Fourier transform pairs

For every time domain waveform, there is a corresponding frequency domain waveform, and vice versa. For example a rectangular pulse in the time domain coincides with the sinc function [i.e., sin(x)/x] in the frequency domain. Duality provides that the reverse is also true; a rectangular pulse in the frequency domain coincides with the sinc function in a time domain. Waveforms that correspond to each other in this manner are called Fourier transforms pairs. In this article, I will present several common pairs. I hope you enjoy. 

Let's look around a delta function in the time domain first. It is a simple waveform, and has an equally simple Fourier transform pair.  Assume that there is a delta function in the time domain, which has impulse at 0. Then, its frequency spectrum shows the constant magnitude, while the phase is entirely zero. Assume that the time domain waveform is shifted a few samples to the right (e.g., the delta function will have impulse at 4 ). Shifting the time domain waveform does not affect the magnitude, but adds a linear component to the phase. The following figures might help you to understand easily. 

Let's present the same information in the rectangular form. The pictures below show delta functions with the frequency domain in rectangular form. 

As is usually the case, the polar form is much easier to understand because the magnitude is just a constant value, while the phase is a simple line.  In comparison, the real and imaginary parts are sinusoidal oscillations which are difficult to understand.

The other interesting feature is the duality of the DFT(i.e. discrete fourier transform). Usually, each sample in the frequency domain corresponds to a sinusoid in the time domain. However, the reverse is also true: each sample in the time domain is corresponds to a sinusoid in the frequency domain. For instance, an impulse at sample number four ( 'd' of above picture) in the time domain results in four cycle of cosine function in the real part of the frequency spectrum, and four cycle of sine function in the imaginary part. 

As you recall, each sample in the time domain results in a cosine wave being added to real part of the frequency spectrum, and a negative sine wave being added to the imaginary part. The amplitude of each sinusoid is come from the amplitude of  each impulse in the time domain. 

* I think it's better to use "understand" than "catch the meaning". Usually, "catch the meaning" is better to use if you don't understand a certain expression in a language or literary stuff. Can be used for films too