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ACCESS IC LAB

Graduate Institute of Electronics Engineering, NTU

Multirate Processing ofMultirate Processing ofDigital Signals: FundamentalsDigital Signals: Fundamentals

VLSI Signal Processing台灣大學電機系吳安宇

ACCESS IC LAB Graduate Institute of Electronics Engineering, NTU

Outline

Introduction

Sampling Rate Conversion

Multistage Implementation

Practice Structure

Polyphase Implementation


Motivation

DefinitionMore than one sampling rate (clock) are used in a system

Module 1 Module 2

clock 1

clock 2

?


Conversion Approach

Analog approach

Digital approach (multirate DSP system)


Analog Approach

nITmtc nTTmhnxtxmy

AdvantagesSimple

Straightforward

Arbitrary sampling rate

DisadvantagesD/A & A/D converter are needed

Ideal (near perfect) lowpass filter is needed

Introduced noise and distortion


Digital Approach

Sampling rate conversionInterpolation

Increase the sampling rate

DecimationDecrease the sampling rate


Sampling TheoryIf the highest frequency component in a signal is fmax, then the signal should be sampled at the rate of at least 2fmax for the samples to describe the signal completely, i.e.,

max2 fFs

For Fs < 2fmax, alias occurs in the sampling process. Alias Distortion (aliasing)


Aliasing

fmax Fs

f

-Fs

X(f)


Interpolation by L

s

s

F

F

LT

T

1

otherwise ,0

,L

GH I

L h(m) nx mw my

sF

ss LFF sF


Interpolation by L

2 2 L/

/L 2

nx mw my

X W Y

L h(m) nx mw my

sF sF


Decimation by M

s

s

F

FM

T

T

otherwise ,0

,1MH I

h(m) M nx mw my

sF

MFF ss sF


h(m) M nx mw my

sF sF

Decimation by M

nx nw my

2 2 2 4 6 8M/


Conversion by a Rational Factor M/L

Cascade of two process

s

s

F

F

L

M

T

T

L h1(m) nx mw my

sFss F

M

LF

'

h2(m) M

ss LFF ''

Interpolation by L Decimation by M


Conversion by a Rational Factor M/L

A more efficiency implementation

L h (m) nx mw my

sF ss FM

LF

'

M

ss LFF ''

mw'

''sF

otherwise ,0

,min ,ML

LH I



I

iiLL

1

L h(m) nx mw my

sF sF

L1 h(m)L2 LI

nx my

L1

nxh1(m) L2 h2(m) L1 h1(m)

my



AdvantagesReduce the complexity

Reduce storage devices (registers)

Simplify (relax) filter design problem

Reduce the finite wordlength effect

DisadvantagesIncrease the control circuit

Difficulty in choosing I and best Lj for 1 i I


Interpolated FIR (IFIR) Approach

Nothing to do with interpolation and decimation

Conceptually similar

Suitable for narrowband FIR filter designLPF

HPF

BPF


p s

2p 2s

p s

p

s

Desired narrowband responseAssume required filter order is N.

Stretched filterRequired filter order is reduced to N/2.

zG

2zG

Interpolated version of stretched filterRequired filter order is still N/2.

Desired Undesired

zI Image suppresserRequired filter order is M.Order (N/2+M) is needed to implement!(N/2+M) << N for small M

Application: Interpolated FIR (IFIR)


Interpolated FIR (IFIR)

(a) G(z) (a) G(z2)

(a) G(z2)I(z)(b) I(z)


Interpolated FIR (IFIR)

Quantity Compared

Filter order

Number of Multipliers

Number of Adders

ConventionalMethod

233

117

233

IFIR Method

131

66

131

G(z) I(z) Total

6

4

6

268

70

137


Some Useful Operations

Duality and TranspositionA dual system is that performs a complementary operation to that of an original system, and it can be constructed form the original system through the process of transposition.

The transposition operation is one in which the direction of all branches in the network are reversed, and the roles of the input and output of the network are interchanged.


Duality and Transpositiontransposition

z-1

z-1

z-1z-1

z-1

z-1

nx nx ny ny


L

Duality and Transposition

They are not true in time-varying system, but can be treated as sampling rate reverse process.

L

M M

Mh(n) Mh(n)

Mh(n)L Mh(n) L

transposition

transposition

transposition

transposition


Practical Structure

Decimation Mh(n)

z-1

z-1

z-1

M

z-1

z-1

z-1

M

M

M

M

z-1

z-1

z-1

M

M

M

M


Practical Structure

Interpolation L h(n)

z-1

z-1

z-1

L L

z-1

z-1

z-1

z-1

z-1

z-1

L

L

L

L


Application: Polyphase FIR Filter

Polyphase decomposition

n

nznhzH

lMnhne

znezE

zEzzH

l

n

nll

M

l

Ml

l1

0

h(n) nx ny

z-1

z-1

z-1

E0(zM)

E1(zM)

EM-1(zM)

nx ny


Polyphase FIR Filter

Noble identity

E (zM) M nx ny

E (z)M nx ny

E (z) L nx ny

E (zM)L nx ny

Noble identity

Noble identity



H (z) 3 nx ny

z-1

z-1

z-1

3

z-1

z-1

h0

h1

h2

h3

h4

h5

h0

z-3

3

z-3

z-3

h3

h1

h4

h2

h5

z-1

z-1

3z-1

z-1

E0(z3)

E1(z3)

E2(z3)



z-1

z-1

E0(z3)

E1(z3)

E2(z3)

3

3

3

h0

z-3

z-3

z-3

h3

h1

h4

h2

h5

z-1

z-1

3

3

3

z-1

z-1

E0(z)

E1(z)

E2(z)

3

3

3

z-1

z-1

3

3

3

h0

z-1

z-1

z-1

h3

h1

h4

h2

h5


Structure Comparison

z-1

z-1

3

3

3

h0

z-1

z-1

z-1

h3

h1

h4

h2

h5

z-1

z-1

z-1

3

z-1

z-1

3

3

3

3

3

h0

h1

h2

h3

h4

h5

Direct implementation Polyphase implementation

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