Date post: | 16-Aug-2015 |
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Engineering |
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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