Continuous-Time Signal

Sources

1. Time Response Analysis
2. Continuous and discrete time signals
3. Signals and Systems
4. Signals and Systems: Part 1
5. Signals and Systems: Part 2
6. What is Unit Step
7. Unit Ramp

Time Response Analysis

REFERENCE 1

Gowda, K. (2020, December 14). Time response analysis and standard test signals. CircuitBread. Retrieved October 18, 2024, from https://www.circuitbread.com/time-resp-an

• Time response analysis examines the changes in the system’s state or variables with respect to time; thus, it is helpful for assessing the performance of the system.
  • They give crucial information that allows for error minimization and the control of response speed.
• There is a period that occurs when a system changes between two states due to the existence of energy storing elements—something that is found in every system. The response that is generated in this period is known as the transient response.
• In contrast, the period where the system settles to its final state is known as the steady state response. Ideally, it is close to the desired response, indicating that the system is stable.
• Another way of examining the system is through the natural and forced response. The natural response is the response of the system to the stored energy without any external input, whereas a forced response is the response of the system to the external input without any stored energy.

Continuous and Discrete Time Signals

REFERENCE 2

Grami, A. (2015). Signals, systems, and spectral analysis. In Elsevier eBooks (pp. 41–150). https://doi.org/10.1016/b978-0-12-407682-2.00003-x

• The system variables that vary over time can also be viewed as signals. According to Grami (2015), a signal pertains to any physical phenomenon that communicate information. This can be defined by functions of time, and, as such, apply to system variables that change with respect to time.
• It is important to classify the types of signals due to the difference in their representation and processing method. One way of doing it is to classify them as either continuous-time or discrete-time signals.
  • Continuous-time signals are defined for all time $t$. They can have a zero value for either particular instants of time or specific time intervals.
  • Discrete-time signals can only posses discrete values for its independent variable; hence, they are limited to discrete instants of time $n$. Although we do not have the values at the non-discrete instants of time, it does not imply that a zero value is present there.

Reference 3

Cuff, P. (2011). Signals and Systems [Slide show]. Princeton University. https://www.princeton.edu/~cuff/ele301/files/lecture1_2.pdf

• Continuous-time signals have values for all points in time. For this reason, a continuous-time system is used when referring to a system with continuous-time inputs and outputs.
• Discrete-time signals have values for only discrete points in time. For this reason, a discrete-time system is used when referring to a system with discrete-time inputs and outputs.

Sinusoids

Reference 4

Oppenheim, A. (2011a). Signals and Systems: Part I. MIT OpenCourseWare. Retrieved October 19, 2024, from https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/defeabeacb2b0ffecfbda27d7cc0b1dd_MITRES_6_007S11_lec02.pdf

• Sinusoid signals will become foundational for more general signals: their representation will be significantly contribute to the representation of signals and the analysis of important class of systems.
• There are notable distinctions found between a continuous-time signal and a discrete-time signal.
  • Continuous-time sinusoids are periodic. Their time shift corresponds to a phase change, and their phase change corresponds to their time shift. Family of continuous-time sinusoids, which have the form $A\cos {\omega }_{0}t$, also have unique corresponding signals.
  • Discrete-time sinusoids, in contrast, are not periodic. Moreover, their time shift does not necessarily correspond to a phase change, and vice versa. Lastly, the family of discrete-time sinusoids $A\cos ({\Omega }_{0}n+\varphi )$, where the frequency differs by an integer multiple of $2\pi$, have two identical sequences.

Unit Step and Unit Impulse

Reference 5

Oppenheim, A. (2011b). Signals and Systems: Part II. MIT OpenCourseWare. Retrieved October 19, 2024, from https://ocw.mit.edu/courses/res-6-007-signals-and-systems-spring-2011/a21157f121ca5ba3d946766a5d387018_MITRES_6_007S11_lec03.pdf

• Unit step and unit impulse are also important basic signals.
• Unit step
  • 0 for negative time, and unity for positive time for both discrete and continuous time.
  • It is a well-defined sequence in discrete time.
  • It is a mathematical complication of a discontinuity at the origin in continuous time.
  • The unit impulse is the derivative of the unit step, and the unit step is the integral of the unit impulse.
• Unit impulse
  • The sequence is 0 except at $n=0$ in discrete time (it is in unity).
  • In continuous time, it has infinite height and zero width but with a finite area.
  • Unit impulse is the first difference of the unit step. Conversely, the unit step is the resulting sum of the unit impulse.

Reference 6

Saini, M. K. (2021, November 11). What is a unit step signal? Tutorialspoint. Retrieved October 19, 2024, from https://www.tutorialspoint.com/what-is-a-unit-step-signal

• The step signal or step function is a type of standard signal that only exists when $t>0$; otherwise, it is 0 when $t<0$. They assist in the analysis of plenty of systems.
  • If it has a unity magnitude, it is known as a unit step signal or unit step function $u(t)$.
  • It is used to determine how fast the system responds to sudden change in the input signal.
  • A continuous-time unit step signal has a unit step signal in every instant of time. It is expressed as $u(t)=\{\text{1 for }t\ge 0\text{ 0 for }t<0\}$.
  • A discrete-time unit step signal only has a unit step signal in discrete instants of time. It is expressed as $u(t)=\{\text{1 for }n\ge 0\text{ 0 for }n<0\}$.

Unit Ramp

Reference 7

Gajic, Z. (2003). Common signals [Slide show]. Rutgers University. https://eceweb1.rutgers.edu/~gajic/solmanual/slides/chapter2.pdf

• A continuous-time ramp signal $r(t)$ is equal to $0$ except at $t\ge 0$, where it is equal to $t$. In this regard, its slope is equal to $1$ at $t>0$.
• The discrete counterpart of the continuous-time unit ramp $r(t)$ is equal to 0 except at $k\ge 0$, where it is equivalent to $k$.
• The main difference between the two is that the continuous-time ramp have values for all instants of time, while the discrete-time ramp only have values for discrete instants of time.