Laplace and Z transforms EE422G Signals and Systems Laboratory Laplace and Z transforms Kevin D. Donohue Electrical and Computer Engineering University of Kentucky

Laplace Transform Fourier Series and Transforms are used for modeling signals in terms of their frequency content, while systems and signal interactions are modeled with Laplace transforms, given by To see relationship with Fourier let s =  + j: Under what conditions does the Laplace Transform become the Fourier Transform? 𝑋 ˆ 𝑠 = −∞ ∞ 𝑥 𝑡 exp −𝑠𝑡 𝑑𝑡 𝑋 ˆ 𝑠 = −∞ ∞ 𝑥 𝑡 exp −σ𝑡 exp −𝑗ω𝑡 𝑑𝑡 𝑋 ˆ 𝑠 = −∞ ∞ 𝑥 𝑡 exp −σ𝑡 cos ω𝑡 −𝑗sin ω𝑡  𝑑𝑡

Convolution Relationships Given a relaxed linear time invariant system, for input x(t) and impulse response h(t) the output is computed through the convolution integral: In the Laplace domain, convolution become multiplication: where is the system transfer function (TF). 𝑦 𝑡 = −∞ ∞ ℎ 𝑡−τ 𝑥 τ 𝑑τ 𝑌 ˆ 𝑠 = 𝑋 ˆ 𝑠 𝐻 ˆ 𝑠 𝐻 ˆ 𝑠 𝐻 ˆ 𝑠 = 𝑌 ˆ 𝑠 𝑋 ˆ 𝑠

Differential Equation and TF Given a relaxed system (zero for initial conditions) described by differential equation: Apply Laplace Transform relationships to find the Transfer function and the impulse response. Show: 𝑥 𝑡 =6 𝑦 𝑡 +2 𝑦 𝑡 +𝑦 𝑡 𝐻 ˆ 𝑠 = 1 6 𝑠 𝑠 2 + 1 3 𝑠+ 1 6 ℎ 𝑡 =exp  −𝑡 6  cos  5 5 𝑡 − 5 6 sin  5 3 𝑡 6

The Z Transform Consider a numerical integration of the Laplace integral Let z = exp(s Ts ), which implies z-n = exp(-n s Ts ) Now let Ts = 1 and define Z-transform as: 𝑋 𝑠 = 𝑛=−∞ ∞ 𝑥 𝑛 𝑇 𝑠 exp −𝑠𝑛 𝑇 𝑠 𝑇 𝑠 𝑋  ln 𝑧 𝑇 𝑠  = 𝑛=−∞ ∞ 𝑥 𝑛 𝑇 𝑠 𝑧 −𝑛 𝑇 𝑠 𝑋 ˆ 𝑧 = 𝑛=−∞ ∞ 𝑥 𝑛 𝑧 −𝑛

z- Transform Example Given impulse response h[n] = an u[n] (where u[] is the unit step), find the TF (Use ZT of impulse response) and comment on stability and convergence. Show 𝐻 ˆ 𝑧 = 𝑧 𝑧−𝑎

z- Transform Example Find TF from difference equation for system in input x[n] and output y[n]: and comment on stability Show 𝑦 𝑛 =𝑥 𝑛 −𝑥 𝑛−1 +0.5𝑦 𝑛−1 −0.25𝑦 𝑛−2 𝐻 ˆ 𝑧 = 𝑧 2 −𝑧 𝑧 2 −0.5𝑧+0.25