Introduction to continuous time LTI systems
Continuous LTI system stands for Linear Time invariant system. LTI systems theory plays a key role in designing most of dynamic system. Most of the practical systems of interest can be modeled as linear time in variant systems or at least approximations of them around nominal operating point because of its(theory of LTI systems) simplicity in dealing complex real systems. The analysis of LTI systems is equipped with well developed theory rich in its content.
Definition of continuous time LTI systems
A continuous time LTI system is one which deals with continuous time signals and satisfies both the principles of linearity and time invariance. The principles of linearity and time invariance can be stated as follows
Principle of Linearity
Consider a linear system S characterized by the transformation operator T[]. Let x_{1}, x_{2} are the inputs applied to it and y_{1}, y_{2} are the outputs. For a system to be linear it has to satisfy both the principles of homogeneity and superposition. The following equations hold for a linear system
y_{1 }= T[x_{1}], y_{2} = T[x_{2}]
Principle of homogeneity: T [a*x_{1}] = a*y_{1}, T [b*x_{2}] = =b*y_{2}
Principle of superposition: T [x_{1}] + T [x_{2}] = a*y_{1}+b*y_{2}
Linearity: T [a*x_{1}] + T [b*x_{2}] = a*y_{1}+b*y_{2}
Where a, b are constants.
Principle of Time invariance
The principle of time variance states that the behavior of the system should not change with time. In a time invariant system if input is delayed by time t_{0} the output will also gets delayed by t_{0}. Mathematically it is specified as follows
y(t-t_{0}) = T[x(t-t_{0})]
Examples of Linear time invariant systems
Consider a system characterized by following input output relationship y(t) = A*x(t)(signal amplifier output in linear range). Now we will check whether this system is linear time invariant or not
linearity:
Homogeneity
If an input of a*x(t) is applied to this system the output = A*a*x(t) which is equal to a*A*x(t)= a*y(t). Hence this system satisfies homogeneity principle.
Superposition
let x_{1}(t) , x_{2}(t) be two distinct inputs applied to this system then
y_{1}(t)=A*x_{1}(t),y_{2}(t)=A*x_{2}(t)
if we apply a input to equal to sum of individual inputs a*x_{1}(t), b*x_{2}(t)
x'(t) = a*x_{1}(t)+b*x_{2}(t) then
output y'(t) = A*a*x_{1}(t)+A*b*x_{2}(t)
y'(t) =a*y_{1}(t)+b*y_{2}(t)
hence the system satisfies superposition principle.
Since the system satisfies both linearity and superposition the system is a linear system.
Time invariance:
If input is delayed by t_{o} i.e. input = x(t-t_{o}) the
output = A*x(t-t_{o}) (equation 1)
If output is delayed by t_{o} i.e. y(t-t_{o})(replace t by t_{o} in the RHS and LHS of the equation
y(t-to) =A* x(t-t_{o}) (equation 2)
From equation 1 and 2 it is clear that this system is time invariant.
Summing up the properties of linearity and time invariance the system characterized by output y(t)=A*x(t) is a linear time invariant system.
Whether this system is LTI system or not
Y(t) = sin(x(t))
linearity:
Homogeneity
If an input of a*x(t) is applied to this system the output = A*sin(a*x(t)) which is not equal to a*y(t) = a*sin(x(t)). Hence this system does not satisfy homogeneity principle.
Superposition
let x_{1}(t) , x_{2}(t) be two distinct inputs applied to this system then
y_{1}(t)=sin(x_{1}(t)),y_{2}(t)=sin(x_{2}(t))
if we apply an input to equal to sum of individual inputs a*x_{1}(t), b*x_{2}(t)
x'(t) = a*x_{1}(t)+b*x_{2}(t) then
output y'(t) =sin(a*x_{1}(t))+sin(b*x_{2}(t)) ≠ a*y1(t)+b*y2(t)
hence the system does not satisfy superposition principle.
Since the system does not satisfy linearity and superposition the system is a Non-linear system.
Time invariance:
If input is delayed by to i.e. input = x(t-to) the
output =sin(x(t-to)) (equation 1)
If output is delayed by to i.e. y(t-to)(replace t by to in the RHS and LHS of the equation
y(t-to) = sin(x(t-to)) (equation 2)
From equation 1 and 2 it is clear that this system is time invariant.
Summing up the properties of non linearity and time invariance the system characterized by output y(t)=sin(x(t)) is a not a linear time invariant system. It is a non linear time invariant system.
A Linear time invariant system in time domain can be described by differential equations of the form
Where x(t) is input to the system, y(t) is output of the system, a_{k} and b_{k} are constant coefficients independent of time.
Example: A RC circuit shown in the figure is characterized by differential equation
dy(t)/dt+R*C*y(t)=x(t)/(R*C)
Where y(t) is output voltage, x(t) is input voltage.
The solution of the above equation is given as
if the constant is set to zero you can easily verify that the system is linear time invariant system.
Note: If a_{k} and b_{k} are functions of time then the system is linear time variant system.
S.NO. System input output relationship LTI system/Not a LTI system | |
1. y(t) = dx/dt Linear Time Invariant system | |
2. y(t) = t*x(t) linear time variant system | |
3. y(t) = t^{2}*x^{2}(t) non linear time variant system | |
4. dy/dt +t*y(t) = x(t) linear time variant system | |
5. y(t) = ln(x(t)) Non linear time invariant system | |
6. y(t) = x(t)+constant Non linear time invariant system |