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

An even signal, either in continuous-time or discrete-time, is defined as a signal that matches its own time-reversed version.
Even signals are symmetrical around the vertical axis, meaning that the signal for negative time values mirrors the signal for positive time values.
A signal is termed odd if it equates to the negative of its time-reversed counterpart.
Odd signals are recognized by their antisymmetrical nature about the vertical axis.
Any continuous-time signal can be expressed as a combination of even and odd components.
Here, the first component on the right side is an even function, while the second component is odd.
A complex signal can be decomposed into even and odd components using conjugate symmetries.
The product of an even function and an odd function is an odd function. Multiplying two functions of the same type, either both even or odd, will result in an even function.
Lastly, adding or subtracting two even functions will produce an even function, and similarly, adding or subtracting two odd functions will yield an odd function.

even and odd signals

An even signal, whether in continuous-time or discrete-time, is defined by its symmetry with its time-reversed version. Mathematically, this is represented as
<img alt="Symmetry in signal processing, equation x(t)=x(-t), mathematical concept illustration." src="/files/ftp_upload/16039/16039_PT%20EQUATION_1.svg" style="text-align: center; height: 14px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
for continuous-time signals and
<img alt="Discrete-time signal equation x[n]=x[-n], symmetry concept, formula depiction." src="/files/ftp_upload/16039/16039_PT%20EQUATION_2.svg" style="text-align: center; height: 14px;" data-altupdatedat="1762173717000" data-alt="Equation2" />
for discrete-time signals. Even signals exhibit symmetry around the vertical axis, meaning the signal for negative time values mirrors that for positive time values.
In contrast, a signal is termed odd if it does not match its time-reversed counterpart, represented by
<img alt="Odd function equation x(-t)=-x(t) in mathematical symmetry concept." src="/files/ftp_upload/16039/16039_PT%20EQUATION_3.svg" style="text-align: center; height: 14px;" data-altupdatedat="1762173717000" data-alt="Equation3" />
for continuous-time signals and
<img alt="Antisymmetry in discrete signals equation, x[-n]=-x[n], mathematical concept." src="/files/ftp_upload/16039/16039_PT%20EQUATION_4.svg" style="text-align: center; height: 14px;" data-altupdatedat="1762173717000" data-alt="Equation4" />
for discrete-time signals. Odd signals are characterized by their antisymmetrical nature about the vertical axis.
Any continuous-time signal can be expressed as a combination of even and odd components. This decomposition is given by:
<img alt="Equation for signal decomposition; x(t) = xe(t) + xo(t), concept of even-odd functions." src="/files/ftp_upload/16039/16039_PT%20EQUATION_5.svg" style="text-align: center; height: 14px;" data-altupdatedat="1762173717000" data-alt="Equation5" />
&#8203;Here, the first term on the right side is an even function, while the second term is an odd function.
Complex signals can also be decomposed into even and odd components using conjugate symmetries. The product of an even function and an odd function results in an odd function. Furthermore, multiplying two functions of the same type&#8212;either both even or both odd&#8212;yields an even function.
Additionally, the algebraic operations of addition and subtraction follow specific rules: adding or subtracting two even functions results in an even function, and adding or subtracting two odd functions produces an odd function. These properties are fundamental in signal analysis, enabling the decomposition and simplification of complex signals into manageable components.

An even signal, either in continuous-time or discrete-time, is defined as a signal that matches its own time-reversed version.
Even signals are ...

Education

JoVE Core

Electrical Engineering

Unit 1

Introduction to Signals and Systems

KEY TERMS AND CONCEPTS

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...

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

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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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