# Fourier Transform vs. Lapace Transform

Background

Time Domain Function: f(t)

Frequency Domain Function:f(w)

Complex Number: S= σ + jω (Real + Imaginary)

• Real part of the coefficient models the constant, decay, or growth behavior of the signal.
• Imaginary part of the coefficient models the constant amplitude oscillation part of the signal
• Complex part (i.e a mix of both) models time varying amplitude and oscillation parts of the function
• Complex Conjugate Pairs: σ ± jω

Laplace Transformation

• Transforms time domain equation f(t) to complex s domain equation F(s)
• S is a complex variable = real + imaginary
• Inverse Laplace Transformation
• Transforms complex s domain equation F(s) to time domain equation f(t)

Fourier Transformation

Same as Laplace Transformation with real part zeroed out (σ = 0) , and the signal is causal (e.g. any real world signal is causal which does not contain data from the future, i.e S=0, T <0 ))

Summary

• Fourier transformation is used for purely frequency analysis, with decaying factor or real component σ removed, leaving the oscillatory part for analysis (i.e frequency analysis).