Notes of Signal Theory

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SYSTEM THEORY A system is defined by a function T{·} which transforms the input sequence x[n] in the output sequence y[n]. LTI systems Linear Time Invariant systems. Z-Transform The Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain representation. Definition Representing the discrete -time signal as the sequence x(n) we obtain: Properties Z-Transform and Laplace Transform The Z-transform can be considered as a discrete-time equivalent of the Laplace transform (s-domain). where 𝑠 =𝜎+ 𝑗𝜔 then 𝑧 = 𝑒𝜎𝑇𝑒𝑗𝜔𝑇 hence |𝑧|= 𝑒𝜎𝑇 ∠𝑧 = 𝜔𝑇 Continuous-time Fourier transform is evaluated on the Laplace s-domain's imaginary line, and it is stable in the left half-plane. Each frequency component of the signal is mapped as a point on the imaginary axes: Discrete-time Fourier transform is evaluated over the unit circle of the z -domain, and it is stable within the unit circle. Each frequency component of the signal is mapped as a point on the unit circle: 𝜔=2𝜋𝑓 → 𝑓=𝜔/2𝜋 and 𝑇 =1/𝑓𝑠 so 𝜔𝑇=2𝜋 𝑓/𝑓𝑆 Linearity 𝑍[𝑎𝑥1(𝑛)+𝑏𝑥2(𝑛)]=𝑎𝑋(𝑧)+𝑏𝑋(𝑧) Time shifting 𝑍[𝑥(𝑛−𝑘)]=𝑧−𝑘𝑋(𝑧) Mult. by exponential 𝑍[𝑎𝑛𝑥(𝑛)]=𝑋( 𝑧 𝑎) Convolution 𝑍[𝑥1(𝑛)⊗𝑥2(𝑛)]=𝑋1(𝑧)𝑋2(𝑧) Differentiation 𝑍[𝑛𝑥(𝑛)]=−𝑧 𝑑𝑋(𝑧) 𝑑𝑧 Complex conjugate 𝑍[𝑥∗(𝑛)]=𝑋∗(𝑧∗) Time reversal 𝑍[𝑥(−𝑛)]=𝑋[𝑧−1] wrt the CONTINUOUS – LAPLACE DOMAIN SIGNAL THEORY Signals are representation of quantitative measurements in function of an independent variable. 1-D SIGNAL defined in time : An ordered sequence of numbers that describes the variations and trends of a quantity in time. ➔ The order of the numbers often determined by the order of measurements (or events) in «time» ➔ time is the axis that identifies the order (= ordering axes) MULTIDIMENSIONAL SIGNALS: Multidimensional sequence of…