Response

Am

plitu

de


-1.5 -1 -0.5 0 0.5 1 1.5 2

-1

Real Part

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-20

-15

-10

-5

0

Normalized Frequency (×π rad/sample)

Mag

nitu

de (

dB)


-16.2522

-12.091

-7.9299

-3.7687

0.3924

Pha

se

(rad

ians

)

0 2 4 6 8 10 12 14 16 18 20-0.2531

-0.2025

-0.1519

Samples

We can´tdesign low and high passfilters (LPF and HPF)

Type IVOdd order – Odd symmetric

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Am

plitu

de

Impulse Response

-0.5

0

0.5

1

Imag

inar

y P

art

21

Pole/Zero Plot


0 50 100 150 200 250 300 350 400

-0.5

-0.4

Time (useconds)

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-1

Real Part

0 5 10 15 20

-35

-30

-25

-20

-15

-10

-5

0

Frequency (kHz)

Mag

nitu

de (

dB)


-22.2571

-19.4695

-16.6818

-13.8942

-11.1066

-8.319

-5.5314

-2.7437

Ph

ase

(rad

ians

)

We can´tdesign lowpass filters(LPF)

Minimun phase FIR filter

-1

-0.5

0

0.5

1

1.5

Imag

inar

y P

art

54

Pole/Zero Plot

-0.5

0

0.5

1

Imag

inar

y P

art

50

Pole/Zero Plot

SASE 2012 Eng. Julian S. Bruno0 5 10 15 20

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency (kHz)

Mag

nitu

de

(dB

)


-19.3495

-16.5521

-13.7546

-10.9571

-8.1597

-5.3622

-2.5647

0.2327

3.0302

5.8277

Ph

ase

(rad

ian

s)

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4-1.5

Real Part

0 5 10 15 20

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency (kHz)

Mag

nit

ud

e (d

B)


-11.2977

-9.2226

-7.1475

-5.0724

-2.9973

-0.9222

1.1529

3.228

5.3031

7.3782

Ph

ase

(rad

ian

s)

-1.5 -1 -0.5 0 0.5 1 1.5

-1

Real Part

Minimun phase FIR filter

-0.05

0

0.05

0.1

0.15

Impulse Response

Am

plit

ud

e

-0.05

0

0.05

0.1

0.15

Impulse Response

Am

plit

ud

e

SASE 2012 Eng. Julian S. Bruno0 5 10 15 20

26.5

26.6

26.7

26.8

26.9

27

27.1

27.2

27.3

27.4

27.5

Frequency (kHz)

Gro

up d

elay

(in

sam

ple

s)

Group Delay

0 0.2 0.4 0.6 0.8 1

-0.15

-0.1

Time (mseconds)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.1

Time (mseconds)

0 5 10 15 2012

14

16

18

20

22

24

26

Frequency (kHz)

Gro

up

del

ay (

in s

amp

les)

Group Delay

Minimum-Phase vs Linear Phase

� L-P FIR filter has zeros inside, on, and outside the unitcircle.

� M-P FIR filter has all of its zeros on or within the unit circle.

� Given optimal M-P and L-P digital FIR filters that meet the same magnitude specificationsame magnitude specification , the M-P filter would have same magnitude specificationsame magnitude specification , the M-P filter would have a reduced filter length reduced filter length of typically one half to three fourths of the L-P filter length, and Minimum group delayMinimum group delay , where energy would be concentrated in the lowlow --delaydelay instead of the medium-delay coefficients

� M-P filters can simultaneously meet constraints on delay constraints on delay and magnitude responseand magnitude response while generally requiring fewer computations and less memory than L-P filters.


FIR filters: Coefficients calculation

� Moving average (MA)

� Windowed Sinc filters

� Remez exchange (Parks McClellan)

method


Moving average filters

� Each output will be the average of the last M samples of the input.

� This kind of filters is very easy to implement, but it has very poor roll-off and attenuation in stopband.

� It is mainly used for smoothing purposes.SASE 2012 Eng. Julian S. Bruno

∑∑−

=

−

=

+−=−=1

0

1

0

]2/[1

][;][1

][M

j

M

j

MjnxM

nyjnxM

ny

MA filters: applications

� Improvements can be achieved if we increase M or if the same filter is passed multiple times.


