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Filtros Digitais
• Diagramas de bloco e grafos de fluxo de sinal
• Estruturas de filtros IIR
• Projeto de filtro FIR
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6.1.1 Filter Transfer Function
• Linear
constant-coefficient
difference
equations (LCCDEs).
k 1
k 0
k 1
k 0
y[n]   ak y[n  k ]  bk x[n  k ]
N
 y[n]   ak y[n  k ]   bk x[n  k ]
M
K
M
• Taking the two sided Z-transform we have


k 1
 1   ak z 

 k  k  0

N
  bk z k 
H ( z )  H 2 ( z ) H1 ( z )  



1

 M


• H(z) is a filter transfer function, and can be used to describe
any digital filter.
– Expressed as a diagram or signal flow graph
– Implemented as a digital circuit
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6.1.2 Block Diagram and Signal Flow Graph
• A signal flow graph representation of LCCDE is same as a block
diagram, and it is a network of directed branches that connect at
nodes which are variables.
Block diagram representation of a firstorder digital filter.
Structure of the signal flow graph.
Structure of the signal flow graph
with the delay branch indicated by z-1.
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6.1.3 Block Diagram: Direct Form I Realization
v[n ]   bk x[n  k ]
k 0
M
N
y[n]   ak y[n  k ]  v[n]
 k 0

k

V ( z )  H ( z ) X ( z )   b z k  X ( z )
M

k 1
1


k 1
 1   ak z 
k 

N
2
Y ( z )  H ( z )V ( z )  
V ( z )
1




This structure (non-canonic, direct form I) can be too sensitive to
finite word-length errors (quantization errors) – errors are summed,
fed back and re-amplified over and over.
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6.1.4 Block Diagram: Direct Form II Realization
w[n]   a w[n  k ]  x[n]
k 1
k
N
N
y[n]   bk w[n  k ]


k 1
 1   ak z 
k 

N
2
W ( z)  H ( z) X ( z)  
 X ( z)
1




k 0
 k 0

Y ( z )  H1 ( z )W ( z )    bk z k W ( z )
M

This structure (canonic direct from, direct form II) requires less
delay elements. The minimum number of delays is max(N, M).
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6.1.5 Example of LTI Implementation
Consider the LTI system with system (transfer) function
1  1.5z 1  0.9 z 2
H ( z) 
1  2 z 1
b0  1, b1  2, a1  1.5, a2  0.9.
we have two implementation as follows
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6.1.6 Signal Flow Graph: Direct Forms
• Given the LCCDE y[n]   a y[n  k ]  b x[n  k ]
k 1
k 0
k
N
Signal Flow Chart of Direct From I
k
M
Signal Flow Chart of Direct From II
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6.2.1 Structure of IIR: Cascade Form
• By factoring the numerator and denominator we can write
I
H ( z)   Hi ( z)
i 1
which can be drawn as a cascade of smaller sections:
H1 ( z )
H 2 ( z)

H I (z )
H (z )
• Advantages: Smaller sections – less feedback error.
• Disadvantages: Errors fed from section-to-section.
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6.2.2 Parallel Realization
• By performing a partial fraction expansion we can write
I
H ( z)   Hi ( z)
i 0
which can be drawn as a parallel sum of smaller sections
H1 ( z )
H1 ( z )

:
H1 ( z )
H (z )
• Advantages: smaller sections- less feedback error, and error
confined to each section.
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6.2.3 Structure of IIR: Example (Cascade)
1
2
1

2
z

z
Given a two-order system H ( z ) 
1  0.75z 1  0.125z 2
• Cascade Structure (Not unique)
 1  z 1  1  z 1 


H ( z )  
1 
1 
 1  0.5z  1  0.25z 
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6.2.4 Structure of IIR: Example (Parallel)
1
2
1

2
z

z
Given a two-order system H ( z ) 
1  0.75z 1  0.125z 2
• Parallel Structure (Not unique)
 7  8 z 1
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H ( z)  8 
8

1
2
1
1  0.75z  0.125z
1  0.5z
1  0.25z 1
Parallel-form structure using secondorder system (form I)
Parallel-form structure using
first-order system
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6.3.1 Digital Filter Design
•
Given a set specifications or stated constraints on
–
–
•
magnitude spectrum H ( e j )
phase spectrum H (e j )
Find {am} and {bk } where,
M
H ( z) 
a
m 0
m
z m
K
1   bk z k
k 1
•
The constraints may include
–
–
–
–
–
–
zero, small, or linear phase
specific bandlimit within a passband
amount of ripple within a passband
amount of ripple within a stopband
sharpness of transitions between passband/stopband
filter order K, M.
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6.3.2 Finite Impulse Response (FIR) Filter Design
• Here it is assumed that b1  b2  b3      bK  0
M
• Hence
m
H ( z )   am z
m 0
And so the unit pulse response of the filter is clearly:
an ; 0  n  M
h( n )  
else
 0;
• Problem: Given specifications on H ( e j ) and H (e j ) ,
find {an ; n  1,...,M }
• FIR filters are often called non-recursive for obvious reasons.
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6.3.3 FIR Filter: Advantages and Disadvantages
•
Advantages:
–
–
–
–
–
•
Always stable (assume non-recursive implementation).
Quantization noise is not much of a problem.
Can be designed to have exact linear phase even when causal,
while meeting a prescribed phase to arbitrary accuracy.
Design complexity generally linear.
Transients have a finite duration.
Disadvantages:
–
A high-order filter is generally needed to satisfy the stated
specification – so more coefficients are needed with more
storage and computation.
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6.3.4 FIR Filter Design: Linear Phase Condition
• Definition: The digital filter
is linear phase if
j
j
H (e )  H (e ) e
jH ( e j )
H (e j )  C for  [ , ]
for some real number C. If C=0, then the filter has zero- phase,
which is only possible when the filter is non-causal.
• Achieving linear phase is quite important in applications where is
desirable not to distort the signal phase much –i.e., where the
frequency locations are critical, such as speech signals.
• Many applications benefit be the linear phase thought as
– shaping frequencies according to the magnitude spectrum.
– Time-shifting the response by an amount -C
x(n  C)  e jC X (e j )
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6.3.5 Linear Phase Condition
• Theorem: a causal FIR filter with unit pulse response
h(n)  0; n  0,...,M
is linear phase if h(n) is even symmetric:
Proof: Suppose M is odd. Then
H ( z) 
( M 1) / 2
 h( n ) z
n 0
n
M

