Signals and System List of Signals

Signals & System List of Signals

Signals and systems can be classified into several categories based on various properties. Here’s a breakdown of the most common types:

1. Causal vs Non-Causal Systems:

• Causal System: The output of the system at any time depends only on present and past inputs, not future inputs.
  Example: $y(t)=x(t)+x(t-1)$
• Non-Causal System: The output of the system depends on future inputs as well.
  Example: $y(t)=x(t+1)+x(t)$

2. Static vs Dynamic Systems:

• Static (Memoryless) System: The output at any time depends only on the input at that exact time.
  Example: $y(t)=2x(t)$
• Dynamic System: The output at any time depends on past or future inputs, i.e., the system has memory.
  Example: $y(t)=x(t)+x(t-1)$

3. Linear vs Non-Linear Systems:

• Linear System: The system obeys the principles of superposition (additivity and homogeneity). Mathematically, a system is linear if: $y(a{x}_1+b{x}_2)=ay(x_1)+by(x_2)$
• Non-Linear System: The system does not satisfy superposition. Non-linear systems may exhibit complex behaviors like chaos or harmonics.
  Example: $y(t)=x^2(t)$

4. Time-Invariant vs Time-Variant Systems:

• Time-Invariant System: The behavior of the system does not change over time. If you shift the input, the output shifts accordingly.
  Example: $y(t)=x(t-1)$
• Time-Variant System: The system’s properties change over time. Shifting the input may result in a different output.
  Example: $y(t)=t\cdot x(t)$

5. Deterministic vs Stochastic (Random) Signals:

• Deterministic Signal: The signal can be exactly described by a mathematical function, and its future values are completely predictable.
  Example: $x(t)=\cos (2\pi ft)$
• Stochastic (Random) Signal: The signal cannot be predicted exactly due to inherent randomness. It is often described using statistical properties like mean and variance.
  Example: White noise

6. Continuous-Time vs Discrete-Time Signals:

• Continuous-Time Signal: The signal is defined for every time instant, and time is considered a continuous variable.
  Example: $x(t)=e^{-t}$
• Discrete-Time Signal: The signal is defined only at specific time instants (usually integer values), meaning time is considered a discrete variable.
  Example: $x[n]=\sin (2\pi fn)$

7. Periodic vs Aperiodic Signals:

• Periodic Signal: The signal repeats itself after a fixed interval of time (period $T$).
  Example: $x(t)=\sin (2\pi ft)$, where $T=\frac{1}{f}$
• Aperiodic Signal: The signal does not exhibit periodic behavior.
  Example: A single pulse or decaying exponential $x(t)=e^{-t}$

8. Even vs Odd Signals:

• Even Signal: A signal is even if it is symmetric about the vertical axis, meaning $x(t)=x(-t)$.
  Example: $x(t)=\cos (t)$
• Odd Signal: A signal is odd if it is anti-symmetric about the vertical axis, meaning $x(t)=-x(-t)$.
  Example: $x(t)=\sin (t)$

9. Energy vs Power Signals:

• Energy Signal: A signal is an energy signal if it has finite energy, meaning: ${\int }_{-\infty }^{\infty }|x(t){|}^{2}dt<\infty$ Example: A decaying exponential signal
• Power Signal: A signal is a power signal if it has finite average power but infinite energy. $P={\lim }_{T\to \infty }\frac{1}{2T}{\int }_{-T}^T|x(t){|}^{2}dt$ Example: A periodic signal like $x(t)=\sin (2\pi ft)$

10. Stable vs Unstable Systems:

• Stable System: A system is stable if bounded inputs always result in bounded outputs (BIBO stability).
  Example: $y(t)=e^{-t}x(t)$
• Unstable System: A system is unstable if bounded inputs can lead to unbounded outputs.
  Example: $y(t)=e^{t}x(t)$

11. Causal vs Non-Causal Signals:

• Causal Signal: A signal is causal if it is zero for all $t<0$.
  Example: $x(t)=u(t)$, where $u(t)$ is the unit step function.
• Non-Causal Signal: A signal that has non-zero values for times before $t=0$.
  Example: $x(t)=e^t$

12. Analog vs Digital Signals:

• Analog Signal: A signal with continuously varying amplitude over time.
  Example: An audio signal from a microphone
• Digital Signal: A signal that takes discrete values in both time and amplitude.
  Example: A binary signal representing a sequence of 0s and 1s

13. Symmetric vs Anti-Symmetric Systems:

• Symmetric System: The system behaves the same when input is reversed in time.
• Anti-Symmetric System: The output reverses sign when the input is reversed in time.

14. Bounded vs Unbounded Signals:

• Bounded Signal: The amplitude of the signal does not exceed a certain finite value.
  Example: $x(t)=\cos (2\pi ft)$
• Unbounded Signal: The amplitude of the signal grows without bound.
  Example: $x(t)=t^2$

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Yes, a signal can be classified as either causal or non-causal based on when it exists in time.

1. Causal Signal:

A signal is said to be causal if it is zero for all time before a specific point, typically before $t=0$. In other words, the signal only exists or starts after time $t=0$. This is relevant in systems where the future input doesn’t affect the current output.

$$x(t)=0\text{for all}t<0$$

Example:

The unit step function $u(t)$ is a causal signal because it is zero for $t<0$ and 1 for $t\ge 0$:

$$u(t)=\{\begin{array}{ll}0,& t<0\\ 1,& t\ge 0\end{array}$$

2. Non-Causal Signal:

A signal is non-causal if it has non-zero values for times before the present (before $t=0$), meaning it can exist in the past, present, or even future. This implies that the signal may be defined for negative time as well.

$$x(t)\ne 0\text{for some}t<0$$

Example:

A signal like $x(t)=e^t$ for all $t$ is non-causal because it is non-zero for $t<0$.

Summary:

• Causal Signal: Exists or has non-zero values only for $t\ge 0$.
• Non-Causal Signal: Can have non-zero values for $t<0$.

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