Signals and Systems Overview

Signals and Systems

• This is a subject covering the study of signals and systems, treating inputs and outputs mathematically.
• Systems map input functions to output functions.

Contents

• Topics covered include Introduction, Overview of System Properties, Signal Operations, Signals: Unit Step and Delta, Euler's Formula and Trigonometry, Trigonometric and Exponential Signals, Periodic Signals, Even and Odd Signals, Energy and Power Signals, Linearity, Time Invariance, System Stability, System State, Characterization of System Response, Linear Time Invariant (LTI) Systems, Impulse Response and Convolution, Convolution Examples, Fourier Series, Complex Exponential Fourier Series, Fourier Series Examples, Fourier Transform, Fourier Transform Examples, Linearity of the Fourier Transform, Fourier Transform: Symmetry Property, Fourier Transform: Scaling Property, Fourier Transform: Shifting Property, Fourier Transform: The Convolution Property, Parseval's Relation, Fourier Transform Examples, Sampling, Sampling Theorem, Quantization, Quantization Examples.
• Page numbers are included for each topic.

Overview of System Properties

• Characteristics of systems involve the number of inputs and outputs, stability, and causality.
• Systems are classified as Single Input Single Output (SISO), Multiple Input Multiple Output (MIMO), Single Input Multiple Output (SIMO), and Multiple Input Single Output (MISO).
• Examples include Cruise control (car) and cell phones (speech conversion to radio signals).

Stability

• A system is stable if a bounded input produces a bounded output.
• Cruise control is an example of a stable system, while an uncontrolled car accelerating without bound is an unstable system.

Causality

• A causal system's output at a time t depends only on the input values up to that time t.
• Image processing is an example of a causal system; predicting future events is a non-causal system.

Linearity

• A system is linear if it satisfies the properties of homogeneity and additivity.
• Linearity is an important concept in signal and system analysis. Homogeneity means output is scaled proportionally to input scaling; additivity means the output of a sum of inputs is the sum of the individual outputs.

Time Invariance

• A time-invariant system's response to a shifted input is the shifted response.
• The principle of superposition holds for linear time-invariant (LTI) systems.

Signal Operations

• Scaling: Multiplying a signal by a constant.
• Addition: Adding two or more signals together.
• Time shifting: Shifting a signal in time.

Signals: Unit Step and Delta

• The unit step function is used to represent the sudden onset of a signal. it is 0 for t<0 and 1 for t>0
• The delta function (impulse function) represents an infinitely tall and thin pulse with unit area.

Euler's Formula and Trigonometry

• Euler's formula establishes a relationship between exponential functions and trigonometric functions.
• Cosine and sine functions are defined using the coordinates on the unit circle.

The Complex Plane

• Complex numbers can be visualized as points in a Cartesian coordinate system.
• The complex conjugate of a number is obtained by changing the sign of its imaginary component.
• The length squared of a complex number is the product of the number and its conjugate, which is always a real number.

Euler's Formula

• Euler's formula relates the exponential function to the trigonometric functions cosine and sine.

Trigonometric Identities

• Trigonometric identities are derived from Euler's formula.
• These relate trigonometric functions of sums and differences of angles to their individual angles.

Trigonometric and Exponential Signals

• Sinusoids are periodic functions that can be expressed in terms of cosine and sine functions.
• Exponential signals, such as e^at are used to model various physical phenomena.

Periodic Signals

• A signal is periodic if it repeats itself after a fixed period (τ).
• Properties of periodicity are discussed.

Even and Odd Signals

• Even signals are symmetric about the vertical axis (t = 0).
• Odd signals are antisymmetric about the vertical axis (t = 0).

Energy and Power Signals

• Energy signals have finite energy.
• Power signals have finite average power.

Linearity: Definition

• A system is linear if it satisfies homogeneity and additivity.
• Demonstrates how to check if a system is linear.

Linearity: Examples

• Shows how to determine if a system is linear, with examples showing a system that is linear (a gain of two) and a system that is non-linear (a squarer).

Time Invariance: Conceptual

• A time-invariant system's response to a shifted input is the shifted response. This concept is analyzed using graphs.

Time Invariance: Mathematics

• The example shows how to check if a system is or is not time invariant based on inputs and corresponding outputs

System Stability

• Bounded-Input Bounded-Output (BIBO) Stability: A system is stable if a bounded input produces a bounded output. This is analyzed using gain.

System State

• Systems can have memory, that is to say their output at time t depends on the past. Information about the state of a system gives relevant information about the system’s output.
• State variables are associated with memory elements in a system. Examples of state variables are capacitor voltage and inductor current in electrical circuits; position and velocity in mechanical systems

Characterization of System Response

• Response of a system to a unit step input is investigated in the context of stability
• Steady state error and settling time are discussed as metrics for response stability

Linear Time Invariant (LTI) Systems

• LTI systems are important in signal processing.
• The impulse response completely characterizes the system.

Impulse Response and Convolution

• The output of an LTI system is the convolution of the input and the impulse response.

Convolution Examples

• Example showing convolution of unit step with exponential functions, two rectangular pulses and Triangle with Rectangle Functions are illustrated

Fourier Series

• Fourier series decomposes periodic functions into a sum of sinusoidal waves (harmonics). Describes how to find the coefficients of the functions

Fourier Transform

• The Fourier transform converts a function of time into a function of frequency and vice-versa. Explains derivation and application of the Fourier Transform.

Fourier Transform Examples

• Illustrates transforms of various functions, including rectangular pulses, the delta function and ejwot.

Fourier Transform: Symmetry Property

• Describes how the transform is symmetric for real-valued functions

Fourier Transform: Scaling, Shifting and Convolution Properties

• Relates properties to scaling, shifting and convolution for Fourier Transforms

Parseval's Relation

• Shows that the energy in a signal can be calculated from either the time domain integral or the frequency domain integral.

Sampling

• Discusses uniform quantization.
• A minimum sampling frequency (Nyquist rate) is required to avoid aliasing. Describes the minimum sample rate and discusses a sampling example
• Sampling theorem is explained, and applies to continuous signals.

Quantization

• Quantization is the process of rounding real-valued signals to a discrete set of values (a finite number of levels) based on bit resolution
• Discusses uniform uniform quantization and quantizer map.
• Analyzing quantization noise (error) introduced by quantization.
• Provides examples of quantization techniques and evaluating their error rates.

This quiz covers the fundamental concepts of Signals and Systems, focusing on the mathematical treatment of input and output functions. Topics include system properties, signal operations, Fourier transforms, and system stability among others. Test your knowledge on various signal types and system responses.