Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Electrical Engineering (EE): Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

The document Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Digital Signal Processing.
All you need of Electrical Engineering (EE) at this link: Electrical Engineering (EE)

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.

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

The triangular signal is an even signal. Since, it is symmetrical about Y-axis. We can say it is mirror image about Y-axis.

Consider another signal as shown in the figure below.

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

We can see that the above signal is even as it is symmetrical about Y-axis.

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.

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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.

  • Even × Even = Even
  • Odd × Odd = Even
  • Even × Odd = Odd
  • Even ± Even = Even
  • Odd ± Odd = Odd
  • Even ± Odd = Neither even nor odd

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.

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Where xe(t) represents the even signal and xo(t) represents the odd signal

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

And

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Example

Find the even and odd parts of the signal Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Solution − From reversing x(n), we get

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Now, according to formula, the even part

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Similarly, according to formula the odd part is\

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Periodic and Non-Periodic 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.

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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.

Non-Periodic Signal

Simply, we can say, the signals, which are not periodic are non-periodic in nature. As obvious, these signals will not repeat themselves after any interval time.

Non-periodic 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 non-zero.

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.

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

For any finite signal x(t) the energy can be symbolized as E and is written as;

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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 non-zero i.e. (0<p<∞). For power type signal, normalized average power is finite and non-zero. Almost all the periodic signals are power signals and their average power is finite and non-zero.

In mathematical form, the power of a signal x(t) can be written as;

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Difference between Energy and Power Signals

The following table summarizes the differences of Energy and Power Signals.

Power signalEnergy Signal
Practical periodic signals are power signals.Non-periodic signals are energy signals.
Here, Normalized average power is finite and non-zero.Here, total normalized energy is finite and non-zero.

Mathematically,

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Mathematically,

 

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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  Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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.

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)
Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)
Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Therefore, 

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)       …Ans.

Example 2 − Test whether the signal given Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE) is conjugate or not?

Solution − Here, the real part being t2 is even and odd part (imaginary) being sintsin⁡t 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;

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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;

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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 −

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)
Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)
Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)
Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)
i.e. Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE) is an integral multiple of 2π

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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;

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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;

Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE)

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

The document Classification of CT & DT Signals Notes | Study Digital Signal Processing - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Digital Signal Processing.
All you need of Electrical Engineering (EE) at this link: Electrical Engineering (EE)

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