dsp slide 1 graph theory x y y = x x y a y = a x x z y y = x and z = x xz y z = x + y z = x - y xz y...
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DSP Slide 1
Graph theoryGraph theory
x y y = x
x ya
y = a x
x
z
yy = x
and z = x
x z
yz = x + y
z = x - y
x z
y
-y = z-1 x
x yz-1
DSP graphs are made up of • points • directed lines• special symbolspoints = signalsall the rest = signal processing systems
splitter = tee connector
unit delay
adder
identity = assignment
gain
DSP Slide 2
Why is graph theory useful ?Why is graph theory useful ?
DSP graphs capture both• algorithms and• data structures
Their meaning is purely topological
Graphical mechanisms for simplifying (lowering MIPS or
memory)
Four basic transformations1. Topological (move points around)2. Commutation of filters (any two filters commute!)3. Unification of identical signals (points) and removal of redundant branches4. Transposition theorem
DSP Slide 3
Basic blocksBasic blocks
yn = a0 xn + a1 xn-1
yn = xn - xn-1
Explicitly draw point only when need to store value (memory point)
DSP Slide 5
General MAGeneral MA
we would like to build
but we only have 2-input adders !
tapped delay line = FIFO
L
l
lnln xay0
DSP Slide 6
General MA (cont.)General MA (cont.)
Instead we can build
We still have tapped delay line = FIFO (data structure)
But now iteratively use basic block D (algorithm)
L
l
lnln xay0
MACs
DSP Slide 7
General MA (cont.)General MA (cont.)
There are other ways to implement the same MA
still have same FIFO (data structure)
but now basic block is A (algorithm)
Computation is performed in reverse
There are yet other ways (based on other blocks)
L
l
lnln xay0
FIFO MACs
DSP Slide 8
Basic AR blockBasic AR block
One way to implement
Note the feedback
Whenever there is a loop, there is recursion (AR)
There are 4 basic blocks here too
1 nnn byxy
DSP Slide 9
General AR filtersGeneral AR filters
M
m
mnmnn ybxy0
There are many ways to implement the general AR
Note the FIFO on outputsand iteration on basic blocks
DSP Slide 10
ARMA filtersARMA filters
M
m
mnm
L
l
lnln ybxay01
The straightforward implementation :
Note L+M memory points
Now we can demonstrate
how to use graph theory
to save memory
DSP Slide 11
ARMA filters (cont.)ARMA filters (cont.)
M
m
mnm
L
l
lnln ybxay01
We can commute
the MA and AR filters
(any 2 filters commute)
Now that there are points representing
the same signal !
Assume that L=M (w.o.l.g.)
DSP Slide 12
ARMA filters ARMA filters (cont.)(cont.)
M
m
mnm
L
l
lnln ybxay01
So we can use only one point
And eliminate redundant branches
DSP Slide 13
Allowed transformationsAllowed transformations
1. Geometrical transformations that do no change topology
2. Commutation of any two filters
3. Unification of identical points (signals)
and elimination of redundant branches
4. Transposition theorem exchange input and output reverse all arrows replace adders with splitters replace splitters with adders
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