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

2. 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

3. 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

4. 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 𝑠+ 1 6
ℎ 𝑡 =exp  −𝑡 6  cos  𝑡 − sin  𝑡 6

5. 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

6. 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

7. 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