Post on 07-Feb-2016
description
15: Subband Processing
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 1 / 12
Subband processing
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 2 / 12
Subband processing
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 2 / 12
• The Hm(z) are bandpass analysis filters and divide x[n] intofrequency bands
Subband processing
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 2 / 12
• The Hm(z) are bandpass analysis filters and divide x[n] intofrequency bands
• Subband processing often processes frequency bands independently
Subband processing
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 2 / 12
• The Hm(z) are bandpass analysis filters and divide x[n] intofrequency bands
• Subband processing often processes frequency bands independently• The Gm(z) are synthesis filters and together reconstruct the output
Subband processing
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 2 / 12
• The Hm(z) are bandpass analysis filters and divide x[n] intofrequency bands
• Subband processing often processes frequency bands independently• The Gm(z) are synthesis filters and together reconstruct the output• The Hm(z) outputs are bandlimited and so can be subsampled
without loss of information
Subband processing
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 2 / 12
• The Hm(z) are bandpass analysis filters and divide x[n] intofrequency bands
• Subband processing often processes frequency bands independently• The Gm(z) are synthesis filters and together reconstruct the output• The Hm(z) outputs are bandlimited and so can be subsampled
without loss of information◦ Sample rate multiplied overall by
∑
1Pi
Subband processing
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 2 / 12
• The Hm(z) are bandpass analysis filters and divide x[n] intofrequency bands
• Subband processing often processes frequency bands independently• The Gm(z) are synthesis filters and together reconstruct the output• The Hm(z) outputs are bandlimited and so can be subsampled
without loss of information◦ Sample rate multiplied overall by
∑
1Pi
∑
1Pi
= 1⇒ critically sampled : good for coding
Subband processing
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 2 / 12
• The Hm(z) are bandpass analysis filters and divide x[n] intofrequency bands
• Subband processing often processes frequency bands independently• The Gm(z) are synthesis filters and together reconstruct the output• The Hm(z) outputs are bandlimited and so can be subsampled
without loss of information◦ Sample rate multiplied overall by
∑
1Pi
∑
1Pi
= 1⇒ critically sampled : good for coding∑
1Pi
> 1⇒ oversampled : more flexible
Subband processing
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 2 / 12
• The Hm(z) are bandpass analysis filters and divide x[n] intofrequency bands
• Subband processing often processes frequency bands independently• The Gm(z) are synthesis filters and together reconstruct the output• The Hm(z) outputs are bandlimited and so can be subsampled
without loss of information◦ Sample rate multiplied overall by
∑
1Pi
∑
1Pi
= 1⇒ critically sampled : good for coding∑
1Pi
> 1⇒ oversampled : more flexible• Goals:
(a) good frequency selectivity in Hm(z)
Subband processing
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 2 / 12
• The Hm(z) are bandpass analysis filters and divide x[n] intofrequency bands
• Subband processing often processes frequency bands independently• The Gm(z) are synthesis filters and together reconstruct the output• The Hm(z) outputs are bandlimited and so can be subsampled
without loss of information◦ Sample rate multiplied overall by
∑
1Pi
∑
1Pi
= 1⇒ critically sampled : good for coding∑
1Pi
> 1⇒ oversampled : more flexible• Goals:
(a) good frequency selectivity in Hm(z)(b) perfect reconstruction: y[n] = x[n− d] if no processing
Subband processing
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 2 / 12
• The Hm(z) are bandpass analysis filters and divide x[n] intofrequency bands
• Subband processing often processes frequency bands independently• The Gm(z) are synthesis filters and together reconstruct the output• The Hm(z) outputs are bandlimited and so can be subsampled
without loss of information◦ Sample rate multiplied overall by
∑
1Pi
∑
1Pi
= 1⇒ critically sampled : good for coding∑
1Pi
> 1⇒ oversampled : more flexible• Goals:
(a) good frequency selectivity in Hm(z)(b) perfect reconstruction: y[n] = x[n− d] if no processing
• Benefits: Lower computation, faster convergence if adaptive
2-band Filterbank
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 3 / 12
2-band Filterbank
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 3 / 12
Vm(z) = Hm(z)X(z)
2-band Filterbank
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 3 / 12
Vm(z) = Hm(z)X(z)
Um(z) = 1K
∑K−1k=0 Vm(e
−j2πk
K z1
K )
[K = 2]
2-band Filterbank
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 3 / 12
Vm(z) = Hm(z)X(z)
Um(z) = 1K
∑K−1k=0 Vm(e
−j2πk
K z1
K ) = 12
{
Vm
(
z1
2
)
+ Vm
(
−z1
2
)}
[K = 2]
2-band Filterbank
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 3 / 12
Vm(z) = Hm(z)X(z)
Um(z) = 1K
∑K−1k=0 Vm(e
−j2πk
K z1
K ) = 12
{
Vm
(
z1
2
)
+ Vm
(
−z1
2
)}
Wm(z) = Um(z2) = 12 {Vm(z) + Vm(−z)} [K = 2]
2-band Filterbank
