bhaskar rao fred harris - university of california, san...

Bhaskar Rao

fred harris

ECE-251D Filter banks and Wavelets

28-October 2014

˃ Sample Rate

+ By embedding Sample Rate Changes in the filter we can form

Linear Time Varying (LTV) or Periodically Time Varying (PTV) filters.

+ In an LTV or PTV filter, we can apply an input signal at one frequency and the response can occur at another frequency.

+ In the sampled data world we call this Aliasing (traveling under a different name).

+ Aliasing is your friend!

˃ As an Aside

+ Most of us have limited experience

with LTV systems.

+ Actually we all share one common

LTV system experience, a swing!

+ Every child on the playground masters

the two modes of swing excitation.

2

.

Rotate Body Mass in

Negative Direction

when

Angular Velocity

is Zero and

Angular

Acceleration

is Negative

Additive

Growth Per Cycle

“How to Pump a Swing”, Stephen Wirkus, Richard Rand, and Andy Ruina,

The College Mathematics Journal, Vol. 29, No. 4, September 1998

Rotate Body Mass in

Positive Direction

when

Angular Velocity

is Zero and

Angular

Acceleration

is Positive

3 3

3

4

M-to-1

fsfs fs/M

M-to-1Filter

H(Z)

x(n) y(n) y(nM)N

Filter Length N Proportional to

1/f

Resample Ratio M Controlled by f

4

fs

fsfs/M

0

0

2BWf

2BW f

f/2f/2

fS> 2BW

Nyquist

fS= 2BW+f

harris

( )

22

2 ; (Protect from Aliasing When Down Sampling)

S

S

f A dBN

f

fBW

M

Filter Length at Input Sample Rate :

2 -Sided Alias Free BW as Fraction of Output Sample Rate :

Aliased Transition BW, Δf as Fraction of Ou

(1 ) ; (Allowable Aliasing When Down Sampling)

( ) 1 ( )

(1 ) / 22 (1 ) / 22

1 ( ) (Op;

(1 ) 22

S

S

S

ff

M

f A dB A dBN

f M M

N A dB N

M

tput Sample Rate :

Substitute in Filter Length at Input Sample Rate :

Dividing both sides by M

s/Output) (Ops)

(Input/Output) (Input)N

M M

N-TapLowpass Filter

M-to-1

f f fs s s

M

x(n) y(n) y(nM)

f /Ms -f /Ms 0

f=(1-f /Ms

2BW = f /M s

f

........

5

N and M are Coupled and Their Ratio has Interesting Units

Reduce BW, Reduce Sample

Rate

Fraction of Spectrum allocated to Aliased Transition BW

5

f

-80 dB

2-Sided BW40 kHz

Transition BW40 kHz

-2 MHz 2 MHz20kHz

60kHz

0

-6dB/Oct

0.1 dB Ripple

-90 dB

6 Nyquist Rate for Filter is 40 kHz+40 kHz = 80 kHz or fs/50

Can Perform 50-to-1 Down Sample and Still Satisfy Nyquist

400 TapFIR Filter

4-MHZ InputSample Rate

4-MHZ Output Sample Rate

400 TapFIR Filter


80-kHZ Output Sample Rate

50-to-1 Ratio

6

0 20-20-40 40kHz kHzkHz kHz

f

Transition Band

Aliased in Transition

BW

fS=2f0+f

2-Sided BW + Transition BW Folding Frequency

Folding Frequency

8

Low Pass Filter

Low PassDerivative Filter

Derivative Operator

x(n)

x(n)

y(n) y(n).

y(t).

h(n)

h(n)

ddn

ddn

9

Low Pass Filter

Re-sampled Low Pass Filter

x(n)

x(n)

y(n) y(nM)

y(nM)

h(n)

h(nM)

M:1

M:1

Extract Delays To First

Non-Zero Coefficient

For M-to-1 resample start at Index r and Increment by M

For 3-to-1 resample start at index r and increment by 3

C0

C0

C0

C3

C3

C3

C4

C4

C4

C5

C5

C5

C6

C6

C6

C7

C7

C7

C8

C8

C8

C9

C9

C9

C10

C10

C10

C11

C11

C11

C1

C1

C1

C2

C2

C2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

This mapping from 1-D to 2-D is used by Cooley-Tukey FFT.

Polyphase Filters and CT-FFT are kissing cousins! 10

H( Z )

x(n) y(n) y(nM)M:1

Z

Z

Z

Z

-1

-2

-(M-2)

-(M-1)

H ( Z )0

M

H ( Z )1

M

H ( Z )2

M

H ( Z )M-2

M

H ( Z )M-1

M

....

