Convolution Ex

X1 (rectangle) convolution with itself 2

The result will be a triangle (X3)

X1

-1

The ending point of the X1(first) signal + the ending point of the X1 (second) signal

1

*

2

X1

-1-1 The starting point of the X1(first) signal + the starting point of the X1 (second) signal

-1

1

1+1 X3

-2

0

2

X1 (rectangle) convolution with itself OR convolution with another rectangle of the same width 2

Then we get the height from Area of X1 multiplied by Area of X1 equals Area of X3

-1

A1 x A2 = A3 2 x 2=4

2 x 2=4

X1

1

*

2

0.5 x 4 x h

-1

X1

1

multiply

h X3

-2

0

2

X1 (rectangle) convolution with X2 (rectangle) 2

The result will be a trapezoid (X3)

X1

-1

The ending of the X1(first) signal + the ending of the X2 (second) signal

1

*

3 X2

-1-2 -2

2

The starting of the X1(first) signal + the starting of the X2 (second) signal X3

1+2

-3

0

3

X1 (rectangle) convolution with X2 (rectangle) 2

We want to get the region of the constant part in the signal

X1

-1

The starting of the X1(first) signal + the ending of the X2 (second) signal

1

*

3 X2

1- 2 -2

2

The ending of the X1(first) signal + the starting of the X2 (second) signal X3

-1+2

-3

-1

0

1

3

X1 (rectangle) convolution with X2 (rectangle) 2

X1

Then we get the height from Area of X1 multiplied by Area of X2 equals Area of X3

-1

1

*

A1 x A2 = A3

3 X2

2 x 2=4

4 x 3=12

0.5 x h x (6+2)

-2

2

multiply

h X3

-3

-1

0

1

3

Another example X1 (rectangle) convolution with X2 (rectangle) X1

The result will be a trapezoid (X3)

The ending of the X1(first) signal + the ending of the X2 (second) signal

2^1/2

-3

3

*

X2

2^1/2

-3-2 -2

2

The starting of the X1(first) signal + the starting of the X2 (second) signal X3

3+2

-5

0

5

Another example X1 (rectangle) convolution with X2 (rectangle) X1

We want to get the region of the constant part in the signal

2^1/2

-3

3

The starting of the X1(first) signal + the ending of the X2 (second) signal

*

X2

2^1/2

3-2 -2

2

The ending of the X1(first) signal + the starting of the X2 (second) signal X3

-3+2

-5

-1

0

1

5

Another example X1 (rectangle) convolution with X2 (rectangle) X1

Then we get the height from Area of X1 multiplied by Area of X2 equals Area of X3

-3

3

*

A1 x A2 = A3 6x(2^0.5)

4x(2^0.5)

2^1/2

X2

2^1/2

0.5 x h x (10+2)

multiply

-2

2

h X3

-5

-1

0

1

5

Area of trapezoid:

A = 0.5 x h x (a + b)

h

Or A = 2 x (0.5 x c x h + d x h)

-5

-1 c

0

d a

b

1

5

• Note that it could be in the opposite way, the trapezoid is given and you need to simplify it to two rectangles. 8

-7

-1

0

1

1.5^1/2

1.5^1/2

-4

4

7

*

-3

3