Discrete linear time invariant(LTI) system

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 x1, x2 are the inputs applied to it and y1, y2 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= T[x1(n)], y2 = T[x2(n)]        

Principle of homogeneity: T [a*x1(n)] = a*y1(n), T [b*x2(n)] = =b*y2(n)

Principle of superposition: T [x1(n)] + T [x2(n)] = a*y1(n)+b*y2(n)

                                Linearity: T [a*x1(n)] + T [b*x2(n)] = a*y1(n)+b*y2(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 t0 the output will also gets delayed by t0. Mathematically it is specified as follows

                                                            y(n-n0) = T[x(n-n0)]

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 x1(n) , x2(n) be two distinct inputs applied to this system then

         y1(t)=A*x1(n),y2(t)=A*x2(n)

if we apply a input to equal to sum of individual inputs a*x1(n), b*x2(n) 

                                                                 x'(n) = a*x1(n)+b*x2(n) then

                                                          output y'(n) = A*a*x1(n)+A*b*x2(n)

                                                                        y'(n) =a*y1(n)+b*y2(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 no(no*Ts) i.e. input = x(n-no) the

        output = A*x(n-no) (equation 1)

If output is delayed by no i.e. y(n-no)(replace n by n-no in the RHS and LHS of the equation

y(n-no) =A* x(n-no) (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 x1(n) , x2(n) be two distinct inputs applied to this system then

y1(n)=sin(x1(n)),y2(t)=sin(x2(n))

if we apply an input to equal to sum of individual inputs a*x1(n), b*x2(n) 

                                        x'(n) = a*x1(n)+b*x2(n) then

                           output y'(n) =sin(a*x1(n))+sin(b*x2(n)) ≠ a*y1(n)+b*y2(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-no) the

                                                             output =sin(x(n-no)) (equation 1)

If output is delayed by to i.e. y(n-no)(replace t by to in the RHS and LHS of the equation

                                       y(n-no) = sin(x(n-no)) (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

                                     \sum_{k=0}^{N}a_{k}*y(n-k) = \sum_{k=0}^{N}b_{p}*x(n-k)

Where x(n) is input to the system, y(n) is output of the system, ak and bk 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

y(n)=\sum_{k=-\infty }^{n}x(k) you can easily verify that the system is linear time invariant system.

Note: If ak and bk are functions of discrete time n*ts then the system is linear time variant system.

S.NO.        System input output relationship             LTI system/Not a LTI system  
1.                      y(n) = an*x(n)                                 Linear Time variant system  
2.                       y(n) = n*x(n)                                  linear time variant system  
3.                        y(n) = n2*x2(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  

 

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