An Overview of Analog Signals and Systems ECE 520 Lecture notes General Overview An input output system A number of important signal processing tasks can be modeled as an input output system The system function is the mapping from the input into the output General Overview Example General Overview A system can be connected to another system in various forms Cascade and Parallel Interconnection A feedback interconnection One dimensional continuous time Systems A block diagram of a multi input multi output system Systems are usually classified into continuous time or discrete time systems depending on whether the inputs are continuous time or discrete time Systems in general can be classified in terms of other properties such as causality linearity time invariance and stability One dimensional continuous time Systems A system is called a memoryless system if its output for each value of time depends on the input at that same time One dimensional continuous time Systems A system is causal if the output at any time depends only on values of the input at the present time and in the past All memoryless systems are causal The inverse is not necessarily true One dimensional continuous time Systems A system is stable if for every bounded input the output is bounded i e finite One dimensional continuous time Systems A system is time invariant if a time shift in the input signal causes a time shift in the output signal One dimensional continuous time Systems A linear system is one that possesses the superposition property Then the system is linear if 1 The response output to 1 2 is 1 2 additivity property 2 The response to 1 is 1 where is a constant scaling or homogeneity property One dimensional continuous time Systems A linear system is one that possesses the superposition property Special Function Definitions The impulse Dirac delta function is defined in terms of the following properties Special Function Definitions The unit step function Consider a system with input and response The impulse response of the system is the output when The step response of the system is the output when The sinusoidal response of the system is the output when is sinusoidal function e g cos Analysis of Linear Systems Analysis of Linear Systems The Impulse Response of Linear Systems Analysis of Linear Systems The Impulse Response of a time invariant linear system convolution integral Analysis of Linear Systems The Impulse Response of a time invariant linear system 1 Usually we have as the desired independent variable i e we have and We want So sketch and 2 Flip or 3 Shift or by multiply by or and integrate Analysis of Linear Systems The Impulse Response of a time invariant linear system Analysis of Linear Systems The Impulse Response of a time invariant linear system convolution integral Analysis of Linear Systems More Properties for LTI Systems Causality for LTI System 0 for 0 Stability of LTI System is bounded input bounded output stable BIBO Stable A LTI System is BIBO stable if and only if The Step Response for LTI Systems the impulse response for LTI System is the derivative of the step response Signal Representation Fourier Series is a representation of periodic signals using trigonometric functions Signal Representation Fourier Series is a representation of periodic signals using trigonometric functions Signal Representation Other Fourier Series is a representations of periodic signals using exponential functions and trigonometric functions Signal Representation Examples of exponential functions representation Signal Representation The Fourier Transform The Fourier transform of a function is defined as The inverse Fourier transform of a function is defined as The Fourier Transform Existence of the Fourier Transform If Then The Fourier Transform Existence of the Fourier Transform In some cases the previous condition might not hold but we still can define the Fourier transform For example the unit step function the sinusoidal functions sin or cos some periodic functions besides the sinusoids etc their Fourier transform exists and is well defined Also some functions which do not exist physically e g the delta function functions of infinite discontinuities e g sin 1 etc also have Fourier Transforms The Fourier Transform Fourier Transform The Fourier Transform Properties of the Continuous time Fourier Transform i Linearity ii Symmetry Properties Let be a real valued function of time then The Fourier Transform Properties of the Continuous time Fourier Transform For a general function we can decompose it into is in general complex i e The Fourier Transform Properties of the Continuous time Fourier Transform Corollary 1 If is real then is even and is odd Corollary 2 If is real and even then is real and even The Fourier Transform Properties of the Continuous time Fourier Transform Time Shifting Consider the Fourier Transform pair then 2 The Frequency Shifting property Consider the Fourier Transform pair then 2 Modulation The Fourier Transform Properties of the Continuous time Fourier Transform Differentiation Consider the Fourier Transform pair then 2 2 and Integration Consider the Fourier Transform pair then 0 1 2 1 2 The Fourier Transform Properties of the Continuous time Fourier Transform Time and Frequency Scaling Consider the Fourier Transform pair then 1 The Fourier Transform Properties of the Continuous time Fourier Transform Duality Consider the Fourier Transform pair then The Fourier Transform Properties of the Continuous time Fourier Transform The Parseval s Theorem Consider the Fourier Transform pair then the energy of a certain signal is the same in the time and the frequency domains EXP Evaluate the energy in the signal The Fourier Transform Properties of the Continuous time Fourier Transform The Convolution Theorem Consider the Fourier Transform pair and then Multiplication Consider the Fourier Transform pair 1 1 and 2 2 then 1 2 1 2 The Fourier Transform Properties of the Continuous time Fourier Transform The Fourier Transform of Some Non realizable Functions The Fourier Transform Properties of the Continuous time Fourier Transform The Fourier Transform of Some Non realizable Functions The Fourier Transform Properties of the Continuous time Fourier Transform The Fourier Transform of Some Non realizable Functions impulse train is periodic in a Fourier Series form The Fourier Transform Properties of the Continuous time Fourier Transform The Fourier Transform of Some Non realizable Functions