Signals and systems come in various flavors, each with unique characteristics. From continuous to discrete, periodic to aperiodic, and energy to power signals, understanding these classifications is key to grasping signal processing fundamentals.

Systems, too, have their own set of properties. Linear or nonlinear, time-invariant or time-varying, causal or non-causal - these traits shape how systems process signals. Knowing these distinctions helps in analyzing and designing effective signal processing systems.

Continuous vs Discrete Signals and Systems

Continuous-time Signals and Systems

• Continuous-time signals are defined for all values of time and are typically represented by a continuous independent variable, such as t
  • Examples include analog signals like speech or music
• Continuous-time systems operate on continuous-time signals and produce continuous-time outputs
  • The input-output relationship is described by differential equations

Discrete-time Signals and Systems

• Discrete-time signals are defined only at specific time instants and are typically represented by a discrete independent variable, such as n
  • Examples include digital signals like sampled audio or video
• Discrete-time systems operate on discrete-time signals and produce discrete-time outputs
  • The input-output relationship is described by difference equations
• The process of converting a continuous-time signal to a discrete-time signal is called sampling, while the reverse process is called reconstruction

Signal Classification

Periodicity

• Periodic signals repeat themselves at regular intervals, satisfying the condition $x(t) = x(t + T)$ for continuous-time signals or $x[n] = x[n + N]$ for discrete-time signals, where $T$ and $N$ are the fundamental periods
  • Examples include sinusoidal waves and square waves
• Aperiodic signals do not exhibit a repeating pattern and do not satisfy the periodicity condition
  • Examples include exponential decays and random noise

Energy and Power

• Energy signals have finite energy, defined as the integral of the squared magnitude of the signal over its entire domain
  • For continuous-time signals, $E = \int|x(t)|^2 dt$, and for discrete-time signals, $E = \sum|x[n]|^2$
  • Examples include transient signals like a single pulse or a damped sinusoid
• Power signals have finite average power, defined as the limit of the average energy over a finite interval as the interval length approaches infinity
  • For continuous-time signals, $P = \lim_{T\to\infty} \frac{1}{T} \int|x(t)|^2 dt$, and for discrete-time signals, $P = \lim_{N\to\infty} \frac{1}{N} \sum|x[n]|^2$
  • Examples include persistent signals like a constant DC value or a periodic waveform
• Signals can be further classified as deterministic or random, depending on whether their future values can be precisely predicted from past values

System Properties

Linearity and Non-linearity

• Linear systems satisfy the properties of superposition and homogeneity
  • Superposition means that the response to a sum of inputs is equal to the sum of the responses to each individual input
  • Homogeneity means that scaling the input by a constant factor scales the output by the same factor
  • Examples include systems described by linear differential or difference equations, such as filters and integrators
• Nonlinear systems do not satisfy the properties of superposition and homogeneity
  • The response to a sum of inputs is not equal to the sum of the responses to each individual input, and scaling the input does not necessarily scale the output by the same factor
  • Examples include systems with saturation, hysteresis, or multiplicative effects, such as amplifiers with clipping or mechanical systems with friction

Time-invariance and Time-variance

• Time-invariant systems have input-output relationships that do not change with time
  • Delaying the input results in an equally delayed output
  • Examples include systems with constant coefficients in their differential or difference equations, such as linear time-invariant (LTI) filters
• Time-varying systems have input-output relationships that change with time
  • Delaying the input does not necessarily result in an equally delayed output
  • Examples include systems with time-varying coefficients or parameters, such as adaptive filters or modulated communication channels
• The properties of linearity and time-invariance are important for the analysis and design of systems using techniques such as Fourier analysis and Laplace transforms

Causality, Stability, and Invertibility

Causality

• Causal systems have outputs that depend only on current and past inputs, not on future inputs
  • For a continuous-time system to be causal, its impulse response $h(t)$ must satisfy $h(t) = 0$ for $t < 0$
  • For a discrete-time system to be causal, its impulse response $h[n]$ must satisfy $h[n] = 0$ for $n < 0$
  • Examples include real-time systems like control systems and audio filters
• Non-causal systems have outputs that depend on future inputs
  • Examples include offline processing systems like audio or video editors, where future samples are available

Stability

• Stable systems have bounded outputs for bounded inputs
  • For a continuous-time system to be stable, its impulse response $h(t)$ must be absolutely integrable, i.e., $\int|h(t)| dt < \infty$
  • For a discrete-time system to be stable, its impulse response $h[n]$ must be absolutely summable, i.e., $\sum|h[n]| < \infty$
  • Examples include systems with decaying impulse responses, such as stable filters and feedback control systems
• Unstable systems may have unbounded outputs even for bounded inputs
  • Examples include systems with growing impulse responses, such as unstable feedback systems or resonant circuits

Invertibility

• Invertible systems have a unique input for each output, allowing the input to be recovered from the output
  • For a system to be invertible, it must be both injective (one-to-one) and surjective (onto) in the mapping between input and output
  • The inverse system, if it exists, has an impulse response $g(t)$ or $g[n]$ such that the convolution of $h(t)$ and $g(t)$ (or $h[n]$ and $g[n]$) equals the Dirac delta function
  • Examples include lossless compression systems and reversible transformations
• Non-invertible systems may have multiple inputs that produce the same output
  • Examples include lossy compression systems and many-to-one mappings