Fourier transform. Spectral analysis brief theoretical review fourier analysis

LABORATORY WORK №3
FOURIER TRANSFORM. SPECTRAL ANALYSIS
BRIEF THEORETICAL REVIEW
FOURIER ANALYSIS
The theory of Fourier series lies in the idea that most signals, and all engineering signals, can be represented as a sum of sine waves (including square waves and triangle waves). This analysis can be expressed as a Fourier series. There are two types of Fourier expansions:

• 
  Fourier series (continuous, periodic signals): If a (reasonably well-behaved) function is periodic, then it can be written as a discrete sum of trigonometric or exponential functions with specific frequencies.
  
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  Fourier transform (continuous, aperiodic signals): A general function that isn’t necessarily periodic (but that is still reasonably well-behaved) can be written as a continuous integral of trigonometric or exponential functions with a continuum of possible frequencies.
  

FOURIER TRANSFORM
The Fourier transform is very similar to the Laplace transform. The Fourier transform uses assumption that any finite time-domain can be broken into a finite sum of sinusoidal (sine and cosine). Under this assumption, the Fourier transform converts a time domain signal into its frequency domain representation as a function of the radial frequency. The frequency domain representation is also called the spectrum of the signal.
The Fourier transform of a signal is defined as follows:
(1)
It is possible to show that the Fourier transform is equivalent to the Laplace transform when the following condition is true:

then
(2)
The formulas (1) and (2) represent direct Fourier transform.
The inverse Fourier transform is defined as follows:
(3)
The Fourier transform exists for all functions that satisfy the following condition
.
For arbitrary signals, the signal must be digitized, and a Discrete Fourier transform (DFT) performed. The standard numerical algorithm used for the DFT is called Fast Fourier Transform (FFT) or Discrete FFT (DFFT). The fast Fourier transform is a mathematical method for transforming a function of time into a function of frequency. Sometimes it is described as transforming from the time domain to the frequency domain. It is very useful for analysis of time-dependent phenomena.