# Digital Signal Processing II Lecture 4: Filter Realization ...

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24 Νοε 2013 (πριν από 4 χρόνια και 7 μήνες)

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DSP
-
CIS

Chapter
-
6: Filter Implementation

Marc Moonen

Dept. E.E./ESAT, KU Leuven

marc.moonen@esat.kuleuven.be

www.esat.kuleuven.be
/
scd
/

DSP
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CIS / Chapter
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6: Filter Implementation / Version 2012
-
2013

p.
2

Filter Design/Realization

Ste
p
-
1

: define filter specs

(pass
-
band, stop
-
band, optimization criterion,…)

Ste
p
-
2

: derive optimal transfer function

FIR or IIR design

Ste
p
-
3

: filter realization
(block scheme/flow graph)

direct form realizations, lattice realizations,…

Ste
p
-
4

: filter implementation
(software/hardware)

finite word
-
length

issues, …

question: implemented filter = designed filter ?

‘You can’t always get what you want’
-
Jagger/Richards (?)

Chapter
-
4

Chapter
-
5

Chapter
-
6

DSP
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CIS / Chapter
-
6: Filter Implementation / Version 2012
-
2013

p.
3

Chapter
-
6 : Filter Implementation

Introduction

Filter implementation & finite
wordlength

problem

Coefficient Quantization

Arithmetic Operations

Scaling

Quantization noise

Limit Cycles

Orthogonal Filters

DSP
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6: Filter Implementation / Version 2012
-
2013

p.
4

Introduction

Filter implementation & finite word
-
length problem :

So far have assumed that signals/coefficients/arithmetic
operations are represented/performed with
infinite

precision.

In practice, numbers can be represented only to a
finite

precision, and hence signals/coefficients/arithmetic
operations are subject to quantization
(truncation/rounding/...) errors.

Investigate impact of…

-

quantization of filter coefficients

-

quantization (& overflow) in arithmetic operations

PS

`A course in digital signal processing

, B.Porat, Wiley 1997

DSP
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CIS / Chapter
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6: Filter Implementation / Version 2012
-
2013

p.
5

Introduction

Filter implementation & finite word
-
length problem :

We consider
fixed
-
point filter implementations
, with a
`short

word
-
length.

In hardware design, with tight speed requirements, finite
word
-
length problem is a relevant problem.

In signal processors with a `sufficiently long

word
-
length,
e.g. with 16 bits (=4 decimal digits) or 24 bits (=7 decimal
digits) precision, or with
floating
-
point

representations and
arithmetic, finite word
-
length issues are less relevant.

DSP
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CIS / Chapter
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6: Filter Implementation / Version 2012
-
2013

p.
6

Q:Why

bother

for one and the same filter?

Introduction

Back to Chapter
-
5…

DSP
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CIS / Chapter
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6: Filter Implementation / Version 2012
-
2013

p.
7

Introduction: Example

Transfer function

%
IIR Elliptic
Lowpass

filter
designed using

%
ELLIP
function.

% All frequency values are in Hz.

Fs

= 48000;

% Sampling
Frequency

L
= 8;

%
Order

Fpass

= 9600; %
Passband

Frequency

Apass

= 60;

%
Passband

Ripple (dB)

Astop

= 160;

%
Stopband

Attenuation (dB)

Poles & zeros

DSP
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6: Filter Implementation / Version 2012
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2013

p.
8

Introduction: Example

Filter outputs…

Direct form realization

@ infinite precision…

Lattice
-

@ infinite precision…

Difference…

DSP
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CIS / Chapter
-
6: Filter Implementation / Version 2012
-
2013

p.
9

Introduction: Example

Filter outputs…

Direct form realization

@ infinite precision…

Direct form realization

@ 10
-
bit precision…

Difference…

DSP
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CIS / Chapter
-
6: Filter Implementation / Version 2012
-
2013

p.
10

Introduction: Example

Filter outputs…

Direct form realization

@ infinite precision…

Direct form realization

@ 8
-
bit precision…

Difference…

DSP
-
CIS / Chapter
-
6: Filter Implementation / Version 2012
-
2013

p.
11

Introduction: Example

Filter outputs…

Direct form realization

@ infinite precision…

Lattice
-

@ 8
-
bit precision…

Difference…

Better select a good realization
!