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 3 / 12
Vm(z) = Hm(z)X(z)
Um(z) = 1K
∑K−1k=0 Vm(e
−j2πk
K z1
K ) = 12
{
Vm
(
z1
2
)
+ Vm
(
−z1
2
)}
Wm(z) = Um(z2) = 12 {Vm(z) + Vm(−z)} [K = 2]
= 12 {Hm(z)X(z) +Hm(−z)X(−z)}
2-band Filterbank
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 3 / 12
Vm(z) = Hm(z)X(z)
Um(z) = 1K
∑K−1k=0 Vm(e
−j2πk
K z1
K ) = 12
{
Vm
(
z1
2
)
+ Vm
(
−z1
2
)}
Wm(z) = Um(z2) = 12 {Vm(z) + Vm(−z)} [K = 2]
= 12 {Hm(z)X(z) +Hm(−z)X(−z)}
Y (z) =[
W0(z) W1(z)]
[
G0(z)G1(z)
]
2-band Filterbank
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 3 / 12
Vm(z) = Hm(z)X(z)
Um(z) = 1K
∑K−1k=0 Vm(e
−j2πk
K z1
K ) = 12
{
Vm
(
z1
2
)
+ Vm
(
−z1
2
)}
Wm(z) = Um(z2) = 12 {Vm(z) + Vm(−z)} [K = 2]
= 12 {Hm(z)X(z) +Hm(−z)X(−z)}
Y (z) =[
W0(z) W1(z)]
[
G0(z)G1(z)
]
= 12
[
X(z) X(−z)]
[
H0(z) H1(z)H0(−z) H1(−z)
] [
G0(z)G1(z)
]
2-band Filterbank
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 3 / 12
Vm(z) = Hm(z)X(z)
Um(z) = 1K
∑K−1k=0 Vm(e
−j2πk
K z1
K ) = 12
{
Vm
(
z1
2
)
+ Vm
(
−z1
2
)}
Wm(z) = Um(z2) = 12 {Vm(z) + Vm(−z)} [K = 2]
= 12 {Hm(z)X(z) +Hm(−z)X(−z)}
Y (z) =[
W0(z) W1(z)]
[
G0(z)G1(z)
]
= 12
[
X(z) X(−z)]
[
H0(z) H1(z)H0(−z) H1(−z)
] [
G0(z)G1(z)
]
=[
X(z) X(−z)]
[
T (z)A(z)
]
2-band Filterbank
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 3 / 12
Vm(z) = Hm(z)X(z)
Um(z) = 1K
∑K−1k=0 Vm(e
−j2πk
K z1
K ) = 12
{
Vm
(
z1
2
)
+ Vm
(
−z1
2
)}
Wm(z) = Um(z2) = 12 {Vm(z) + Vm(−z)} [K = 2]
= 12 {Hm(z)X(z) +Hm(−z)X(−z)}
Y (z) =[
W0(z) W1(z)]
[
G0(z)G1(z)
]
= 12
[
X(z) X(−z)]
[
H0(z) H1(z)H0(−z) H1(−z)
] [
G0(z)G1(z)
]
=[
X(z) X(−z)]
[
T (z)A(z)
]
We want (a) T (z) = 12 {H0(z)G0(z) +H1(z)G1(z)} = z−d
2-band Filterbank
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 3 / 12
Vm(z) = Hm(z)X(z)
Um(z) = 1K
∑K−1k=0 Vm(e
−j2πk
K z1
K ) = 12
{
Vm
(
z1
2
)
+ Vm
(
−z1
2
)}
Wm(z) = Um(z2) = 12 {Vm(z) + Vm(−z)} [K = 2]
= 12 {Hm(z)X(z) +Hm(−z)X(−z)}
Y (z) =[
W0(z) W1(z)]
[
G0(z)G1(z)
]
= 12
[
X(z) X(−z)]
[
H0(z) H1(z)H0(−z) H1(−z)
] [
G0(z)G1(z)
]
=[
X(z) X(−z)]
[
T (z)A(z)
]
[X(−z)A(z) is “aliased” term]
We want (a) T (z) = 12 {H0(z)G0(z) +H1(z)G1(z)} = z−d
and (b) A(z) = 12 {H0(−z)G0(z) +H1(−z)G1(z)} = 0
Perfect Reconstruction
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 4 / 12
For perfect reconstruction without aliasing, we require
12
[
H0(z) H1(z)H0(−z) H1(−z)
] [
G0(z)G1(z)
]
=
[
z−d
0
]
Perfect Reconstruction
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 4 / 12
For perfect reconstruction without aliasing, we require
12
[
H0(z) H1(z)H0(−z) H1(−z)
] [
G0(z)G1(z)
]
=
[
z−d
0
]
Hence:[
G0(z)G1(z)
]
=
[
H0(z) H1(z)H0(−z) H1(−z)
]
−1 [2z−d
0
]
Perfect Reconstruction
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 4 / 12
For perfect reconstruction without aliasing, we require
12
[
H0(z) H1(z)H0(−z) H1(−z)
] [
G0(z)G1(z)
]
=
[
z−d
0
]
Hence:[
G0(z)G1(z)
]
=
[
H0(z) H1(z)H0(−z) H1(−z)
]
−1 [2z−d
0
]
= 2z−d
H0(z)H1(−z)−H0(−z)H1(z)
[
H1(−z) −H1(z)−H0(−z) H0(z)
] [
10
]
Perfect Reconstruction
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 4 / 12
For perfect reconstruction without aliasing, we require
12
[
H0(z) H1(z)H0(−z) H1(−z)
] [
G0(z)G1(z)
]
=
[
z−d
0
]
Hence:[
G0(z)G1(z)
]
=
[
H0(z) H1(z)H0(−z) H1(−z)
]
−1 [2z−d
0
]
= 2z−d
H0(z)H1(−z)−H0(−z)H1(z)
[
H1(−z) −H1(z)−H0(−z) H0(z)
] [
10
]
= 2z−d
H0(z)H1(−z)−H0(−z)H1(z)
[
H1(−z)−H0(−z)
]
Perfect Reconstruction
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 4 / 12
For perfect reconstruction without aliasing, we require
12
[
H0(z) H1(z)H0(−z) H1(−z)
] [
G0(z)G1(z)
]
=
[
z−d
0
]
Hence:[
G0(z)G1(z)
]
=
[
H0(z) H1(z)H0(−z) H1(−z)
]
−1 [2z−d
0
]
= 2z−d
H0(z)H1(−z)−H0(−z)H1(z)
[
H1(−z) −H1(z)−H0(−z) H0(z)
] [
10
]
= 2z−d
H0(z)H1(−z)−H0(−z)H1(z)
[
H1(−z)−H0(−z)
]
For all filters to be FIR, we need the denominator to be
H0(z)H1(−z)−H0(−z)H1(z) = cz−k
Perfect Reconstruction
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 4 / 12
For perfect reconstruction without aliasing, we require
12
[
H0(z) H1(z)H0(−z) H1(−z)
] [
G0(z)G1(z)
]
=
[
z−d
0
]
Hence:[
G0(z)G1(z)
]
=
[
H0(z) H1(z)H0(−z) H1(−z)
]
−1 [2z−d
0
]
= 2z−d
H0(z)H1(−z)−H0(−z)H1(z)
[
H1(−z) −H1(z)−H0(−z) H0(z)
] [
10
]
= 2z−d
H0(z)H1(−z)−H0(−z)H1(z)
[
H1(−z)−H0(−z)
]
For all filters to be FIR, we need the denominator to be
H0(z)H1(−z)−H0(−z)H1(z) = cz−k , which implies[
G0(z)G1(z)
]
= 2czk−d
[
H1(−z)−H0(−z)
]
Perfect Reconstruction
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 4 / 12
For perfect reconstruction without aliasing, we require
12
[
H0(z) H1(z)H0(−z) H1(−z)
] [
G0(z)G1(z)
]
=
[
z−d
0
]
Hence:[
G0(z)G1(z)
]
=
[
H0(z) H1(z)H0(−z) H1(−z)
]
−1 [2z−d
0
]
= 2z−d
H0(z)H1(−z)−H0(−z)H1(z)
[
H1(−z) −H1(z)−H0(−z) H0(z)
] [
10
]
= 2z−d
H0(z)H1(−z)−H0(−z)H1(z)
[
H1(−z)−H0(−z)
]
For all filters to be FIR, we need the denominator to be
H0(z)H1(−z)−H0(−z)H1(z) = cz−k , which implies[
G0(z)G1(z)
]
= 2czk−d
[
H1(−z)−H0(−z)
]
d=k= 2
c
[
H1(−z)−H0(−z)
]
Quadrature Mirror Filterbank (QMF)
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 5 / 12
Quadrature Mirror Filterbank (QMF)
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 5 / 12
QMF satisfies:(a) H0(z) is causal and real
Quadrature Mirror Filterbank (QMF)
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 5 / 12
QMF satisfies:(a) H0(z) is causal and real(b) H1(z) = H0(−z): i.e.
