SIGNALS & SYSTEMS (ENT 281) Chapter 1 (Part 3): Signals & Systems Modelling Concepts Dr. Hasimah Ali School of Mechatronic Engineering University Malaysia Perlis

What is System? There are various type of systems: electrical systems, mechanical systems, biological systems and so on. The actual physical structure of the system determines the exact relation between the input x(t). Physical devices such as motor, amplifier, filter, boiler and turbine are examples of the systems.

Block diagram representation of a system. What is System? A system is defined as an entity that acts on an input signal and transforms it into an output signal. A system = is a set of elements or functional blocks which are connected together and produces an output in response to an signal. The response or output of the system depends upon the transfer function of the system. It is a cause-and-effect relation between two or more signals. Block diagram representation of a system.

Classification of Systems Elements of a communication system Feedback of control system

1. Continuous-time and discrete-time case: Example 1.12 Moving-average system Consider a discrete-time system whose output signal y[n] is the average of the three most recent values of the input signal x[n], that is Formulate the operator H for this system; hence, develop a block diagram representation for it.

1. Discrete-time-shift operator Sk: Shifts the input x[n] by k time units to produce an output equal to x[n  k]. 2. Overall operator H for the moving-average system: (a): cascade form; (b): parallel form.

Properties of Systems ★ Stability ★ Memory ★ Causality ★ Invertibility

Both Mx and My represent some finite positive number 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 operator H is BIBO stable if the output signal y(t) satisfies the condition whenever the input signals x(t) satisfy the condition Both Mx and My represent some finite positive number One famous example of an unstable system:

The moving-average system is stable. Example 2 Moving-average system (continued) Show that the moving-average system described in Example 1.12 is BIBO stable. Sol. 1. Assume that: 2. Input-output relation: The moving-average system is stable.

Example 3 Unstable system Consider a discrete-time system whose input-output relation is defined by where r > 1. Show that this system is unstable. Sol. 1. Assume that: 2. We find that: With r > 1, the multiplying factor rn diverges for increasing n. The system is unstable.

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 possess memoryless if its output signal depends only on the present values of the input signal. Ex.: Resistor Memoryless ! Ex.: Inductor Memory !

Ex.: Moving-average system Memory ! Ex.: A system described by the input-output relation Memoryless !

Causality A system is said to be causal if its present value of the output signal depends only on the present or past values of the input signal. A system is said to be noncausal if its output signal depends on one or more future values of the input signal. Ex.: Moving-average system Causal ! Ex.: Moving-average system Noncausal !  A causal system must be capable of operating in real time.

H inv = inverse operator Invertibility A system is said to be invertible if the input of the system can be recovered from the output. 1. Continuous-time system: The notion of system invertibility. The second operator H inv is the inverse of the first operator H. Hence, the input x(t) is passed through the cascade correction of H and H inv completely unchanged. x(t) = input; y(t) = output H = first system operator; H inv = second system operator 2. Output of the second system: H inv = inverse operator

3. Condition for invertible system: I = identity operator Example 4 Inverse of System Consider the time-shift system described by the input-output relation where the operator S t0 represents a time shift of t0 seconds. Find the inverse of this system. Sol. 1. Inverse operator S t 0: 2. Invertibility condition: Time shift of t0

Example 5 Non-Invertible System Show that a square-law system described by the input-output relation is not invertible. Sol. Since the distinct inputs x(t) and  x(t) produce the same output y(t). Accordingly, the square-law system is not invertible.

★ Time Invariance A system is said to be time invariance if a time delay or time advance of the input signal leads to an identical time shift in the output signal.  A time-invariant system do not change with time. The notion of time invariance. (a) Time-shift operator St0 preceding operator H. (b) Time-shift operator St0 following operator H. These two situations are equivalent, provided that H is time invariant.

★ Time Invariance 1. Continuous-time system: 2. Input signal x1(t) is shifted in time by t0 seconds: S t 0 = operator of a time shift equal to t0 3. Output of system H: 4. For Fig (b), the output of system H is y1(t  t0): 5. Condition for time-invariant system:

Example 7 Inductor x1(t) = v(t) y1(t) = i(t) The inductor shown in figure is described by the input-output relation: where L is the inductance. Show that the inductor so described is time invariant. Solution: 1. Let x1(t) x1(t  t0) Response y2(t) of the inductor to x1(t  t0) is (A) 2. Let y1(t  t0) = the original output of the inductor, shifted by t0 seconds: (B) 3. Changing variables: (A) Inductor is time invariant.

Example 8 Thermistor x1(t) = v(t) y1(t) = i(t) Let R(t) denote the resistance of the thermistor, expressed as a function of time. We may express the input-output relation of the device as Show that the thermistor so described is time variant. Solution: 1. Let response y2(t) of the thermistor to x1(t  t0) is 2. Let y1(t  t0) = the original output of the thermistor due to x1(t), shifted by t0 seconds: 3. Since R(t)  R(t  t0) Time variant!

★ Linearity A system is said to be linear in terms of the system input (excitation) x(t) and the system output (response) y(t) if it satisfies the following two properties of superposition and homogeneity: 1. Superposition: 2. Homogeneity: a = constant factor  Linearity of continuous-time system 1. Operator H represent the continuous-tome system. 2. Input: x1(t), x2(t), …, xN(t)  input signal; a1, a2, …, aN  Corresponding weighted factor 3. Output:

Superposition and homogeneity where 4. Commutation and Linearity:  Linearity of discrete-time system Same results, see Example 9

Example 9 Linear Discrete-Time system Consider a discrete-time system described by the input-output relation Show that this system is linear. Solution 1. Input: 2. Resulting output signal: where Linear system!

Example 10 Nonlinear Continuous-Time System Consider a continuous-time system described by the input-output relation Show that this system is nonlinear. Solution: 1. Input: 2. Output: Here we cannot write Nonlinear system!

Example 1.21 Impulse Response of RC Circuit For the RC circuit shown in Fig. determine the impulse response y(t). Solution 1. Recall: Unit step response RC circuit for Example 1.20, in which we are given the capacitor voltage y(t) in response to the step input x(t) = y(t) and the requirement is to find y(t) in response to the unit-impulse input x(t) = (t). 2. Rectangular pulse input:. x(t) = x(t) 1/ /2 /2 Rectangular pulse of unit area, which, in the limit, approaches a unit impulse as Δ0. 3. Response to the step functions x1(t) and x2(t):

REFERENCES Simon Haykin and Barry Van Veen, “Signals and Systems”, Wiley, 2nd Edition, 2002 Charles L. Phillips, John M. Parr, Eve A. Riskin; “Signals, Systems and Transforms”, Prentice Hall, Fourth Edition, 2009 M.J. Roberts, “Signals and Systems”, International Edition, McGraw Hill, 2nd Edition 2012