From Continuous Time System to Discrete Time System

ES400Jack 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 timeTechniques: 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 system4. Iterate the process until close correlation between the measuredand 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”=0VD0=“1”, D1=“0”, D2=“0”What is Vout?

Comparator

If Vi>Vr, then Vo=“1”Else zero.

A Simple Op-Amp Based Comparator

A Analog-to-digital Converter

(Inverter)

(NAND)

EOC: End of ConversionA binary output approximately equal to Vx will be when EOC=1

Sampling in Telephone Systems