Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
dsp - dsp - dsp - dsp
Typology: Lecture notes
1 / 30
This page cannot be seen from the preview
Don't miss anything!
1
filters with either a common input or a
summed output
below
2
bank are known as analysis filtersanalysis filters
decompose the input signal x [ n ] into a set of
subband signals subband signals with each subband
signal occupying a portion of the original
frequency band
H ( z ) k
v [ n ] k
4
bank are known as synthesis filterssynthesis filters
a set of subband signalssubband signals (typically
belonging to contiguous frequency bands)
into one signal y [ n ] at its output
F ( z ) k
v [ n ] k
^
5
banks with equal passband widths is outlined
next
filter with a real impulse response :
without any loss of generality
0
H z
n
n H ( z ) h [ n ] z 0 0
0
h n
0
H z
7
whose impulse response is given by
where we have used the notation
H ( z ) k
h [ n ] k
0
2 /
0
kn
M
j kn M
k
h n h n e h n W
j M
M
W e
2 /
0 k M 1
n
n k
n M
n
k k
H z h n z h n zW
0 k M 1
8
by
obtained by shifting the response of to
the right by an amount 2 k / M
0
k
k (^) M
H z H zW 0 k M 1
( 2 / )
0
j j k M
k
H e H e
0 k M 1
H ( z ) k
0
H z
10
general complex, and hence does
not necessarily exhibit symmetry with
respect to = 0
previous slide can be seen to be uniformly
shifted version of the response of the basic
prototype filter
h [ n ] k
j
k
H e
0
H z
11
could be used as the analysis filters in the
analysis filter bank or as the synthesis filters
in the synthesis filter bank
filters are uniformly shifted version of that
of the prototype filter, the filter bank
obtained is called a uniform filter bankuniform filter bank
0
k
k (^) M
H z H zW 0 k M 1
13
we arrive at the M -band polyphase
decomposition of :
the identity
H ( z ) 0
k
M
zW
1
0
M (^) kM
M
k M
k M
H z z W E z W
( ) ( )
H ( z ) k
1
0
z W E z k M
M (^) k M
^ M
( ),
kM
M
14
written in matrix form as
( ) ( ) [
M k
M
k
M
k
k M
H z W W W
2 1 1
( )
( )
( )
( )
( )
M
M
M
M
M
M
z E z
z E z
z E z
E z
1
1
2
2
1
1
0
0 k M 1
. . .
16
uniform analysis filter bank, more
commonly known as the uniform DFTuniform DFT
analysis filter bank analysis filter bank , is then as shown below
17
uniform DFT filter bank is much smaller than
that of a direct implementation as shown
below
19
derive the structure for a uniform DFTuniform DFT
synthesis filter bank synthesis filter bank as shown below
Type I uniform DFT Type II uniform DFT
synthesis filter bank synthesis filter bank
20
determine the polyphase components of an
IIR transfer function
( )
( )
( )
( )
( )
M
M
M
M
M
M
z E z
z E z
z E z
E z
1
1
2
2
1
1
0
M
1
( )
( )
( )
( )
H z
H z
H z
H z
M 1
2
1
0
D
. . .
. . .
M
i
E z
H ( z ) 0
