(New page: == Linearity == A system is said to be linear if it satisfies the properties of scaling and superposition. Thus, the following holds true for all linear systems: :Suppose there are two in...)
 
Line 12: Line 12:
 
:A linear system must satisfy the condition
 
:A linear system must satisfy the condition
 
::<math>\,ay1(t) + by2(t) = C\left\{ax1(t) + bx2(t)\right\}</math>
 
::<math>\,ay1(t) + by2(t) = C\left\{ax1(t) + bx2(t)\right\}</math>
 +
 +
== Example of a Linear System ==
 +
::<math>\,x1(t) = sin(t)</math>
 +
::<math>\,x2(t) = cos(t)</math>
 +
 +
::<math>\,y1(t) = C\left\{x1(t)\right\} = C(sin(t))</math>
 +
::<math>\,y2(t) = C\left\{x2(t)\right\} = C(cos(t))</math>
 +
 +
::<math>\,ay1(t) + by2(t) = a*C*sin(t) + b*C*cos(t) = C\left\{asin(t) + bcos(t)\right\} = C\left\{ax1(t) + bx2(t)\right\}</math>

Revision as of 04:44, 9 September 2008

Linearity

A system is said to be linear if it satisfies the properties of scaling and superposition. Thus, the following holds true for all linear systems:

Suppose there are two inputs
$ \,x1(t) $
$ \,x2(t) $
with outputs
$ \,y1(t) = C\left\{x1(t)\right\} $
$ \,y2(t) = C\left\{x2(t)\right\} $
A linear system must satisfy the condition
$ \,ay1(t) + by2(t) = C\left\{ax1(t) + bx2(t)\right\} $

Example of a Linear System

$ \,x1(t) = sin(t) $
$ \,x2(t) = cos(t) $
$ \,y1(t) = C\left\{x1(t)\right\} = C(sin(t)) $
$ \,y2(t) = C\left\{x2(t)\right\} = C(cos(t)) $
$ \,ay1(t) + by2(t) = a*C*sin(t) + b*C*cos(t) = C\left\{asin(t) + bcos(t)\right\} = C\left\{ax1(t) + bx2(t)\right\} $

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva