Just like Continuous time signals, Discrete time signals can be classified according to the conditions or operations on the signals.

Even and Odd Signals

Even Signal

A signal is said to be even or symmetric if it satisfies the following condition;

$$x(-n) = x(n)$$

Here, we can see that x(-1) = x(1), x(-2) = x(2) and x(-n) = x(n). Thus, it is an even signal.

Odd Signal

A signal is said to be odd if it satisfies the following condition;

$$x(-n) = -x(n)$$

From the figure, we can see that x(1) = -x(-1), x(2) = -x(2) and x(n) = -x(-n). Hence, it is an odd as well as anti-symmetric signal.

Periodic and Non-Periodic Signals

A discrete time signal is periodic if and only if, it satisfies the following condition −

$$x(n+N) = x(n)$$

Here, x(n) signal repeats itself after N period. This can be best understood by considering a cosine signal −

$$x(n) = A cos(2pi f_{0}n+theta)$$
$$x(n+N) = Acos(2pi f_{0}(n+N)+theta) = Acos(2pi f_{0}n+2pi f_{0}N+theta)$$
$$= Acos(2pi f_{0}n+2pi f_{0}N+theta)$$

For the signal to become periodic, following condition should be satisfied;

$$x(n+N) = x(n)$$
$$Rightarrow Acos(2pi f_{0}n+2pi f_{0}N+theta) = A cos(2pi f_{0}n+theta)$$

i.e. $2pi f_{0}N$ is an integral multiple of $2pi$

$$2pi f_{0}N = 2pi K$$
$$Rightarrow N = frac{K}{f_{0}}$$

Frequencies of discrete sinusoidal signals are separated by integral multiple of $2pi$.

Energy and Power Signals

Energy Signal

Energy of a discrete time signal is denoted as E. Mathematically, it can be written as;

$$E = displaystyle sumlimits_{n=-infty}^{+infty}|x(n)|^2$$

If each individual values of $x(n)$ are squared and added, we get the energy signal. Here $x(n)$ is the energy signal and its energy is finite over time i.e $0

Power Signal

Average power of a discrete signal is represented as P. Mathematically, this can be written as;

$$P = lim_{N to infty} frac{1}{2N+1}displaystylesumlimits_{n=-N}^{+N} |x(n)|^2$$

Here, power is finite i.e. 0<P<∞. However, there are some signals, which belong to neither energy nor power type signal.