MA filters: Recursive Implementation

The cost of implementing a MA filter is M sums and � The cost of implementing a MA filter is M sums and 1 multiplication.

� This can be improved if a recursive approach is taken.

� So any MA filter can be implemented by 2 sums and one accumulator.

� Keep in mind that a higher M implies a higher delay.


MA Filter: Example

0.5

1

1.5

input filter

output MA filteroutput FIR filter


0 100 200 300 400 500 600-1.5

-1

-0.5

0

MA filter: M=50 , An=10% , FIR filter: Equiripple order 39

Windowed Sinc FIR filters

� A sinc of infinite duration is the time domain equivalent of an ideal filter response.As it is impossible to � As it is impossible to store this signal, it is truncated to M samples.

� This truncation causes a severe degradation to the ideal frequency response.


Windowed Sinc FIR filters

� In order to obtain better filter features, the truncated sinc can be multiplied by a window function.

x

function.� These windows

achieve either better smoothness, roll-off or stopband attenuation than the sinc.

� For this purposes, many windows were designed.

=


Windowed Sinc FIR filters: Design

� Windowed Sinc filters are very easy to design.

� We only have to multiply a window formula to the sinc signal.

� One problem of window filters is that they are they are not optimal in the not optimal in the kernel length sensekernel length sense .

� The frequency of the sinusoidal oscillation is approximately equal to Fc


Windowed Sinc FIR filters: Windows

Hamming:

Blackman:

Hanning:


Optimal FIR filters: Remez exchange (Parks McClellan) method

� This design algorithm allows filter designs that that are optimal in the kernel are optimal in the kernel length senselength sense .

� The usage of the method is as simple as passing the filter specification to the algorithm and receiving its filter kernel.


FIR filters: Design method

Design method

Passband StopbandTransition

bandMatlab

Function

Equiripple Ripply Ripply Narrow remez(lf)

Least Squares Flat Ripply Narrow firls(lf)Least Squares Flat Ripply Narrow firls(lf)

Max. Flat (LP) Flat Flat Wide maxflat(*)

Least Pth-norm Flat Ripply Narrow firlpnorm(*)

Constrained Equiripple

Flat Ripply Narrow firceqrip(*)

Window ** ** ** fir1


FIR filters: Implementation

� Direct form or transversal structure.� Symmetry (Linear phase) optimization.� Fast convolution (overlap add/save).


FIR filters: Direct Form

z -1

x

z -1

x

z -1

x

+

x

+ + +

x(n)

y(n)

h0 h1 hM-1 hM

x(n+1) x(n+M-2) x(n+M-1)

� Most DSP architectures are optimized for implementing this structure.

M MAC operations

M RAM positions (signal)M RAM positions (filter)Computational cost:


FIR filters: Transversal Form

x x x

+

x

+ +

x(n)

y(n)

h0 h1 hM-1 hM

x(n+1) x(n+M-2) x(n+M-1)

z -1 z -1 z -1

� Most DSP architectures are optimized for implementing this structure.

M MAC operations

M RAM positions (signal)M RAM positions (filter)Computational cost:


FIR filters: Linear phase form

z -1 z -1 z -1x(n)

z -1 z -1 z -1

+ + +

x(n+1) x(n+half-1) x(half)

x(n+M-2)x(n+half+1)

x(n+M-1)

h(n)

half n

M odd

x

++ + + y(n)

h0

M/2 (even) M/2+1 (odd) MAC operations

M RAM positions (signal)M/2 (even) M/2+1 (odd) RAM positions (filter)Computational cost:

+ + +

xh1 xhhalf-1 xhhalf h(n)

half n

M even


FIR filters: Fast convolution

FFT iFFTx(n) X y(n)

H(w) M : 2N multiple

M

M

� Overlap add/save algorithm.

M/2 complex multiplications

M (2M for performance) RAM positions (signal)M RAM positions (filter)

Computational cost:

H(w)

2 FFT (M . log2(M) )


FIR filters: Wordlength effects

� ADC quantization noise.