∣
∣H0(ejω)
∣
∣ is reflected around ω = π2
Quadrature Mirror Filterbank (QMF)
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 5 / 12
QMF satisfies:(a) H0(z) is causal and real(b) H1(z) = H0(−z): i.e.
∣
∣H0(ejω)
∣
∣ is reflected around ω = π2
(c) G0(z) = 2H1(−z) = 2H0(z)
Quadrature Mirror Filterbank (QMF)
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 5 / 12
QMF satisfies:(a) H0(z) is causal and real(b) H1(z) = H0(−z): i.e.
∣
∣H0(ejω)
∣
∣ is reflected around ω = π2
(c) G0(z) = 2H1(−z) = 2H0(z)(d) G1(z) = −2H0(−z) = −2H1(z)
Quadrature Mirror Filterbank (QMF)
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 5 / 12
QMF satisfies:(a) H0(z) is causal and real(b) H1(z) = H0(−z): i.e.
∣
∣H0(ejω)
∣
∣ is reflected around ω = π2
(c) G0(z) = 2H1(−z) = 2H0(z)(d) G1(z) = −2H0(−z) = −2H1(z)
QMF is alias-free:A(z) = 1
2 {H0(−z)G0(z) +H1(−z)G1(z)}
Quadrature Mirror Filterbank (QMF)
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 5 / 12
QMF satisfies:(a) H0(z) is causal and real(b) H1(z) = H0(−z): i.e.
∣
∣H0(ejω)
∣
∣ is reflected around ω = π2
(c) G0(z) = 2H1(−z) = 2H0(z)(d) G1(z) = −2H0(−z) = −2H1(z)
QMF is alias-free:A(z) = 1
2 {H0(−z)G0(z) +H1(−z)G1(z)}
= 12 {2H1(z)H0(z)− 2H0(z)H1(z)} = 0
Quadrature Mirror Filterbank (QMF)
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 5 / 12
QMF satisfies:(a) H0(z) is causal and real(b) H1(z) = H0(−z): i.e.
∣
∣H0(ejω)
∣
∣ is reflected around ω = π2
(c) G0(z) = 2H1(−z) = 2H0(z)(d) G1(z) = −2H0(−z) = −2H1(z)
QMF is alias-free:A(z) = 1
2 {H0(−z)G0(z) +H1(−z)G1(z)}
= 12 {2H1(z)H0(z)− 2H0(z)H1(z)} = 0
QMF Transfer Function:T (z) = 1
2 {H0(z)G0(z) +H1(z)G1(z)}
Quadrature Mirror Filterbank (QMF)
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 5 / 12
QMF satisfies:(a) H0(z) is causal and real(b) H1(z) = H0(−z): i.e.