.... ....

x(n) y(n) y(nM)M:1

H( Z )

x(n) y(n) y(nM)M:11

0

( ) ( )N

n

n

H Z h n Z

1 1( )

0 0

( ) ( )M N

r nM

r n

H Z h r nM Z

1 1

0 0

( ) ( )M N

r nM

r n

H Z Z h r nM Z

M-Path Partition Supports M-to-1 Down Sample

11

21 1

0 0

( ) ( )M Nj k

r nMM

r n

G Z e Z h r nM Z

1 1

0 0

( ) ( ) ( ) ( ) ( )k k k

N Nj n j jn n

n n

G Z h n e Z h n e Z H e Z

1 1( ) ( )

0 0

( ) ( ) k

M Nj r nM r nM

r n

G Z h r nM e Z

1 1

0 0

( ) ( )k k

M Nj j nMr nM

r n

G Z e Z h r nM e Z

k

k

M θ = k 2π

2πor θ = k

M

Modulation Theorem of Z-Transform

12

Z

Z

Z

Z

-1

-2

-(M-2)

-(M-1)

H ( Z )0

M

H ( Z )1

M

H ( Z )2

M

H ( Z )M-2

M

H ( Z )M-1

M

....

.... ....

x(n) y(n) y(nM)M:1e

e

e

e

e

j k0

j k1

j k2

j k(M-2)

j k(M-1)

2

2

2

2

2

M

M

M

M

M

13

M Delays

1 Delay

M Delays

1 Delay

Input Clock, T

Output Clock, MT

Z Z-M -1

M:1 M:1

M-Units of Delay at Input Rate Same as 1-Unit of Delay at Output Rate

H(Z ) H(Z )-M -1

M:1 M:1

14

Z

Z

Z

Z

-1

-2

-(M-2)

-(M-1)

H ( Z )0

H ( Z )1

H ( Z )2

H ( Z )M-2

H ( Z )M-1

....

.... ....

x(n) y(nM,k)

M:1

M:1

M:1

M:1

M:1

e

e

e

e

e

j k0

j k1

j k2

j k(M-2)

j k(M-1)

2

2

2

2

2

M

M

M

M

M

We Reduce Sample Rate

M-to-1 Prior to Reducing Bandwidth

(Nyquist is Raising His Eyebrows!)

We Intentionally Alias the Spectrum.

(Were you Paying Attention when

they discussed the importance of

anti-aliasing filters?)

M-fold Aliasing!

M-Unknowns!

M-Paths Supply M-Equations

We can the Separate Aliases!

15

Z

Z

Z

Z

-1

-2

-(M-2)

-(M-1)

H ( Z )0

H ( Z )1

H ( Z )2

H ( Z )M-2

H ( Z )M-1

....

.... ....

x(n) y(nM,k)

M:1

M:1

M:1

M:1

M:1

e

e

e

e

e

j k0

j k1

j k2

j k(M-2)

j k(M-1)

2

2

2

2

2

M

M

M

M

M

Want Phase Spinners as far away from Resampler as Possible

16

H ( Z )0

H ( Z )1

H ( Z )2

H ( Z )M-2

H ( Z )M-1

.... ....

x(n) y(nM,k)e

e

e

e

e

j k0

j k1

j k2

j k(M-2)

j k(M-1)

2

2

2

2

2

M

M

M

M

M

Note: By moving the Phase Spinner

To the Filter Output We don’t

Select Desired Center Frequency

Till after Down Sampling

And Path Processing

This Means that

The Processing for every Channel

is the same till the Phase Spinner

No longer LTI, Filter now has

M-Different Impulse Responses!

Now LTV or PTV Filter.

17

18

....

4-MHz 80-kHz

50 PathPolyphase Filter


80-kHZ Output Sample Rate

50-to-1

8 taps

400 Weights

4-MHz 80-kHz8-Tap

Filter

50-PathCoefficient Bank

Select Path Weights

400 Weights

N/M = 400/50 = 8-ops/Input

18

19

....

....

4-MHz

4-MHz80-kHZ

8 taps 8 taps





80-kHZ Internal Sample Rate

50-to-1 1-to-50

400 Weights 400 Weights

400-Tap Filter Implemented with

8 Ops/Input

400-Tap Interpolator Implemented with

8 Ops/Output

19

400-Tap Filter Implemented

with 16 Ops/In-Out

f

Transition BW, 40 kHz

Transition BW, 4 kHz

-2 MHz 2 MHz 20 kHz 60 kHz

400 Tap Filter

4000 Tap Filter

....

....

