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A Closer Look at Fourier Theory

Fourier theory is formulated on the idea that waveforms can be represented in terms of cosines and sines of different frequency. In general, Fourier theory states that any kind of signals can be composed as a sum of many sinusoids. This theory is used in antenna studies, random process modeling, linear systems analysis, quantum physics, probability theory, and boundary-value programs.

Fourier Theory

Fourier theory is the fundamental idea for frequency analysis in engineering and science, and this is the mathematical basis to learn wave optics. There are different types of Fourier theory based on whether the problem is digital, analytical, analog, and signal processing. The different types are Fourier series, Fourier Transform, Discrete-Time Fourier Transform, Discrete Fourier Transform, and Z Transform.

Fourier Transform

The Fourier transform separates or decomposes a function or waveform into sinusoids of different frequency which can be summed to the original waveform. It differentiates or identifies the different frequency of the waveforms and their relative amplitudes. 

The definition for the Fourier transform and the inverse properties is given by:

F (a, b) = SUM {F (c, d) * exp (-j *2*pi*(a*c + b*d)/N)} and

F (c, d) = SUM {F (a, b) * exp (+j *2*pi*(a*c + b*d)/N)}

Where the value of a, b, c, and d can be 0 to N -1, the value of j is SQRT (-1) and SUM is given by the double summation values of a, b or c, d.

Fourier Transform Properties

The Fourier transform has the basic properties such as linearity, translation, modulation, scaling, convolution, and conjugation.

Scaling: This property states that if the width of a signal is decreased, keeping height at a constant value, the Fourier transform becomes shorter and wider. By increasing the width, the signal becomes taller and narrower.

This is defined as {f (ax)} = |a|?¹ F (s/a) where the value of ‘a’ can be a nonzero, real constant.

 Shifting: This property states that the Fourier transform of the shifted signal is equal to the value of the unshifted signal with an exponential value that has a linear phase.

This is defined as {f (x-x?)} = F (s) exp (i2x?s) where x? is a real number.

Convolution: The convolution property is the product of the separate transforms.

This is denoted by G (w) = {f (a) h (a)}.

Correlation: The correlation integral forms a pair of Fourier transform.

 This is defined as h(s) = f (a) g (s + a) du

Parseval’s Theorem

Parseval’s theorem defines that square value of a function is proportional to the square value of its transform. This theorem is also known as Rayleigh’s Identity or Rayleigh’s energy theorem.

This theorem is given as M = ∑ pnxn and m = ∑ qnxn  

Where the value of x is substituted with cos u + i sin u to form the real and imaginary parts.

The final expression is given as M = a + ib and m = c + id.

Sampling Theorem

The sampling theorem defines that a bandlimited real signal can be reconstructed perfectly only if the sampling frequency is larger than the frequency of the sampled signal. The minimum sampling frequency is called as Nyquist frequency or Nyquist rate. A signal that is sampled below the Nyquist frequency is called undersampled.

Aliasing

Aliasing is the effect that makes different signals to become identical when they are sampled. Also, it refers to the reconstruction of a signal from samples and the resultant signal differs from the original signal.

Discrete Fourier Transform

The discrete Fourier transform (DFT) is used in the Fourier analysis and signal processing and this also changes one function into another. The real numbers are given as input to the DFT and can be easily computed by Fast Fourier Transform algorithms.

The equation is given as:

Fx=1N(1−p)2n=0N−1fnWN−kn     where x =012N−1, p is a real number, and q can be either 1 or -1.

Fast Fourier Transform

The Fast Fourier Transform (FFT) is an algorithm for DFT, formulated by Cooley and Tukey in 1965. There are two types of FFT algorithms: decimation-in-frequency and decimation-in-time.

Fourier theory is a great tool in engineering and science. The Fourier transform is used in the restoration of the astronomical data. This tool is used in most fields of science to change a problem into another form such that it can be easily solved.