DSP
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CIS / Chapter
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6: Filter Implementation / Version 2012
-
2013

p.
12

Coefficient Quantization

The coefficient quantization problem

:

15 decimal digits (such that filter meets specifications)

For implementation, have to quantize coefficients to the
word
-
length used for the implementation.

As a result, implemented filter may fail to meet
specifications… ??

DSP
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6: Filter Implementation / Version 2012
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2013

p.
13

Coefficient Quantization

Coefficient quantization effect on
pole locations

:

example : 2nd
-
order system

`triangle of stability

: denominator polynomial is stable (i.e.

roots inside unit circle) iff coefficients lie inside triangle…

Proof: Apply Schur
-
Cohn stability test (see Chapter
-
5).

2
1
2
1
.
.
1
.
.
1
)
(

z
z
z
z
z
H
i
i
i
i
i

i

i

-
1

1

-
2

2

DSP
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6: Filter Implementation / Version 2012
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2013

p.
14

Coefficient Quantization

example (continued) :

with 5 bits per coefficient, all possible `quantized

pole positions are...

Low density of `quantized

pole locations at z=1, z=
-
1,
hence problem for narrow
-
band LP and HP filters (see Chapter
-
4).

-1.5
-1
-0.5
0
0.5
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
end
end

)
plot(poles

1
:
0625
.
0
:
1
for

2
:
1250
.
0
:
2
for
stable)

(if

i
i

DSP
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CIS / Chapter
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6: Filter Implementation / Version 2012
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2013

p.
15

Coefficient Quantization

example (continued) :

possible remedy: `coupled realization

poles are where are realized/quantized

hence

quantized

pole locations are (5 bits)

-1.5
-1
-0.5
0
0.5
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5

.
j

1
,
1

+

+

+

-

y[k]

u[k]

coefficient precision = pole precision

DSP
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6: Filter Implementation / Version 2012
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2013

p.
16

Coefficient Quantization

Coefficient quantization effect on
pole locations

:

example : higher
-
order systems (first
-
order analysis)

tightly spaced poles (e.g. for narrow band filters) imply

high sensitivity of pole locations to coefficient quantization

hence preference for low
-

p
o
l
y
n
o
m
i
a
l

:

1

a
1
.
z

1

a
2
.
z

2

.
.
.

a
L
.
z

L
r
o
o
t
s

a
r
e

:

p
1
,
p
2
,
.
.
.
,
p
L
`
q
u
a
n
t
i
z
e
d
'

p
o
l
y
n
o
m
i
a
l
:

1

ˆ
a
1
.
z

1

ˆ
a
2
.
z

2

.
.
.

ˆ
a
L
.
z

L
`
q
u
a
n
t
i
z
e
d
'

r
o
o
t
s

a
r
e
:

ˆ
p
1
,
ˆ
p
2
,
.
.
.
,
ˆ
p
L
ˆ
p
l

p
l

p
l
L

i
(
p
l

p
j
)
j

l

.
(
ˆ
a
i

a
i
)
i

1
L

DSP
-
CIS / Chapter
-
6: Filter Implementation / Version 2012
-
2013

p.
17

Coefficient Quantization

PS
: Coefficient quantization in lossless lattice realizations

In lossless lattice, all coefficients are sines and cosines, hence all

values between

1 and +1…, i.e. `dynamic range

and coefficient

quantization error well under control.

o = original transfer function

+ = transfer function after 8
-
bit

truncation

of lossless lattice

filter coefficients

-

= transfer function after 8
-
bit

truncation

of direct
-
form

coefficients (bi

s)

DSP
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CIS / Chapter
-
6: Filter Implementation / Version 2012
-
2013

p.
18

Arithmetic Operations

Finite word
-
length effects in arithmetic operations :

In

linear filters, have to consider additions & multiplications

if, two B
-
bit numbers are added, the result has (B+1) bits.

Multiplication:

if a B1
-
bit number is multiplied by a B2
-
bit number, the

result has (B1+B2
-
1) bits.

