ELEN E4810: Digital Signal Processing
Topic 7:
Filter types and structures
1. More filter types
2. Minimum and maximum phase
3. Filter implementation structures
Dan Ellis
2013-10-30
1
1. More Filter Types


We have seen the basics of filters
and a range of simple examples
Now look at a couple of other classes:


Dan Ellis
Comb filters - multiple pass/stop bands
Allpass filters - only modify signal phase
2013-10-30
2
Comb Filters

Replace all system delays z-1 with
longer delays z-L
+
x[n]
y[n]
z-L
z-L
z-L
z-L
→ System that behaves ‘the same’ at a
longer timescale
Dan Ellis
2013-10-30
3
Comb Filters
‘Parent’ filter impulse response h[n]
becomes comb filter output as:
g[n] = {h[0] 0 0 0 0 h[1] 0 0 0 0 h[2]..}

L-1 zeros

Thus, G (z ) = n g[n]z
=  h[n]z
n
Dan Ellis
2013-10-30
n
nL
( )
=H z
L
4
Comb Filters

Hence frequency response:
( ) = H (e )
Ge
j
H(ej!)

jL
parent frequency response
compressed
& repeated L times
G(ej!)
Low-pass response →
L copies
of H(ej!)

pass ! = 0, 2 π/L, 4 π/L...

cut ! = π/L, 3 π/L, 5 π/L... useful to enhance
Dan Ellis
a harmonic series
2013-10-30
5
Allpass Filters


Allpass filter has |A(ej!)|2 = K for all !
i.e. spectral energy is not changed
Phase response is not zero (else trivial)

phase correction

special effects
5
e.g.
0
|H(ω)|
Magnitude (dB)

−5
−10
−15
−20
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Normalized Frequency (×π rad/sample)
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Normalized Frequency (×π rad/sample)
0.8
0.9
1
θ(ω)
Phase (degrees)
0
−100
−200
−300
−400
Dan Ellis
2013-10-30
6
Allpass Filters

Allpass has special form of system fn:
d M + d M 1z 1 + ...+ d1z  ( M 1) + z M
AM (z ) = ±
 ( M 1)
1
1+ d1z + ...+ d M 1z
+ d M z M
= ±z

M
( )
DM z 1
=
DM (z )
mirror-image
polynomials
AM(z) has poles λ where DM(λ) = 0
→ AM(z) has zeros ≥ = 1/λ = λ-1
Dan Ellis
2013-10-30
7
Allpass Filters
AM (z ) = ±z
DM (z )
Any (stable) DM can be used:
1
poles
from 1/DM(z)
0
−1
−3
−2
−1
Phase is always
decreasing:
→ -Mº at ! = º

Dan Ellis
0
1
arg{H(z)}
reciprocal
zeros
from DM(z-1)
Im{z}

( )
DM z
M
1
2
Re{z}
peak
group
delay
0
−π
−2π
−3π
0
2013-10-30
0.2
0.4
0.6
0.8
ω/π
1
8
M
Allpass Filters
Why do mirror-img poly’s give const gain?
 Conj-sym system fn can be factored as:
AM (z ) =
(
K i z  
* 1
i
)
i(z  i )
* 1
*
1
K i i z ( i  z )
=
i(z   i )

o *
i
ZP
λi
ej!
*i
e-j!
+ complex
conjugate p/z
z = ej! → z-1 = e-j! also on u.circle...
Dan Ellis
2013-10-30
1
9
2. Minimum/Maximum Phase

In AP filters, reciprocal roots have..



same effect on magnitude (modulo const.)
different effect on phase
In normal filters, can try
substituting reciprocal roots


reciprocal of stable pole will be unstable ×
reciprocals of zeros?
→ Variants of filters with same magnitude
response, different phase
Dan Ellis
2013-10-30
10
Minimum/Maximum Phase
Hence:

z-b
H1(z) = z - a
H2(z) =
a
|H(ej
b(z - 1/b)
z-a
a
b
3
3
2
2
1
1
0
0
reciprocal
zero..
1/b
.. same
mag..
)|
0.5
(H(ej )) 0
0
-0.5
-
.. added
phase
lag
0.5
-0.5
0
Dan Ellis
0.2
0.4
0.6
0.8
/
-
0
2013-10-30
0.2
0.4
0.6
0.8
/
11
Minimum/Maximum Phase

For a given magnitude response




All zeros inside u.circle → minimum phase
All zeros outside u.c. → maximum phase
(greatest phase dispersion for that order)
Otherwise, mixed phase
i.e. for a given magnitude response
several filters & phase fns are possible;
minimum phase is canonical, ‘best’
Dan Ellis
2013-10-30
12
Minimum/Maximum Phase
Note:
Min. phase + Allpass

= Max. phase
o
o
polezero
cancl’n
o
o
(z   )(z  
z
Dan Ellis
) x (z )(z ) = (z )(z )
*
z
(z   )(z   )
o
*
 1
*
1

