This chapter extends the Fourier transform to the realm of generalized functions or distributions, enhancing its ability to handle a wider range of mathematical constructs. It begins by highlighting the Fourier transform’s universality and application to functions that decrease rapidly, paving the way for a deeper understanding of how generalized functions behave under transformation. The chapter introduces Dirac’s delta function, examining its essential properties and use in modelling impulse responses. It further explores how functions can induce distributions and cover the derivatives of distributions, allowing the Fourier transform to capture complex, non-standard behaviors often encountered in both theory and practical applications. Key distribution properties are discussed, including the Dirac comb (or Sha function), which models periodic sampling and is crucial for understanding the discrete aspects of sampled signals.