SNRADC = 1.76 + 6.02 . ADCBits

� Coefficients quantization noise.� Coefficients quantization noise.

� Roundoff quantization noise.

� Arithmetic overflow noise


FIR filters: Roundoff quantization

12

22 qQ =σx

+

B bitsB bits

2B bits

Non linear model

Q

Linear model

xB bits

B bits

2B bits

e(n)

(roundoff noise power)

q : ADC quantization step (V)+Q

B bits B bits

e(n) q : ADC quantization step (V)

� Each quantization process (multiplication) degrades system’s noise figure.


IIR

Infinite Impulse Response


IIR filters: General characteristics

� IIR filter output sample dependsdepends on previous inputinputsamples and previous filter outputoutput samples.

� Implemented using a recursion equation, instead a convolution and dondon ´́t have linear phase responsest have linear phase responses .

� IIR filters have poles poles and zeros around the Z planezeros around the Z plane , they are not always stableare not always stable .

� IIR filters have poles poles and zeros around the Z planezeros around the Z plane , they are not always stableare not always stable .

� Finite duration of nonzero input values / Infinite duration of nonzero output values. Infinite Impulse Infinite Impulse ResponseResponse

� IIR filters achieves better performance better performance than FIR for the same kernel length.

� Implementation is more complicatedmore complicated , and requires floating point or large word lengths.


IIR filters: Non linear phase response

� As in analog domain, linear phase could be achieved at expense of loosing other desirable features.

� In other words, IIR filters are much faster than FIR filters but filter stability and phase distortion is a VERY important issue.


IIR filters: Linear phase with bidirectional filtering

� Bidirectional filtering is a way of forcing an IIR filter to have perfect linear phase, at expense of speed since, at least, the computing cost is duplicated.


IIR filters: the recursion equation

( ) ( )( )

∑

∑

−

−

=

+==

Mk

kN

kk

Za

Zb

zX

zYzH 0

)(

1

x +

x(n) y(n)b0

z -1

x +

b1

x+

z -1

+

-a1


� Each output is a linear combination of past input and output samples

∑∑==

−−−=M

kk

N

kk knyaknxbny

1)(

0)( )(.)(.)(

( )∑

=

−+k

kk Za

zX

1)(1

The recursive coefficients are scaled to yield a0 = 1

x +

z -1 bN

x+

z -1-aM

IIR filters: Coefficients calculation

� Pole-zero placement.

� Impulse invariance.

Matched z-transform.� Matched z-transform.

� Bilinear z-transform.


IIR filters: Analog & Digital

� As in the analog domain, simple filters could be implemented with a few coefficients.

For HP: And for LP: And for both: (Time decay)

(Cutoff frequency)


IIR filters implementation: Direct Form 1

� DF1 is the most direct way of implementing an IIR filter.

� It uses two buffers (delay lines) and

∑∑==

−−−=M

kk

N

kk knyaknxbny

1)(

0)( )(.)(.)(

x(n) y(n)b0

x + +


(delay lines) and 2(M+1) multiplications and sums.

� More sophisticated forms optimizes memory usage, coefficients precision sensibility and stability.

x +

z -1

x +

z -1

b1

bN

x+

z -1

x+

z -1

-a1

-aM

IIR filters implementation:Direct Form 2 (canonic)

� DF2 Reduces memory usage by two, since the delay line is shared. ∑

∑

=

=

−=

−−=

N

kk

M

kk

knwbny

knwanxnw

0)(

1)(

)(.)(

)(.)()(


line is shared.

x +

x(n) y(n)b0

z -1

x +

x +

z -1

b1

bN

x+

+

x+

-a1

-aM

w(n)

IIR filters: SNR in DF1 and DF2

x +

x(n) y(n)b0

z -1

x +

b1

x+

z -1

+

-a1

x +

x(n) y(n)b0

z -1

x +

b1

x+

+

-a1

w(n)

quantization point quantization pointsInternal node

� The canonic section is susceptible to internal self-sustaining overflow.

