Name: signal symmetry and decomposition
Uploaded: 2024-09-26T13:16:53.000Z
Duration: 1 min 17 s
Channel: JoVE

K12 Standards

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South Carolina College and Career Ready Science Standards

High School

Physics

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Waves and Their Applications in Technologies for Information Transfer

P-PS4-5

Even and Odd Signals

Una señal uniforme, ya sea en tiempo continuo o en tiempo discreto, se define como una señal que coincide con su propia versión invertida en el tiempo.
Las señales pares son simétricas alrededor del eje vertical, lo que significa que la señal para valores de tiempo negativos refleja la señal para valores de tiempo positivos.
Una señal se denomina impar si equivale al negativo de su contraparte invertida en el tiempo.
Las señales impares se reconocen por su naturaleza antisimétrica alrededor del eje vertical.
Cualquier señal de tiempo continuo se puede expresar como una combinación de componentes pares e impares.
Aquí, el primer componente en el lado derecho es una función par, mientras que el segundo componente es impar.
Una señal compleja se puede descomponer en componentes pares e impares utilizando simetrías conjugadas.
El producto de una función par y una función impar es una función impar. La multiplicación de dos funciones del mismo tipo, ya sea par o impar, dará como resultado una función par.
Por último, sumar o restar dos funciones pares producirá una función par, y de manera similar, sumar o restar dos funciones impares producirá una función impar.

señales pares e impares

Una señal par, ya sea en tiempo continuo o discreto, se define por su simetría con su versión invertida en el tiempo. Matemáticamente, esto se representa como 
<img alt="Equation1" src="/files/ftp_upload/16039/16039_PT%20EQUATION_1.svg" style="text-align: center; height: 14px;">
para señales de tiempo continuo y
<img alt="Equation2" src="/files/ftp_upload/16039/16039_PT%20EQUATION_2.svg" style="text-align: center; height: 14px;">
para señales de tiempo discreto. Las señales pares muestran simetría alrededor del eje vertical, lo que significa que la señal para valores de tiempo negativos refleja la de valores de tiempo positivos.
Por el contrario, una señal se denomina impar si no coincide con su contraparte invertida en el tiempo, representada por 
<img alt="Equation3" src="/files/ftp_upload/16039/16039_PT%20EQUATION_3.svg" style="text-align: center; height: 14px;">
para señales de tiempo continuo y
<img alt="Equation4" src="/files/ftp_upload/16039/16039_PT%20EQUATION_4.svg" style="text-align: center; height: 14px;">
para señales de tiempo discreto. Las señales impares se caracterizan por su naturaleza antisimétrica respecto del eje vertical. 
Cualquier señal de tiempo continuo puede expresarse como una combinación de componentes pares e impares. Esta descomposición se da mediante: 
<img alt="Equation5" src="/files/ftp_upload/16039/16039_PT%20EQUATION_5.svg" style="text-align: center; height: 14px;">
Aquí, el primer término del lado derecho es una función par, mientras que el segundo término es una función impar. 
Las señales complejas también pueden descomponerse en componentes pares e impares mediante simetrías conjugadas. El producto de una función par y una función impar da como resultado una función impar. Además, la multiplicación de dos funciones del mismo tipo (ambas pares o ambas impares) da como resultado una función par.
Además, las operaciones algebraicas de suma y resta siguen reglas específicas: sumar o restar dos funciones pares da como resultado una función par, y sumar o restar dos funciones impares produce una función impar. Estas propiedades son fundamentales en el análisis de señales, ya que permiten la descomposición y simplificación de señales complejas en componentes manejables.

Señales pares e impares

Una señal uniforme, ya sea en tiempo continuo o en tiempo discreto, se define como una señal que coincide con su propia versión invertida en el tiempo.
Incluso las señales son ...

Education

Educación

JoVE Core

Electrical Engineering

Ingeniería Eléctrica

Unit 1

Unidad 1

Introduction to Signals and Systems

Introducción a Señales y Sistemas

KEY TERMS AND CONCEPTS

TÉRMINOS Y CONCEPTOS CLAVE

signal symmetry and decomposition

Signal symmetry helps classify continuous-time and discrete-time signals as even or odd. An even signal matches its time-reversed version. It mirrors around the vertical axis, so the values at negative time match the values at positive time.

An odd signal does not match its time-reversed counterpart. It has antisymmetry about the vertical axis. These symmetry rules are used in both continuous-time and discrete-time signal analysis.

Any continuous-time signal can be split into even and odd parts. The decomposition writes the signal as a sum of one even function and one odd function. Complex signals can also be separated into even and odd components by using conjugate symmetry.

These signal types also follow clear algebra rules. The product of an even function and an odd function is odd. The product of two functions of the same type, whether both even or both odd, is even. Addition and subtraction also stay consistent: even with even gives even, and odd with odd gives odd. These patterns make complex signals easier to analyze and simplify.

