Spectral Analysis of Signals Using Fourier Transforms: Frequency Spectrum, Power Spectrum, and Energy Spectrum

Abstract

Signal analysis in the frequency domain occupies a central position in applied mathematics and engineering. This work presents a structured examination of three fundamental spectrum types derived through Fourier transform theory: the frequency spectrum, the power spectrum, and the energy spectrum. For each type, precise mathematical definitions are established, computational procedures are outlined, and illustrative numerical examples are worked through using the Continuous Fourier Transform (CFT), the Discrete Fourier Transform (DFT), and the Fast Fourier Transform (FFT). The internal relationships among these three spectral representations are discussed, and Parseval’s theorem is invoked to validate the equivalence between energy quantified in the time domain and energy quantified in the frequency domain. All numerical results obtained through digital computation accord with analytically derived expectations, confirming the theoretical framework and illustrating the broad engineering utility of spectral methods.