Fourier Transforms, symmetry, real/imaginary

1. What is real and imaginary part of Fourier transform?
2. Is Fourier transform of real function symmetric?
3. What is symmetry property of Fourier transform?
4. Is the Fourier transform of a real function real?

What is real and imaginary part of Fourier transform?

The real portion of an FFT result is how much each frequency component resembles a cosine wave, the imaginary component, how much each component resembles a sine wave.

Is Fourier transform of real function symmetric?

When we take the the Fourier Transform of a real function, for example a one-dimensional sound signal or a two-dimensional image we obtain a complex Fourier Transform. This Fourier Transform has special symmetry properties that are essential when calculating and/or manip- ulating Fourier Transforms.

What is symmetry property of Fourier transform?

Symmetry Properties

Represent x(t) as the sum of an even function and an odd function (recall that any function can be represented as the sum of an even part and an odd part). x(t)=xo(t)+xe(t) Express the Fourier Transform of x(t), substitute the above expression and use Euler's identity for the complex exponential.

Is the Fourier transform of a real function real?

Theorem 5.3 The Fourier transform of a real even function is real. Theorem 5.4 The Fourier transform of a real odd function is imaginary.