Interconnections of LTI systems

Interconnections of the LTI systems

The results for the continuous-time system and the discrete-time system are obtained by using nearly identical approaches, therefore we derive the continuous-time results and then simply state the discrete-time results. 

Parallel connections of the LTI Systems

Let us consider two LTI ( Linear time invariant) systems with impulse responses h1(t) and h2(t) connected in parallel. The output of this connection of systems, y(t) , is the sum of the outputs of the two systems:

y(t)=y1(t)+y2(t)
    =x(t)*h1(t)+x(t)*h2(t)

We can then substitute the integral representation of each convolution:

y(t) = ∫ x(T)h1(t-T) dT +  ∫ x(T)h2(t-T) dT. 

As x(T) is a common input the two integrals are combined to obtain:

y(t)= ∫ x(T)h1(t-T) dT +  ∫ x(T)h1(t-T) dT. 
     =  ∫ x(T)h(t-T) dT 
     = x(t)*h(t). 

Where, h(t) = h1(t) + h2(t).

We identify the h(t) as the impulse response of the equivalent system representing the parallel connection of the two systems. The impulse response of the overall system represented by the two LTI systems that are interconnected in parallel is the sum of their individual impulse responses. 

Mathematically, the preceding result implies that convolution possesses the distributive property. 

x(t)*h1(t) + x(t)*h2(t) = x[n]* {h1[n] + h2[n]} 

Cascade connection of systems

Let us next consider the cascade connection of two LTI systems. 

Let z(t) be the output of the first system and therefore the input to the second system in the cascade. 

The output can therefore be expressed as:

y(t) = z(t) * h2(t),
Or y(t)= ∫ z(T)h2(t-T) dT. 

As z(T) is the output of the first system, it is expressed in the terms of the input that is x(T) as:

z(T) = x(T)*h1(T)

It can be also represented as:

∫ v(T)h1(T-v) dv.

Here, v is used as the variable of integration in the convolution integral, where after substituting the values it will come to:

y(t)= ∫ ∫ x(v)h1(T-v)h2(T-v) dv dT. 

The integral is identified as the convolution of h1(t) with h2(t), evaluated at t-v,

If we define h(t) = h1(t) * h2(t), then it will be:

h(t-v) = ∫ h1(n)h2(T-v-n) dn. 

Substituting the values,

y(t) =  ∫ x(v)h1(t-v) dv. 
= x(t)*h(t). 

Therefore, the impulse response of an equivalent system representing two LTI systems connected in cascade is the convolution of their individual impulse responses. The cascade-connection is an input-output equivalent to the single system represented by the impulse response h(t). 

Substituting z(t) = x(t)* h1(t) into the expression for y(t) and h(t)=h1(t)*h2(t) into the alternative expression for y(t) given in the above mentioned equation, it establishes the fact that convolution possesses the associative property, that is:

{x(t)*h1(t)} * h2(t) = x(t) * {h1(t) * h2(t)}. 

The second important property for the cascade connection of the LTI systems concerns the ordering of the systems. 

We write h(t) = h1(t) * h2(t) as the integral:

 h(t) = ∫ h1(T)h2(T-v) dT. 

And also perform the change of variable v=t-T  to obtain 

h(t) = ∫ h1(t-v) h2(v) dv. 
= h2(t) * h1(t) 

Therefore the convolution of h1(t) and h2(t) can be performed in either of the order. This corresponds to interchanging.the order of the LTI systems in the cascade without affecting the result. 

Since, x(t) * {h1(t) * h2(t)} = x9t) * h2(t) * h1(t). 

We can conclude that the output of a cascade combination of the LTI systems is independent of the order in which the systems are connected. 

Mathematically, it can be said that the convolution operation possesses the commutative property, and can be represented as:

h1(t) * h2(t) = h2(t) * h1(t). 

The commutative property is often used to simplify the evaluation or interpretation of the convolution integral. 

Discrete-time LTI systems and convolutions have properties that are identical to their continuous-time counterparts. 

For example, the impulse response of a cascade connected to one of the LTI systems is given by convocation of the individual impulse responses, and the output of that s=cascade combination of LTI systems is independent of the order in which the other systems are connected. 

Alongside, the discrete-time convolution is associative, therefore:

{ x[n] * h1[n] * h2[n] = x[n] * {h1[n] * h2 [n]}. 

And commutative too, as represented below:

h1[n] * h2[n] = h2[n] * h1[n]. 

Reference 

Interconnections of LTI systems