From Continuous Time System to Discrete Time System ES400 Jack Ou, Ph.D. Chapter 1

Outline Modeling – Signal – System Continuous Time System ADC Discrete Time System

Signals Signals are divided into two natural categories – Continuous time signal: Discrete time signal: defined at only defined for all values of time – Discrete Time Signal: Only defined at certain instants of time.

Example of a Continuous Time Signal Criterion: The signal defined for all values of time Techniques: Fourier series, Fourier transform, Laplace transform

Example of a Discrete Time Signal Criterion: The signal is defined for at only certain instants of time Technique: Z transform, DFT, FFT

Tinkering

Mathematical Modeling of a Physical System

Mathematical Solutions of Physical Problems 1.Formulate a math model for physical signal and system involved. 2.Equations are solved for typical excitation function. 3.Compare math solution with the response of the physical system 4.Iterate the process until close correlation between the measured and model is achieved.

Continuous Time Example

Mathematical Modeling of Elementary Circuits

Use KVL to formulate the mathematical representation of a physical system KVL: The algebraic sum of Voltages around any closed loop in an electric circuit is zero.

Describe the input signal mathematically

Represent Input Voltage Source using Laplace Transform

Solve the problem in the Laplace Domain Laplace transform the KVL expression Solve the variable of interest Inverse Laplace Transform

Not All Systems Can Be Represented Using a Continuous Time Representation

Convert a Signal From Continuous Time System to Discrete Time System Operational Amplifier Digital to Analog Converter Comparator Counter-Ramp Analog to Digital Converter

Operational Amplifier If used in a feedback configuration, V + =V -. Large input impedance!

A Simple Voltage Amplifier

Digital-to-analog Converter Purpose: Convert a binary number to a voltage.

A Simple Digital to Analog Converter Assume “1”=5V, “0”=0V D0=“1”, D1=“0”, D2=“0” What is Vout?

Comparator If V i >V r, then V o =“1” Else zero.

A Simple Op-Amp Based Comparator

A Analog-to-digital Converter (Inverter) (NAND) EOC: End of Conversion A binary output approximately equal to Vx will be when EOC=1

Sampling in Telephone Systems