∣
∣H0(ejω)
∣
∣ is reflected around ω = π2
(c) G0(z) = 2H1(−z) = 2H0(z)(d) G1(z) = −2H0(−z) = −2H1(z)
QMF is alias-free:A(z) = 1
2 {H0(−z)G0(z) +H1(−z)G1(z)}
= 12 {2H1(z)H0(z)− 2H0(z)H1(z)} = 0
QMF Transfer Function:T (z) = 1
2 {H0(z)G0(z) +H1(z)G1(z)}
= H20 (z)−H2
1 (z) = H20 (z)−H2
0 (−z)
Polyphase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 6 / 12
Polyphase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 6 / 12
Polyphase decomposition:H0(z) = P0(z
2) + z−1P1(z2)
Polyphase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 6 / 12
Polyphase decomposition:H0(z) = P0(z
2) + z−1P1(z2)
H1(z) = H0(−z) = P0(z2)− z−1P1(z
2)
Polyphase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 6 / 12
Polyphase decomposition:H0(z) = P0(z
2) + z−1P1(z2)
H1(z) = H0(−z) = P0(z2)− z−1P1(z
2)G0(z) = 2H0(z) = 2P0(z
2) + 2z−1P1(z2)
Polyphase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 6 / 12
Polyphase decomposition:H0(z) = P0(z
2) + z−1P1(z2)
H1(z) = H0(−z) = P0(z2)− z−1P1(z
2)G0(z) = 2H0(z) = 2P0(z
2) + 2z−1P1(z2)
G1(z) = −2H0(−z) = −2P0(z2) + 2z−1P1(z
2)
Polyphase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 6 / 12
Polyphase decomposition:H0(z) = P0(z
2) + z−1P1(z2)
H1(z) = H0(−z) = P0(z2)− z−1P1(z
2)G0(z) = 2H0(z) = 2P0(z
2) + 2z−1P1(z2)
G1(z) = −2H0(−z) = −2P0(z2) + 2z−1P1(z
2)
Polyphase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 6 / 12
Polyphase decomposition:H0(z) = P0(z
2) + z−1P1(z2)
H1(z) = H0(−z) = P0(z2)− z−1P1(z
2)G0(z) = 2H0(z) = 2P0(z
2) + 2z−1P1(z2)
G1(z) = −2H0(−z) = −2P0(z2) + 2z−1P1(z
2)
Transfer Function:T (z) = H2
0 (z)−H21 (z) = 4z−1P0(z
2)P1(z2)
Polyphase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 6 / 12
Polyphase decomposition:H0(z) = P0(z
2) + z−1P1(z2)
H1(z) = H0(−z) = P0(z2)− z−1P1(z
2)G0(z) = 2H0(z) = 2P0(z
2) + 2z−1P1(z2)
G1(z) = −2H0(−z) = −2P0(z2) + 2z−1P1(z
2)
Transfer Function:T (z) = H2
0 (z)−H21 (z) = 4z−1P0(z
2)P1(z2)
we want T (z) = z−d ⇒ P0(z) = a0z−k, P1(z) = a1z
k+1−d
Polyphase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 6 / 12
Polyphase decomposition:H0(z) = P0(z
2) + z−1P1(z2)
H1(z) = H0(−z) = P0(z2)− z−1P1(z
2)G0(z) = 2H0(z) = 2P0(z
2) + 2z−1P1(z2)
G1(z) = −2H0(−z) = −2P0(z2) + 2z−1P1(z
2)
Transfer Function:T (z) = H2
0 (z)−H21 (z) = 4z−1P0(z
2)P1(z2)
we want T (z) = z−d ⇒ P0(z) = a0z−k, P1(z) = a1z
k+1−d
⇒ H0(z) has only two non-zero taps ⇒ poor freq selectivity
Polyphase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 6 / 12
Polyphase decomposition:H0(z) = P0(z
2) + z−1P1(z2)
H1(z) = H0(−z) = P0(z2)− z−1P1(z
2)G0(z) = 2H0(z) = 2P0(z
2) + 2z−1P1(z2)
G1(z) = −2H0(−z) = −2P0(z2) + 2z−1P1(z
2)
Transfer Function:T (z) = H2
0 (z)−H21 (z) = 4z−1P0(z
2)P1(z2)
we want T (z) = z−d ⇒ P0(z) = a0z−k, P1(z) = a1z
k+1−d
⇒ H0(z) has only two non-zero taps ⇒ poor freq selectivity∴ Perfect reconstruction QMF filterbanks cannot have good freq selectivity
QMF Options
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 7 / 12
QMF Options
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 7 / 12
A(z) = 0⇒ no alias term
QMF Options
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 7 / 12
A(z) = 0⇒ no alias termT (z) = H2
0 (z)−H21 (z) = H2
0 (z)−H20 (−z)
QMF Options
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 7 / 12
Polyphase decomposition:
A(z) = 0⇒ no alias termT (z) = H2
0 (z)−H21 (z) = H2
0 (z)−H20 (−z)
QMF Options
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 7 / 12
Polyphase decomposition:
A(z) = 0⇒ no alias termT (z) = H2
0 (z)−H21 (z) = H2
0 (z)−H20 (−z) = 4z−1P0(z
2)P1(z2)
QMF Options
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 7 / 12
Polyphase decomposition:
A(z) = 0⇒ no alias termT (z) = H2
0 (z)−H21 (z) = H2
0 (z)−H20 (−z) = 4z−1P0(z
2)P1(z2)
Options:• Perfect Reconstruction: T (z) = z−d ⇒H0(z) is a bad filter.
QMF Options
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 7 / 12
Polyphase decomposition:
A(z) = 0⇒ no alias termT (z) = H2
0 (z)−H21 (z) = H2
0 (z)−H20 (−z) = 4z−1P0(z
2)P1(z2)
Options:• Perfect Reconstruction: T (z) = z−d ⇒H0(z) is a bad filter.
• T (z) is Linear Phase FIR:⇒ Tradeoff:
∣
∣T (ejω)∣
∣ ≈ 1 versus H0(z) stopband attenuation
QMF Options
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 7 / 12
Polyphase decomposition:
A(z) = 0⇒ no alias termT (z) = H2
0 (z)−H21 (z) = H2
0 (z)−H20 (−z) = 4z−1P0(z
2)P1(z2)
Options:• Perfect Reconstruction: T (z) = z−d ⇒H0(z) is a bad filter.
