Lecture3 Signal and Systems
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This document introduces concepts related to signals and systems. It defines a system as a mathematical model that relates an input signal to an output signal. Systems can be continuous-time or discrete-time, depending on whether the input and output signals are continuous or discrete. Other properties discussed include whether a system is linear or nonlinear, time-invariant or time-varying, causal or noncausal, stable, and whether it involves feedback. Examples are provided to illustrate different types of systems.
![Time-Invariant and Time-Varying
Systems
• A system is called time-invariant if a time shift (delay
or advance) in the input signal causes the same time
shift in the output signal.
• Thus, for a continuous-time system, the system is
time-invariant if
• For a discrete-time system, the system is time-
invariant (or shift-invariant ) if
• Else the systems are known as the time varying
systems
EE-2027 SaS, L3: 10/20
{ ( )} ( )x t y tτ τ− = −T
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Lecture3 Signal and Systems
- 1. EE-2027 SaS, L3: 1/20 Lecture 3: Signals & Systems Concepts Systems, signals, mathematical models. Continuous- time and discrete-time signals. Energy and power signals. Some system properties. Specific objectives: • Introduction to systems • Continuous and discrete time systems • Properties of a system
- 2. System • A system is a mathematical model of a physical process that relates the input (or excitation) signal to the output (or response) signal • Let x and y be the input and output signals, respectively, of a system. • Then the system is viewed as a transformation (or mapping) of x into y. • This transformation is represented by the mathematical notation EE-2027 SaS, L3: 2/20 y = xT
- 3. Continued…. • where T is the operator representing some well- defined rule by which x is transformed into y • Multiple input and/or output signals are possible • We will restrict our attention for the most part in this text to the single-input, single-output case EE-2027 SaS, L3: 3/20
- 4. Continuous;Time and Discrete-Time Systems • If the input and output signals x and y are continuous- time signals, then the system is called a continuous- time system • If the input and output signals are discrete-time signals or sequences, then the system is called a discrete-time system Fig. (a) Continuous-time system; (b) discrete-time system. EE-2027 SaS, L3: 4/20
- 5. Systems with Memory and without Memory • A system is said to be memoryless if the output at any time depends on only the input at that same time. Otherwise, the system is said to have memory • An example of a memoryless system is a resistor R with the input x(t) taken as the current and the voltage taken as the output y(t) . The input-output relationship (Ohm's law) of a resistor is EE-2027 SaS, L3: 5/20 ( ) ( )y t Rx t=
- 6. Causal and Noncausal Systems: • A system is called causal if its output y ( t ) at an arbitrary time depends on only the input x ( t ) for . • That is, the output of a causal system at the present time depends on only the present and/or past values of the input, not on its future values. • Thus, in a causal system, it is not possible to obtain an output before an input is applied to the system. A system is called noncausal if it is not causal. EE-2027 SaS, L3: 6/20 ot t= ot t≤
- 7. Examples of causal systems are • • EE-2027 SaS, L3: 7/20 ( ) ( )y t x t= ( ) ( 1)y t x t= −
- 8. Linear Systems and Nonlinear Systems • If the operator T in satisfies the following two conditions, then T is called a linear operator and the system represented by a linear operator T is called a linear system: 1. Additivity: Given that , and , then 2. Homogeneity (or Scaling): for any signals x and any scalar . • Can be combined into a single condition as EE-2027 SaS, L3: 8/20 y = xT 1 1x y=T 2 2x y=T 1 2 1 2{ }x x y y+ = +T { }x yα α=T 1 1 2 2 1 1 2 2{ }x x y yα α α α+ = +T α
- 9. Examples of Linear and nonlinear systems are • (Non linear) • (Non linear) • (Linear) • Note that a consequence of the homogeneity (or scaling) property of linear systems is that a zero input yields a zero output. This follows readily by setting .This is another important property of linear systems. EE-2027 SaS, L3: 9/20 2 y x= cosy x= y x= { }x yα α=T 0α =
- 10. Time-Invariant and Time-Varying Systems • A system is called time-invariant if a time shift (delay or advance) in the input signal causes the same time shift in the output signal. • Thus, for a continuous-time system, the system is time-invariant if • For a discrete-time system, the system is time- invariant (or shift-invariant ) if • Else the systems are known as the time varying systems EE-2027 SaS, L3: 10/20 { ( )} ( )x t y tτ τ− = −T { [ ]} [ ]x n k y n k− = −T
- 11. Linear Time-Invariant Systems • If the system is linear and also time-invariant, then it is called a linear time-invariant (LTI) system. • Will be discussed in detail in next chapter EE-2027 SaS, L3: 11/20
- 12. Stable Systems • A system takes bounded-input and produces bounded-output (BIBO) stable , known as Stable System EE-2027 SaS, L3: 12/20
- 13. Feedback Systems • A special class of systems of great importance consists of systems having feedback. • In a feedback system, the output signal is fed back and added to the input to the system EE-2027 SaS, L3: 13/20
- 14. •End EE-2027 SaS, L3: 14/20