This document provides an overview of multirate digital signal processing using filter banks and subband processing. It describes: 1) How signals are split into multiple frequency bands or subbands using an analysis filter bank and then downsampled. 2) Individual subband processing such as coding/compression can then be applied. 3) The subbands are then reconstructed using upsampling and a synthesis filter bank to combine the signals back into the original sampling rate. 4) Perfect reconstruction filter banks can be designed to remove aliasing introduced during downsampling to accurately reconstruct the original signal up to a delay.

1. AIKTC
Multirate Digital Signal
Processing
By
 Anjali Yadav
Uzma Shaikh
 Aasiya Mundasad

2. INTRODUCTION
•Digital Filter Banks
•Filter Banks and Subband Processing
•Applications
•Advantages

3. •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)

4. 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

5. 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

6. 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)
π2
H1(z)
H2(z)
H3(z)
H4(z)
IN
π2
H1 H4H3H2
H1 H4H3H2
H1 H4H3H2
π2
K=4
π2
Filter Banks and Subband Processing [2/6]

7. 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

8. 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]

9. 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

10. 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)
π2
G1 G4G3G2
π2
G1 G4G3G2
π2
G1 G4G3G2G1(z)
G2(z)
G3(z)
G4(z)
+
OUT
K=4
π2

11. 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)
3
3
3
3 3H4(z)
u[k]
G1(z)
G2(z)
G3(z)
G4(z)
+
y[k]=u[k-d]?
output=input
output=input
output=input