subband adsp

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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


Subband processing





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


• The Hm(z) are bandpass analysis filters and divide x[n] intofrequency bands

Subband processing





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



• Subband processing often processes frequency bands independently

Subband processing





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



• Subband processing often processes frequency bands independently• The Gm(z) are synthesis filters and together reconstruct the output

Subband processing





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



• 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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas




without loss of information◦ Sample rate multiplied overall by

∑

1Pi

Subband processing





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas





∑

1Pi

∑

1Pi

= 1⇒ critically sampled : good for coding

Subband processing





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas





∑

1Pi

∑

1Pi

= 1⇒ critically sampled : good for coding∑

1Pi

> 1⇒ oversampled : more flexible

Subband processing





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas





∑

1Pi

∑

1Pi


1Pi

> 1⇒ oversampled : more flexible• Goals:

(a) good frequency selectivity in Hm(z)

Subband processing





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas





∑

1Pi

∑

1Pi


1Pi


(a) good frequency selectivity in Hm(z)(b) perfect reconstruction: y[n] = x[n− d] if no processing

Subband processing





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas





∑

1Pi

∑

1Pi


1Pi


(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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


2-band Filterbank





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


Vm(z) = Hm(z)X(z)

2-band Filterbank





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


Vm(z) = Hm(z)X(z)

Um(z) = 1K

∑K−1k=0 Vm(e

−j2πk

K z1

K )

[K = 2]

2-band Filterbank





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


For perfect reconstruction without aliasing, we require

12

[

H0(z) H1(z)H0(−z) H1(−z)

] [

G0(z)G1(z)

]

=

[

z−d

0

]






• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



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

]






• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



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

]






• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



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)

]






• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



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






• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



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)

]


H0(z)H1(−z)−H0(−z)H1(z) = cz−k , which implies[

G0(z)G1(z)

]

= 2czk−d

[

H1(−z)−H0(−z)

]






• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



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)

]


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)





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas







• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


QMF satisfies:(a) H0(z) is causal and real






• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


QMF satisfies:(a) H0(z) is causal and real(b) H1(z) = H0(−z): i.e.

∣

∣H0(ejω)

∣

∣ is reflected around ω = π2






• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



∣

∣H0(ejω)

∣


(c) G0(z) = 2H1(−z) = 2H0(z)






• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



∣

∣H0(ejω)

∣


(c) G0(z) = 2H1(−z) = 2H0(z)(d) G1(z) = −2H0(−z) = −2H1(z)






• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



∣

∣H0(ejω)

∣


(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)}






• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



∣

∣H0(ejω)

∣


(c) G0(z) = 2H1(−z) = 2H0(z)(d) G1(z) = −2H0(−z) = −2H1(z)


2 {H0(−z)G0(z) +H1(−z)G1(z)}

= 12 {2H1(z)H0(z)− 2H0(z)H1(z)} = 0






• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



∣

∣H0(ejω)

∣


(c) G0(z) = 2H1(−z) = 2H0(z)(d) G1(z) = −2H0(−z) = −2H1(z)


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)}






• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



∣

∣H0(ejω)

∣


(c) G0(z) = 2H1(−z) = 2H0(z)(d) G1(z) = −2H0(−z) = −2H1(z)


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


Polyphase QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


Polyphase decomposition:H0(z) = P0(z

2) + z−1P1(z2)

Polyphase QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



2) + z−1P1(z2)

H1(z) = H0(−z) = P0(z2)− z−1P1(z

2)

Polyphase QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



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)


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



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)


0 (z)−H21 (z) = 4z−1P0(z

2)P1(z2)


k+1−d

⇒ H0(z) has only two non-zero taps ⇒ poor freq selectivity

Polyphase QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



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)


0 (z)−H21 (z) = 4z−1P0(z

2)P1(z2)


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


QMF Options





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


A(z) = 0⇒ no alias term

QMF Options





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


A(z) = 0⇒ no alias termT (z) = H2

0 (z)−H21 (z) = H2

0 (z)−H20 (−z)

QMF Options





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


Polyphase decomposition:


0 (z)−H21 (z) = H2

0 (z)−H20 (−z)

QMF Options





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas




0 (z)−H21 (z) = H2

0 (z)−H20 (−z) = 4z−1P0(z

2)P1(z2)

QMF Options





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas




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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas




0 (z)−H21 (z) = H2

0 (z)−H20 (−z) = 4z−1P0(z

2)P1(z2)


• T (z) is Linear Phase FIR:⇒ Tradeoff:

∣

∣T (ejω)∣

∣ ≈ 1 versus H0(z) stopband attenuation

QMF Options





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas




0 (z)−H21 (z) = H2

0 (z)−H20 (−z) = 4z−1P0(z

2)P1(z2)


• 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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


Linear Phase QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


H0(z) order M , linear phase

Linear Phase QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


H0(z) order M , linear phase ⇒ H0(ejω) = ±e−jωM

2

∣

∣H0(ejω)

