**Definition:** A signal is a function representing a physical quantity or variable, typically containing information about the behavior or nature of a phenomenon. There are various types of signals classified based on the nature of the information they contain.

**For example**, in an RC circuit, the signal may represent the voltage across the capacitor or the current flowing in the resistor.

**Representation of Signal:** Mathematically, a signal is represented as a function of an independent variable t, where t usually represents time. Therefore, a signal is denoted by x(t).

Also read: What is a Signal?

## Classification of Signals

Signals are classified into different types, and the representation and processing of a signal depend on its type.

The different types of signals are as below.

- Continuous Time and Discrete Time Signals
- Analog and Digital Signals
- Real and Complex Signals
- Deterministic and Random Signals
- Even and Odd Signals
- Periodic and Not Periodic Signals
- Energy and Power Signals
- Causal and Not Causal Signals

### 1. Continuous Time Signals and Discrete Time Signals

A **continuous-time signal**, denoted as **x(t)**, has a continuous variable t. If t is a discrete variable, meaning** x(t) **is defined at discrete times, then it’s termed as a** discrete-time signal**. Since a discrete-time signal is defined at discrete times, it’s often represented as a sequence of numbers, denoted by **{x _{n}}** or

**x[n]**, where n is an integer.

**Continuous-time signals**→**x(t)**, vary continuously over time.**Discrete-time signals**→**x[n]**, are defined at specific discrete times and represented as sequences of numbers.

A **discrete-time signal x[n]** might represent a phenomenon where the independent variable is inherently discrete. For example, the daily closing stock market average inherently evolves at discrete points in time, namely at the close of each day.

#### Sampling

A discrete-time signal x[n] can be obtained by **sampling** a continuous-time signal x(t), such as…

x(t_{0}),x(t_{1}), …,x(t_{n}),…

or in a shorter form as

x[0],x[1], …,x[n], …

or x_{0},x_{1},…,x_{n},…

where we understand that

x_{n} = x[n] = x(t_{n})

and x_{n}‘s are called **samples **and the time interval between them is called the **sampling interval**.

When the sampling intervals are equal (uniform sampling), then

x_{n} = x[n] = x(nT_{s})

where the constant T_{s} is the sampling interval

### 2. Analog and Digital Signals

If a continuous-time signal x(t) can assume any value in the continuous interval (a, b), where a may be -∞ and b may be +∞, then the continuous-time signal x(t) is termed an **analog signal**.

On the other hand, if a discrete-time signal x[n] can only take on a finite number of distinct values, then we refer to this signal as a **digital signal**.

### 3. Real and Complex Signals

A signal x(t) is classified as a** real signal** if its value is a real number, and it is categorized as a** complex signal** if its value is a complex number.

A general complex signal x(t) can be represented as a function of the form:

**x(t) = x _{1}(t) + jx_{2}(t)**

where x_{1}(t) and x_{2}(t) are real signals and j = √-1

### 4. Even and Odd Signals

**Even Signals**: A signal x(t) or x[n] is referred to as an **even signal** if

**x(-t) = x(t)x[ -n] = x[n]**

Examples of even signals are given in the image below.

**Odd Signals: **A signal x(t) or x[n] is referred to as an **odd signal** if

**x(-t) = -x(t)x[-n] = -x[n]**

Examples of odd signals are given in the image below.

Any signal x(t) or x[n] can be represented as the sum of two signals, one of which is even and the other odd. That is,

**x(t) = x _{e}(t) + x_{o(}t) **

x[n] = x_{e}[n] + x_{o}[n]

Where,

- The even part of x(t) is calculated as:
**x**_{e}(t) = 1/2 * {x(t) + x(-t)} - The odd part of x(t) is calculated as:
**x**_{o}(t) = 1/2 * {x(t) – x(-t)} - The even part of x[n] is calculated as:
**x**_{e}[n] = 1/2 * {x[n] + x[-n]} - The odd part of x[n] is calculated as:
**x**_{o}[n] = 1/2 * {x[n] – x[-n]}

It’s worth noting that:

- The product of two even signals or of two odd signals results in an even signal.
- The product of an even signal and an odd signal yields an odd signal.

### 5. Periodic and Nonperiodic Signals

#### Continuous Time Periodic Signal

A continuous-time signal x(t) is considered **periodic **with period T if there exists a positive nonzero value of T such that:

**x(t + T) = x(t)** for all t

An example of such a signal is shown below.

From equation x(t + T) = x(t) or figure above, it can be deduced that:

**x(t + mT) = x(t)** for all t and any integer m.

The **fundamental period** T_{0} of x(t) is defined as the smallest positive value of T for which equation x(t + T) = x(t) holds.

It’s essential to understand that this definition doesn’t extend to a constant signal x(t), often referred to as a DC signal. For a constant signal x(t), determining a fundamental period is not feasible because x(t) repeats indefinitely for any choice of T, rendering the concept of a smallest positive value irrelevant.

Any continuous-time signal that is not periodic is referred to as a **nonperiodic (or aperiodic) signal**.

#### Discrete Time Periodic Signal

Periodic discrete-time signals are defined similarly. A sequence (discrete-time signal) x[n] is considered periodic with period N if there exists a positive integer N such that:

**x[n + N] = x[n]** for all n

An example of such a sequence is illustrated in figure below.

