Properties of discrete-time linear convolution and system properties

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 , 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. Memoryless system: From equation (4.4) we see that an LTI system is memory less if and only if. 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  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 , for n < 0. 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.  note:- the above all information is produced by IIT under NPtel programme