AIKTCMultirate Digital Signal Processing

By Anjali YadavUzma Shaikh Aasiya Mundasad

INTRODUCTION

•Digital Filter Banks

•Filter Banks and Subband Processing

•Applications

•Advantages

•Decimation: decimator (Down-samplerDown-sampler)

example : u[k]: 1,2,3,4,5,6,7,8,9,…

2-fold down-sampling: 1,3,5,7,9,...

•Interpolation: expander (Up-samplerUp-sampler)

example : u[k]: 1,2,3,4,5,6,7,8,9,…

2-fold up-sampling: 1,0,2,0,3,0,4,0,5,0...

L u[0], u[N], u[2N]...u[0],u[1],u[2]...

M u[0],0,..0,u[1],0,…,0,u[2]...u[0], u[1], u[2],...

Down-sampler and up-sampler (Revisited)

Basic Sampling Rate Alteration DevicesBasic Sampling Rate Alteration Devices Up-samplerUp-sampler - Used to increase the

sampling rate by an integer factor

Down-samplerDown-sampler - Used to decrease the sampling rate by an integer factor

General `subband processing’ set-up/overview:

- signals split into frequency channels/subbands (`analysis bank’)

- per-channel/subband processing

- reconstruction (`synthesis bank’)

- multi-rate structure: down-sampling / up-sampling

Filter Banks and Subband Processing [1/6]

subband processing 3H1(z)

subband processing 3H2(z)

subband processing 3H3(z)

3

3

3

3 subband processing 3H4(z)

IN

G1(z)

G2(z)

G3(z)

G4(z)

+

OUT

Step-1: Analysis filter bank

- collection of M filters (`analysis filters’, `decimation filters’) with a

common input signal

- ideal (but non-practical) frequency responses = ideal bandpass filters

- typical frequency responses (overlapping, marginally overlapping,

non-overlapping)

2

H1(z)

H2(z)

H3(z)

H4(z)

IN2

H1 H4H3H2

H1 H4H3H2

H1 H4H3H2

2

K=4

2

Filter Banks and Subband Processing [2/6]

Step-2: Decimators (down-samplers)

- subband sampling rate reduction by factor N

- critically decimatedcritically decimated filter banks (= maximally down-sampled filter banks):

N = K (where, K = number filters/subbands)

this sounds like maximum efficiency, but aliasing problem arises!

- over-sampled filter banks (= non-critically down-sampled filter banks):

N < K

Filter Banks and Subband Processing [3/6]

H1(z)

H2(z)

H3(z)

H4(z)

IN

3

3

3

3

N=3K=4

Step-3: Subband processing

- Example :

coding (=compression) + (transmission or storage) + decoding

- Filter bank design mostly assumes subband processing has `unit

transfer function’ (output signals = input signals), i.e. mostly ignores

presence of subband processing

subband processingH1(z)

subband processingH2(z)

subband processingH3(z)

3

3

3

3 subband processingH4(z)

IN

N=3K=4

Filter Banks and Subband Processing [4/6]

Step-4: Expanders (up-samplers)

- restore original fullband sampling rate by N-fold up-sampling

(= insert N-1 zeros in between every two samples)

Filter Banks and Subband Processing [5/6]

subband processing 3H1(z)

subband processing 3H2(z)

subband processing 3H3(z)

3

3

3

3 subband processing 3H4(z)

IN

K=4 N=3 N=3

Filter Banks and Subband Processing [6/6]

Step-5: Synthesis filter bank

- collection of K filters (`synthesis filters’, `interpolation filters’) with a

`common’ (summed) output signal

- frequency responses : preferably `matched’ to frequency responses of

the analysis filters, e.g., to provide perfect reconstruction (see below)

2

G1 G4G3G2

2

G1 G4G3G22

G1 G4G3G2G1(z)

G2(z)

G3(z)

G4(z)

+

OUT

K=4

2

Aliasing versus Perfect Reconstruction

Assume subband processing does not modify subband signals

(e.g. lossless coding/decoding)

- The overall aim could be to have y[k]=u[k-d], i.e. that the output signal is

equal to the input signal up to a certain delay

- But: down-sampling introduces ALIASING, especially in maximally

decimated (but even so in non-maximally decimated) filter banks

- Question : Can y[k]=u[k-d] be achieved in the presence of aliasing?

- Answer = YES, see below: PERFECT RECONSTRUCTION banks with

synthesis bank designed to remove aliasing effects !

output=input 3H1(z)

3H2(z)

3H3(z)

3333 3H4(z)

u[k]

G1(z)

G2(z)

G3(z)

G4(z)

+y[k]=u[k-d]?

output=input

output=input

output=input