What is the Fourier Transform and why is it important?

The Fourier Transform decomposes a signal into its constituent frequencies, converting time-domain representation to frequency-domain: X(f) = integral of x(t)*e^(-j2*pi*f*t)dt. It reveals frequency content, bandwidth, and spectral characteristics. Important for: filter design (frequency-selective systems), modulation analysis, spectral analysis, and understanding system frequency response. The inverse transform reconstructs time signal from spectrum. Variants include DTFT (discrete-time), DFT (discrete frequency), and FFT (efficient algorithm).