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Why does Fourier transform give complex numbers?
6 Answers. The FFT is fundamentally a change of basis. The basis into which the FFT changes your original signal is a set of sine waves instead. In order for that basis to describe all the possible inputs it needs to be able to represent phase as well as amplitude; the phase is represented using complex numbers.
What is the complex conjugate of a Fourier transform?
The Fourier transform of f∗(x) (the complex conjugate) is g∗(−u). If f(x) is real, then g(−u) = g∗(u) (i.e. the Fourier transform of a real function is not necessarily real, but it obeys g(−u) = g∗(u)).
Which is the conjugated expression of the Fourier transform?
[Note that they are not the respective Fourier transforms of the real and imaginary parts of the time domain signal.] Also, the notation X ∗ ( j ω) is to be understood as [ X ( j ω)] ∗, i.e., the whole expression for X ( j ω) is conjugated.
Which is the most important complex matrix in Fourier transform?
The most important complex matrix is the Fourier matrix Fn, which is used for Fourier transforms. Normally, multiplication by Fn would require n2 mul tiplications. The fast Fourier transform (FFT) reduces this to roughly n log. 2 n multiplications, a revolutionary improvement.
Which is the complex conjugate of x ( j ω )?
X ∗ ( j ω) is the complex conjugate of X ( j ω). So if where X R ( ω) and X I ( ω) refer to the real and imaginary parts of X ( j ω), respectively. [Note that they are not the respective Fourier transforms of the real and imaginary parts of the time domain signal.]
How is Fourier transform used to write periodic functions?
Fourier series is a way of writing a periodic function or signal as a sum of functions of different frequencies: (x)= a0 + a1 cos x + b1 sin x + a2 cos 2x + b2 sin 2x + ···. When working with finite data sets, the discrete Fourier transform is the key to this decomposition.