Introduction to Discrete LTI systems
Discrete LTI system stands for discrete Linear Time invariant system. Discrete LTI systems theory plays a key role in designing most of discrete time dynamic system. The continuous LTI system theory can be applied to Discrete LTI systems by replacing continuous time variable t by discrete time.
Definition of Discrete time LTI systems
A discrete time LTI system is one which deals with Discrete time signals and satisfies both the principles of linearity and time invariance. The principles of linearity and time invariance for a discrete time system 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}(n)], y_{2} = T[x_{2}(n)]
Principle of homogeneity: T [a*x_{1}(n)] = a*y_{1}(n), T [b*x_{2}(n)] = =b*y_{2}(n)
Principle of superposition: T [x_{1}(n)] + T [x_{2}(n)] = a*y_{1}(n)+b*y_{2}(n)
Linearity: T [a*x_{1}(n)] + T [b*x_{2}(n)] = a*y_{1}(n)+b*y_{2}(n)
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(n-n_{0}) = T[x(n-n_{0})]
Examples of Discrete Linear time invariant systems
Consider a system characterized by following input output relationship y(t) = A*x(n). Now we will check whether this system is linear time invariant or not
linearity:
Homogeneity
If an input of a*x(n) is applied to this system the output = a*A*x(n) which is equal to a*A*x(n)= a*y(n). Hence this system satisfies homogeneity principle.
Superposition
let x_{1}(n) , x_{2}(n) be two distinct inputs applied to this system then
y_{1}(t)=A*x_{1}(n),y_{2}(t)=A*x_{2}(n)
if we apply a input to equal to sum of individual inputs a*x_{1}(n), b*x_{2}(n)
x'(n) = a*x_{1}(n)+b*x_{2}(n) then
output y'(n) = A*a*x_{1}(n)+A*b*x_{2}(n)
y'(n) =a*y_{1}(n)+b*y_{2}(n)
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 n_{o}(n_{o}*T_{s}) i.e. input = x(n-n_{o}) the
output = A*x(n-n_{o}) (equation 1)
If output is delayed by n_{o} i.e. y(n-n_{o})(replace n by n-n_{o} in the RHS and LHS of the equation
y(n-n_{o}) =A* x(n-n_{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(n) is a linear time invariant system.
Whether this system is LTI system or not
Y(n) = sin(x(n))
linearity:
Homogeneity
If an input of a*x(n) is applied to this system the output = A*sin(a*x(n)) which is not equal to a*y(n) = a*sin(x(n)). Hence this system does not satisfy homogeneity principle.
Superposition
let x_{1}(n) , x_{2}(n) be two distinct inputs applied to this system then
y_{1}(n)=sin(x_{1}(n)),y_{2}(t)=sin(x_{2}(n))
if we apply an input to equal to sum of individual inputs a*x_{1}(n), b*x_{2}(n)
x'(n) = a*x_{1}(n)+b*x_{2}(n) then
output y'(n) =sin(a*x_{1}(n))+sin(b*x_{2}(n)) ≠ a*y_{1}(n)+b*y_{2}(n)
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(n-n_{o}) the
output =sin(x(n-n_{o})) (equation 1)
If output is delayed by to i.e. y(n-n_{o})(replace t by to in the RHS and LHS of the equation
y(n-n_{o}) = sin(x(n-n_{o})) (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(n)=sin(x(n)) 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(n) is input to the system, y(n) is output of the system, a_{k} and b_{k} are constant coefficients independent of time.
Example:consider a discrete time LTI system characterized by difference equation y(n) – y(n-1) = x(n) (accumulator analogous to integrator in continuous time)
Where y(n) is output voltage, x(n) is input voltage.
The solution of the above equation is given as
you can easily verify that the system is linear time invariant system.
Note: If a_{k} and b_{k} are functions of discrete time n*t_{s} then the system is linear time variant system.
S.NO. System input output relationship LTI system/Not a LTI system | |
1. y(n) = a^{n}*x(n) Linear Time variant system | |
2. y(n) = n*x(n) linear time variant system | |
3. y(n) = n^{2}*x^{2}(n) non linear time variant system | |
4. (n+1)*y(n) – y(n-1) = x(n) linear time variant system | |
5. y(n) = ln(x(n)) Non linear time invariant system | |
6. y(n) = x(n)+constant Non linear time invariant system |