Page 1
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
Page 2
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
Page 3
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
Page 4
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
Page 5
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
Page 6
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
Page 7
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
Page 8
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
Page 9
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
Page 10
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)
Page 11
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
Page 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 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
Page 13
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
Page 14
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)
Page 15
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]
Page 16
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]
Page 17
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]
Page 18
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)}
Page 19
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)
]
Page 20
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)
]
Page 21
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)
]
Page 22
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
Page 23
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
Page 24
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
]
Page 25
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
]
Page 26
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
]
Page 27
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)
]
Page 28
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
Page 29
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)
]
Page 30
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)
]
Page 31
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
Page 32
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
Page 33
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
Page 34
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)
Page 35
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)
Page 36
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)}
Page 37
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
Page 38
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)}
Page 39
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)
Page 40
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
Page 41
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)
Page 42
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)
Page 43
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)
Page 44
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)
Page 45
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)
Page 46
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)
Page 47
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
Page 48
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
Page 49
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
Page 50
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
Page 51
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
Page 52
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)
Page 53
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)
Page 54
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)
Page 55
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.
Page 56
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
Page 57
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
Page 58
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
Page 59
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
Page 60
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)
∣
∣
Page 61
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)
Page 62
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
Page 63
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)
Page 64
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 /
Page 65
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
Page 66
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
Page 67
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
Page 68
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
Page 69
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
Page 70
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
Page 71
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
Page 72
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
Page 73
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:
Page 74
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)
Page 75
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
Page 76
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)
Page 77
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.
Page 78
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)
Page 79
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
Page 80
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:
Page 81
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
Page 82
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
Page 83
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.
Page 84
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.
Page 85
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.
Page 86
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.
Page 87
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
Page 88
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
Page 89
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
Page 90
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
Page 91
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
Page 92
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
Page 93
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
Page 94
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
Page 95
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)
Page 96
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).
Page 97
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
Page 98
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.
Page 99
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).
Page 100
Merry Xmas
DSP and Digital Filters (2015-5583) Subband Processing: 15 – 12 / 12