Signals in Natural Domain
Chapter 4 :  Time-Domain Representation for Linear Time-Invariant Systems
 

Properties of discrete-time linear convolution and system properties

If and are sequences, then the following useful properties of the discrete time convolution can be shown to be true
1. Commutativity
2. Associativity
`
3. Distributivity over sequence addition
4. The identity sequence
5. Delay operation
6. Multiplication by a constant

Note that these properties are true only if the convolution sum (4.4) exists for every n.

If the input output relation is defined by convolution i.e. if

For a given sequence a, then the system is linear and time invariant. This can be verified using the properties of the convolution listed above. The impulse response of the systems is obviously.

In terms of LTI system, commutative property implies that we can interchange input and impulse response.

 


Fig 4.5

 

The distributive property implies that parallel interconnection of two LTI system is an LTI system with impulse response as sum of two impulse responses.

 


Fig 4.6

 

The associativity property implies that series connection of two LTI system is an LTI system. Where impulse response is convolution of individual responses. The commutativity property implies that we can interchange the order of the two system in series.

 


Fig 4.7

 

Since an LTI system is completely characterized by its impulse response, we can specify system- properties in terms of impulse response.

  1. Memoryless system: From equation (4.4) we see that an LTI system is memory less if and only if.
  2. Causality for LTI system: The output of a causal system depends only on preset and past-values of the input. In order for a system to be causal must not depend on a for. From equation (4.4) we see that for this to be true, all of the terms that multiply values of for must be zero.

    put   to get

    or
    Thus impulse response for a causal LTI system must satisfy the condition h[n] = 0 for n < 0.
    If the impulse response satisfies this condition, the system is causal. For a causal system we can write        

    or             
     We say a sequence is causal if a, for n < 0.
  3. Stability for LTI system: A system is stable if every bounded input produces a bonded output. Consider  input such that    for all n.

     

Taking absolute value

From triangle inequality for complex numbers we get

        

Using property that

Since each we get

        

If the impulse response is absolutely summable, that is

                                                                                                                                    (4.5)

then     

and is bounded for all n, and hence system is stable. Therefore equation (4.5) is sufficient condition for system to be stable. This condition is also necessary. This is prove by showing that if condition (4.5) is violated then we can find a bounded input which produces an unbounded output. Let

Let     

         

This is a bounded sequence

                                    
                              

So y[0] is unbounded. Thus, the stability of a discrete time linear time invariant system is equivalent to absolute summability of the impulse response.