• T (z) is Linear Phase FIR:⇒ Tradeoff:
∣
∣T (ejω)∣
∣ ≈ 1 versus H0(z) stopband attenuation
• T (z) is Allpass IIR: H0(z) can be Butterworth or Elliptic filter⇒ Tradeoff: ∠T (ejω) ≈ τω versus H0(z) stopband attenuation
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
H0(z) order M , linear phase
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
H0(z) order M , linear phase ⇒ H0(ejω) = ±e−jωM
2
∣
∣H0(ejω)
∣
∣
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
H0(z) order M , linear phase ⇒ H0(ejω) = ±e−jωM
2
∣
∣H0(ejω)
∣
∣
T (ejω) = H20 (e
jω)−H21 (e
jω) = H20 (e
jω)−H20 (−ejω)
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
H0(z) order M , linear phase ⇒ H0(ejω) = ±e−jωM
2
∣
∣H0(ejω)
∣
∣
T (ejω) = H20 (e
jω)−H21 (e
jω) = H20 (e
jω)−H20 (−ejω)
= e−jωM∣
∣H0(ejω)
∣
∣
2− e−j(ω−π)M
∣
∣H0(ej(ω−π))
∣
∣
2
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
H0(z) order M , linear phase ⇒ H0(ejω) = ±e−jωM
2
∣
∣H0(ejω)
∣
∣
T (ejω) = H20 (e
jω)−H21 (e
jω) = H20 (e
jω)−H20 (−ejω)
= e−jωM∣
∣H0(ejω)
∣
∣
2− e−j(ω−π)M
∣
∣H0(ej(ω−π))
∣
∣
2
= e−jωM(
∣
∣H0(ejω)
∣
∣
2− (−1)M
∣
∣H0(ej(π−ω))
∣
∣
2)
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
H0(z) order M , linear phase ⇒ H0(ejω) = ±e−jωM
2
∣
∣H0(ejω)
∣
∣
T (ejω) = H20 (e
jω)−H21 (e
jω) = H20 (e
jω)−H20 (−ejω)
= e−jωM∣
∣H0(ejω)
∣
∣
2− e−j(ω−π)M
∣
∣H0(ej(ω−π))
∣
∣
2
= e−jωM(
∣
∣H0(ejω)
∣
∣
2− (−1)M
∣
∣H0(ej(π−ω))
∣
∣
2)
M even ⇒ T (ejπ2 ) = 0 /
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
H0(z) order M , linear phase ⇒ H0(ejω) = ±e−jωM
2
∣
∣H0(ejω)
∣
∣
T (ejω) = H20 (e
jω)−H21 (e
jω) = H20 (e
jω)−H20 (−ejω)
= e−jωM∣
∣H0(ejω)
∣
∣
2− e−j(ω−π)M
∣
∣H0(ej(ω−π))
∣
∣
2
= e−jωM(
∣
∣H0(ejω)
∣
∣
2− (−1)M
∣
∣H0(ej(π−ω))
∣
∣
2)
M even ⇒ T (ejπ2 ) = 0 / so choose M odd ⇒ − (−1)
M= +1
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
H0(z) order M , linear phase ⇒ H0(ejω) = ±e−jωM
2
∣
∣H0(ejω)
∣
∣
T (ejω) = H20 (e
jω)−H21 (e
jω) = H20 (e
jω)−H20 (−ejω)
= e−jωM∣
∣H0(ejω)
∣
∣
2− e−j(ω−π)M
∣
∣H0(ej(ω−π))
∣
∣
2
= e−jωM(
∣
∣H0(ejω)
∣
∣
2− (−1)M
∣
∣H0(ej(π−ω))
∣
∣
2)
M even ⇒ T (ejπ2 ) = 0 / so choose M odd ⇒ − (−1)
M= +1
Select h0[n] by numerical iteration to minimize
α∫ π
π2+∆
∣
∣H0(ejω)
∣
∣
2dω + (1− α)
∫ π
0
(∣
∣T (ejω)∣
∣− 1)2
dω
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
H0(z) order M , linear phase ⇒ H0(ejω) = ±e−jωM
2
∣
∣H0(ejω)
∣
∣
T (ejω) = H20 (e
jω)−H21 (e
jω) = H20 (e
jω)−H20 (−ejω)
= e−jωM∣
∣H0(ejω)
∣
∣
2− e−j(ω−π)M
∣
∣H0(ej(ω−π))
∣
∣
2
= e−jωM(
∣
∣H0(ejω)
∣
∣
2− (−1)M
∣
∣H0(ej(π−ω))
∣
∣
2)
M even ⇒ T (ejπ2 ) = 0 / so choose M odd ⇒ − (−1)
M= +1
Select h0[n] by numerical iteration to minimize
α∫ π
π2+∆
∣
∣H0(ejω)
∣
∣
2dω + (1− α)
∫ π
0
(∣
∣T (ejω)∣
∣− 1)2
dω
α → balance between H0(z) being lowpass and T (ejω) ≈ 1
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
H0(z) order M , linear phase ⇒ H0(ejω) = ±e−jωM
2
∣
∣H0(ejω)
∣
∣
T (ejω) = H20 (e
jω)−H21 (e
jω) = H20 (e
jω)−H20 (−ejω)
= e−jωM∣
∣H0(ejω)
∣
∣
2− e−j(ω−π)M
∣
∣H0(ej(ω−π))
∣
∣
2
= e−jωM(
∣
∣H0(ejω)
∣
∣
2− (−1)M
∣
∣H0(ej(π−ω))
∣
∣
2)
M even ⇒ T (ejπ2 ) = 0 / so choose M odd ⇒ − (−1)
M= +1
Select h0[n] by numerical iteration to minimize
α∫ π
π2+∆
∣
∣H0(ejω)
∣
∣
2dω + (1− α)
∫ π
0
(∣
∣T (ejω)∣
∣− 1)2
dω
α → balance between H0(z) being lowpass and T (ejω) ≈ 1
Johnston filter(M = 11):
h0[n] M=11
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
H0(z) order M , linear phase ⇒ H0(ejω) = ±e−jωM
2
∣
∣H0(ejω)
∣
∣
T (ejω) = H20 (e
jω)−H21 (e
jω) = H20 (e
jω)−H20 (−ejω)
= e−jωM∣
∣H0(ejω)
∣
∣
2− e−j(ω−π)M
∣
∣H0(ej(ω−π))
∣
∣
2
= e−jωM(
∣
∣H0(ejω)
∣
∣
2− (−1)M
∣
∣H0(ej(π−ω))
∣
∣
2)
M even ⇒ T (ejπ2 ) = 0 / so choose M odd ⇒ − (−1)
M= +1
Select h0[n] by numerical iteration to minimize
α∫ π
π2+∆
∣
∣H0(ejω)
∣
∣
2dω + (1− α)
∫ π
0
(∣
∣T (ejω)∣
∣− 1)2
dω
α → balance between H0(z) being lowpass and T (ejω) ≈ 1
Johnston filter(M = 11):
h0[n] M=11
0 1 2 3-60
-40
-20
0H
0H
1
ω
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
H0(z) order M , linear phase ⇒ H0(ejω) = ±e−jωM
2
∣
∣H0(ejω)
∣
∣
T (ejω) = H20 (e
jω)−H21 (e
jω) = H20 (e
jω)−H20 (−ejω)
= e−jωM∣
∣H0(ejω)
∣
∣
2− e−j(ω−π)M
