Fourier Transform

Last modified: July 21, 2026

Fourier Transform โ€” simple explanation with equations ๐ŸŽง๐Ÿ“

This note keeps the math clean and minimal, and every equation is paired with intuition.

1. Big picture (again, but mathematically)

The Fourier Transform (FT) converts a signal from:

  • Time domain โ†’ x(t)
  • Frequency domain โ†’ X(f) or X(ฯ‰)

It answers:

โ€œHow much of each frequency is present in the signal?โ€

2. Fourier Series (where everything starts)

For a periodic signal with period T, we write it as a sum of sinusoids:

$$ x(t) = \sum_{k=-\infty}^{\infty} c_k e^{j 2\pi k f_0 t} $$

where

$$f_0 = \frac{1}{T}$$

and the coefficients are:

$$ c_k = \frac{1}{T} \int_{T} x(t), e^{-j 2\pi k f_0 t}, dt

$$

๐Ÿ”น Meaning:
Each c_k tells how much of frequency k fโ‚€ is present.

3. From Fourier Series โ†’ Fourier Transform

What if the signal is not periodic?

Idea:

  • Let the period T โ†’ โˆž
  • Frequency spacing fโ‚€ โ†’ 0
  • Discrete frequencies become continuous

That converts the sum into an integral.

4. Fourier Transform (continuous-time)

Forward Fourier Transform

$$X(f) = \int_{-\infty}^{\infty} x(t)\, e^{-j 2\pi f t}\, dt$$

or using angular frequency ฯ‰ = 2ฯ€f:

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

๐Ÿ”น Interpretation:

  • Multiply the signal by a complex sinusoid
  • Integrate โ†’ measure similarity
  • Result = strength of frequency f

Inverse Fourier Transform

\[ x(t) = \int_{-\infty}^{\infty} X(f)\, e^{j 2\pi f t}\, df \]

(or)

\[ x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega)\, e^{j \omega t}\, d\omega \]

๐Ÿ”น Meaning:
Add back all frequencies to reconstruct the signal.

5. What is X(f) exactly?

X(f) is complex:

$$ X(f) = |X(f)| e^{j \angle X(f)} $$
  • |X(f)| โ†’ Magnitude spectrum (strength)
  • โˆ X(f) โ†’ Phase spectrum (timing info)

Most plots show only |X(f)|.

6. Discrete Fourier Transform (DFT)

Computers work with samples, not continuous signals.

Given N samples x[n], the DFT is:

$$ X[k] = \sum_{n=0}^{N-1} x[n]\, e^{-j \frac{2\pi}{N} k n} $$

where:

  • n โ†’ time index
  • k โ†’ frequency bin
  • Total bins = N

Inverse DFT

\[ x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k]\, e^{j \frac{2\pi}{N} k n} \]

7. FFT (important clarification โšก)

  • FFT is NOT a new formula
  • It is a fast algorithm to compute the DFT

Mathematically:

\[ \text{FFT} \equiv \text{DFT} \]

Difference is only speed, not result.

8. Frequency resolution (very important)

For DFT:

\[ \Delta f = \frac{f_s}{N} \]

where:

  • f_s = sampling frequency
  • N = number of samples

๐Ÿ”น More samples โ‡’ better frequency resolution
๐Ÿ”น Zero-padding improves visual smoothness, not true resolution

9. One-line intuition for the equations ๐Ÿง 

\[ X(f) = \int x(t)\, e^{-j 2\pi f t}\, dt \]

=
โ€œSlide a sine wave of frequency f over the signal and measure how well it matches.โ€

10. Ultra-short summary

  • Fourier Series โ†’ periodic signals
  • Fourier Transform โ†’ non-periodic signals
  • DFT โ†’ sampled signals
  • FFT โ†’ fast way to compute DFT

If you want next:

  • ๐Ÿ” Derivation: Fourier Series โ†’ FT step-by-step
  • ๐Ÿ” Physical meaning of phase
  • ๐Ÿงฎ DFT bins, leakage, windowing
  • ๐Ÿง  FFT as filtering operation

Just say th