2013-10-30
 1
o
*
1

13
Inverse Systems


hi[n] is called the inverse of hf[n] iff
hi [n]  h f [n] =  [n]
( )
( )
j
j
H
e

H
e
=1
Z-transform: f
i
x[n]
Hf(z)
y[n]
Hi(z)
w[n]
W (z ) = H i (z )Y (z ) = H i ( z )H f ( z ) X ( z ) = X (z )

 w[n] = x [n]
i.e. Hi(z) recovers x[n] from o/p of Hf(z)
Dan Ellis
2013-10-30
14
Inverse Systems


What is Hi(z)?
H i (z)H f (z) = 1
 H i (z ) = 1/ H f (z )
Hi(z) is reciprocal polynomial of Hf(z)
P (z)
D( z )
H f (z) =
 H i (z) =
D( z )
P (z )

Just swap
poles and
Hf(z)
zeros:
Dan Ellis
poles of fwd
→ zeros of bwd
zeros of fwd
→ poles of bwd
o
o
o
2013-10-30
Hi(z)
15
Inverse Systems
When does Hi(z) exist?
 Causal+stable → all H (z) poles inside u.c.
i
→ all zeros of Hf(z) must be inside u.c.
→ Hf(z) must be minimum phase
 H (z) zeros outside u.c. → unstable H (z)
f
i
 H (z) zeros on u.c. → unstable H (z)
f
i
( )
( )
H i e j = 1/ H f e j = 1/0  =
→ only invert if min.phase,
Dan Ellis
2013-10-30
lose...
!
Hf(ej!) ≠ 0
16
System Identification
x[n]



H(z)
y[n]
Inverse filtering = given y and H, find x
System ID = given y (and ~x), find H
Just run convolution backwards?

y[n] =  h[k ] x [n  k ]
k=0
y[0] = h[0] x [0]  h[0]
deconvolution
but: errors
accumulate
y[1] = h[0] x [1] + h[1] x [0]  h[1]...
Dan Ellis
2013-10-30
17
System Identification
x[n]



H?(z)
y[n] + noise
Better approach uses correlations;
Cross-correlate input and output:
rxy [] = y[]  x [] = h? []  x []  x []
= h? []  rxx []
If rxx is ‘simple’, can recover h?[n]...
e.g. (pseudo-) white noise:
rxx []   []  h? [n]  rxy []
Dan Ellis
2013-10-30
18
System Identification


Can also work in frequency domain:
S xy ( z ) = H ? (z )  S xx ( z )
make a const.
x[n] is not observable → Sxy unavailable,
but Sxx(ej!) may still be known, so:
j
j
*
j
Syy (e ) = Y (e )Y (e )
= H (e
) X (e )H (e ) X (e )
= H (e )  S (e )
j
j
j
2
*
j
*
j
j
xx

Use e.g. min.phase to rebuild H(ej!)...
Dan Ellis
2013-10-30
19
3. Filter Structures


Many different implementations,
representations of same filter
Different costs, speeds, layouts, noise
performance, ...
Dan Ellis
2013-10-30
20
Block Diagrams
Useful way to illustrate implementations
 Z-transform helps analysis:
y[n] Y ( z ) = G1 ( z )[ X ( z ) + G2 ( z )Y ( z )]
G1(z)
 Y (z )[1  G1 (z )G2 (z )] = G1 (z ) X (z )
G2(z)
Y (z)
G1 (z )
 H (z) =
=
X (z ) 1  G1 ( z )G2 ( z )
 Approach

x[n]
+


Dan Ellis
Output of summers as dummy variables
Everything else is just multiplicative
2013-10-30
21
Block Diagrams

More complex example:
+
-Æ
x[n]
+
y[n]
w1
+
-±
Ø
∞
z-1
W1 = X  z 1W3
w2
+
w3
1
"
z-1
1
Y = z W3 + W1
Y  + z ( +  ) + z ( )
 =
X 1+ z 1 ( +  ) + z 2 ( )
1
2
stackable
2nd order section
Dan Ellis
2013-10-30
W2 = W1  z W2
1
W3 = z W2 + W2
W1
W2 =
1+ z 1
1
z +  W1
W3 =
1
1+ z
(
)
22
Delay-Free Loops
Can’t have them!
+
u
Β
y


Α
+
v
y = B(v + Au)
u=x+y
y = B(v + A(x + y))
At time n = 0, setup inputs x and v ;
need u for y, also y for u →can’t calculate
1
Algebra:
1  BA
y(1
Dan Ellis
BA) = Bv + BAx
Bv + BAx
y=
1 BA
x
BA
1  BA
y
2013-10-30
+
x
+

B
1  BA
u
B
1  BA
v
can simplify...
23
Equivalent Structures

Modifications to block diagrams that do
not change the filter
e.g. Commutation H = AB = BA
A