∣
∣H0(ej(ω−π))
∣
∣
2
= e−jωM(
∣
∣H0(ejω)
∣
∣
2− (−1)M
∣
∣H0(ej(π−ω))
∣
∣
2)
M even ⇒ T (ejπ2 ) = 0 / so choose M odd ⇒ − (−1)
M= +1
Select h0[n] by numerical iteration to minimize
α∫ π
π2+∆
∣
∣H0(ejω)
∣
∣
2dω + (1− α)
∫ π
0
(∣
∣T (ejω)∣
∣− 1)2
dω
α → balance between H0(z) being lowpass and T (ejω) ≈ 1
Johnston filter(M = 11):
h0[n] M=11
0 1 2 3-60
-40
-20
0H
0H
1
ω0 1 2 3
-0.04
-0.02
0
0.02
0.04
ω
Linear Phase QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 8 / 12
T (z) ≈ 1
H0(z) order M , linear phase ⇒ H0(ejω) = ±e−jωM
2
∣
∣H0(ejω)
∣
∣
T (ejω) = H20 (e
jω)−H21 (e
jω) = H20 (e
jω)−H20 (−ejω)
= e−jωM∣
∣H0(ejω)
∣
∣
2− e−j(ω−π)M
∣
∣H0(ej(ω−π))
∣
∣
2
= e−jωM(
∣
∣H0(ejω)
∣
∣
2− (−1)M
∣
∣H0(ej(π−ω))
∣
∣
2)
M even ⇒ T (ejπ2 ) = 0 / so choose M odd ⇒ − (−1)
M= +1
Select h0[n] by numerical iteration to minimize
α∫ π
π2+∆
∣
∣H0(ejω)
∣
∣
2dω + (1− α)
∫ π
0
(∣
∣T (ejω)∣
∣− 1)2
dω
α → balance between H0(z) being lowpass and T (ejω) ≈ 1
Johnston filter(M = 11):
h0[n] M=11
0 1 2 3-60
-40
-20
0H
0H
1
ω0 1 2 3
-0.04
-0.02
0
0.02
0.04
ω
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
Choose P0(z) and P1(z) to be allpass IIR filters:
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
Choose P0(z) and P1(z) to be allpass IIR filters:H0,1(z) =
12
(
P0(z2)± z−1P1(z
2))
, G0,1(z) = ±2H0,1(z)
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
Choose P0(z) and P1(z) to be allpass IIR filters:H0,1(z) =
12
(
P0(z2)± z−1P1(z
2))
, G0,1(z) = ±2H0,1(z)
A(z) = 0⇒ No aliasing
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
Choose P0(z) and P1(z) to be allpass IIR filters:H0,1(z) =
12
(
P0(z2)± z−1P1(z
2))
, G0,1(z) = ±2H0,1(z)
A(z) = 0⇒ No aliasingT (z) = H2
0 −H21 = . . . = z−1P0(z
2)P1(z2)
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
Choose P0(z) and P1(z) to be allpass IIR filters:H0,1(z) =
12
(
P0(z2)± z−1P1(z
2))
, G0,1(z) = ±2H0,1(z)
A(z) = 0⇒ No aliasingT (z) = H2
0 −H21 = . . . = z−1P0(z
2)P1(z2) is an allpass filter.
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
Choose P0(z) and P1(z) to be allpass IIR filters:H0,1(z) =
12
(
P0(z2)± z−1P1(z
2))
, G0,1(z) = ±2H0,1(z)
A(z) = 0⇒ No aliasingT (z) = H2
0 −H21 = . . . = z−1P0(z
2)P1(z2) is an allpass filter.
H0(z) and H1(z) are power complementary :∣
∣H0(ejω)
∣
∣
2+∣
∣H1(ejω)
∣
∣
2= H0(e
jω)H0(e−jω)+H1(e
jω)H1(e−jω)
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
Choose P0(z) and P1(z) to be allpass IIR filters:H0,1(z) =
12
(
P0(z2)± z−1P1(z
2))
, G0,1(z) = ±2H0,1(z)
A(z) = 0⇒ No aliasingT (z) = H2
0 −H21 = . . . = z−1P0(z
2)P1(z2) is an allpass filter.
H0(z) and H1(z) are power complementary :∣
∣H0(ejω)
∣
∣
2+∣
∣H1(ejω)
∣
∣
2= H0(e
jω)H0(e−jω)+H1(e
jω)H1(e−jω)
= . . . = 12
∣
∣P0(ejω)
∣
∣
2+ 1
2
∣
∣P1(ejω)
∣
∣
2= 1
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
Choose P0(z) and P1(z) to be allpass IIR filters:H0,1(z) =
12
(
P0(z2)± z−1P1(z
2))
, G0,1(z) = ±2H0,1(z)
A(z) = 0⇒ No aliasingT (z) = H2
0 −H21 = . . . = z−1P0(z
2)P1(z2) is an allpass filter.
H0(z) and H1(z) are power complementary :∣
∣H0(ejω)
∣
∣
2+∣
∣H1(ejω)
∣
∣
2= H0(e
jω)H0(e−jω)+H1(e
jω)H1(e−jω)
= . . . = 12
∣
∣P0(ejω)
∣
∣
2+ 1
2
∣
∣P1(ejω)
∣
∣
2= 1
H0(z) can be made a Butterworth or Elliptic filter with MH = 4MP + 1:
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
Choose P0(z) and P1(z) to be allpass IIR filters:H0,1(z) =
12
(
P0(z2)± z−1P1(z
2))
, G0,1(z) = ±2H0,1(z)
A(z) = 0⇒ No aliasingT (z) = H2
0 −H21 = . . . = z−1P0(z
2)P1(z2) is an allpass filter.
H0(z) and H1(z) are power complementary :∣
∣H0(ejω)
∣
∣
2+∣
∣H1(ejω)
∣
∣
2= H0(e
jω)H0(e−jω)+H1(e
jω)H1(e−jω)
= . . . = 12
∣
∣P0(ejω)
∣
∣
2+ 1
2
∣
∣P1(ejω)
∣
∣
2= 1
H0(z) can be made a Butterworth or Elliptic filter with MH = 4MP + 1:
0 1 2 3
-8
-6
-4
-2
0
P0(z2)
z-1P1(z2)
MP=1
A0=1+0.236z-1
A1=1+0.715z-1
ω
∠
Phase cancellation: ∠z−1P1 = ∠P0 + π
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
Choose P0(z) and P1(z) to be allpass IIR filters:H0,1(z) =
12
(
P0(z2)± z−1P1(z
2))
, G0,1(z) = ±2H0,1(z)
A(z) = 0⇒ No aliasingT (z) = H2
0 −H21 = . . . = z−1P0(z
2)P1(z2) is an allpass filter.
