![ Discrete-time systems process data samples – normally regularly sampled at T
Continuous-time input and output are x(t) and y(t)
Discrete-time input and output samples are x[nT] and y[nT] when n is an integer
and- n +
Continuous-Time and Discrete-Time Systems](https://image.slidesharecdn.com/lec-250801201119-63c5fe79/85/Lec-2-Basics-properties-system-for-your-information-11-320.jpg)
![Continuous & Discrete-Time Mathematical Models
of Systems
Continuous-Time Systems
Most continuous time systems
represent how continuous
signals are transformed via
differential equations
.
E.g. circuit, car velocity
Discrete-Time Systems
Most discrete time systems
represent how discrete signals
are transformed via difference
equations
E.g. bank account, discrete
car velocity system
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Lec.2 Basics properties system for your information
- 1. CECE 3030 Signals & Systems Lec.2 Basic Properties of Systems
- 2. What are Systems ? Systems are used to process signals to modify or extract information Physical system – characterized by their input-output relationships E.g. electrical systems are characterized by voltage-current relationships for components and the laws of interconnections (i.e. Kirchhoff’s laws) From this, we derive a mathematical model of the system “Black box” model of a system: SYSTEM MODEL
- 3. Classification of Systems Systems may be classified into: 1. Linear and non-linear systems 2. Constant parameter and time-varying-parameter systems 3. Instantaneous (memoryless) and dynamic (with memory) systems 4. Causal and non-causal systems 5. Continuous-time and discrete-time systems 6. Analogue and digital systems 7. Invertible and noninvertible systems 8. Stable and unstable systems
- 4. A linear system exhibits the additivity property: if then It also must satisfy the homogeneity or scaling property: if then These can be combined into the property of superposition: if then A non-linear system is one that is NOT linear (i.e. does not obey the principle of superposition) Linear Systems (1)
- 5. Show that the system described by the equation (1) is linear. Let x1(t) y1(t) and x2(t) y2(t), then and Multiply 1st equation by k1, and 2nd equation by k2, and adding them yields: This is the system described by the equation (1) with and Linear Systems (4)
- 6. Time-invariant system is one whose parameters do not change with time: Time-Invariant Systems TI System delay by T seconds TI System delay by T seconds
- 7. In general, a system’s output at time t depends on the entire past input. Such a system is a dynamic (with memory) system A system whose response at t is completely determined by the input signals over the past T seconds is a finite-memory system Networks containing inductors and capacitors are infinite memory dynamic systems If the system’s past history is irrelevant in determining the response, it is an instantaneous or memoryless systems Instantaneous and Dynamic Systems
- 8. Systems are generally composed of components (sub-systems) We can use our understanding of the components and their interconnection to understand the operation and behaviour of the overall system . Series/cascade Parallel Feedback System Structures System 1 System 2 x y System 1 System 2 x y + System 2 System 1 x y +
- 9. Consider the following simple RC circuit: Output y(t) relates to x(t) by: The second term can be expanded: This is a single-input, single-output (SISO) system. In general, a system can be multiple-input, multiple-output (MIMO). Linear Systems (2)
- 10. Causal system – output at t0 depends only on x(t) for t t0 I.e. present output depends only on the past and present inputs, not on future inputs Any practical REAL TIME system must be causal. Noncausal systems are important because: 1. Realizable when the independent variable is something other than “time” (e.g. space) 2. Even for temporal systems, can prerecord the data (non-real time), mimic a non- causal system 3. Study upper bound on the performance of a causal system Causal and Noncasual Systems
- 11. Discrete-time systems process data samples – normally regularly sampled at T Continuous-time input and output are x(t) and y(t) Discrete-time input and output samples are x[nT] and y[nT] when n is an integer and- n + Continuous-Time and Discrete-Time Systems
- 12. Continuous & Discrete-Time Mathematical Models of Systems Continuous-Time Systems Most continuous time systems represent how continuous signals are transformed via differential equations . E.g. circuit, car velocity Discrete-Time Systems Most discrete time systems represent how discrete signals are transformed via difference equations E.g. bank account, discrete car velocity system ) ( 1 ) ( 1 ) ( t v RC t v RC dt t dv s c c ) ( ) ( ) ( t f t v dt t dv m First order differential equations ] [ ] 1 [ 01 . 1 ] [ n x n y n y ] [ ] 1 [ ] [ n f m n v m m n v First order difference equations ) ) 1 (( ) ( ) ( n v n v dt n dv
- 13. Invertible and Noninvertible Systems Let a system S produces y(t) with input x(t), if there exists another system Si, which produces x(t) from y(t), then S is invertible Essential that there is one-to-one mapping between input and output For example if S is an amplifier with gain G, it is invertible and Si is an attenuator with gain 1/G Apply Si following S gives an identity system (i.e. input x(t) is not changed) System S System Si x(t) x(t) y(t)
- 14. Stable and Unstable Systems Externally stable systems: Bounded input results in bounded output (system is said to be stable in the BIBO sense
- 15. Linear Differential Systems (1) Many systems in electrical and mechanical engineering where input x(t) and output y(t) are related by differential equations For example:
- 16. Linear Differential Systems (2) In general, relationship between x(t) and y(t) in a linear time-invariant (LTI) differential system is given by (where all coefficients ai and bi are constants): and Use compact notation D for operator d/dt, i.e etc. We get: o r
- 17. How Are Signal & Systems Related (i) ? How to design a system to process a signal in particular ways ? Design a system to restore or enhance a particular signal – Remove high frequency background communication noise – Enhance noisy images from spacecraft Assume a signal is represented as x(t) = d(t) + n(t) Design a system to remove the unknown “noise” component n(t), so that y(t) d(t) System ? x(t) = d(t) + n(t) y(t) d(t)
- 18. How Are Signal & Systems Related (ii) ? How to design a system to extract specific pieces of information from signals – Estimate the heart rate from an electrocardiogram – Estimate economic indicators (bear, bull) from stock market values Assume a signal is represented as x(t) = g(d(t)) Design a system to “invert” the transformation g(), so that y(t) = d(t) System ? x(t) = g(d(t)) y(t) = d(t) = g-1 (x(t))
- 19. How Are Signal & Systems Related (iii) ? How to design a (dynamic) system to modify or control the output of another (dynamic) system – Control an aircraft’s altitude, velocity, heading by adjusting throttle, rudder, ailerons – Control the temperature of a building by adjusting the heating/cooling energy flow . Assume a signal is represented as x(t) = g(d(t)) Design a system to “invert” the transformation g(), so that y(t) = d(t) dynamic system ? x(t) y(t) = d(t)
- 23. THANK YOU