From Equation **x[n + N] = x[n]** and figure above, it can be concluded that:

**x[n + mN] = x[n]** for all n and any integer m.

The fundamental period N_{0} of x[n] is defined as the smallest positive integer N for which **x[n + N] = x[n]** holds.

Any sequence that lacks periodicity is termed a **nonperiodic (or aperiodic) sequence.**

It’s worth noting that

- A sequence obtained by uniformly sampling a periodic continuous-time signal may not exhibit periodicity.
- The sum of two continuous-time periodic signals may not result in a periodic signal.
- Conversely, the sum of two periodic sequences always yields a periodic sequence.

### 6. Energy and Power Signals

Consider v(t) to be the voltage across a resistor R producing a current i(t). The instantaneous power p(t) per ohm is defined as

p(t)= $ \frac {v(t)i(t)}{R} $ = $ i^ {2} $ (t)

Total energy E and average power P on a per-ohm basis are

E= $ \int _ {-\infty }^ {\infty } $ $ i^ {2} $ (t)dt joules

P= $ \lim _ {T\rightarrow \infty } $ $ \frac {1}{T} $ $ \int _ {-T/2}^ {T/2} $ $ i^ {2} $ (t)dt watts

**For an arbitrary continuous-time signal x(t), **

- The normalized energy content E of x(t) is defined as:
- E= $ \int _ {-\infty }^ {\infty } $ |x(t) $ |^ {2} $ $ dt $

- The normalized average power P is defined as:
- P= $\lim_{T\to\infty}\frac1T\int_{-T/2}^{T/2}\Bigl|x(t)\Bigr|^2dt$

Similarly, for a **discrete-time signal x[n]**,

- The normalized energy content E is defined as:
- E= $ \sum _ {n=-\infty}^ {\infty } $ |x|n] $ ^ {2} $

- The normalized average power P is defined as:
- P= $\lim_{N\to\infty}\frac1{2N+1}\sum_{n\operatorname{=}-N}^N\left|x[n]\right|^2$

Signal Type | Normalized Energy Content (E) | Normalized Average Power (P) |
---|---|---|

Continuous-time (x(t)) | E= $ \int _ {-\infty }^ {\infty } $ |x(t) $ |^ {2} $ $ dt $ | P= $\lim_{T\to\infty}\frac1T\int_{-T/2}^{T/2}\Bigl|x(t)\Bigr|^2dt$ |

Discrete-time (x[n]) | E= $ \int _ {-\infty }^ {\infty } $ |x(t) $ |^ {2} $ $ dt $ | P= $\lim_{N\to\infty}\frac1{2N+1}\sum_{n\operatorname{=}-N}^N\left|x[n]\right|^2$ |

Based on above definitions, the following classes of signals are defined:

- x(t) (or x[n]) is considered an
**energy signal**(or sequence) if and only if 0<*E*<∞, and consequently*P*=0. - x(t) (or x[n]) is classified as a
**power signal**(or sequence) if and only if 0<*P*<∞, thereby implying that*E*=∞. - Signals that do not meet either of these conditions are referred to as
**neither energy nor power signals****(NENP)**.

Signal Type | Condition | Energy Content (E) | Average Power (P) |
---|---|---|---|

Energy Signal (or Sequence) | 0<E<∞ | 0<E<∞ | P=0 |

Power Signal (or Sequence) | 0<P<∞ | E=∞ | 0<P<∞ |

Neither Energy nor Power | E=∞ or P=∞ or both | E=∞ or P=∞ or both | E=∞ or P=∞ or both |

It’s worth noting that

- A
**periodic signal**is classified as a power signal if its energy content per period is finite. - In such cases, the average power of the signal only needs to be calculated over a period.

## Summary

Signal Type | Description |
---|---|

Continuous-time (x(t)) | Signals varying continuously over time, represented by functions of an independent variable t. |

Discrete-time (x[n]) | Signals defined at discrete points in time, represented as sequences of numbers. |

Analog Signal | Continuous-time signals that can take on any value in a continuous interval. |

Digital Signal | Discrete-time signals that can only take on a finite number of distinct values. |

Real Signal | Signals whose values are real numbers. |

Complex Signal | Signals whose values are complex numbers. |

Even Signal | Signals where x(-t) = x(t) (or x[-n] = x[n]). |

Odd Signal | Signals where x(-t) = -x(t) (or x[-n] = -x[n]). |

Periodic Signal | Signals that repeat themselves over a period of time. |

Nonperiodic (Aperiodic) Signal | Signals that do not repeat themselves. |

Energy Signal | Signals with finite energy content (0 < E < ∞) and zero average power (P = 0). |

Power Signal | Signals with finite average power (0 < P < ∞) and infinite energy content (E = ∞). |

Neither Energy nor Power Signal | Signals with infinite energy content or infinite average power, or both (E = ∞ or P = ∞ or both). |

Sampling | The process of obtaining discrete-time signals from continuous-time signals. |

Energy and Power Calculation | Calculating energy and average power for continuous-time and discrete-time signals, involving integration or summation. |

Periodic Signal Power | Periodic signals can be power signals if their energy content per period is finite, and their average power is calculated over a period. |

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