H0(z) and H1(z) are power complementary :∣
∣H0(ejω)
∣
∣
2+∣
∣H1(ejω)
∣
∣
2= H0(e
jω)H0(e−jω)+H1(e
jω)H1(e−jω)
= . . . = 12
∣
∣P0(ejω)
∣
∣
2+ 1
2
∣
∣P1(ejω)
∣
∣
2= 1
H0(z) can be made a Butterworth or Elliptic filter with MH = 4MP + 1:
0 1 2 3
-8
-6
-4
-2
0
P0(z2)
z-1P1(z2)
MP=1
A0=1+0.236z-1
A1=1+0.715z-1
ω
∠
0 1 2 3-60
-40
-20
0H
0H
1M
H=5
ω
Phase cancellation: ∠z−1P1 = ∠P0 + π
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
Choose P0(z) and P1(z) to be allpass IIR filters:H0,1(z) =
12
(
P0(z2)± z−1P1(z
2))
, G0,1(z) = ±2H0,1(z)
A(z) = 0⇒ No aliasingT (z) = H2
0 −H21 = . . . = z−1P0(z
2)P1(z2) is an allpass filter.
H0(z) and H1(z) are power complementary :∣
∣H0(ejω)
∣
∣
2+∣
∣H1(ejω)
∣
∣
2= H0(e
jω)H0(e−jω)+H1(e
jω)H1(e−jω)
= . . . = 12
∣
∣P0(ejω)
∣
∣
2+ 1
2
∣
∣P1(ejω)
∣
∣
2= 1
H0(z) can be made a Butterworth or Elliptic filter with MH = 4MP + 1:
0 1 2 3
-8
-6
-4
-2
0
P0(z2)
z-1P1(z2)
MP=1
A0=1+0.236z-1
A1=1+0.715z-1
ω
∠
0 1 2 3-60
-40
-20
0H
0H
1M
H=5
ω
Phase cancellation: ∠z−1P1 = ∠P0 + π ; Ripples in H0 and H1 cancel.
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
Choose P0(z) and P1(z) to be allpass IIR filters:H0,1(z) =
12
(
P0(z2)± z−1P1(z
2))
, G0,1(z) = ±2H0,1(z)
A(z) = 0⇒ No aliasingT (z) = H2
0 −H21 = . . . = z−1P0(z
2)P1(z2) is an allpass filter.
H0(z) and H1(z) are power complementary :∣
∣H0(ejω)
∣
∣
2+∣
∣H1(ejω)
∣
∣
2= H0(e
jω)H0(e−jω)+H1(e
jω)H1(e−jω)
= . . . = 12
∣
∣P0(ejω)
∣
∣
2+ 1
2
∣
∣P1(ejω)
∣
∣
2= 1
H0(z) can be made a Butterworth or Elliptic filter with MH = 4MP + 1:
0 1 2 3
-8
-6
-4
-2
0
P0(z2)
z-1P1(z2)
MP=1
A0=1+0.236z-1
A1=1+0.715z-1
ω
∠
0 1 2 3-60
-40
-20
0H
0H
1M
H=5
ω
H0(z)
Phase cancellation: ∠z−1P1 = ∠P0 + π ; Ripples in H0 and H1 cancel.
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
Choose P0(z) and P1(z) to be allpass IIR filters:H0,1(z) =
12
(
P0(z2)± z−1P1(z
2))
, G0,1(z) = ±2H0,1(z)
A(z) = 0⇒ No aliasingT (z) = H2
0 −H21 = . . . = z−1P0(z
2)P1(z2) is an allpass filter.
H0(z) and H1(z) are power complementary :∣
∣H0(ejω)
∣
∣
2+∣
∣H1(ejω)
∣
∣
2= H0(e
jω)H0(e−jω)+H1(e
jω)H1(e−jω)
= . . . = 12
∣
∣P0(ejω)
∣
∣
2+ 1
2
∣
∣P1(ejω)
∣
∣
2= 1
H0(z) can be made a Butterworth or Elliptic filter with MH = 4MP + 1:
0 1 2 3
-8
-6
-4
-2
0
P0(z2)
z-1P1(z2)
MP=1
A0=1+0.236z-1
A1=1+0.715z-1
ω
∠
0 1 2 3-60
-40
-20
0H
0H
1M
H=5
ω
H0(z)
0 1 2 3
5
10
15
T(z)
ω (rad/sample)
Phase cancellation: ∠z−1P1 = ∠P0 + π ; Ripples in H0 and H1 cancel.
IIR Allpass QMF
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 9 / 12
|T (z)| = 1
Choose P0(z) and P1(z) to be allpass IIR filters:H0,1(z) =
12
(
P0(z2)± z−1P1(z
2))
, G0,1(z) = ±2H0,1(z)
A(z) = 0⇒ No aliasingT (z) = H2
0 −H21 = . . . = z−1P0(z
2)P1(z2) is an allpass filter.
H0(z) and H1(z) are power complementary :∣
∣H0(ejω)
∣
∣
2+∣
∣H1(ejω)
∣
∣
2= H0(e
jω)H0(e−jω)+H1(e
jω)H1(e−jω)
= . . . = 12
∣
∣P0(ejω)
∣
∣
2+ 1
2
∣
∣P1(ejω)
∣
∣
2= 1
H0(z) can be made a Butterworth or Elliptic filter with MH = 4MP + 1:
0 1 2 3
-8
-6
-4
-2
0
P0(z2)
z-1P1(z2)
MP=1
A0=1+0.236z-1
A1=1+0.715z-1
ω
∠
0 1 2 3-60
-40
-20
0H
0H
1M
H=5
ω
H0(z)
0 1 2 3
5
10
15
T(z)
ω (rad/sample)
Phase cancellation: ∠z−1P1 = ∠P0 + π ; Ripples in H0 and H1 cancel.