For instance, two B
-
bit numbers yield a (2B
-
1)
-
bit product

Typically (especially so in an IIR (feedback) filter), the result of an
addition/multiplication has to be represented again as a B

-
bit number
(e.g. B

=B). Hence have to remove most significant bits and/or least
significant bits…

DSP
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CIS / Chapter
-
6: Filter Implementation / Version 2012
-
2013

p.
19

Arithmetic Operations

Option
-
1:
Most significant bits

(MSBs)

If the result is known to be
(almost)
always upper bounded such that 1 or
more MSBs are
(almost)
always redundant, these MSBs can be dropped
without loss of accuracy
(mostly)
. Dropping MSBs then leads to better
usage of available word
-
length, hence better SNR.

This implies we have to monitor potential
overflow
(=dropping MSBs
that are non
-
scaling

to
avoid overflow.

Option
-
2 :
Least significant bits

(LSBs)

Rounding/truncation/… to B

bits introduces
quantization noise
.

The effect of quantization noise is usually analyzed in a
statistical

manner.

Quantization, however, is a
deterministic non
-
linear
effect, which may
give rise to
limit cycle oscillations
.

DSP
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CIS / Chapter
-
6: Filter Implementation / Version 2012
-
2013

p.
20

Quantization Noise

Quantization mechanisms:

Rounding Truncation Magnitude Truncation

mean=0 mean=(
-
0.5)LSB (biased!) mean=0

variance=(1/12)LSB^2 variance=(1/12)LSB^2 variance=(1/6)LSB^2

input

probability

error

output

DSP
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CIS / Chapter
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6: Filter Implementation / Version 2012
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2013

p.
21

Quantization Noise

Statistical analysis is based on the following
assumptions

:

-

each quantization error is random, with uniform probability distribution
function (see previous slide)

-

quantization errors at the output of a given multiplier are
uncorrelated/independent (=white noise assumption)

-

quantization errors at the outputs of different multipliers are
uncorrelated/independent (=independent sources assumption)

One noise source is inserted

for each

Since the filter is a
linear filter

the output

noise generated by each noise source can be

computed and added to the output signal…

y[k]

u[k]

+

x

-
.99

+

e[k]

DSP
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CIS / Chapter
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6: Filter Implementation / Version 2012
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2013

p.
22

Limit Cycles

Statistical analysis is simple/convenient, but quantization is
truly a
non
-
linear

effect, and should be analyzed as a
deterministic

process.

Though very difficult, such analysis may reveal odd behavior:

Example:

y[k] =
-
0.625.y[k
-
1]+u[k]

4
-
bit rounding arithmetic

input u[k]=0, y[0]=3/8

output y[k] = 3/8,
-
1/4, 1/8,
-
1/8, 1/8,
-
1/8, 1/8,
-
1/8, 1/8,..

Oscillations in the absence of input (u[k]=0) are called

`
zero
-
input limit cycle oscillations

DSP
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CIS / Chapter
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6: Filter Implementation / Version 2012
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2013

p.
23

Limit Cycles

Example:

y[k] =
-
0.625.y[k
-
1]+u[k]

4
-

input u[k]=0, y[0]=3/8

output y[k] = 3/8,
-
1/4, 1/8, 0, 0, 0,.. (no limit cycle!)

Example:

y[k] = 0.625.y[k
-
1]+u[k]

4
-
bit rounding

input u[k]=0, y[0]=3/8

output y[k] = 3/8, 1/4, 1/8, 1/8, 1/8, 1/8,..

Example:

y[k] = 0.625.y[k
-
1]+u[k]

4
-
bit truncation

input u[k]=0, y[0]=
-
3/8

output y[k] =
-
3/8,
-
1/4,
-
1/8,
-
1/8,
-
1/8,
-
1/8,..

Conclusion: weird, weird, weird,… !

DSP
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CIS / Chapter
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6: Filter Implementation / Version 2012
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2013

p.
24

Limit Cycles

Limit cycle oscillations are clearly
unwanted

(e.g. may be
audible in speech/audio applications)

Limit cycle oscillations can only appear if the filter has
feedback. Hence
FIR filters

cannot have limit cycle
oscillations.

