Signal and Systems Introduction to Signals and Systems

3 July 2015Veton Këpuska2 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.

3 July 2015Veton Këpuska3 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? 3 July 2015Veton Këpuska4

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. 3 July 2015Veton Këpuska5

Example of Signals: Speech 3 July 2015Veton Këpuska6

Example of Signals EKG: 3 July 2015Veton Këpuska7

Example of Signals: EEC 3 July 2015Veton Këpuska8

Categories of Signals  Signals can be: 1.Continuous, or 2.Discrete:  T – sampling rate  f – sampling frequency – 1/T   – radial sampling frequency – 2f= 2/T 3 July 2015Veton Këpuska9

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 3 July 2015Veton Këpuska10 Deterministic signal Corrupting, stochastic noise signal

What is a System? 3 July 2015Veton Këpuska11

3 July 2015Veton Këpuska12 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)

3 July 2015Veton Këpuska13 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

3 July 2015Veton Këpuska14 Anatomical Structures for Speech Production

3 July 2015Veton Këpuska15 Uniform Tube Model  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

3 July 2015Veton Këpuska16 RLC Circuit  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)

3 July 2015Veton Këpuska17 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.

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 3 July 2015Veton Këpuska18

Example:  Single Input/Single Output (SISO) System  Multiple Input/Multiple Output (MIMO) System 3 July 2015Veton Këpuska19 V in V out Electrical Network x 1 (t) System … x 2 (t) x p (t) y 1 (t) … y 2 (t) y p (t)

Example:  Alternate Block Diagram Representation of a Multiple Input/Multiple Output (MIMO) System 3 July 2015Veton Këpuska20 System x(t)y(t)

System Modeling 3 July 2015Veton Këpuska21 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 3 July 2015Veton Këpuska22

Model Types 1.Continuous Models 2.Discrete Models 3 July 2015Veton Këpuska23

End 3 July 2015Veton Këpuska24