Fourier transform frequency and time relation

Question

From Fourier transform:

$$X(\omega) = \int_{-\infty}^{+\infty} x(t)e^{-j\omega t} \,dt$$

how does the frequency $\omega$ and time $t$ are related? and why half of frequencies will be negative and half - positive?

Answer 1

do you what does this equation do and how will it convert time domain signal ${x(t)}$ to frequency domain ${X(w)}$? it takes every fundamental frequency and matches(sum over (integrate)) with time domain signal, and find how much quantity of a given time domain signal ${x(t)}$ relates with it. so if a complex time domain signal has some amount of 10Hz, some amount of 50 Hz , fourier transform will tell you how much of that frequency component is present in a given time signal. if there is no component of say 35 Hz present in a given signal, its co-efficient will be zero. I'll try to add more info later sometime.

To Analyze signals/images, it can be extremely useful to decompose them into a sum of more elementary pieces or patterns, and then operate on decomposed version, piece by piece. these simpler pieces are basic waveforms. they are essential building blocks for signals. In context of Fouriers analysis for analog signals these basic waveforms are simply sines and cosines. Specifically, the two basic waveforms of interest are cos(wt) and sin(wt), or their complex exponential equivalent ${e^{iwt}}$, where w acts as a frequency parameter.

$${e^{i \theta}=cos(\theta)+isin(\theta)}$$

now, assuming you the time convolution theorem and how convolution in one domain is multiplication in another domain. we are dealing with two domain 1. Time Domain 2. Frequency domain. DFT/DCT converts time domain signal into frequency domain. which in layman term is give you an idea how much of each frequency component is present in the given complex signal.

FT can be better understand by DFT.

$${X_k= \Sigma _{(n=0..N-1)} x_n \cdot e^{-i2\pi kn/N} }$$ where

• N = number of time samples we have
• n = current sample we're considering (0 .. N-1)
• ${x_n}$ = value of the signal at time n
• k = current frequency we're considering (0 Hertz up to N-1 Hertz)
• ${X_k}$ = amount of frequency k in the signal (amplitude and phase, a complex number)

So, analogous to this is, if you are given food item say pizza and you want to find out how much salt , chilly, and other ingredients are present in a given (complex food), run a Fourier transform and get the answer. so if your signal ${x(t)}$ is the combination of various fundamental frequencies with a proper quantity of each, then Fourier transform will tell you strength of a given frequency. A topic from DSP

• And if you are interested in real world examples then JPEG, MPEG and all digital TV compression are possible because of this (or its variants like DCT/FFT). where you know after decomposing an image, how to compress/drop high frequency components which are not easily visible to eye.