du broff convolution

Tau-Vee Convolution

An alternative to the “sliding function” method of convolution

Contents

• What is Convolution (Slides 3-5)

• Preliminaries (Slides 6-7)

• A Detailed Example (Slides 8-47)

• Additional Examples (Slides 48-62)

• Summary (Slides 63-77)

Convolution with Impulses

... 0 0 ...y t x h t x h t

Approximating Continuous Data

0.5

0.5

ˆk

k

x k x t dt x k

x t

t

t

h t

x

x

h t

Overlap

0 0

ˆ

k

k

y t x h t x h t

x k h t k

y t x k h t k x h t d

Disclaimer

• This presentation is free, without any restrictions, to anyone who wants to use it.

• There is no copyright on this presentation.

What you will need

• A pencil (possibly with an eraser if you make mistakes).

• A printer because you may find the presentation easier to follow if you print out a few slides.

Let’s get started

Let d where

0 1 0 0

1 1 2 0 1 and

2 2 3 0 1

0 3

Find .

y t f t g t f g t

t t

t t tf t g t

t t

t

y t

Change of variables

Define so that

0 1 0 0

1 1 2 0 1 and

2 2 3 0 1

0 3

v t

v

v vf g v

v

Breakpoints

The points where the definitions of and change

are called breakpoints. Breakpoints occur at 1,2,3

and 0,1 . As shown in the next slide, these breakpoints

divide the plane into a number of

f g v

v

v

subdomains. In the

next series of slides you will be guided through the process of

constructing a map of these subdomains. If you would like

to follow along with this part of the presentation please

begin by printing a copy of the next slide.

Print this slide

Follow Along if You Dare

Copy the next series of lines and points onto your printed slide.

Done!

So what?

You have now divided the plane into 6 diagonal bands with each band corresponding to one range of time:

Band 1: t less than -1

Band 2: t between -1 and 0

Band 3: t between 0 and 2

Band 4: t between 2 and 3

Band 5: t between 3 and 4

Band 6: t greater than 4

Band 1:Tau-Vee Method

Band 1: Sliding Function Method

Band 2:Tau-Vee Method

Band 2: Sliding Function Method

Band 3:Tau-Vee Method

Band 3: Sliding Function Method

Band 4: Tau-Vee Method

Band 4: Sliding Function Method

Band 5: Tau-Vee Method

Band 5: Sliding Function Method

Band 6:Tau-Vee Method

Band 6: Sliding Function Method

Further Comments

Question: Do these drawings have to be very carefully drawn in order to work?

Answer: Probably as long as you have the breakpoints in order it will still work. The next slide shows a pretty messy free-hand version of the diagram for the same example.

Another fine mess

This drawing was pretty crummy when I drew it. To make it worse, I spilled weak coffee on it , rode over it with a bicycle, crumpled it up, uncrumpled it, and stomped it on the ground. Only then did I scan it into this presentation. You can still see the 6 diagonal bands and should be able to figure out the integrands and integration limits for all 6 bands! It is still usable! Amazing!

I also deliberately did not use the correct scale. You can see that the distance from tau=-1 to tau=2 is almost equal to the distance between tau=2 and tau=3. That’s part of the reason why the lines of constant t are deformed into curves instead of nice straight lines at a 45 degree angle. Just try to avoid “time catastrophes”—two different constant time lines intersecting!

Can this method work for discrete convolution?

Let

1,2, 1, 3 for 0,1,2,3

0 for all other values of

2,4,6 for 5,6,7

0 for all other values of

Let m

nx n

n

ny n

n

z n x n y n x m y n m

k n m

Therefore

2,8,12,2, 18, 18 for 5,6,7,8,9,19

0 for all other values of

nz

n

Reference

The tau-vee method for discrete convolution is essentially identical to a method previously described in the following reference:

Enders A. Robinson, “The Minimum Delay Concept in System Design, Part I”, Digital Electronics, Dec. 1963.

Functions that Extend to Infinity

All functions in the previous examples were nonzero only over a finite interval of time. Does this method work for functions that start and or stop at infinity?

Example

Let

for 0 and for 1

1 for 0 1

and let 0 for all 1 while g 0 whenever t 1.

Note that the definition of extends back towards .

Find *

te tf t g t t t t

t t

f t t t

f t

f g

Summary of Tau-Vee Method

Lines of constant v

Breakpoints in are represented by

horizontal lines.

g v

Lines of constant v

Lines of constant tau

Breakpoints in are represented by

vertical lines.

f

Lines of constant tau

Corners

Corners occur at each intersection of a constant

line with a constant line. The value of at a corner

is given by

v

t

t v

Corners

Lines of Constant t

Lines of constant are diagonal lines (or curves) along which

the value of is constant. Lines of constant are drawn

through each corner. These lines then divide the plane into

a number of d

t

t v t

v

iagonal bands.

Lines of Constant t

Integration Path

Each band corresponds to a range of values of . The

convolution within a band is represented by a path on

the diagram. For the convolution of piecewise defined

functions, each time the path crosses a

t

line of constant

the definition of changes. Each time the path

crosses a line of constant the definition of changes.

v g v

f

Integration Path

Integration Path

1 2

2 2 2 1

For values of in this band:

*

t

f g f g v d

f g v d f g v d

Integration limits

When the path crosses a line of constant , the

corresponding integration limit is simply the value of

. When the path crosses a line of constant the

corresponding integration limit is found by sol

v

ving

the equation for the value of , i.e. .

In this case the integration limit will be in the form of

a number. For example:

v t t v

t

Integration Limits

Eliminating the variable, v

6

1 2

?

3 ?

2 2 2 1

6 3

*

? Depends on intersections not shown on the

previous diagram

t

t

f g f g t d

f g v d f g v d