A signal can be anything which conveys information. For example a picture of a person gives you information regarding whether he is short or tall, fair or black e.t.c Mathematically signal is defined as a function of one or more dependent variables that conveys information about the state of a system. For example;

a) A speech signal is a function of time. Here independent variable is time and dependent variable is amplitude of speech signal.

b) A picture containing varying brightness is a function of two spatial variables. Here independent variables are spatial coordinates (X, Y) and dependent variable is brightness or amplitude of picture.

**Classification of signals**

Signals can be classified based on parameter used to classify them such as

a) **Nature of independent variable** such as time as

- Continuous time signal
- Discrete time signal

b) **Nature of dependent variable** or signal

- Analog signal
- Digital signal

c) **Number of independent variables**

- One dimensional signal
- Two dimensional signal
- Multi dimensional signal

c) Based on **periodicity of signal** as

- Periodic signal
- Aperiodic signal

d) Based on **nature of indeterminancy**

- Deterministic signal
- random signal

e) Based on **causality**

- Casual signal
- Anti casual signal
- Non casual signal

f) Based on **energy content in the signal**

- Energy signal
- Power signal
- Neither energy nor power signals.

**Continuous time signal, discrete time signal**

Continuous time signals are the signals that are defined at a continuum of times i.e. time can assume any value from (-∞, ∞). It is a one to one mapping of signal for every value time assumes from (-∞, ∞), for every instance of time there exists a unique and single value of function f (t). The signal also can have continuum of amplitude values. These signals are also called as **analog signals**.

If we sample the signal at discrete intervals of time ignoring the values signal takes for times other than sampling times then the signal is defined as **discrete signal**. The signal amplitudes are continuous and analog in nature.

Signals strictly speaking are all continuous time in nature; in discrete signals we are just ignoring the unwanted information in the signal by taking signal amplitudes at discrete instants of time.

**Analog signals, digital signals**

Signals which are continuous in time and amplitude are called analog signals. That is both independent and dependent variables are continuous in amplitude. All Continuous time signals are analog signals and vice versa.

Digital signals are one in which time is discrete in nature and amplitude of signals are quantized i.e. they are allowed to take values from a fixed set of amplitudes. For example a binary signal can have only two values zero or one. Digital signals are widely used in communications as they are less prone to noise.

**One dimensional signal, two dimensional signal, Multi dimensional signal**

If the signal is a function of only one independent variable such signal is referred to as one dimensional signal. For example a noisy voice signal shown in the figure is a one dimensional signal is a function of only time.

Similarly if the signal is a function of two dependent variables variable such signal is referred to as one dimensional signal. For example a simple black and white picture is a function of intensity shown in the figure is a two dimensional signal is a function of spatial coordinates X and Y. At each point (X, Y) an intensity value is assigned and mapped onto computer screen as a 2D image.

Multidimensional signal is a function of more than two variables. For example a video signal is a function of three independent variables which are time and two spatial coordinates(X, Y).

**Periodic signal, Aperiodic signal**

A signal can be classified as periodic signal if it repeats itself after a time interval of T, Where T is called period of the signal. Mathematically a signal f(t) is said to be periodic if f(t+T) = f(t), where the T is the smallest positive non zero value of all possible values of constants T for which the equality holds then T is said to be period of f(t). For example a sine wave is periodic wave which satisfies sin (θ) = sin (2*n*Π+ θ) where n =0, +-1, +-2… But as per definition of period only 2*pi qualifies as period. Hence sin (θ) is said to be periodic with period 2* Π.

A signal which is not periodic is said to be Aperiodic signal. Mathematically it can be defined as a periodic signal with infinite period. Here infinite period signifies that the signal never repeats itself. Most of the signals we will be dealing are aperiodic in nature.

**Deterministic signal,****random signal**

A deterministic signal is a signal about which there is no uncertainty with respect to its value at any time. Deterministic signals are modelled uniquely and completely specified functions of time. For example consider a carrier sine wave f (t) = A_{m}*cos (ω_{c}*t) its value at any instant is completely defined and can be determined with certainty.

A random signal is a signal about which there is some degree of uncertainty before it actually shows up or before it actually occurs. For example an outcome of a flipped coin can be heads or tails we don’t know with 100 % certainty what will be the outcome of the event of flipping a coin before the coin is flipped. That’s why we assign probabilities on the possible outcomes based on our past experiences. In the example of flipping a coin we can say that the outcome can be heads with 50% probability and outcome can be tails with 50% probability. Noise is a random signal which can be defined in terms of probability.

**Casual signal, Anti casual signal, Non casual signal**

The notion of causality actually is more apt for systems nevertheless it can be applied to signals also. A signal is causal if its amplitude is zero for negative instants of time i.e. t < 0. The causality of a signal depends on time reference (t= 0) at what instant time is initialized to zero.

A signal is Anti casual signal if its amplitude is zero for positive instants of time t > 0. A non casual signal is one whose amplitude is non zero for t < 0 and t > 0.

**Energy signal, Power signal**

Consider a resistor of R ohms with the current passing through it is i (t), then the voltage across the resistor is V (t) = i (t)*R. The instantaneous power dissipated in the resistor is

P = V^{2}(t)/R in terms of voltage signal,

P = i^{2}(t)*R in terms of current signal.

To remove the dependence of resistance and ease the analysis it is customary in signal analysis to work with one ohm resistor for which both reduce to same form P = V^{2}(t) = i^{2}(t). Hence the

instantaneous power associated with signal f(t) is defined as P = |f(t)|^{2}.

Total energy of a signal f(t) is defined as E = as T tends to infinity.

Average power is defined as P = as T tends to infinity.

A signal f(t) is an energy signal if its total energy if finite and non zero. The average power associated with a energy signal is zero.

A signal f (t) is a power signal if its average power if finite and non zero. The energy associated with a power signal is infinite.

Signals which have infinite power and infinite energy are classified neither as energy signals or power signals.

Typically periodic signals and random signals are power signals.