Laplace transform (unilateral) of a function g(t)

= L(g(t))

s ... complex variable (s = s + jw)

Shorthand notation: g(t) ↔ G(s)

Key properties of Laplace transform:

• Linear operation
• If g(t) ↔ G(s), then dg(t)/dt ↔ sG(s) - g(0)
• If g(t) ↔ G(s), then
• If g₁(t) ↔ G₁(s) and g₂(t) ↔ G₂(s), then

* ... convolution

Z-transform

z ... complex variable

z-transform bears the same kind of relationship to the discrete time Fourier transform as the Laplace transform does to the Fourier integral transform.

It has similar properties with respect to difference equations as the Laplace transform does with respect to differential equations.

z-transform can be seen as a shorthand notation for Laplace transform of discrete-time signals by defining z ≡ esT.

Properties of z-transform:

• If x(n) ↔ X(z), then x(n-k) ↔ z-kX(z)
• Convolution in the time domain is equivalent to multiplication in the frequency domain
• Multiplication in the time domain is equivalent to convolution in the frequency domain

Related Reading

Gregorian, R., Temes G. C., Analog MOS integrated circuits for signal processing, Wiley, 1986.

Johns, D. A., Martin, K., Analog integrated circuit design, Wiley, 1997.

Razavi, B., RF Microelectronics, Prentice Hall, 1998.

Bendat, J. S., Piersol, A. G., Random Data: Analysis & Measurement Procedures, J. Wiley, 2000.