Electric Circuit 2
(ELE 211)
Name:
Mohamed Abdelrazek Ibrahim
Abou-Shahin
Sec: 3
Academic number:
220353
(Fourier Transformation Report)
• Fourier Transformation
The Fourier transform is a mathematical tool used to analyze signals. It essentially
takes a signal, which can be represented as a function of time (t), and breaks it down
into its constituent frequencies. In other words, it tells you how much of each
frequency is present in the original signal.
The difference between the Fourier transform and the Fourier series is that the
Fourier transform is applicable for non-periodic signals, while the Fourier series is
applicable to periodic signals.
• Periodic Function:
A periodic function repeats itself every T seconds, so T is the period of the
function. A periodic function satisfies the relationship,
f (t) = f (t ± nT)
Some examples of periodic functions
Fourier series representation of a periodic function
av, an, and bn are known as the Fourier coefficients,
The effect of symmetry on the fourier coefficients
Four types of symmetry make the task of evaluating the Fourier
coefficients easier.
• even-function symmetry
• odd-function symmetry
• half-wave symmetry
• quarter-wave symmetry.
Even-Function Symmetry
f (t) = f (-t)
Odd-Function Symmetry
f (t) = - f (-t)
Half-wave symmetry
f (t) = - f (t – T / 2)
a periodic function has half-wave symmetry if after it is shifted one-half period and
inverted.
Choosing where t 0 = can make a periodic function even, odd, or neither. (a) A
periodic triangular wave that is neither even nor odd. (b) The triangular wave of (a)
made even by shifting the function along the t axis. (c) The triangular wave of (a)
made odd by shifting the function along the t axis.
Quarter-Wave Symmetry
The term quarter-wave symmetry describes a periodic function that has half-wave
symmetry and, in addition, symmetry about the midpoint of the positive and
negative half-cycles.
If the function is made even
If the function is made odd
Fourier Transformation
The Fourier transform extends this frequency-domain representation to functions
that are not periodic. the Fourier transform is not a new transform. It is a special
case of the bilateral Laplace transform, with the real part of the complex frequency
set to zero.
Hence σ = zero,
Inverse Fourier transform
Applications
Frequency Analysis of Signals: Electric fields can carry signals, such as those in
electromagnetic waves or transmission lines. By applying the Fourier transform,
we can determine the frequency spectrum of the electric field, revealing the
different frequencies present in the signal. This information is crucial for tasks like
filtering out unwanted noise, identifying specific signal components, or
understanding the bandwidth requirements of a system.
Transient Analysis: When dealing with time-varying electric fields, like those
generated by pulses or sudden changes in voltage, the Fourier transform helps
understand the behavior over time. By analyzing the frequency content, we can
predict how the electric field changes and how long it takes for transients to settle.
Circuit Analysis: In analyzing complex circuits, the Fourier transform can be used
to study the frequency response of filters, transmission lines, and other
components. By understanding how these components behave at different
frequencies, engineers can design circuits with desired filtering characteristics or
predict signal distortion.
Spectral Analysis of Light: In the field of optics, the Fourier transform is used to
analyze the spectral content of light. By analyzing the electric field of light, we can
determine its color components or identify the presence of specific spectral lines
emitted by atoms or molecules.
A Fourier Series has many applications in mathematical analysis as it is defined as
the sum of multiple sines and cosines. Thus, it can be easily differentiated and
integrated, which usually analyses the functions such as saw waves which are
periodic signals in experimentation. It also provides an analytical approach to solve
the discontinuity problem. In calculus, this helps in solving complex differential
equations.
• Fourier Transformation Report:
• References:
NO. Reference URL
1- Electric circuits https://www.pearson.com/en-us/subject-
nilsson and riedel catalog/p/electric-
12th edition circuits/P200000003451/9780137648160
2- Byjus site https://byjus.com
3- Chat GPT https://gemini.google.com
