Signals & Systems ,Fourier, Laplace, Z Transforms

Continuous and discrete representation of LTI systems:

Fourier transform:

The Fourier transform decomposes a function of time into its constituent frequencies. The Fourier Transform is an important image processing tool used to decompose an image into its sine and cosine components. The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is used to represent a general nonperiodic function by a continuous or integral superposition of complex exponentials.

Fourier transforms are used to convert/represent a time-varying function in the frequency domain. Fourier is mainly used for steady-state signal analysis, while Fourier is good for continuous signals. Laplace and z transforms: The Laplace transform converts integral and differential equations into algebraic equations. This is like phasors, but. • applies to general signals, not just sinusoids.

while Laplace is used for the analysis of transient signals. Laplace is good at finding pulse response, step functions, delta functions,  The Z transform converts a discrete time signal,

Z transforms are very similar to Laplace transforms but are discrete conversions of time intervals, closer to digital implementations. They all look the same because the methods used to convert are very similar.

Time and frequency analysis:

Time-frequency analysis identifies the time at which various signal frequencies are present, usually by calculating a spectrum at regular time intervals.

Sonogram Also called a “short-time Fourier transform”, an ultrasound is a two-dimensional image created by calculating Fourier spectra using a sliding time window: