Some background on Digital Filter Banks
Analysis Bank
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Digital Filter Banks (cont.)
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Discrete Fourier transform (DFT) as a
Filter Bank
W* below is the MxM DFT matrix with elements
conjugated. This is the IDFT matrix. Note also that
the samples are in reverse time order from top
to bottom.
From diagram: si(n) ≡ x(n-i)
(current sample)
(oldest sample)
Note: not exactly a serial-to-parallel
converter. Input fed successively to all filter bank inputs
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This is the k-th freq. sample
Not a function of z since
not a function of n.
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DFT as a Filter Bank: Determine Filters Hk(z)
What are these filters doing?
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DFT as a Filter Bank
averaging
-
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DFT as a Filter Bank (cont.)
Frequency shift factor
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DFT as a Filter Bank (cont.)
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DFT as a Filter Bank (cont.)
M-1 frequency-shifted versions
of prototype filter H0(z).
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DFT as a Filter Bank (cont.)
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This is the k-th freq. sample
xk (n) =
M
X1
x(n
i)W
ki
i=0
xk (n + M
1) =
M
X1
x(n + M
i=0
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i)W
ki
shifted by
M-1 in time
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DFT as a Filter Bank (cont.)
for k=0,1,... M-1.
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Interpretation of DFT Filter Analysis Bank
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DFT Analysis Bank
x0(n)
x(n)
z-1
x1(n)
z-1
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W*
•••
•••
z-1
xM-1(n)
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DFT Synthesis Bank
y0(n)
x0(n)
y1(n)
x1(n)
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W
•••
z-1
•••
xM-1(n)
z-1
yM-1(n)
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z-1
x̂(n)
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DFT Combined Analysis/Synthesis Bank
x0(n)
x(n)
z-1
x1(n)
z-1
6
6
6
4
xM-1(n)
y0 (n)
y1 (n)
..
.
yM
W
•• ••
••
••
•
W*
y1(n)
1 (n)
)x̂(n) =
3
2
7
6
7
⇤6
7 = WW 6
5
4
M
X1
m=0
2
yM
)x̂(n) = M x̂(n
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x(n)
x(n 1)
..
.
x(n
1 m (n
(M
(M
m)
1))
1))
yM-1(n)
2
7
6
7
6
7 = MI 6
5
4
z-1
(M
z-1
3
x(n)
x(n 1)
..
.
x(n
z-1
••
•
z-1
2
y0(n)
1))
7
7
7
5
The bandpass filters in the filter bank
have overlapping frequency responses…
But perfect reconstruction (no distortion)
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x̂(n)
Recall Synthesis Bank
The synthesis filters are related to the
analysis filters via the relationship:
Fp (z) = W
p
H0 (zW p ), p = 0, 1, . . . , M
and where, as defined previously, W ⌘ e
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j2 /M
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