≡
B
B
A
Factoring AB+CB = (A+C)·B
x1
x1
A(z)B(z)
+
x2
C(z)B(z)
fewer blocks
Dan Ellis
A(z)
y
+

x2
B(z)
C(z)
less computation
2013-10-30
24
y
Equivalent Structures
x
Transpose

z-1
b3
input output
y
1
2
Y = b1 X + b2 z X + b3 z X
1
1
= b1 X + z b2 X + z b3 X
(
Dan Ellis
b2
)
2013-10-30
≡
z-1
b1
+

reverse paths
adders nodes
z-1
+

b1
b2
z-1
b3
+

y
x
25
FIR Filter Structures
Direct form “Tapped Delay Line”
h0
h2
y[n] = h0 x [n] + h1 x [n 1] + ...
z-1
h3
h4
y
=
4
h x
k=0 k
[n  k ]
Transpose
+
h3
z-1
h2
z-1
+
+
h4
z-1

h1
z-1
h1
z-1
+
x
+

z-1
+
z-1
+
x
+

h0
y
Re-use delay line if several inputs xi
for single output y ?
Dan Ellis
2013-10-30
26
FIR Filter Structures

Cascade

factored into e.g. 2nd order sections
H (z ) = h0 + h1z 1 + h2 z 2 + h3 z 3
= h0 1   0 z 1 1   1z 1 1  1* z 1
(
)(
)(
)
2 2
1
1
= h0 (1   0 z )(1  2 Re{1 }z + 1 z )
h0
+
z-1
≥0
+
x
z-1
z-1
Dan Ellis
2013-10-30
y
-2Re{≥1}
|≥1|2
27
FIR Filter Structures

Linear Phase:
n
Symmetric filters with h[n] = (-)h[N - n]
z-1
z-1
b0
b1
...
+
+
+

z-1
+
y[n] = b0 ( x [n] + x [n  4 ])
+b1 ( x [n 1] + x [n  3])
+b2 x [n  2]
z-1
x
b2
y
half as many
multiplies
Also Transpose form:
gains first, feeding folded delay/sum line
Dan Ellis
2013-10-30
28
IIR Filter Structures

IIR: numerator + denominator
p0 + p1z 1 + p2 z 2 + ...
H (z) =
1+ d1z 1 + d2 z 2 + ...
1
= P (z) 
D( z )
Dan Ellis
z-1
p1
p2
+
FIR
z-1
+
p0
-d1
z-1
-d2
z-1
2013-10-30
all-pole
IIR
29
IIR Filter Structures
Hence, Direct form I
p0
p1
z-1
p2
-d1
z-1
-d2
z-1
Commutation → Direct form II (DF2)
p0
+
Dan Ellis
-d1
z-1
p1
-d2
z-1
p2
+

z-1
+

2013-10-30
• same signal
∴ delay lines merge
• “canonical”
= min. memory usage
30
IIR Filter Structures
Use Transpose on FIR/IIR/DF2
p1
z-1
p2
z-1
+

p0
+
x
y
+

-d1
-d2
“Direct Form II Transpose”
Dan Ellis
2013-10-30
31
Factored IIR Structures

Real-output filters have
Æ
conjugate-symm roots:
−Ø
1
H (z) =
1
1
1  ( + j )z 1  (  j )z
(

Ø
)(
Can always group into 2nd order terms
with real coefficients:
(
)(
)
H (z) =
1
1
2
2 2
1


z
1

2

z
+
(

+

(
)
(
1
2
2
2 )z )...
real root
p0 1  1z 1 1  2 2 z 1 + ( 22 +  22 )z 2 ...
Dan Ellis
2013-10-30
32
)
Cascade IIR Structure
+ +
+
p0
+ +
x
Implement as cascade of
second order sections (in DFII)
z-1
-∞1
Æ1
2Æ2
fwd gain
factored
out
y
+

z-1
z-1
z-1 -2∞2
z-1
−(Æ22+Ø22) ∞22+±22

Second order sections (SOS):


Dan Ellis
modular - any order from optimized block
well-behaved, real coefficients (sensitive?)
2013-10-30
33
Second-Order Sections

‘Free’ choices:



Optimize numerical properties:



grouping of pole pairs with zero pairs
order of sections
avoid very large values (overflow)
avoid very small values (quantization)
e.g. Matlab’s zp2sos


Dan Ellis
attempt to put ‘close’ roots in same section
intersperse gain & attenuation?
2013-10-30
34
Second Order Sections

Factorization affects intermediate values
Original System
(2 pair poles, zeros)
Dan Ellis
Factorization 1
Factorization 2


2013-10-30
35
Parallel IIR Structures
Can express H(z) as sum of terms (IZT)
N

  = (1    z 1 )F (z ) z= 
H (z) = consts + 
=11   z 1

 Or, second-order terms:

 0 k + 1k z 1
H (z) =  0 + 
k 1+  z 1 +  z 2
1k
2k

Suggests parallel realization...
Dan Ellis
2013-10-30
36

Parallel IIR Structures
∞0
+
∞01
+
-Æ11
z-1
+
x

∞12
+
∞02
+
-Æ22 z-1

∞11
-Æ21 z-1
-Æ12 z-1
y


Sum terms become
parallel paths
Poles of each SOS
are from full TF
System zeros arise
from output sum
Why do this?


Dan Ellis
stability/sensitivity
reuse common terms
2013-10-30
37