� In practice, the poles of the canonic section tend to amplify noise generated during computation more.


x +

x +

z -1 bN

x+

x+

z -1-aM

x +

x +

z -1 bN

x+

x+

-aM

IIR filters implementation: Design Pitfalls

� IIR implementation is conditioned mainly by finite wordlength effects.

� Coefficient quantization, overflow and roundoff errors are most common problems.errors are most common problems.

� To go through these problems, some precautions must be taken into account.


IIR filters: Overflow

� The final result overflows.� Intermediate results overflow. (Modulo math)� Unity gain filters can have outputs that are

greater than ±1 with a bounded input.greater than ±1 with a bounded input.


IIR filters: Noise

� Noise is generated by truncatingtruncating high resolution results.

� The noise generated by truncation often resembles the spectrum of the signal being filtered.

� The generated noise is most harmful when it re-� The generated noise is most harmful when it re-circulates through the recursive coefficientsthrough the recursive coefficients .

� Force correlated truncation noise truncation noise to be random (white) by using ditherusing dither .

� If the truncation process is deterministic then correction can be applied to the filter.

�� Fixed point truncation is deterministicFixed point truncation is deterministic .�� Floating point truncation is not deterministicFloating point truncation is not deterministic .


IIR filters implementation: Coefficient quantization sensibility

� Frequency response are severely affected for small wordlengths.

� The sensibility grows with the order of the with the order of the system.

� Second order partitioning mitigate this problem by diminishing speed.

� Instability can occur for very sharp responses.



Mkbbbzb

za

zbzHzazyzbzxzy

kkq

Mk

kq

N

k

kk

M

k

kkN

k

kk

M

k

kk

,,2,1

1)(,)()()(

1

0

10

K=∆+=

+=−=

∑

∑

∑∑∑

−

=

−

=

−

=

−

=

−


Nkaaa

Mkbbb

zazH

kkkq

kkkq

N

k

k

kq

kkq

q ,,2,1

,,2,1

1)(

1

0

K

K

=∆+==∆+=

+=

∑

∑

=

−

=

∑

∑

∑

=

−

==

−

−

==

−

∆∂∂=∆∆+=

−Π=+==

−Π=+==

N

kk

k

iikkkq

kq

N

k

N

k

k

kqqq

qq

k

N

k

N

k

kk

ap

ppppp

zpzazDzD

zNzH

zpzazDzD

zNzH

1

1

11

1

11

,

)1(1)(,)(

)()(

)1(1)(,)(

)()(

∑∏=

≠=

−

∆−

−=∆N

kkN

ill

li

kNi

i app

pp

1

1

)(


� Second order sections (SOS) can be grouped in cascade or parallel.

� When cascading SOS’s the

H1x(n) y(n)HN

� When cascading SOS’s the order is chosen to maximize SNR.

� Cascade is the most typically used in DSP processors.

H1x(n) y(n)

HN

+


IIR Filter: SOS Forms

SASE 2012 Eng. Julian S. BrunoMark Allie comp.dsp 2004

Direct Form I

Z-1

Z-1

b0

b1 -a1

x(n) y(n)

Direct Form I

N

N

2N

2N

2N

2N N

Q

N(lo)

N(hi)yt(n)

e(n)

N

N N

N

Qyt(n)


Z-1

Z-1

b2 -a2N 2N 2N N

Fixed Point:

Modulo math useful. One summer before saturator.

No Input scaling required to prevent internal overflow.

One truncation feedback error term.

Truncation noise affects recursive terms.

Truncation noise negligible for N large enough.

Floating point:

No input scaling required.

No overflow concerns.

Truncation error is coefficient dependant and not

correctible.


Number of bits in mantissa critical.

N N

( ) )(1

1)(

1 12

11

12

11

22

110 zE

zazazX

zaza

zbzbbzYt −−−−

−−

++−

++++

=

Mark Allie comp.dsp 2004

Direct Form II

Z-1

b0x(n)

y(n)

Direct Form II

Q

2N

e1(n)

2N

Q

N(hi) N(hi)

N(lo)

e2(n)

w(n)N

N(lo)N

N N

N

yt(n)


Z-1

b1

b2

-a1

-a2

2N

2N

2N

2N

w(n-1)

w(n-2)

Fixed Point:

Modulo math not useful. There are 2 sums

separated by a saturator.

