Famous convolution signals

* From X1 (rectangle) convolution with itself -2 2 The result will be a triangle (X3) X1 -1 1 2 From The ending point of the X1(first) signal + the ending point of the X1 (second) signal * X1 -1 1 2 - 1 - 1 The starting point of the X1(first) signal + the starting point of the X1 (second) signal 1 + 1 X3 -2 2

* h = 8 X1 (rectangle) convolution with itself OR convolution with another rectangle of the same width X1 -1 1 2 Then we get the height from Area of X1 multiplied by Area of X1 equals Area of X3 A1 x A2 = A3 * X1 -1 1 2 2 x 2=4 2 x 2=4 0.5 x 4 x h multiply h X3 h = 8 -2 2

* From h = 16 X1 (rectangle) convolution with X2 (rectangle) -3 3 The result will be a trapezoid (X3) X1 -1 1 2 From The ending of the X1(first) signal + the ending of the X2 (second) signal * X2 -2 2 3 - 1 - 2 The starting of the X1(first) signal + the starting of the X2 (second) signal X3 1 + 2 -3 3 h = 16

* Then h = 16 X1 (rectangle) convolution with X2 (rectangle) -3 -1 1 3 We want to get the region of the constant part in the signal The starting of the X1(first) signal + the ending of the X2 (second) signal * X2 -2 2 3 1- 2 The ending of the X1(first) signal + the starting of the X2 (second) signal X3 - 1 + 2 -3 -1 1 3 h = 16

* h = 12 h = 16 X1 (rectangle) convolution with X2 (rectangle) h -3 -1 Then we get the height from Area of X1 multiplied by Area of X2 equals Area of X3 A1 x A2 = A3 * X2 -2 2 3 2 x 2=4 4 x 3=12 0.5 x h x (6+2) multiply h X3 h = 12 -3 -1 1 3 h = 16

* From h = 16 Another example X1 (rectangle) convolution with X2 (rectangle) The result will be a trapezoid (X3) X1 -3 3 2^1/2 From The ending of the X1(first) signal + the ending of the X2 (second) signal * 2^1/2 X2 -2 2 - 3 - 2 The starting of the X1(first) signal + the starting of the X2 (second) signal X3 3 + 2 -5 5 h = 16

* Then h = 16 Another example X1 (rectangle) convolution with X2 (rectangle) X1 -3 3 2^1/2 Then We want to get the region of the constant part in the signal 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 1 5 h = 16

Another example X1 (rectangle) convolution with X2 (rectangle) X1 -3 3 2^1/2 Then we get the height from Area of X1 multiplied by Area of X2 equals Area of X3 A1 x A2 = A3 * X2 2^1/2 6x(2^0.5) 4x(2^0.5) 0.5 x h x (10+2) multiply -2 2 h X3 h = 8 -5 -1 1 5 h = 16

h = 16 Area of trapezoid: A = 0.5 x h x (a + b) Or A = 2 x (0.5 x c x h + d x h) h -5 -1 5 1 c d a b h = 16

* Note that it could be in the opposite way, the trapezoid is given and you need to simplify it to two rectangles. -1 1 -7 7 8 1.5^1/2 1.5^1/2 -4 4 -3 3 *