1. 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 - y = z -1 x x y z -1 DSP graphs are made up of points directed lines special symbols points = signals all the rest = signal processing systems splitter = tee connector unit delay adder identity = assignment gain

2. DSP Slide 2 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 transformations 1.Topological (move points around) 2.Commutation of filters (any two filters commute!) 3.Unification of identical signals (points) and removal of redundant branches 4.Transposition theorem

3. DSP Slide 3 Basic blocks y n = a 0 x n + a 1 x n-1 y n = x n - x n-1 Explicitly draw point only when need to store value (memory point)

4. DSP Slide 4 Basic MA blocks y n = a 0 x n + a 1 x n-1

5. DSP Slide 5 General MA we would like to build but we only have 2-input adders ! tapped delay line = FIFO

6. DSP Slide 6 General MA (cont.) Instead we can build We still have tapped delay line = FIFO (data structure) But now iteratively use basic block D (algorithm) MACs

7. DSP Slide 7 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) FIFOMACs

8. DSP Slide 8 Basic 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

9. DSP Slide 9 General AR filters There are many ways to implement the general AR Note the FIFO on outputs and iteration on basic blocks

10. DSP Slide 10 ARMA filters The straightforward implementation : Note L+M memory points Now we can demonstrate how to use graph theory to save memory

11. DSP Slide 11 ARMA filters (cont.) 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.)

12. DSP Slide 12 ARMA filters (cont.) So we can use only one point And eliminate redundant branches

13. DSP Slide 13 Allowed 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