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Module 2

System Classification and Properties, Time Domain Representation of LTI systems (Convolution Sum and Convolution Integral)

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Syllabus

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Systems Viewed as Interconnection of Operations

  • A system may be viewed as interconnection of operations that transforms an input signal into an output signal with properties different from those of the input signal.

H

 

 

H

 

 

 

(a)

(b)

 

 

 

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Sol:

 

 

 

 

 

 

 

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Fig 3: Parallel form of implementation of discrete-time system

Ex 2: Express the operator that describes the input-output relation in terms of shift of operator S.

 

 

Sol:

 

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Properties of Systems

 

1. Stability

  • A system is said to be bounded-input, bounded-output (BIBO) stable if and only if every bounded input results in a bounded output.
  • The output of such a system do not diverge if the input does not diverge.

Consider a continuous-time system whose input-output relation is described as

 

 

 

 

whenever the input signal satisfy the condition

 

 

 

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Ex 3: Show that the system described by the output equation is BIBO stable.

 

Sol:

Assume that

 

 

Using the input-output relation

 

 

 

 

 

 

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Ex 4: Consider a discrete-time system whose input-output relation is defined by

 

 

Sol:

 

 

 

 

 

 

 

This does not guarantee a bounded output signal. Hence the system is unstable.

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2. Memory

  • A system is said to possess memory if its output signal depends on past or future values of the input signal.

A system is said to be memoryless if its output signal depends only on the present values of the input signal.

 

 

 

 

The memory of the inductor extends into the infinite past.

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Ex 5: The moving average system described by the input-output relation

 

 

Ex 6: A system described by the input-output relation

 

 

Ex 7: How far does the memory of the moving average system described by the input-output relation

 

extend in to the past?

Sol: Four time units

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Ex 8: The input-output relation of a semiconductor diode is represented by

 

 

Sol: No

Ex 9: The input-output relation of a capacitor is described by

 

What is the extent of the capacitor’s memory?

 

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3. Causality

  • A system is said to be causal if the present value of the output signal depends only on the present or the past values of the input signal.
  • The moving average system described by

 

causal

  • By contrast, the moving average system described by

 

noncausal

  • Causality is required for a system to be capable of operating in real time while noncausal system can only operate in non-real time fashion.

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4. Invertiblity

  • A system is said to be invertible if the input of the system can be recovered from the output.

H

 

 

 

 

Fig 4: Notion of system invertibility

 

 

 

 

 

 

 

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  • In any event, a system is not invertible unless distinct inputs applied to a system produce distinct outputs.

Ex 10: Show that the square law described by the input-output relation

 

is not invertible.

Sol:

 

Hence square law is not invertible.

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Ex 11: Consider the time-shift system described by the input-output relation

 

 

Sol:

 

 

 

 

 

 

 

 

 

 

 

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5. Time Invariance

  • A system is said to be time invariant if a time delay or time advance of the input signal leads to identical time shift in the output signal.
  • This implies that a time invariant system responds identically no matter when the input signal is applied.
  • The characteristics of a time invariant system do not change with time. Otherwise, the system is said to be time-variant.
  • Consider a continuous-time system whose input-output relation is described by

 

 

 

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Sol:

 

 

 

 

 

 

 

 

 

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Let

 

 

 

 

 

 

 

 

 

It follows that ordinary inductor is time-invariant.

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Ex 13: Is a discrete-time system described by input-output relation

 

time invariant?

Sol:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Hence

The given discrete-time system is not time invariant.

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6. Linearity

 

1. Superposition:

Consider a system that is initially at rest.

 

 

 

 

 

 

 

 

 

 

2. Homogeneity:

 

 

 

 

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  • When a system violates either the principle of superposition or the property of homogeneity, the system is said to be nonlinear.

 

 

 

  • The resulting output signal is written as

 

 

  • If system is linear, we may express output signal of system as

 

 

 

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Output

Output

 

 

Fig 6: Linear property of a system

 

 

 

 

 

 

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Ex 14: Consider a discrete-time system described by the input-output relation

 

Show that this system is linear.

Sol:

 

 

We may then express the resulting output signal of the system as

 

 

 

 

 

Thus given system satisfies both superposition and homogeneity and is therefore linear.

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Ex 15: Consider a continuous-time system described by the input-output relation

 

Check if this system is linear or not.

Sol:

 

 

Correspondingly, the output signal of the system is given by double summation

 

 

It is in the different from that describing the

input signal.

 

Thus, the system violates the principle of superposition and is nonlinear.

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then

If

 

Hence the system does not satisfy the principle of superposition which makes it nonlinear.

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Ex 16: Consider a discrete-time system described by the input-output relation

 

Check if this system is linear or not.

Sol:

 

 

We may then express the resulting output signal of the system as

 

 

 

 

Thus given system satisfies both superposition and homogeneity and is therefore linear.

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Ex 17: The system has the input-output relation

 

Check if the system is (i) memoryless (ii) stable (iii) causal (iv) linear and (v) time invariant

Sol:

 

 

 

 

 

 

 

 

This guarantee a bounded output signal.

Hence the system is stable.

 

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Correspondingly, the output signal of the system is given by

 

 

 

Thus, the system violates the principle of superposition and is nonlinear.

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Hence

The given continuous-time system is time invariant.

