11.10 Filter Banks
1/7
What Are Filter Banks?
Often need to slice up a “wideband” signal into various “subbands”
Figure from Porat’s Book
2/7
Filter Banks Application: Cell Phone Basestation
FDMA = Frequency-Division Multiple Access
Each user (or set of users) is assigned a different band
…
f1
…User M
f2 f
1 GHz
-π
Antenna
Demod
LPE
Converter
& ADC
Filter
Bank
Demod
…
BPF
f1 to
f2
Demod
User 1
-π
…
LPE
0
0
θ
User 1
Message
…
User 1
User M
Message
…User M
π
θ
0
π θ3/7
Filter Banks & Subband Processing
Sometimes we want to:
• Split a signal into subbands using an “analysis” filter bank
• Process each subband
• Then… re-assemble subbands using a “synthesis” filter bank
u0[n]
v0[n]
Subband Processing
Analysis u1[n] Subband Processing
Filter
Bank
…
x[n]
v1[n] Synthesis
Filter
Bank
y[n]
Subband Processing
uM-1[n]
vM-1[n]
Usual Design Goal: Design so that if the subband processing does
nothing (i.e., imagine that vi[n] = ui[n]) we get:
y[n] = c x[n – l]
“Perfect Reconstruction (PR) Property”
4/7
Example: Subband Data Compression
u0[n]
Analysis
Filter
Bank
…
x[n]
Compress
& Code bit
stream
M
U
X
bit stream
Compress
& Code bit
uM-1[n]
stream
Data Link
v0[n]
bit stream
D
E
M
U
X
De-Code
Synthesis
Filter
Bank
…
Design Goal
Reduce Data Rate
While Ensuring
xr[n] ≈ x[n]
xr[n]
De-Code
vM-1[n]
5/7
Decimated Filter Banks
Each output channel of a filter bank spans only a fraction
of the input BW:
Whole digital BW = 2π
Each of M subbands has BW = 2π/M
Î Can Decimate Each Channel by M
BPF
f1 to
f2
User 1
-π
…
LPE
Converter
& ADC
LPE
0
Filter
Bank
…
Recall Cell Phone Basestation Example:
Each Channel’s Signal
Is Oversampled
Î Can Be Decimated
…User M
π
θ
0
π θ
6/7
Decimated Filter Banks (cont.)
Fig. 12.26
From Porat’s
Book
Necessary: Decimation Factor K ≤ M (M = # Channels)
If K = M, called a “Maximally Decimated Filter Bank”
Our
Focus
Maximally Decimated FB’s are the most computationally efficient
… but the filters must meet strict requirements.
Î sometimes better to use K < M
7/7