4-MHz

4-MHz





4-MHZ OutputSample Rate

4-MHZ OutputSample Rate

80-kHZ Internal Rate

80-kHZ 80-kHZ

50-to-1 1-to-50

8 taps

80 taps

8 taps

400 Weights 400 Weights

H2 Filter

H2

H1 Filter

f = 4 kHz, f = 4000 kHzS

N = 80 Taps2

f = 4 kHz, f = 80 kHzS

N = 4000 Taps1

1

50Clock Rate

4000 tap Filter Implemented with

18 Ops/Input

20

H (Z)0H (Z)0

H (Z)1 H (Z)1

H (Z)2H (Z)2

H (Z)(M-1)H (Z)(M-1)

....

....

x(n,k )1

y(n,0)

y(nM,0)

f ff

0 0 0

21

f ff

k1 k1

H (Z)0H (Z)0

H (Z)1 H (Z)1

H (Z)2H (Z)2

H (Z)(M-1)H (Z)(M-1)

e e

e e

e e

e e

j j

j j

j j

j j

0k1 0k1

1k1 1k1

2k1 2k1

(M-1)2k1 (M-1)2k1

2 2

2 2

2 2

2 2

M M

M M

M M

M M....

....

x(n,k )1

y(n,k )1

y(nM,k )1

22

H (Z)0H (Z)0

H (Z)1 H (Z)1

H (Z)2H (Z)2

H (Z)(M-1)H (Z)(M-1)

e e

e e

e e

e e

j j

j j

j j

j j

0k1 0k2

1k1 1k2

2k1 2k2

(M-1)k1 (M-1)k2

2 2

2 2

2 2

2 2

M M

M M

M M

M M....

....

x(n,k )1

y(n,k )2

y(nM,k )1

f ff

k1 k2

2

3

f ff

f

k1 k1k2 k2

H (Z)0 H (Z)0

H (Z)1 H (Z)1

H (Z)2 H (Z)2

H (Z)(M-1) H (Z)(M-1)

ej0k2

2M

ej0k1

2M e

j0k12M

ej0k2

2M

....

....

x(n,k )1

y(n,k )+y(n,k )1 2

y(nM,k )1

y(nM,k )2

24

We have a method to reduce workload when the bandwidth is a small fraction of the sample rate. That won’t help us when we have a bandwidth that is large fraction of the sample rate. Possibly worse, the sample rate is extremely high. Hmm??? The solution: We simply change the problem that we can’t solve to one we can and solve that one instead. (The Star Trek Kobayashi Scenario, Thinking Out of the Box)). We partition the bandwidth into multiple, parallel, reduced bandwidth segments so that each derives benefit from the down-sampling, up-sampling process.

f

f

f

f

f

f

Input Bandwidth

Output Filter51 Multiplies

Traditional 600 Taps I 600 Multipliesnput Filter

Channelizer Filter 600 Taps

Channelized Bandwidth

Enabled ChannelsDisabled Channels Disabled Channels

8.5% of Workload91.5% Reduction

Sum Over

Row Output

Time Samples

nM-th Output

Time Sample

From r-th Row

Phase Rotator

On r-th Row,

Selecting Alias

From k-th

Nyquist Zone nM-th

Time Sample

From K-th

Nyquist Zone

Recognize Sum as Scaled

IFFT of Row Vector of Time Samples at Time nM

2

7

21

0

( , ) ( )M j rk

Mr

r

y nM k y nM e

M-Channel Analysis Channelizer: M-Narrowband Filter Spans

Centered on Multiples of Output Sample Rate

Aliased to Baseband with Unique Phase Profiles

Separated by Phase Rotators in M-Point IFFT

Hmm... this is verygood stuff....

28

M-Pnt IFFT

Alias Multiple Narrow Band Signals

from Different Center Frequencies

to Baseband By Down Sampling

Alias Multiple Narrow Band

Baseband Signals to Different

Center Frequencies By Up Sampling

29

H ( Z )0

H ( Z )1

H ( Z )2

H ( Z )M-2

H ( Z )M-1

....

....

....

x(n)

x(nM+0)

x(nM+1)

x(nM+2)

x(nM+M-2)

x(nM+M-1)

y(n,0)

y(n,k)

y(n,1)

y(n,2)

y(n,M-1)

y(n,M-2)

M-Point IFFT

H ( Z )0

H ( Z )1

H ( Z )2

H ( Z )M-2

H ( Z )M-1

....

....

