pulssisarjaksi

Pulssisarjaksi is a term used in signal processing to describe representing a time-domain signal as a sum of shifted and scaled pulses of a chosen shape. The idea is to express a signal as a pulse series, where a basic pulse p(t) is repeated at times t = kT with amplitudes a_k, yielding a function x(t) = sum_k a_k p(t − kT). Here T is the pulse repetition interval and {a_k} is a sequence of amplitudes. The pulse shape p(t) can be rectangular, Gaussian, or another localized function, and the choice of p(t) together with T determines the time and frequency characteristics of the resulting signal.

In analysis, the frequency-domain representation follows X(ω) = P(ω) · ∑_k a_k e^(−jωkT), where P(ω) is the Fourier transform of p(t) and the sum is the discrete-time Fourier transform of the amplitude sequence. This form shows how the pulse shape acts as a spectral envelope for the spectral components created by the pulse sequence. The pulssisarjaksi representation is particularly convenient for understanding synthesis, filtering, and modulation schemes, and it is sparse when many a_k are zero.

Applications of pulssisarjaksi include digital communications (for modeling pulse-amplitude and pulse-position modulation), radar and ultrasound imaging, and time-domain signal reconstruction. Variants arise from using different pulse shapes, which produce characteristic spectral features (for example, rectangular pulses yield sinc-like spectra, while Gaussian pulses produce smooth spectra). The concept is related to impulse trains, basis expansions, and convolutional representations in signal processing.