∣

∣

Linear Phase QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



2

∣

∣H0(ejω)

∣

∣

T (ejω) = H20 (e

jω)−H21 (e

jω) = H20 (e

jω)−H20 (−ejω)

Linear Phase QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



2

∣

∣H0(ejω)

∣

∣

T (ejω) = H20 (e

jω)−H21 (e

jω) = H20 (e


= e−jωM∣

∣H0(ejω)

∣

∣

2− e−j(ω−π)M

∣

∣H0(ej(ω−π))

∣

∣

2

Linear Phase QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



2

∣

∣H0(ejω)

∣

∣

T (ejω) = H20 (e

jω)−H21 (e

jω) = H20 (e


= e−jωM∣

∣H0(ejω)

∣

∣


∣

∣H0(ej(ω−π))

∣

∣

2

= e−jωM(

∣

∣H0(ejω)

∣

∣

2− (−1)M

∣

∣H0(ej(π−ω))

∣

∣

2)

Linear Phase QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



2

∣

∣H0(ejω)

∣

∣

T (ejω) = H20 (e

jω)−H21 (e

jω) = H20 (e


= e−jωM∣

∣H0(ejω)

∣

∣


∣

∣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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



2

∣

∣H0(ejω)

∣

∣

T (ejω) = H20 (e

jω)−H21 (e

jω) = H20 (e


= e−jωM∣

∣H0(ejω)

∣

∣


∣

∣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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



2

∣

∣H0(ejω)

∣

∣

T (ejω) = H20 (e

jω)−H21 (e

jω) = H20 (e


= e−jωM∣

∣H0(ejω)

∣

∣


∣

∣H0(ej(ω−π))

∣

∣

2

= e−jωM(

∣

∣H0(ejω)

∣

∣

2− (−1)M

∣

∣H0(ej(π−ω))

∣

∣

2)


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



2

∣

∣H0(ejω)

∣

∣

T (ejω) = H20 (e

jω)−H21 (e

jω) = H20 (e


= e−jωM∣

∣H0(ejω)

∣

∣


∣

∣H0(ej(ω−π))

∣

∣

2

= e−jωM(

∣

∣H0(ejω)

∣

∣

2− (−1)M

∣

∣H0(ej(π−ω))

∣

∣

2)


M= +1


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



2

∣

∣H0(ejω)

∣

∣

T (ejω) = H20 (e

jω)−H21 (e

jω) = H20 (e


= e−jωM∣

∣H0(ejω)

∣

∣


∣

∣H0(ej(ω−π))

∣

∣

2

= e−jωM(

∣

∣H0(ejω)

∣

∣

2− (−1)M

∣

∣H0(ej(π−ω))

∣

∣

2)


M= +1


α∫ π

π2+∆

∣

∣H0(ejω)

∣

∣

2dω + (1− α)

∫ π

0

(∣

∣T (ejω)∣

∣− 1)2

dω


Johnston filter(M = 11):

h0[n] M=11

Linear Phase QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



2

∣

∣H0(ejω)

∣

∣

T (ejω) = H20 (e

jω)−H21 (e

jω) = H20 (e


= e−jωM∣

∣H0(ejω)

∣

∣


∣

∣H0(ej(ω−π))

∣

∣

2

= e−jωM(

∣

∣H0(ejω)

∣

∣

2− (−1)M

∣

∣H0(ej(π−ω))

∣

∣

2)


M= +1


α∫ π

π2+∆

∣

∣H0(ejω)

∣

∣

2dω + (1− α)

∫ π

0

(∣

∣T (ejω)∣

∣− 1)2

dω



h0[n] M=11

0 1 2 3-60

-40

-20

0H

0H

1

ω

Linear Phase QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



2

∣

∣H0(ejω)

∣

∣

T (ejω) = H20 (e

jω)−H21 (e

jω) = H20 (e


= e−jωM∣

∣H0(ejω)

∣

∣


∣

∣H0(ej(ω−π))

∣

∣

2

= e−jωM(

∣

∣H0(ejω)

∣

∣

2− (−1)M

∣

∣H0(ej(π−ω))

∣

∣

2)


M= +1


α∫ π

π2+∆

∣

∣H0(ejω)

∣

∣

2dω + (1− α)

∫ π

0

(∣

∣T (ejω)∣

∣− 1)2

dω



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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


T (z) ≈ 1


2

∣

∣H0(ejω)

∣

∣

T (ejω) = H20 (e

jω)−H21 (e

jω) = H20 (e


= e−jωM∣

∣H0(ejω)

∣

∣


∣

∣H0(ej(ω−π))

∣

∣

2

= e−jωM(

∣

∣H0(ejω)

∣

∣

2− (−1)M

∣

∣H0(ej(π−ω))

∣

∣

2)


M= +1


α∫ π

π2+∆

∣

∣H0(ejω)

∣

∣

2dω + (1− α)

∫ π

0

(∣

∣T (ejω)∣

∣− 1)2

dω



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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


IIR Allpass QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


Choose P0(z) and P1(z) to be allpass IIR filters:

IIR Allpass QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



12

(

P0(z2)± z−1P1(z

2))

, G0,1(z) = ±2H0,1(z)

A(z) = 0⇒ No aliasing

IIR Allpass QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



12

(

P0(z2)± z−1P1(z

2))

, G0,1(z) = ±2H0,1(z)


0 −H21 = . . . = z−1P0(z

2)P1(z2) is an allpass filter.