....

x(n)

y(nM,0)

y(nM,k)

y(nM,1)

y(nM,2)

y(nM,M-1)

y(nM,M-2)

M-Point IFFT

Center frequencies, hence channel spacing, and the number of paths in filter partition defined by length M of IFFT. Channel bandwidth and spectral characteristics, in-band ripple, out-of band attenuation, and transition BW defined by prototype low-pass filter in polyphase partition. Channelizer output sample rate determined by input commutator span of P inputs per M- point IFFT output. Three parameters are independent and adjustable.

30

f

f

Crossover BW

Channel BWTransition BW

Transition BW

Channel Spacing

Channel Spacing

Channelizer for High Quality Spectrum Analyzer

Channelizer for High Quality FDM Receiver

31

32

The Winner!

f =fs/MBW

f =fs/MBW

f =fsSMPL

f =fs/MSMPL

f =fs/MBW

f =2(fs/M)SMPL

Band Edge Alias

f

f

f

Filter Spectral Shapeat Input Sample rate

Filter Spectral Shapeat Output Sample RateMaximally Decimated

Filter Spectral Shape at Output Sample RateNon-Maximally Decimated

f f

BW BW

Sample Rate=2 Channel Spacing Sample Rate =Channel SpacingChannel Spacing

Channel Spacing

AliasedTransition

Bandwidth

Filter Sampled at Rate to Avoid Band Edge Aliasing

34

Prototype Low-Pass Filter for 120 Channel Channelizer

35

0.5T T 1.5T 2T

1.5T

T

T

0.5T

0.5T

0mT

(m+1)T

fs

2 fs/M M Path PolyPhaseFilter in Z

M-PNT IFFT

...

..

flg=0

flg=0

flg=

1

flg=

1

Circular Buffer

Phase Continuity

2

Perform M/2-to-1 Resampling in M-Path Filter

Insert Circular Buffer Between Polyphase Filter and IFFT

to Align Input Data Shifting Origin with IFFT’s Origin

3

6

0 0

1 1

2 2

3

M/2-1

M-1

M/2

M/2+1

M-2

M-1

M MM

....

......

..

….

M-P

ath

Po

lyp

ha

se F

ilte

r

M-P

oin

t IF

FT

FDM

TDM

M-P

ath

Input D

ata

Buffer

Circ

ula

r Outp

ut B

uffer

State Engine

M-Path Analysis Filter Bank with M/2 to 1 Down Sampling

Non Maximally Decimated Perfect Reconstruction Filter Bank

3

7

00

11

22

3

M/2-1

M-1

M/2

M/2+1

M-2

M-1

MM M

....

......

..

….

M

-Pa

th P

oly

pha

se F

ilte

r

M-P

oin

t IF

FT

FDM

TDM

M-P

ath

Inp

ut D

ata

Buffer

Circ

ula

r O

utp

ut Buffer

State Engine

M-Path Synthesis Filter Bank with 1 to M/2 Up Sampling Non Maximally Decimated Perfect Reconstruction Filter Bank

3

8

....

.... .... .... ....

....

....

MPoint IFFT

MPoint IFFT

MPathFilter

MPathFilter

nM/2 Point Shift M-PointCircular Buffer


xx

xx

Filtering in the Cascade Channelizer Domain

Looks Like Fast Convolution. It Isn’t!

Fast Convolution Occurs in Frequency Domain,

Alters Spectrum and Returns to Time Domain.

Channelizer Never Leaves Time Domain! 3

9

Impulse Response and Frequency Response

25-Enabled Ports: 5.0 MHz Bandwidth

Sample Rate 24 MHz.

IFFT Size

120 Points.

FFT Bin Width 200 kHz.

Bandwidth Resolution

Multiples of 200 kHz.

4

0

Impulse Response and Frequency Response

39-Enabled Ports: 7.8 MHz Bandwidth

vary_bw_channelizer_120_3

Sample Rate 24 MHz.

IFFT Size

120 Points.

FFT Bin Width 200 kHz.

Bandwidth Resolution Multiples of 200 kHz.

4

1

0 5 10 15 20 25 30-100

-80

-60

-40

-20

0

Spectrum: 600-Tap, 60 Channel Analysis and Synthesis Filter, 29-Enabled Bins, 51 Real Multiplies per Input Sample and 600 Tap Filter With Same Spectrum

Frequency (in Channel Number)

Lo

g M

ag

nitu

de

(d

B)

Red 600 Tap filter ResponseBlue Cascade 600 Tap 60-Path Filter Response

14 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 15-100

-80

-60

-40

-20

0

Spectrum: Detail of Transition Bandwidth


Lo

g M

ag

nitu

de

(d

B)

13 13.5 14 14.5 15-2

-1

0

1

2x 10

-3 Spectrum: Detail of In Band Ripple and Band Edge


Lo

g M

ag

nitu

de

(d

B)

