Even and Odd Signals
Even Signal
A signal is said to be even if it satisfies the following condition;
x(−t) = x(t)
Time reversal of the signal does not imply any change on amplitude here. For example, consider the triangular wave shown below.
The triangular signal is an even signal. Since, it is symmetrical about Yaxis. We can say it is mirror image about Yaxis.
Consider another signal as shown in the figure below.
We can see that the above signal is even as it is symmetrical about Yaxis.
Odd Signal
A signal is said to be odd, if it satisfies the following condition
x(−t) = −x(t)
Here, both the time reversal and amplitude change takes place simultaneously.
In the figure above, we can see a step signal x(t). To test whether it is an odd signal or not, first we do the time reversal i.e. x(t) and the result is as shown in the figure. Then we reverse the amplitude of the resultant signal i.e. –x(t) and we get the result as shown in figure.
If we compare the first and the third waveform, we can see that they are same, i.e. x(t)= x(t), which satisfies our criteria. Therefore, the above signal is an Odd signal.
Some important results related to even and odd signals are given below.
Representation of any signal into even or odd form
Some signals cannot be directly classified into even or odd type. These are represented as a combination of both even and odd signal.
Where x_{e}(t) represents the even signal and x_{o}(t) represents the odd signal
And
Example
Find the even and odd parts of the signal
Solution − From reversing x(n), we get
Now, according to formula, the even part
Similarly, according to formula the odd part is\
Periodic and NonPeriodic Signals
Periodic Signals
Periodic signal repeats itself after certain interval of time. We can show this in equation form as −
x(t) = x(t) ± nT
Where, n = an integer (1,2,3……)
T = Fundamental time period (FTP) ≠ 0 and ≠∞
Fundamental time period (FTP) is the smallest positive and fixed value of time for which signal is periodic.
A triangular signal is shown in the figure above of amplitude A. Here, the signal is repeating after every 1 sec. Therefore, we can say that the signal is periodic and its FTP is 1 sec.
NonPeriodic Signal
Simply, we can say, the signals, which are not periodic are nonperiodic in nature. As obvious, these signals will not repeat themselves after any interval time.
Nonperiodic signals do not follow a certain format; therefore, no particular mathematical equation can describe them.
Energy and Power Signals
A signal is said to be an Energy signal, if and only if, the total energy contained is finite and nonzero (0<E<∞). Therefore, for any energy type signal, the total normalized signal is finite and nonzero.
A sinusoidal AC current signal is a perfect example of Energy type signal because it is in positive half cycle in one case and then is negative in the next half cycle. Therefore, its average power becomes zero.
A lossless capacitor is also a perfect example of Energy type signal because when it is connected to a source it charges up to its optimum level and when the source is removed, it dissipates that equal amount of energy through a load and makes its average power to zero.
For any finite signal x(t) the energy can be symbolized as E and is written as;
Spectral density of energy type signals gives the amount of energy distributed at various frequency levels.
Power type Signals
A signal is said to be power type signal, if and only if, normalized average power is finite and nonzero i.e. (0<p<∞). For power type signal, normalized average power is finite and nonzero. Almost all the periodic signals are power signals and their average power is finite and nonzero.
In mathematical form, the power of a signal x(t) can be written as;
Difference between Energy and Power Signals
The following table summarizes the differences of Energy and Power Signals.
Power signal  Energy Signal 

Practical periodic signals are power signals.  Nonperiodic signals are energy signals. 
Here, Normalized average power is finite and nonzero.  Here, total normalized energy is finite and nonzero. 
Mathematically,  Mathematically,

Existence of these signals is infinite over time.  These signals exist for limited period of time. 
Energy of power signal is infinite over infinite time.  Power of the energy signal is zero over infinite time. 
Solved Examples
Example 1 − Find the Power of a signal
Solution − The above two signals are orthogonal to each other because their frequency terms are identical to each other also they have same phase difference. So, total power will be the summation of individual powers.
Therefore,
…Ans.
Example 2 − Test whether the signal given is conjugate or not?
Solution − Here, the real part being t^{2} is even and odd part (imaginary) being sintsint is odd. So the above signal is Conjugate signal.
Example 3 − Verify whether X(t)=sinωt is an odd signal or an even signal.
Solution − Given X(t)=sinωt
By time reversal, we will get sin(−ωt)
But we know that sin(−ϕ)=−sinϕ.
Therefore,
sin(−ωt) = −sinωt
This is satisfying the condition for a signal to be odd. Therefore, sinωt is an odd signal.
Even and Odd Signals
Even Signal
A signal is said to be even or symmetric if it satisfies the following condition;
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;
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 antisymmetric signal.
Periodic and NonPeriodic 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 −
For the signal to become periodic, following condition should be satisfied;
i.e. is an integral multiple of 2π
Frequencies of discrete sinusoidal signals are separated by integral multiple of 2π.
Energy and Power Signals
Energy Signal
Energy of a discrete time signal is denoted as E. Mathematically, it can be written as;
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<E<∞
Power Signal
Average power of a discrete signal is represented as P. Mathematically, this can be written as;
Here, power is finite i.e. 0<P<∞. However, there are some signals, which belong to neither energy nor power type signal.
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