Continuous -time Fourier Transform

The Fourier series represents periodic functions as a sum of sinusoids using discrete frequency components. When the period approaches infinity, resulting in a single nonperiodic pulse, the discrete summation evolves into a continuous integral called the Fourier transform. This transformation allows analysis of nonperiodic functions in the frequency domain.

Dirichlet conditions establish criteria for representing periodic functions with Fourier series. A function must have finite discontinuities, a finite number of maxima and minima, and be absolutely integrable over its period. If a function fails these conditions, it cannot be represented by a Fourier series.

As the period of a pulse-train waveform increases, the frequency spacing of the line spectra decreases. When the period extends to infinity, the fundamental frequency approaches zero, and discrete frequency components merge into a continuous frequency spectrum represented by the Fourier transform.

The Fourier transform converts images into the frequency domain, enabling filtering techniques to enhance details and reduce noise. After processing, the inverse Fourier transform converts the filtered image back to the spatial domain. This approach is foundational in medical imaging, remote sensing, and digital photography applications.

In Fourier series, a periodic function decomposes into discrete sinusoids at frequencies nω₀. As period extends to infinity, the fundamental frequency ω₀ approaches zero, and the discrete summation over nω₀ becomes a continuous integral over frequency ω. This integral defines the Fourier transform X(ω) in the frequency domain.

Fourier series only represent periodic functions, but many real-world signals are nonperiodic. The Fourier transform extends this capability by treating nonperiodic functions as periodic functions with infinite period, converting them into continuous frequency representations. This enables frequency-domain analysis of any signal meeting integrability requirements.

When a pulse-train period becomes infinite, resulting in a single isolated pulse, the discrete line spectra transform into a continuous spectrum. The summation in the Fourier series evolves into a continuous integral, and the frequency components no longer appear as isolated lines but as a continuous distribution across the frequency domain.