Input scaling required to prevent internal overflow.

One truncation feedback error term.

Truncation noise affects all terms.


Floating point:




correctible.

Truncation noise affects all terms.


N N


Direct Form I Transposed

SASE 2012 Eng. Julian S. BrunoMark Allie comp.dsp 2004

Direct Form II Transposed

Z-1

b0

y(n)

x(n)

Direct Form II Transposed

N 2N

N

2N

N

Q

N(hi)

N(hi)

yt(n)

Qyt(n)

N(lo)

N(lo)

e3(n)

Fixed Point:

Modulo math useful. One summer before saturator.

No Input scaling required to prevent internal overflow.

Multiple truncation feedback error terms*.


Z-1

b1

b2

-a1

-a2

Q

Q

N(hi)

e1(n)

e2(n)

2N

N

2NN

2N

N

2N

N

2N

N

2N

N

N(lo)

(lo)Multiple truncation feedback error terms*.



* One if Q1 and Q2 are removed. Dattorro

Floating point:




correctible.




Tabulated Results

Fixed PointDFI DFII DFIT DFIIT

Floating PointDFI DFII DFIT DFIIT

Input scaling + + + + + +

+ +


Modulo math + +

Truncation noise + + + - - - -Performs well with 32 bit mantissa + + + + + +

Error compensation + + + - - - -

+ Good Attribute + Qualified Good Attribute - Poor Attribute


IIR Filters: Good News

� DFI and DFIIT both work.� A processor with 32 bit integer native word

size capabilities doesn’t need truncation error compensation.error compensation.

� Some modern floating point processors can do this inexpensively and fast.

� Some modern fixed point processors can do this inexpensively and more slowly.


IIR Filters: Floating Point

� All DF IIR filters can work.� Complicated analysis and addend

evaluation. Floating point processors with extended � Floating point processors with extended precision results may work well.

� Analog devices has processors with 32 bit mantissa capabilities.

� Fixed point processing at 32 bits a sure bet


Filters examples


DC Blocker


-1.5 -1 -0.5 0 0.5 1 1.5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Real PartIm

agin

ary

Par

t

Pole/Zero Plot

DC Blocker: BF53X Fixed Point

1.15 1.159.31

Q


1.31

Coefficient quantization error != 0

Product and Accumulator error = 0

Product Roundoff error != 0

Envelope detection


Envelope detection

h(1) = h(3) = h(5) = h(7) = h(9) = h(11) = h(13) = 0


Hilbert Filter

Decimation

)()(,,, Mnxmx

M

ff oldnew

oldsnews ==

Fs,new ≥ 2B

Polyphase implementation


Fs,new < 2B

Interpolation

oldsnews fLf ,, ⋅=

SASE 2012 Eng. Julian S. BrunoPolyphase implementation

Efficient D/A conversion in compact hi-fi system


Equalizing Techniques Flatten DAC Frequency Response


Application in the acquisition of high quality data


Recommended bibliography

� EC Ifeachor, BW Jervis. Digital Signal Processing. A Practical approach. Second Edition. Prentice Hall 2002.� Ch6: A framework for filter design.� Ch7 & 8: FIR & IIR filter design.� Ch13: Analysis of wordlength effects in fixed point DSP systems.

� RG Lyons, Understanding Digital Signal Processing. Third � RG Lyons, Understanding Digital Signal Processing. Third Edition. Prentice Hall 2010.� Ch5 & 6: FIR & IIR filters.

� SW Smith, The Scientist and Engineer’s guide to DSP. California Tech. Pub. 1997.� Ch14-21: FIR & IIR filter design.

� J. Dattorro, “The Implementation of Recursive Digital Filters for High Fidelity Audio,” J. Audio Eng. Soc., vol 36, pp 851-878 (1988 Nov)

NOTE: Many images used in this presentation were extracted from the recommended bibliography.SASE 2012 Eng. Julian S. Bruno

Thank you!

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