Name: discrete-time signal basics and shapes
Uploaded: 2024-09-26T13:16:53.000Z
Duration: 1 min 16 s
Channel: JoVE

The unit step sequence is expressed as 1 for zero and positive values of integer &#39;n&#39;. This can be graphically displayed using a set of eight sample points.
The unit impulse or sample sequence is mathematically expressed for all discrete values of n. It&#39;s zero for all &#39;n&#39; values except at zero.
The unit impulse is the first difference of the unit step, while the unit step is the cumulative sum of the unit sample. This correlation is visually demonstrated through a graph.
The unit impulse sequence can effectively sample the signal value at n equal to zero since it is non-zero only at that point.
The unit ramp sequence exhibits a linear escalation in value with the increase in sample number. For a sequence of 12 samples on a unit ramp, the graph shows a linear increase in the amplitude with the sample number.
A sinusoidal sequence is defined by its amplitude and phase parameters.
The exponential sequence is defined using complex numbers, with exponentially decaying and increasing sequences represented on a graph.

basic discrete time signals

Basic Discrete Time Signals

The unit step sequence is defined as 1 for zero and positive values of the integer n. This sequence can be graphically displayed using a set of eight sample points, showing a step function starting from n=0 and remaining constant thereafter.
The unit impulse or sample sequence is mathematically expressed as zero for all n values except at n=0, where it is one. The unit impulse sequence, denoted by &#948;(n), is the first difference of the unit step sequence, while the unit step sequence u(n) is the cumulative sum of the unit impulse sequence. This relationship can be visually demonstrated through a graph, highlighting how the unit impulse can effectively sample the signal value at n=0.
The unit ramp sequence exhibits a linear increase in value with the increase in the sample number. For instance, a sequence of 12 samples on a unit ramp will show a linear increase in amplitude with each sample number, represented graphically as a straight line.
A sinusoidal sequence is defined by its amplitude and phase parameters. This sequence can be represented as,
<img alt="Sine wave equation x[n]=A sin(&#969;n+&#981;) diagram, relevant to signal processing analysis." src="/files/ftp_upload/16043/16043_PT%20EQUATION_1.svg" style="text-align: center; height: 14px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
where A is the amplitude, &#969; is the angular frequency, and &#934; is the phase.
The exponential sequence is defined using complex numbers, with exponentially decaying and increasing sequences represented on a graph. An exponentially decaying sequence can be written as,
<img alt="Discrete-time signal equation \(x[n]=A&#945;^n\), mathematical symbol for exponential growth." src="/files/ftp_upload/16043/16043_PT%20EQUATION_2.svg" style="text-align: center; height: 14px;" data-altupdatedat="1762173717000" data-alt="Equation2" />
while an exponentially increasing sequence is expressed as,
<img alt="Exponential decay formula x[n]=A&#945;^(-n), equation for discrete signal processing analysis." src="/files/ftp_upload/16043/16043_PT%20EQUATION_3.svg" style="text-align: center; height: 14px;" data-altupdatedat="1762173717000" data-alt="Equation3" />
where A is the initial amplitude and &#945; is a positive constant. These sequences are fundamental in analyzing various signal-processing applications due to their unique properties and behaviors.

The unit step sequence is expressed as 1 for zero and positive values of integer &#39;n&#39;. This can be graphically displayed using a set of eight ...

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Electrical Engineering

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Introduction to Signals and Systems

KEY TERMS AND CONCEPTS

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 cumulative sum of the unit impulse sequence. A graph can show how the impulse samples the signal value at n = 0.

The unit ramp sequence increases linearly as the sample number increases. For example, a 12-sample ramp rises by the same amount at each sample, so its graph appears as a straight line. This makes the ramp easy to recognize on a plot.

A sinusoidal sequence is set by its amplitude and phase. In the expression for this sequence, A is the amplitude, ω is the angular frequency, and Φ is the phase. An exponential sequence uses complex numbers. It can describe both exponentially decaying and exponentially increasing sequences. The decaying form and the increasing form are both shown on graphs, with A as the initial amplitude and α as a positive constant. These basic sequences are important because they help describe many signal-processing applications.

Discrete-Time Signal Basics and Shapes

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...

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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 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...

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

rectangular-and-triangular-pulse-function

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...

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...

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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 the key discrete-time signal shapes used in signal processing, including the unit step, impulse, ramp, sinusoidal, and exponential sequences.

Discrete-Time Signal Basics and Shapes | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

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Discrete-Time Signal Basics and Shapes

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