IIR Allpass QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



12

(

P0(z2)± z−1P1(z

2))

, G0,1(z) = ±2H0,1(z)


0 −H21 = . . . = z−1P0(z


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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



12

(

P0(z2)± z−1P1(z

2))

, G0,1(z) = ±2H0,1(z)


0 −H21 = . . . = z−1P0(z



∣H0(ejω)

∣

∣

2+∣

∣H1(ejω)

∣

∣

2= H0(e


jω)H1(e−jω)

= . . . = 12

∣

∣P0(ejω)

∣

∣

2+ 1

2

∣

∣P1(ejω)

∣

∣

2= 1

IIR Allpass QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



12

(

P0(z2)± z−1P1(z

2))

, G0,1(z) = ±2H0,1(z)


0 −H21 = . . . = z−1P0(z



∣H0(ejω)

∣

∣

2+∣

∣H1(ejω)

∣

∣

2= H0(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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



12

(

P0(z2)± z−1P1(z

2))

, G0,1(z) = ±2H0,1(z)


0 −H21 = . . . = z−1P0(z



∣H0(ejω)

∣

∣

2+∣

∣H1(ejω)

∣

∣

2= H0(e


jω)H1(e−jω)

= . . . = 12

∣

∣P0(ejω)

∣

∣

2+ 1

2

∣

∣P1(ejω)

∣

∣

2= 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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



12

(

P0(z2)± z−1P1(z

2))

, G0,1(z) = ±2H0,1(z)


0 −H21 = . . . = z−1P0(z



∣H0(ejω)

∣

∣

2+∣

∣H1(ejω)

∣

∣

2= H0(e


jω)H1(e−jω)

= . . . = 12

∣

∣P0(ejω)

∣

∣

2+ 1

2

∣

∣P1(ejω)

∣

∣

2= 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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



12

(

P0(z2)± z−1P1(z

2))

, G0,1(z) = ±2H0,1(z)


0 −H21 = . . . = z−1P0(z



∣H0(ejω)

∣

∣

2+∣

∣H1(ejω)

∣

∣

2= H0(e


jω)H1(e−jω)

= . . . = 12

∣

∣P0(ejω)

∣

∣

2+ 1

2

∣

∣P1(ejω)

∣

∣

2= 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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



12

(

P0(z2)± z−1P1(z

2))

, G0,1(z) = ±2H0,1(z)


0 −H21 = . . . = z−1P0(z



∣H0(ejω)

∣

∣

2+∣

∣H1(ejω)

∣

∣

2= H0(e


jω)H1(e−jω)

= . . . = 12

∣

∣P0(ejω)

∣

∣

2+ 1

2

∣

∣P1(ejω)

∣

∣

2= 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)


IIR Allpass QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



12

(

P0(z2)± z−1P1(z

2))

, G0,1(z) = ±2H0,1(z)


0 −H21 = . . . = z−1P0(z



∣H0(ejω)

∣

∣

2+∣

∣H1(ejω)

∣

∣

2= H0(e


jω)H1(e−jω)

= . . . = 12

∣

∣P0(ejω)

∣

∣

2+ 1

2

∣

∣P1(ejω)

∣

∣

2= 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)


IIR Allpass QMF





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


|T (z)| = 1


12

(

P0(z2)± z−1P1(z

2))

, G0,1(z) = ±2H0,1(z)


0 −H21 = . . . = z−1P0(z



∣H0(ejω)

∣

∣

2+∣

∣H1(ejω)

∣

∣

2= H0(e


jω)H1(e−jω)

= . . . = 12

∣

∣P0(ejω)

∣

∣

2+ 1

2

∣

∣P1(ejω)

∣

∣

2= 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)


Tree-structured filterbanks


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




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




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




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




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





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





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





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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas


• Half-band filterbank:◦ Reconstructed output is T (z)X(z) +A(z)X(−z)◦ Unwanted alias term is A(z)X(−z)

Summary





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas



• Perfect reconstruction: imposes strong constraints on analysisfilters Hi(z) and synthesis filters Gi(z).

Summary





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas




• 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





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas





• Can iterate to form a tree structure with equal or unequalbandwidths.

Summary





• Polyphase QMF

• QMF Options


• IIR Allpass QMF


• Summary

• Merry Xmas





• Can iterate to form a tree structure with equal or unequalbandwidths.

See Mitra chapter 14 (which also includes some perfect reconstructiondesigns).

Merry Xmas


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