(10 intermediate revisions by the same user not shown)
Line 1: Line 1:
 
== Definition ==
 
== Definition ==
  
A system is called 'time invariant' if for any input signal x(t) and for any time to that is a real number, the response to the shifted input x(t-To) is the shifted output y(t-To).   
+
A system is called 'time invariant' if for any input signal x(t) and for any time to that is a real number, the response to the shifted input x(t-T) is the shifted output y(t-T).   
  
 
This is saying that for order for a signal to be considered 'time invariant' i must be able to put any signal through the system that has gone through a time shift, and i should get out another signal with the same time shift.
 
This is saying that for order for a signal to be considered 'time invariant' i must be able to put any signal through the system that has gone through a time shift, and i should get out another signal with the same time shift.
  
Another way to look at time invariance is that if I had a signal x(t) and i put i through a time delay of To, then through the system, I should get the same output if i put x(t) through the system first, and then shifted the output function of the system by To.
+
Another way to look at time invariance is that if I had a signal x(t) and i put i through a time delay of T, then through the system, I should get the same output if i put x(t) through the system first, and then shifted the output function of the system by T.
  
 
== Example of Time Invariant System ==
 
== Example of Time Invariant System ==
  
x(t) \to
+
Input signal x(t) and output which equals 3+2*x(t-T)
  
 +
* Send through system first then time shift
 +
** x(t) <math>\to</math> (system) y(t) = 3+2*x(t) <math>\to</math> (time shift by T) z(t) = y(t-T) = 3+2*x(t-T)
 +
 +
* Time shift first, then send through system
 +
** x(t) <math>\to</math> (time shift by T) y(t) = x(t-T) <math>\to</math> (system) w(t) = 3+2*y(t) = 3+2*x(t-T)
 +
 +
*Since the two outputs are equal in this case, then it is safe to say that the system is time invariant.
  
 
== Example of Time Variant System ==
 
== Example of Time Variant System ==
 +
 +
Input signal x(t) with a system of y(t) = x(2t) and time shift of T
 +
 +
*Send through system first and then time shift
 +
** x(t) <math>\to</math> (system) y(t) = x(2t) <math>\to</math> (Time shift by T) z(t) = y(t-T) = x(2(t-T)) = x(2t-2T)
 +
 +
*Time shift first, then send through system
 +
** x(t) <math>\to</math> (Time shift by T) y(t) = x(t-T) <math>\to</math> (system) w(t) = y(2t) = x(2t-T)
 +
 +
*Since the two outputs of z(t) and w(t) are NOT EQUAL, then the system is considered to be time variant.

Latest revision as of 18:42, 10 September 2008

Definition

A system is called 'time invariant' if for any input signal x(t) and for any time to that is a real number, the response to the shifted input x(t-T) is the shifted output y(t-T).

This is saying that for order for a signal to be considered 'time invariant' i must be able to put any signal through the system that has gone through a time shift, and i should get out another signal with the same time shift.

Another way to look at time invariance is that if I had a signal x(t) and i put i through a time delay of T, then through the system, I should get the same output if i put x(t) through the system first, and then shifted the output function of the system by T.

Example of Time Invariant System

Input signal x(t) and output which equals 3+2*x(t-T)

  • Send through system first then time shift
    • x(t) $ \to $ (system) y(t) = 3+2*x(t) $ \to $ (time shift by T) z(t) = y(t-T) = 3+2*x(t-T)
  • Time shift first, then send through system
    • x(t) $ \to $ (time shift by T) y(t) = x(t-T) $ \to $ (system) w(t) = 3+2*y(t) = 3+2*x(t-T)
  • Since the two outputs are equal in this case, then it is safe to say that the system is time invariant.

Example of Time Variant System

Input signal x(t) with a system of y(t) = x(2t) and time shift of T

  • Send through system first and then time shift
    • x(t) $ \to $ (system) y(t) = x(2t) $ \to $ (Time shift by T) z(t) = y(t-T) = x(2(t-T)) = x(2t-2T)
  • Time shift first, then send through system
    • x(t) $ \to $ (Time shift by T) y(t) = x(t-T) $ \to $ (system) w(t) = y(2t) = x(2t-T)
  • Since the two outputs of z(t) and w(t) are NOT EQUAL, then the system is considered to be time variant.

Alumni Liaison

Sees the importance of signal filtering in medical imaging

Dhruv Lamba, BSEE2010