Signal Symmetry and Decomposition

signal and system

Signals and Systems in Everyday Data

Signals and systems help describe how data moves through the world. A signal x(t) is a set of data or a time function that represents a variable of interest. It can carry information about atmospheric temperature, humidity, human voice, television images, a dog’s bark, or birdsongs. A signal can also depend on more than one independent variable. An image, for example, depends on horizontal and vertical position, so it can be treated as a two-dimensional signal. In this course, the focus is on...

classification of signals

Signal Types by Time and Amplitude

Signals in signal processing are grouped by time behavior, amplitude, and whether they start before or after time zero. These signal classes help students describe and compare signals before they are analyzed or processed. A continuous-time signal has a value at every instant in time. A discrete-time signal has values only at specific moments, often written as x(n), where n is an integer. Discrete-time signals often come from systems with naturally discrete values, such as digital audio...

energy and power signals

Power and Energy in Electrical Signals

Power and energy in electrical signals are measured from voltage and current across a resistor. These signals help describe how much energy a signal carries and how much power it delivers over time. For a continuous-time signal, total energy is found by integrating the square of the signal's magnitude over a chosen time interval. Time-averaged power is then the total energy divided by the length of that interval. For a discrete-time signal, total energy is found by summing the squares of the...

even and odd signals

Signal symmetry helps classify continuous-time and discrete-time signals as even or odd. An even signal matches its time-reversed version. It mirrors around the vertical axis, so the values at negative time match the values at positive time. An odd signal does not match its time-reversed counterpart. It has antisymmetry about the vertical axis. These symmetry rules are used in both continuous-time and discrete-time signal analysis. Any continuous-time signal can be split into even and odd...

basic continuous time signals

Step, Impulse, and Ramp Signals

Step, impulse, and ramp signals are the main basic continuous-time signals. They are also called singularity functions. These signals are known for discontinuities or for derivatives that are discontinuous. The unit step function, written as u(t), is zero for negative time values and one for positive time values. It has a jump at t = 0. In many cases, it can model an abrupt change, such as the step voltage that appears when a car ignition key is turned. The derivative of the unit step...

rectangular and triangular pulse function

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Modeling Pulses with Rectangles and Triangles

Rectangular and triangular pulse functions are basic signal shapes used to model waveforms in signal processing. The rectangular pulse has a flat top and is centered at the origin with a height of one unit. It is described by two parameters: T, which gives the center location on the time axis, and τ, which sets the pulse duration. A rectangular pulse can also represent a real signal with a chosen amplitude and length. For example, a pulse with a 5 V amplitude, a 3-second duration, and a center...

exponential and sinusoidal signals

Complex Exponentials in Signal Behavior

Exponential and sinusoidal signals are used to describe how waveforms change over time. The continuous-time exponential function uses constants α and A. When both are real, it shows exponential growth or decay. Graphs of these signals make the rise or drop easy to see. When α is purely imaginary, the signal becomes a complex exponential. In this form, j is the imaginary unit and ω0 is the angular frequency. A complex exponential is periodic when its magnitude is unity. That periodic behavior...

basic discrete time signals

Discrete-Time Signal Basics and Shapes

Discrete-time signals include several basic sequences used in signal processing. The unit step sequence is 1 for zero and positive values of the integer n. A graph of eight sample points shows it as a step that begins at n = 0 and stays constant after that. The unit impulse, also called the sample sequence, is 0 for every n except at n = 0, where it equals 1. This sequence is written as δ(n). It is the first difference of the unit step sequence, and the unit step sequence u(n) is the...

basic operations on signals

Signal Transformations in Time and Amplitude

Signal transformations in time and amplitude are key tools in signal processing. They include time reversal, time scaling, time shifting, and amplitude changes. These operations help students describe how a continuous-time signal changes when it is mirrored, stretched, moved, or resized. Time reversal flips a continuous-time signal about the vertical axis at t = 0. It is done by replacing t with -t, so x(t) becomes x(-t). A graph of the signal shows this mirrored shape clearly. Time scaling...

classification of systems-i

System Properties: Linearity, Causality, Memory

System properties such as linearity, causality, and memory help describe how a system responds to input. These ideas are useful for studying and designing systems in engineering. A system can be checked by looking at how its output changes with different inputs. Linearity means the input-output relationship follows homogeneity and additivity. Homogeneity says that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Additivity means the response...

classification of systems-ii

Continuous and Discrete System Models

Continuous-time and discrete-time systems are two common ways to model signals and their outputs. In continuous-time systems, both input and output signals change smoothly as time moves continuously. These systems are often written with differential or algebraic equations. An RC circuit is one example of a continuous-time system. Its input and output voltage can be related by a differential equation based on Ohm’s law and the capacitor relation. Discrete-time systems work differently. Their...

key terms and concepts

Learn how even and odd signals are identified by symmetry, how any signal can be split into even and odd parts, and the rules for combining them.

Signal Symmetry and Decomposition | JoVE K12

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Signal Symmetry and Decomposition

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