Fourier series[otw_is sidebar=otw-sidebar-3]
The electrical signals can be obtained from a variety of sources. The most common source of the electrical signals is the transducer which measures the physical quantity and converts those physical quantities to the corresponding electrical signals. Thus the electrical signals obtained from the transducers or the sensors contain information about the physical quantities such as light, sound, heat etc. The electrical signals are visualized and represented graphically in the form of the waveform.
Fourier series Waveform:
The waveform of the signal is defined as the graphical representation of the signal.
These signals on the basis of their periodicity are classified in two major classes:
- Periodic Signals.
- Aperiodic Signals.
Periodic Signals are defined as the signals which repeat their waveform after a definite interval of time. This definite interval of time after which the signal repeats its waveform is called the time period of the signal.
Aperiodic Signals:[otw_is sidebar=otw-sidebar-3]
Aperiodic signals are the signals which do not have repetitive behavior and does not repeat its waveform for any time.
For the sake of relevance here only the Periodic Signals are considered. Any Periodic Signal can be represented in terms of the Fourier series on the other hand the Fourier transform is used to represent the frequency content of the Aperiodic Signals.
Examples of the Periodic Waveform:
fourier series square wave
Periodic Signals and Fourier series:
As described in the precious discussion that the Periodic Signals can be represented in the form of the Fourier series. So let us now develop the concept about the Fourier series, what does this series represent, why there is a need to represent the periodic signal in the form of its Fourier series. It has been found that the signal is composed a large number of harmonically related sinusoid signals. That is any periodic signal can be represented as the sum of harmonically related sinusoid signals. Each of the sinusoid signals in the Fourier series has particular frequency, phase and amplitude. Since all the sinusoid signals are harmonically related it means that all the frequencies of the sinusoids in the Fourier are the integer multiple of the basic fundamental frequency. The fundamental frequency is the actual frequency of the signal whose Fourier series representation is to be found. So the Fourier series of the Periodic Signal represents all the sinusoidal signals that combine to form that signal. The Fourier series can be represented as follows:[otw_is sidebar=otw-sidebar-3]
It can be noted in the above figure that the signal is shown as the summation of the harmonically related sinusoids. The component with the frequency 2f0 is called the second harmonic of the signal and the component with the frequency 3f0 is called the third harmonic thus nf0 is called the nth harmonic of the signals where f0 is called the fundamental frequency of the signal. The amplitude and the phase that each sinusoidal component in the Fourier series representation bears are determined by the Fourier coefficient.
The Fourier coefficient is the core of the Fourier series which determines the amplitude and the phase that each component in the Fourier series would carry. These Fourier coefficients are determined by the help of the equation which is shown as follows.
The first expression determines the DC component of the signal. The second and the third expressions determine the amplitude and the phase of the components that compose the periodic signal. The Fourier series can be represented in the compact form as shown follows:
So it can be noted here that the Fourier series of the signal can be calculated by evaluating the Fourier coefficients. The derivation of the Fourier coefficients is out of the scope of this post.
Convergence of the Fourier series:[otw_is sidebar=otw-sidebar-1]
It can be noted from the expression of the Fourier series that this series is basically infinite so the concept of convergence becomes essential while considering the Fourier series of the periodic signals. The Fourier series of the signal is convergent if it follows the following conditions:
Dirichlet Conditions: The Dirichlet conditions are as follows:
- The signal should be single valued.
- The integral of the signal over the time period is finite.
- The discontinuities in the signal in the time period should be finite.
If the signal fulfills the above mentioned conditions then its Fourier series would be convergent.
Spectrum of the Signal:
Spectrum of the signal can defined as the set of the sinusoidal signals which compose the periodic signal. So it should have noted that the Fourier Series of the Periodic Signal represents the spectrum of the signal.
Amplitude spectrum of the signal is the set of all the amplitudes of the sinusoidal signals that comprise the periodic signal. The amplitude of each component in the Fourier series can be represented in the form of the graph as follows:
The above figure shows the amplitude of the sinusoidal component corresponding to the frequency of the component.
Phase spectrum of the signal is the set of all phases of the sinusoidal components that comprise the periodic signal. The phase of each component in the Fourier series can be represented in the form of the graph as follows:
The above figure shows the phase of the sinusoidal component corresponding to the frequency of the component.
That is all for now I hope this post would be helpful for you. In the next post I will come up with more interesting educational topics till then stay connected, keep reading and enjoy learning.