The Fourier Transform

Fourier Analysis · Axiom Academy

Discover what happens when we extend Fourier analysis beyond periodic functions. The Limitation of Fourier Series Try to visualize a simple pulse signal using the Fourier Series. Watch what happens. What if We Make the Period Longer? Use the slider to increase the period T. Watch what happens to the "ghost" repetitions. The Frequency Spectrum Goes Continuous Now let's look at what happens in the frequency domain as we increase T. ∫ The Fourier Transform Formula From this limiting process, we derive the Fourier Transform integral. The Fourier Transform is the natural extension of the Fourier Series when we let the period T → ∞. It allows us to analyze non-periodic signals. As the period increases, the spacing between frequency components shrinks. The discrete frequency spectrum becomes a continuous frequency spectrum. The Fourier Transform and Inverse Fourier Transform form a perfect pair, allowing us to move freely between time and frequency domains. The Fourier Transform works for virtually any signal: pulses, transients, noise, audio, images, and more. It's one of the most powerful tools in mathematics and engineering.

This is the written version of the interactive lesson above. See the full Fourier Analysis course.