Tree-structured filterbanks
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 10 / 12
A half-band filterbank divides the full band into two equal halves.
0 1 2 30
0.5
1
X=Y
0 1 2 3
0 1 2 3
0 1 2 3
Tree-structured filterbanks
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 10 / 12
A half-band filterbank divides the full band into two equal halves.
0 1 2 30
0.5
1
X=Y
0 1 2 30
0.5
1
V1
U1
0 1 2 3
0 1 2 3
Tree-structured filterbanks
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 10 / 12
A half-band filterbank divides the full band into two equal halves.
You can repeat the process on either or both of the signals u1[p]and v1[p]. 0 1 2 3
0
0.5
1
X=Y
0 1 2 30
0.5
1
V1
U1
0 1 2 3
0 1 2 3
Tree-structured filterbanks
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 10 / 12
A half-band filterbank divides the full band into two equal halves.
You can repeat the process on either or both of the signals u1[p]and v1[p]. 0 1 2 3
0
0.5
1
X=Y
0 1 2 30
0.5
1
V1
U1
0 1 2 30
0.5
1
V1
V2
U2
0 1 2 3
Tree-structured filterbanks
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 10 / 12
A half-band filterbank divides the full band into two equal halves.
You can repeat the process on either or both of the signals u1[p]and v1[p].
Dividing the lower band in half repeatedly results in an octaveband filterbank .
0 1 2 30
0.5
1
X=Y
0 1 2 30
0.5
1
V1
U1
0 1 2 30
0.5
1
V1
V2
U2
0 1 2 3
Tree-structured filterbanks
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 10 / 12
A half-band filterbank divides the full band into two equal halves.
You can repeat the process on either or both of the signals u1[p]and v1[p].
Dividing the lower band in half repeatedly results in an octaveband filterbank .
0 1 2 30
0.5
1
X=Y
0 1 2 30
0.5
1
V1
U1
0 1 2 30
0.5
1
V1
V2
U2
0 1 2 30
0.5
1
V1
V2
V3
U3
Tree-structured filterbanks
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 10 / 12
A half-band filterbank divides the full band into two equal halves.
You can repeat the process on either or both of the signals u1[p]and v1[p].
Dividing the lower band in half repeatedly results in an octaveband filterbank . Each subband occupies one octave (= a factorof 2 in frequency).
0 1 2 30
0.5
1
X=Y
0 1 2 30
0.5
1
V1
U1
0 1 2 30
0.5
1
V1
V2
U2
0 1 2 30
0.5
1
V1
V2
V3
U3
Tree-structured filterbanks
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 10 / 12
A half-band filterbank divides the full band into two equal halves.
You can repeat the process on either or both of the signals u1[p]and v1[p].
Dividing the lower band in half repeatedly results in an octaveband filterbank . Each subband occupies one octave (= a factorof 2 in frequency).
The properties “perfect reconstruction” and “allpass” arepreserved by the iteration.
0 1 2 30
0.5
1
X=Y
0 1 2 30
0.5
1
V1
U1
0 1 2 30
0.5
1
V1
V2
U2
0 1 2 30
0.5
1
V1
V2
V3
U3
Summary
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 11 / 12
• Half-band filterbank:◦ Reconstructed output is T (z)X(z) +A(z)X(−z)◦ Unwanted alias term is A(z)X(−z)
Summary
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 11 / 12
• Half-band filterbank:◦ Reconstructed output is T (z)X(z) +A(z)X(−z)◦ Unwanted alias term is A(z)X(−z)
• Perfect reconstruction: imposes strong constraints on analysisfilters Hi(z) and synthesis filters Gi(z).
Summary
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 11 / 12
• Half-band filterbank:◦ Reconstructed output is T (z)X(z) +A(z)X(−z)◦ Unwanted alias term is A(z)X(−z)
• Perfect reconstruction: imposes strong constraints on analysisfilters Hi(z) and synthesis filters Gi(z).
• Quadrature Mirror Filterbank (QMF) adds an additional symmetryconstraint H1(z) = H0(−z).◦ Perfect reconstruction now impossible except for trivial case.◦ Neat polyphase implementation with A(z) = 0◦ Johnston filters: Linear phase with T (z) ≈ 1◦ Allpass filters: Elliptic or Butterworth with |T (z)| = 1
Summary
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 11 / 12
• Half-band filterbank:◦ Reconstructed output is T (z)X(z) +A(z)X(−z)◦ Unwanted alias term is A(z)X(−z)
• Perfect reconstruction: imposes strong constraints on analysisfilters Hi(z) and synthesis filters Gi(z).
• Quadrature Mirror Filterbank (QMF) adds an additional symmetryconstraint H1(z) = H0(−z).◦ Perfect reconstruction now impossible except for trivial case.◦ Neat polyphase implementation with A(z) = 0◦ Johnston filters: Linear phase with T (z) ≈ 1◦ Allpass filters: Elliptic or Butterworth with |T (z)| = 1
• Can iterate to form a tree structure with equal or unequalbandwidths.
Summary
15: Subband Processing
• Subband processing
• 2-band Filterbank
• Perfect Reconstruction• Quadrature MirrorFilterbank (QMF)
• Polyphase QMF
• QMF Options
• Linear Phase QMF
• IIR Allpass QMF
• Tree-structured filterbanks
• Summary
• Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 11 / 12
• Half-band filterbank:◦ Reconstructed output is T (z)X(z) +A(z)X(−z)◦ Unwanted alias term is A(z)X(−z)
• Perfect reconstruction: imposes strong constraints on analysisfilters Hi(z) and synthesis filters Gi(z).
• Quadrature Mirror Filterbank (QMF) adds an additional symmetryconstraint H1(z) = H0(−z).◦ Perfect reconstruction now impossible except for trivial case.◦ Neat polyphase implementation with A(z) = 0◦ Johnston filters: Linear phase with T (z) ≈ 1◦ Allpass filters: Elliptic or Butterworth with |T (z)| = 1
• Can iterate to form a tree structure with equal or unequalbandwidths.
See Mitra chapter 14 (which also includes some perfect reconstructiondesigns).
Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 12 / 12