signal and systems introduction to signals and systems

Signal and Systems

Introduction to Signals and Systems

April 19, 2023 Veton Këpuska 2

Introduction to Signals and Systems

Introduction to Signals and Systems as related to Engineering Modeling of physical signals by

mathematical functions Modeling physical systems by

mathematical equations Solving mathematical equations

when excited by the input functions/signals.


Modeling

Engineers model two distinct physical phenomena:1. Signals are modeled by mathematical

functions.2. Physical systems are modeled by

mathematical equations.

What are Signals?


Signals

Signals, x(t), are typically real functions of one independent variable that typically represents time; t.

Time t can assume all real values: -∞ < t < ∞,

Function x(t) is typically a real function.


Example of Signals: Speech


Example of Signals EKG:


Example of Signals: EEC


Categories of Signals

Signals can be:1. Continuous, or2. Discrete:

T – sampling rate f – sampling

frequency – 1/T – radial

sampling frequency – 2f= 2/T


Signal Processing

Signals are often corrupted by noise.

s(t) = x(t)+n(t)

Want to ‘filter’ the measured signal s(t) to remove undesired noise effects n(t).

Need to retrieve x(t).

Signal Processing


Deterministic signal

Corrupting, stochastic

noise signal

What is a System?



Modeling Examples

Human Speech Production is driven by air (input signal) and produces sound/speech (output signal)

Voltage (signal) of a RLC circuit Music (signal) produced by a musical

instrument Radio (system) converts radio

frequency (input signal) to sound (output signal)


Speech Production

Human vocal tract as a system: Driven by air (as input signal) Produces Sound/Speech (as output signal)

It is modeled by Vocal tract transfer function: Wave equations, Sound propagation in a uniform acoustic tube

Representing the vocal tract with simple acoustic tubes

Representing the vocal tract with multiple uniform tubes


Anatomical Structures for Speech Production


Uniform Tube Model

cos,

cos

sin,

cos

j tg

j tg

l x cu x t U e

l c

l x ccp x t j U e

A l c

Volume velocity, denoted as u(x,t), is defined as the rate of flow of air particles perpendicularly through a specified area.

Pressure, denoted as p(x,t), and

tjg eUtu )(),0(


RLC Circuit

v(t)

L R

C

i(t)

Voltage, v(t) input signal Current, i(t) output signal Inductance, L (parameter of the system) Resistance, R (parameter of the system) Capacitance, C (parameter of the system)

t

tvdiC

tRidt

tdiL )()(

1)(

)(


Newton’s Second Law in Physics

The above equation is the model of a physical system that relates an object’s motion: x(t), object’s mass: M with a force f(t) applied to it: f(t), and x(t) are models of physical signals. The equation is the model of the physical

system.

2

2 )()(

dt

txdMtf

What is a System?

A system can be a collection of interconnected components: Physical Devices and/or Processors

We typically think of a system as having terminals for access to the system: Inputs and Outputs


Example:

Single Input/Single Output (SISO) System

Multiple Input/Multiple Output (MIMO) System


Vin Vout

Electrical Network

+

-

+

-

x1 (t)System

…

x2 (t)

xp (t)

y1 (t)

…

y2 (t)

yp (t)

Example:

Alternate Block Diagram Representation of a Multiple Input/Multiple Output (MIMO) System


Systemx(t) y(t)

1

2

1

pp tx

tx

tx

t

x

1

2

1

qq ty

ty

ty

ty

System Modeling


Physical System

Mathematical Model

Model Analysis

Model Simulation

Design Procedure

Model Types

1. Input-Output Description Frequency-Domain Representations:

Transfer Function - Typically used on ideal Linear-Time-Invariant Systems

Fourier Transform Representation Time-Domain Representations

Differential/Difference Equations Convolution Models

2. State-Space Description Time-Domain Representation


Model Types

1. Continuous Models2. Discrete Models


End


signal and systems introduction to signals and systems

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