Implement 600 Tap Filter with 60 Path, 600 Tap Cascade Analysis and Synthesis Filter

42

0

0

51.2

51.2

-51.2

-51.2

102.4

47.5-47.548.0-48.0

-102.4

f (MHz)

f (MHz)

Channel-BChannel-A

0.1 dB In-Band Ripple

80.0 dB Out-of-Band Attenuation

1401 Tap Filter

Filter Specifications

Signal Specifications; Complex Input Samples, Independent Upper and Lower 95 MHZ Channels

43

Filters for Next Generation *DOCSIS Standard

*Data Over Cable Service Interface Specification

0 10 20 30 40 50 60 70 80 90 100-100

-80

-60

-40

-20

0

Spectrum: 1401-Tap Filter

Frequency (MHz

Lo

g M

ag

(d

B)

47 47.5 48 48.5-100

-80

-60

-40

-20

0

Spectrum: Detail of Transition Bandwidth

Frequency (MHz)

Lo

g M

ag

(d

B)

45 45.5 46 46.5 47 47.5 48-0.1

-0.05

0

0.05

0.1Spectrum: Detail of In Band Ripple and Band Edge

Frequency (MHz)

Lo

g M

ag

(d

B)

Complex Input Signal Require 2-Filters for Upper Band

2800 ops/Input

44

35 40 45 50 55 60-100

-80

-60

-40

-20

0

Spectrum Direct Implementation (Red), Synthesized Implementation (Blue), Outer Tier Channel (Dashed Black), Inner Tier Channel (Black)

Frequency (MHz)

Lo

g M

ag

(d

B)

Outer Tier Filter 30

Inner TierFilter

35 40 45 50-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Spectrum, Zoom to Pass Band Ripple

Frequency (MHz)

Lo

g M

ag

(d

B)

1401 Tap Direct Filter Response, (5600 Mults for Complex Input)

60-Path Ripple 10-Path Ripple

ChannelResponse

104 Mult

44 45 46 47 48 49 50 51-100

-80

-60

-40

-20

0

Spectrum, Zoom to Transition BW

Frequency (MHz)

Lo

g M

ag

(d

B)

Outer Tier Filter 30 Inner

TierFilter

60-Path Outer 10-Path Inner

45

....

.... .... .... ....

....

.... MPoint IFFT

MPoint IFFT

MPathFilter

MPathFilter



xx

xx

Frequency Domain Filtering With Cascade

M/2-to-1 Analysis and 1-to-M/2 Synthesis Channelizers

600 Tap Prototype Filter

60 Path Polyphase Partition

10 Coefficients per Path

Input Filter:

20 Operations Per Input Sample

60 Point Good-Thomas Nested Winograd FFT

165 Real Multiples

Amortized over 30 Input Samples:

5.5 Operations per Input Sample

For Input and Output Polyphase Filter and IFFT:

51 Operations per Input-Output Sample Pair

Workload: 8.5% of 600 Coefficient Tapped Delay Line Filter

With same Frequency Response! 46

520 530 540 550 560 570 580 590 600 610 620

-0.1

0

0.1

0.2

0.3

0.4

0.5

Impulse Response, 60 Path, 600 Tap, Cascade Analysis and Synthesis Filter Bank Filter

Time Index

Am

plitu

de

250 260 270 280 290 300 310 320 330 340 350

-0.1

0

0.1

0.2

0.3

0.4

0.5

Impulse Response, 600 Tap Direct Implementation Filter

Time Index

Am

plitu

de

51 Multiplies per Input Sample

600 Multiplies per Input Sample

4

7

......

......

......

..

..

....

......

......

MPoint FFT

PPoint FFT

MPoint IFFT

MPoint IFFT

MPathFilter

PPathFilter

MPathFilter

MPathFilter

M-PointCircular Buffer



P-PointCircular Buffer

x

x

x

x

x

x

Change Bandwidth with Binary Mask, Change Sample Rate with Transform Size

4

8

49

..

........

..

P1Point IFFT

P2Point IFFT

MPoint IFFT

P1PathFilter

P2PathFilter

MPathFilter


P1-PointCircular Buffer

P2-PointCircular Buffer

Reassembled Wide Band Channels from Short Synthesis Channelizers

....

.... .... ....

....

....

MPoint IFFT

MPoint IFFT

MPathFilter

MPathFilter



Cascade M/2-to-1 Analysis and 1-to-M/2 Synthesis Channelizers

Frequency Domain Filtering and Spectral Shuffle

52

Professor harris, may I be excused?

My brain is full.54

Is Open For Questions

55

bhaskar rao fred harris - university of california, san...

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