Fourier Transform and Superposition of Sinusoidal Functions

Abstract

The Fourier transformation theory provides the theoretical basis for understanding the representation of any signal as a superposition of sinusoidal functions in time and frequency planes. This chapter deals with the Fourier and inverse Fourier transforms of sequences in the time and frequency domains. A single pulse in the time (frequency) domain is transformed to a sinusoidal function in the frequency (time) domain. Thus, a sequence composed of pulses is represented as a superposition of sinusoidal functions with magnitude and phase. The auto-correlation of a sequence in the time domain, whose Fourier transform discarding phase information determines the power spectral function in the frequency domain, gives an estimate of the period of the sequence. In contrast, the auto-correlation of the spectral function, including the magnitude and phase in the frequency domain, yields the time sequence of the squared original samples that show the change in the signal power or signal dynamics in the time domain. The change in the signal power is formally defined by the envelope for the analytic signal. As well as the envelope the periodic property represents the signature of a waveform in the time domain. The periodicity is lost as the spectral function is spread over the line spectral components. This loss of periodicity can be represented by the envelope of the auto-correlation function of the waveform. The envelope as a limit is a sinc function when the power spectral function is uniformly spread around the discrete spectral components.