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Module 3

Syllabus: LTI system properties in terms of impulse response,

Fourier Representation of periodic signals

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Interconnection of LTI systems

  • We develop the relationships between the impulse response of an interconnection of an LTI systems and the impulse responses of the constituent systems.
  • The results for continuous and discrete-time systems are obtained by using nearly identical approaches.
  • We derive the continuous-time results and then simply state the discrete-time results.

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Parallel connection of LTI systems

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig 1: Interconnection of two LTI systems. (a) Parallel connection of two systems (b) Equivalent systems

 

 

 

 

 

 

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Identical results hold for discrete-time case

 

 

 

Distributive property

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Cascade connection of LTI systems

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig 1: Interconnection of two LTI systems. (a) Cascade connection of two systems (b) Equivalent systems

(c) Equivalent system: Interchange system order

 

 

 

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Interchange the order of integration to obtain,

 

 

 

 

 

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Substituting this relationship in to (3), we get

 

 

Hence the impulse response of an equivalent system representing two LTI systems connected in cascade is the convolution of their individual impulse responses.

 

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

 

 

Perform the change of variable,

 

 

 

 

 

 

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Hence the convolution can be performed in either order. This corresponds to interchanging the order of LTI systems in cascade without affecting the result as shown in Fig 2(c).

 

Mathematically, we can say that convolution operation possesses the commutative property, or

 

Discrete time LTI systems and convolutions have properties that are identical to their continuous-time counterparts. Also, discrete-time convolution is associative, so that

 

and commutative, or

 

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Sol:

Using the convolution sum,

 

 

 

Hence

 

 

 

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Sol:

Using the convolution sum,

 

 

 

Hence

 

 

 

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Sol:

 

 

 

 

 

 

 

 

Hence

 

 

 

 

 

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Ex 3: Consider the interconnection of four LTI systems, as depicted in Fig 1. The impulse responses of the system are

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sol:

 

 

Fig 3:

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Fig 3:

 

 

 

 

 

 

 

 

 

 

 

Sol:

The overall impulse response of the system is

 

 

 

 

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Sol:

 

 

 

 

---------------> (2)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Now,

 

 

 

 

 

 

 

 

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Common overlap interval

 

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Using the standard geometric series,

 

 

 

 

 

 

 

 

 

The overall impulse response of the system is

 

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Relations between LTI system properties and impulse response

  • The impulse response completely characterizes the input-output behavior of an LTI system.
  • Hence, the properties of the system such as memory, causality and stability are related to system’s impulse response.

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Memoryless LTI systems

  • Recall that the output of a memoryless LTI system depends only on the current input.
  • The output of a discrete-time LTI system is expressed as

 

(by exploiting the commutative property of convolution)

 

Expand the sum term by term

 

 

 

 

 

 

 

 

 

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  • A discrete-time LTI system is memoryless if and only if

 

 

 

  • Writing the output of a continuous-time system as

 

  • We see that analogously to the discrete-time case, a continuous-time LTI system is memoryless if and only if

 

 

All memoryless LTI systems simply perform scalar multiplication on the input.

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Causal LTI systems

  • The output of a causal LTI system depends only on past or present values of the input.
  • Again, we can write the convolution sum as

 

 

Expand the sum term by term

 

 

 

 

 

 

 

 

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  • Hence for a discrete-time causal LTI system,

 

  • The convolution sum takes the new form

 

  • The causality condition for a continuous-time system follows in an analogous manner from the convolution integral.

 

  • A causal continuous-time LTI system has an impulse response that satisfies the condition

 

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  • The output of a continuous-time LTI system is thus expressed as the convolution integral

 

Note that causal systems are nonanticipatory; that is they cannot generate the output before the input is applied.

Requiring the impulse response to be zero for negative time is equivalent to saying that the system cannot respond with an output prior to the application of input.

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Stable LTI systems

  • We recall that the system is bounded input-bounded output (BIBO) stable if the output is guaranteed to be bounded for every bounded input.

 

  • The magnitude of the output is given by

 

 

 

 

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  • We conclude that the impulse response of stable LTI system must satisfy the bound

 

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  • A similar steps may be used to establish the fact that a continuous-time LTI system is BIBO stable if and only if the impulse response is absolutely integrable- that is , if and only if

 

Property

Discret-time LTI system

Continuous-time LTI system

Memoyless

​

​

Causality

​

​

Stability

​

​

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Ex 5: For each of the following impulse responses, determine whether the corresponding systems is (i) memoryless, (ii) causal, and (iii) stable. Justify your answers.

 

 

 

 

 

 

 

Sol:

 

Hence the system is not memoryless.

 

Hence the system is not causal.

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Hence the system is stable.

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Sol:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hence the system is not memoryless.

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Hence the system is not causal.

 

 

 

 

 

 

Recall:

 

 

 

 

 

Hence the system is stable.

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Sol:

 

 

 

 

 

 

 

Hence the system is not memoryless.

 

Hence the system is causal.

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Hence the system is unstable.

 

 

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Sol:

 

 

 

 

Hence the system is not memoryless.

 

Hence the system is not causal.

 

 

 

 

 

 

Hence the system is unstable.

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Sol:

 

 

 

 

 

 

 

 

Hence the system is not memoryless.

 

Hence the system is not causal.

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Hence the system is stable.

 

 

 

 

 

 

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We emphasize that the system can be unstable even though the impulse response has finite value.

 

Sol:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hence the system is not memoryless.

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Hence the system is not causal.

 

 

 

 

 

 

Hence the system is not stable.

 

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Invertible Systems and Deconvolution

  • A system is invertible if the input to the system can be recovered from the output except for a constant scale factor.
  • This implies that the existence of the inverse system that takes the output of the original system as its input and produces the input of the original system.

 

 

 

 

 

 

 

 

 

 

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This requirement implies that

 

 

 

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Ex 7: The discrete-time model for a two path communication model is

 

Sol:

 

 

 

 

 

 

 

 

 

 

 

 

 

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to obtain causal inverse system

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hence inverse system has impulse response

 

 

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Step Response

  • Step input signals are often used to characterize the response of an LTI system to sudden changes in the input.

 

 

 

Now, since

 

 

 

 

 

 

 

 

 

Step response is the running sum of the impulse response.

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Now, since

 

 

 

Hence the step response is

 

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Step response of a continuous-time system is expressed as the running integral of the impulse response as

 

We may invert these relationship to express the impulse response in terms of the step response as

 

and

 

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Ex 8: Find the step response of the first-order recursive system with impulse response

 

 

Sol:

 

Step Response,

 

 

 

 

 

 

 

 

 

 

Recall:

 

 

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Sol:

 

Step response,

 

 

 

 

 

 

 

 

 

 

 

 

 

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Hence the step response,

 

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Recall:

 

 

Hence the step response,

 

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Sol:

 

Step Response,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Sol:

 

Step response,

 

 

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Hence the step response

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