Introduction to Signals and SystemsDefinitionsTwo Types of Signals1. Continuous-Time Signals2. Discrete-Time SignalsDiscrete-Time ≠ DigitalSystemsSystem ExamplesTwo Types of SystemsHybrid SystemsAudio Delay System ExampleCreating Useable SystemsLTI SystemIntroduction to Signals and SystemsDavid W. GrahamEE 3272DefinitionsSignal• Conveys information• Function of time• The way it evolves in time is what encodes the information• Examples– Voltage or current in an electronic circuit– Speech and music– Bioelectric signals (e.g. ECG)3Two Types of Signals1. Continuous-time signals2. Discrete-time signals41. Continuous-Time Signals• Signal that has a value for all points in time• Function of time– Written as x(t) because the signal “x” is a function of time• Commonly found in the physical world– ex. Human speech• Displayed graphically as a linex(t)t52. Discrete-Time Signals• Signal that has a value for only specific points in time• Typically formed by “sampling” a continuous-time signal– Taking the value of the original waveform at specific intervals in time• Function of the sample value, n– Write as x[n]– Often called a sequence• Commonly found in the digital world– ex. wav file or mp3• Displayed graphically as individual values– Called a “stem” plotx[n]n12345678910Sample number6Discrete-Time ≠ Digital• Discrete-time sequences are continuous in values, but discrete only in time• Digital signals are discrete in values and discrete in time (they can only take on specific values)7SystemsSystem• A collection of items that together performs a function• Modifies / transforms an input to give an output• Represented bySystemT{ }x(t)y(t) = T{x(t)}T = input2x(t)y(t) = T{x(t)} = x2(t)ex. A squaring system8System ExamplesReal-World Examples of Systems• Circuits• Car– Inputs Æ Steering wheel, force on accelerometer and brakes– Outputs Æ Position of car, velocity• Chemical processes• Electromechanical systems (motors)• Economics, stock market• Biological processes (heart)9Two Types of Systems• Continuous-time systems– Operate on continuous-time signals– Commonly found in the physical world– Represented mathematically using differential equations– System parameters are defined on a continuum of time• Discrete-time systems– Operate on discrete-time signals– ex. Computer algorithms– Represented mathematically by difference equations– System parameters are defined only at discrete points in time10Hybrid Systems• Most real systems are hybrid systems– Use both CT and DT systems• ex. Audio delay systemCTSystemADCDTSystemDACCTSystema(t) b(t) c[n] d[n] e(t) f(t)AmplificationAnalog-to-Digital converter-Sampler-CT to DTex. Delay Digital-to-Analog converter-DT to CTex. Smoothing filter11Audio Delay System ExampleCTSystemADCDTSystemDACCTSystema(t) b(t) c[n] d[n] e(t) f(t)AmplificationAnalog-to-Digital converter-Sampler-CT to DTex. Delay Digital-to-Analog converter-DT to CTex. Smoothing filtera(t)tb(t)td[n]ne(t)te(t)te(t)te(t)tc[n]nSample at a constant rate (time period)12Creating Useable Systems• Purpose of system design– To create an I/O relationship that is predictable• The system I/O relationship must be the same every time … the same today as it was yesterday and will be tomorrow– Time Invariant• Given any type of input, the output follows a reasonable relationship– Easiest is LinearOutIn?LinearOutInNonlinear13LTI System• Most engineering systems are designed to be linear and time invariant (LTI) so that they are predictable• The rest of this course focuses on LTI systems– How to analyze– How to use– How to