Periodic Signals

Periodic signals are a type of electrical signals which repeat their behavior after a certain interval of time known as time period. It is also know as a cyclic function and the part of function which repeats is called a cycle. For example, a sine wave, in which the signal has a unique behavior from t=0s to t=2π s .After this we observe that the waveform begins to repeat . This means that sine wave is a periodic signal with a time period of 2π seconds.

Periodic Signal Graph

If we plot the graph representing the magnitude of periodic signal with time ,the graph of periodic signal looks like this:

As we observe, we can see that a triangular signal is an example of a periodic signal. We see that the signal repeats after a fixed interval of time i.e after T seconds hence it is a periodic signal.

If we carefully observe the shape of graph we see that the graph shows a unique behaviour for first T seconds . It rises linearly for the first half of the cycle and then decreases linearly for another half of the cycle. After this interval a cycle is completed and the graph begins repeating its behaviour in cycles of T seconds.

Fourier Series

It is important to study about general formulas of Fourier series before expressing a periodic signal using Fourier series.

Fourier Series is an expansion of periodic signal in terms of infinite sums of sines and cosines.

The Fourier Series summation is used to represent any period function . All real periodic signals satisfy the Dirichlet conditions and can be represented by an infinite trigonometric .The general representation is

A periodic signal can be represented using Trigonometric Fourier Series as

s(t)= a₀/2+ ∑(a_ncos(ω_n.t)+ b_nsin(ω_n.t)) where a_n,b_n are Fourier coefficients and can be derived using

• a_n= 2/T ∫ s(t) .cos(ω_nt) ????t
• bn= 2/T ∫ s(t) .sin(ω_nt) ????t
• a₀= 2/T ∫ s(t) .????t

The angular frequency can be written as

ω_n=Δωn=( 2????/T).n

Mathematical Expression of Periodic Signal

Let there be a periodic signal y(t) of continuous time t , which repeats with a period T seconds. It means that the value of signal after a time interval which is multiple of time period T will be same as the value of signal at time t.

Then we can say,

y(t+nT)= y(t)

where n is an arbitrary integer

This representation of periodic signals helps us to calculate and plot and measure certain important parameters related to the signal.

Signal Analysis

Signal analysis involves studying the property of signal to extract useful information from it. This generally involves understanding their frequency content, amplitude, phase, and other characteristics.

Let's understand this through example. Consider the signal below and its analysis

y(t)= A cos(2π.f.t+π/4)

where:

• A is the amplitude,
• f is the frequency in Hertz (Hz),
• t is time, and
• ϕ is the phase angle.

Frequency Analysis

Frequency is used to tell the number of cycles a signal completes in one second.

In case the frequency of signal is 1 Hz the above signal becomes

y(t)= A cos(2π.t+π/4)

The relation between time period and frequency is

T=1/f

Amplitude Analysis

Since amplitude is the maximum displacement from origin, Amplitude of this signal is A

Phase Analysis

ϕ is used to represent the starting phase of the signal a time t=0.

In this signal ϕ= π/4 meaning signal starts with an initial phase of π/4

Fourier Analysis

To represent the signal in frequency domain, we will do a Fourier analysis.

For a sine signal

X(f)= 1/2j × [ δ(f−1) − δ(f+1)]

Here, X(f) is Fourier transform and δ is dirac-delta function

Types of Periodic Signals

There are various types of periodic signals, each having a different graphical representation:

• Sinusoidal Wave
• Square Wave
• Triangular Wave
• Sawtooth Wave

Sinusoidal Wave

It is a type of waveform which repeats after constant interval of time and has shape in the form of sine wave as shown below. It is the most common signal used in communication circuits of electronic industry.

Square Wave

It is a type of waveform which repeats after constant interval of time and has shape in the form of uniform squares as shown below. It is the most common signal used in design and analysis circuits of electronic industry.

Triangular Wave

It is a type of waveform which repeats after constant interval of time and has shape in the form of uniform triangles as shown below. It is the most common signal used in DC-CD convertor circuits of electronic industry.

Sawtooth Wave

It is a type of waveform which repeats after constant interval of time and has shape in the form of a sawtooth. It is the most common signal used to create sounds with subtractive analog and virtual analog music synthesizers.

Difference Between Periodic Signal and Non Periodic Signal

The difference between period and aperiodic signals:

Properties of Periodic Signal

Here are some properties of periodic signal:

• One important property of periodic signals is their periodicity .Periodic signals have this property since they show a repeating pattern after a constant interval of time called Time period.
• Another important property if Periodic Signals is Amplitude of periodic signals. It is the maximum absolute value of the signal. which represents magnitude or strength of the signal.
• We can define Frequency of periodic signals as number of cycles per unit time. It is the reciprocal of time period and is denoted in Hertz.
• Orientation or shape or form of a periodic signal is another important property. Common periodic waveforms include sine waves, square waves, sawtooth waves, and triangular waves.

Advantages of Periodic Signals

The advantages of periodic signals are stated below:

• Periodic signals are bandwidth-efficient as compared to non-periodic signals. Frequency modulated and Amplitude modulated signals generally use periodic signals due to their low bandwidth requirement.
• It is easier to analyse periodic signals over aperiodic signals due to their easier mathematical representation. Tools like Fourier series can be easily used for dealing with periodic signals.
• Periodic signals are efficiently transferred as compared to aperiodic signals. Due to their repetitive nature , simple modulation techniques can be used for transmission and storage of information.

Disadvantages of Periodic Signals

The limitations of periodic signal are stated below:

• These signals become complex in case of non-linear systems where we can't apply normal and simple techniques associated with periodic signals.
• Periodic signals lack the application in real-life world since most signals in everyday life are not periodic rather they have some irregularities and have non-repeating behaviour in their patterns.
• Periodic signals are sensitive to noise . This means that addition of noise might disrupt the original wave affection the accuracy and transmission of the signals.

Applications of Periodic Signals

Periodic signal is used in various applications like:

• Periodic signals are abundantly used in communication system. Signals like sine wave are fundamental signals used in various modulation techniques.
• Periodic signals are widely used in audio processing. This is because musical notes, speech signals have to exhibit periodic properties.
• Periodic signals are used in control theory where systems are often modelled and analysed using sine signals.
• Periodic signals are essentially used in signal processing. Periodic signals are analyse using tools like Fourier series to understand various signals.

Solved Examples

Find the Fundamental Frequency of signal v(t)=30sin(100t)+10cos(300t)+6sin(500t+????/4)

We have : ω₁=100rads ω₂=300rads ω₃=500rads

The respective time periods are
T₁= 2π/100 T₂=2π/300 T₃=2π/500
So the fundamental time period(T)= L.C.M( T₁,T₂,T₃)

On calculating T=2π/100
So fundamental Frequency= 2π/T= 100 rad/s

Conclusion

As we have seen that periodic signals play an important role in modern electronics. We have already discussed the cyclic nature of these signals along with other properties. As we discuss the uses of this signal we realize it is used in modulating signals and as an input signal to various circuits in communication system for various purposes. These applications signify the need to find appropriate methods to generate periodic signals. There are different methods to generate these signals and they have been stated to the readers. There are various examples of periodic signals each used for a different use.