Mathematical analysis is very difficult

Truncation often helps to avoid limit cycles (e.g.
magnitude
truncation
, where absolute value of quantizer output is
never larger than absolute value of quantizer input
(=`passive quantizer

)).

Some filter realizations can be made limit cycle free, e.g.
coupled realization,
orthogonal filters

(see below).

DSP
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6: Filter Implementation / Version 2012
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2013

p.
25

Orthogonal Filters

Orthogonal filter

=

state
-
space realization with

orthogonal realization matrix:

PS

: lattice filters and lattice
-
part of lattice
-

Strictly speaking, these are not state
-
space realizations (cfr. supra), but
orthogonal

R is realized as a product of matrices, each of which is
again
orthogonal
, such that useful properties of orthogonal states
-
space realizations indeed carry over (see p. 38)

Why should we be so fond of orthogonal filters ?

I
R
R
R
R
D
C
B
A
R
T
T

.
.
DSP
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CIS / Chapter
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6: Filter Implementation / Version 2012
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2013

p.
26

Orthogonal Filters

Scaling

:

-

in a state
-
space realization, have to monitor overflow in internal states

-

transfer functions from input to internal states (`
scaling functions

) are

defined by impulse response sequence (see p. 22)

-

for an orthogonal filter (R

.R=I), it is proven (next slide) that

which implies that all scaling functions (rows of Delta) have L2
-
norm=1

(=diagonal elements of )

conclusion: orthogonal filters are `
scaled in L2
-
sense

]
[
.
]
[
.
]
1
[
k
u
B
k
x
A
k
x

...

.
2
B
A
AB
B
I
T

T

.

...
0
3
2
B
A
B
A
AB
B
DSP
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CIS / Chapter
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6: Filter Implementation / Version 2012
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2013

p.
27

Orthogonal Filters

Proof

:

First :

Then:

I
B
B
A
A
I
R
R
T
T
T

.
.
.

...
0
0
0

...
...
)
.

.
(
)
.
.
(
)
(

...
.

.
.
.
.
.
.

T
T
T
T
T
T
T
T
A
A
I
T
T
A
A
I
T
A
A
I
T
T
A
A
AA
A
A
AA
A
A
A
A
I
I
A
A
B
B
AA
A
B
B
A
B
B
I
I
T
T
T
DSP
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CIS / Chapter
-
6: Filter Implementation / Version 2012
-
2013

p.
28

Orthogonal Filters

Quantization noise

:

-

in a state
-
space realization, quantization noise in internal states may be

represented by equivalent noise source (vector) Ex[k]

-

transfer functions from noise sources to output (`
noise transfer

functions

) are defined by impulse response sequence

-

for an orthogonal filter (R

.R=I), it is proven that (…try it)

which implies that all noise gains are =1 (all noise transfer functions

have L2
-
norm = 1 = diagonal elements of ).

-

This is then proven to correspond to
minimum possible output noise

variance

for given transfer function, under L2 scaled conditions
(i.e.

when states are scaled such that scaling functions have L2
-
norm=1)

!!
(details omitted)

]
[
]
[
.
]
[
.
]
1
[
k
E
k
u
B
k
x
A
k
x
x

...
)
(
)
(

.
2
T
T
T
T
T
CA
CA
C
I

.
T

...
)
(
)
(
2
T
T
T
CA
CA
C
DSP
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CIS / Chapter
-
6: Filter Implementation / Version 2012
-
2013

p.
29

Orthogonal Filters

Limit cycle oscillations

:

if
magnitude truncation

is used (=`passive quantization

),

orthogonal filters are
guaranteed to be free of limit cycles

(details omitted)

intuition:

quantization consumes energy/power, orthogonal
filter does not generate power to feed limit cycle.

DSP
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6: Filter Implementation / Version 2012
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2013

p.
30

Orthogonal Filters

Orthogonal filters

=
L2
-
scaled, minimum output noise,
limit cycle oscillation free filters !

It can be shown that these statements also hold for

:

* lossless lattice realizations of a general IIR filter

* lattice
-
part of a general IIR filter in lattice
-