 h( n ) z
n
n ( M 1) / 2
( M 1) / 2

j
 H (e )  e
 h( n ) z
n

n 0
( M 1) / 2
 h( M  n ) z
( M  n )
n 0
n
( N  n )
h
(
n
)[
z

z
]

n 0
z e j

( M 1) / 2
 jM / 2
( M 1) / 2
 2h(n) cos[ (n  M / 2)]
n 0
which finishes the proof, why?
• Consider case of M even to be an exercise.
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6.4.1 FIR Filter Design: Windowing
• Goal: Design an FIR digital filter h(n)  H (e j ) with M+1
coefficients that approximates a desired frequency response
D(e j )  D(e j ) e jD ( e
with
1
d (n) 
2



j
)
D ( e j )e jn d
• Usually d(n) cannot be realized for some reasons.
–
–
–
–
d(n) has infinite duration if D ( e j ) contains discontinuities;
If d(n) is non-causal and we want it causal;
If d(n) is longer than can be computed efficiently;
It’s generally desirable to have few coefficients;
• Windowing is the simplest approach to FIR filter design. One can
proceed naively, and thus obtain poor results. But with little care
(basic windowing strategies), windowing can be very effective.
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6.4.2 General Windowing Approach
• Define
h(n)  w(n)  d (n)
w(n)  0 for n {0,1,...,M }
• Then designed filter then has frequency response
where
M
H (e )   w(n)  d (n)e  jn
j
n 0
1

2



D(e jv )W [e j ( v ) ]dv where w(n )  W (e j ).
• Observations: We desire conflicting goals
– w(n ) be time-limited to n {0,1,...,M }
– W (e j ) be spectrally localized – impulse-like, if
j
W (e )  2

 (  2n)
then H (e j )  D(e j )
n  
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6.4.3 Truncation Windowing
• Rectangular window:
1; 0  n  M
w(n)  
else
0;
• The designed filter has frequency response
M
H rec (e )   d (n) w(n)e j  D(e j ) *W (e j )
j
where
n 0
sin[ ( M  1) / 2]  jM / 2
e
 /2
– Is the frequency response Hrec (e j ) a good approximation to the desired
frequency response D(e j ) ?
W (e j ) 
– Actually, for M give Hrec (e j ) is optimal in the mean square sense (MSE).
– However, while rectangular windowing does the best global MSE job, it
suffers dramatically at frequencies.
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6.4.4 Triangle (Bartlett) Windowing
• Suppose that
which
 2n / M ; 0  n  M / 2

w( n )  2  2n / M ; M / 2  n  M

0;
else

2
 sin[ ( M  1) / 4   jM / 2
w( e j )  
2
 e
sin
(

/
4
)


• Note that (ignoring the shift) W (e j ) is a positive function, hence
Htri (e j )  D(e j ) *W (e j )
must rise monotonically at a jump discontinuity (why?).
• In the prior example, using the triangular window gives an
approximation with smooth, but wider transition.
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6.4.5 Windowing: Trade-off
Ripples vs. Transition Width
• Rectangular window has a sharp transition but severe ripple.
• Triangular window has no ripple but a very wide transition.
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6.4.6 Other Windows
• Other windows attempt to optimize this trade-off. Widely used
windows that give intermediate results are:
0.54  0.46cos(2n / M ); 0  n  M
w
(
n
)

– Hamming Window:

0;
else

– Hanning Window:
0.5  0.5 cos(2n / M ); 0  n  M
w(n)  
0;
else

– Blackman Window:
0.42  0.5 cos(2n / M )  0.08cos(4n / M ); 0  n  M
w(n)  
0;
else

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6.4.7 Windowing Comparisons
Rectangular: transition width is optimized.
Blackman: Ripple is minimized.
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6.4.8 Kaiser Window Design
• Here
 I { 1  [(n   ) /  ]2 }
 0
, 0n M
w(n)  
I0 (  )

0
otherwise

Where   M / 2, and I 0 () represents the zeroth-order modified
Bessel function of the first kind, and there are two important
parameters: M, .
– For M held constant, increasing  reduces sidelobe but increase
mainlobe width.
– For  held constant, increasing M reduces mainlobe width but does
not affect sidelobes much.
• Kaiser developed an empirical but careful design procedure for
windowing a filter having sharp discontinuity (e.g. an ideal LPF).
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