Biosignal Processing - Introduction to Biomedical Engineering - page 701

Input Output

x ð t Þ! y ð t Þ

1 1

x ð t Þ! y ð t Þ ,

2 2

we can easily determine the output of a linear system to any arbitrary combination of these
inputs. More generally, a linear superposition and scaling of the input signals produces a
linear superposition and scaling of the output signals

Input Output

ð 11 : 27 Þ

x ð t Þþ k x ð t Þ! k y ð t Þþ k y ð t Þ

k

1 1 2 2 1 1 2 2

where k 2 are arbitrary amplitude scaling constants. These constants scale the input

amplitudes by making them larger (

k 1 and

1). This produces a comparable
change in the net outputs, which are likewise scaled by the same constants.

k > 1) or smaller ( k <

EXAMPLE PROBLEM 11.16
The following information is given for a linear system

Input Output

x 1 ð t Þ¼ cos ð t Þ ! y 1 ð t Þ¼ cos ð t þ p= 2 Þ

x 2 ð t Þ¼ cos ð t Þþ sin ð 2 t Þ! y 2 ð t Þ¼ cos ð t þ p = 2 Þþ 5 sin ð 2 t Þ

x 3 ð t Þ¼ cos ð 3 t Þ ! y 3 ð t Þ¼ 2 cos ð 3 t Þ

Find the output if the input is: x ð t Þ¼ 3 sin ð 2 t Þþ 1 = 2 cos ð 3 t Þ

Solution
The input is represented as a superposition of x 1 , x 2 , and x 3

x 3 ð t Þ
Applying the superposition and scaling properties produces an output

x ð t Þ¼ 3 ð x 2 ð t Þ x 1 ð t Þ Þ þ 1 = 2 x 3 ð t Þ¼ 3 x 2 ð t Þ 3 x 1 ð t Þþ 1 = 2

y ð t Þ¼ 3 y 2 ð t Þ 3 y 1 ð t Þþ 1 = 2 y 3 ð t Þ¼ 3 cos ð ð t þ p= 2 Þþ 5 sin ð 2 t Þ Þ 3 cos ð ð t þ p= 2 Þ Þ þ 1 = 2 2 cos ð ð 3 t Þ Þ

¼ 15 sin ð 2 t Þþ cos ð 3 t Þ

EXAMPLE PROBLEM 11.17
Consider the system given by the expression

y ð t Þ¼ fx ð t fg¼ A x ð t Þþ B :

Determine if this is a linear system.

Solution
To solve this problem, consider a superposition of two separate inputs, x 1 ( t ) and x 2 ( t ), that

independently produce outputs y 1 ( t ) and y 2 ( t ). Apply the input x ð t Þ¼ k 1 x 1 ð t Þþ k 2 x 2 ð t Þ . If the

system is linear, the output obeys

Continued