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Ex 18: The system has the input-output relation

 

Check if the system is (i) memoryless (ii) stable (iii) causal (iv) linear and (v) time invariant

Sol:

 

 

 

 

 

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Ex 19: The system has the input-output relation

Check if the system is (i) memoryless (ii) stable (iii) causal (iv) linear and (v) time invariant

Sol:

 

 

The given equation can be rewritten as

 

 

 

 

 

 

 

This guarantee a bounded output signal.

Hence the system is stable.

 

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Correspondingly, the output signal of the system is given by

 

 

 

 

where

 

Thus, the system satisfies the principle of superposition and homogeneity is thus linear.

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Hence

The given discrete-time system is time invariant.

Ex 20: The system has the input-output relation

 

Check if the system is (i) memoryless (ii) stable (iii) causal (iv) linear and (v) time invariant

Sol:

 

 

 

 

 

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Time Domain Representation of Linear Time-Invariant Systems

  • We examine methods for describing the relationship between the input and output signals for linear time-invariant (LTI) systems.

 

  • Given impulse response, we determine output due to arbitrary input signal by expressing input as a weighted superposition of time-shifted impulses.
  • The weighted superposition is termed the convolution sum for discrete-time systems and the convolution integral for continuous-time systems.

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The Convolution Sum

  • We begin by considering the discrete-time case.
  • An arbitrary signal is expressed as a weighted superposition of shifted impulses.

 

 

 

 

 

 

 

  • We see that the multiplication of a signal by a time-shifted impulse results in a time-shifted impulse with amplitude given by the value of the signal at the time impulse occurs.

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-----------------------> (1)

Equation (1) represents the signal as weighted sum of basis functions, which are time-shifted versions of unit impulse signal. The weights are the values of the signal at the corresponding time shifts.

The graphical illustration of equation (1) is given in Fig 1.

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

Equation (2) indicates that the system output is a weighted sum of response of the system to time-shifted impulses.

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If we assume that the system is time-invariant, then a time shift in the input results in time-shift in the output.

 

 

The response of the system is determined by the system impulse response.

 

-----------------------> (4)

-----------------------> (3)

 

 

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Ex 1: Consider a discrete-time LTI system with the input-output relation given by

 

Determine the output of the system in response to the input

 

Sol:

 

Let

 

we find the impulse response as

 

 

 

 

 

 

 

 

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The system may be written as

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Summing the weighted and shifted impulse responses.

 

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Procedure: Reflect and Shift Convolution Sum Evaluation

 

 

 

 

 

 

 

 

 

 

 

 

 

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Geometric Series

 

 

 

 

1.

 

 

 

2.

 

3.

 

 

4.

 

 

5.

 

 

6.

 

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Determine the output of the system when the input is rectangular pulse defined as

 

 

 

 

 

 

 

 

 

 

 

 

 

Sol:

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Evaluation of sums is simplified by noting that

 

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Observations:

 

 

 

 

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Tabulation Method:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Ex 3: Evaluate the following discrete convolution sums:

 

Sol:

 

 

 

 

---------------> (1)

 

 

 

 

 

 

 

 

 

 

 

 

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Now,

 

 

constant

 

 

 

 

 

 

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Common overlap interval

 

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Hence

 

 

 

 

 

 

 

 

 

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Sol:

 

 

 

 

---------------> (1)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Now,

 

 

 

 

 

 

 

 

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Common overlap interval

 

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Using the standard geometric series,

 

 

 

 

 

 

 

 

 

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Convolution Integral

  • The output of a continuous-time LTI system may also be determined solely from the knowledge of input and system’s impulse response.

 

 

---------------> (1)

 

 

 

 

 

 

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If the system is also time invariant, then

 

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We see that the output of an LTI system in response to an input of a form in (1) can be expressed as:

 

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

 

 

 

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and

 

Sol:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

---------------> (1)

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From (1)

 

 

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overlap

 

From (1)

 

 

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overlap

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From (1)

 

 

 

 

 

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From (1)

 

 

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Hence

 

 

 

 

 

 

 

 

 

 

 

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Sol:

 

 

 

 

 

 

 

 

---------------> (1)

 

 

 

 

 

 

 

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From (1)

 

 

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overlap

 

 

From (1)

 

 

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overlap

 

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From (1)

 

 

 

 

 

 

 

Hence

 

 

 

 

 

 

 

 

 

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and

 

Find the output of the system.

Sol:

 

 

---------------> (1)

 

 

 

 

 

 

 

 

 

 

 

 

 

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1.

2.

 

 

 

 

 

3.

 

 

 

 

 

 

 

 

 

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From (1)

 

 

 

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overlap

 

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From (1)

 

 

 

 

 

 

 

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overlap

 

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From (1)

 

 

 

 

 

 

 

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overlap

 

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From (1)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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overlap

 

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From (1)

 

 

 

 

 

 

 

Hence

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Sol:

 

 

 

 

 

 

 

---------------> (1)

 

 

 

 

 

 

 

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From (1)

 

 

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Common overlap interval

 

 

From (1)

 

 

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Hence

 

 

 

 

Or

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Sol:

 

 

---------------> (1)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Common overlap interval

 

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From (1)

 

 

 

 

 

 

 

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Common overlap interval

 

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From (1)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Hence

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