Advanced Math 2

Arithmetic Sequences

Geometric Sequences

Fibonacci Sequence

Power Series

Taylor Series

Maclaurin Series

• Taylor Series centered @ x = 0

Laplace Transform

• Applied to real (causal) application

• Transformation from continuous-time domain to S-domain (complex frequency domain)

Equation of Laplace Transform

S-domain equation

Neper Frequency

Controls the magnitude of the signal

Radian Frequency

Controls the rotation of the signal

Neper frequency greater than 0

Neper frequency equal to 0

Aim: a signal to die out or go back to its initial state

Neper frequency less than 0

unit step function

Dictates the limits of the transform

LAPLACE TRANSFORMS

• It is named for Pierre-Simon Laplace who introduced the transform in his work on probability theory.

• In physics and engineering, it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems.

Laplace transform of 1

Laplace transform of t

Laplace transform of

FIRST SHIFTING THEOREM

shifts the s-variable

Second Shifting Theorem

shifts the time-variable

transform of

Transform of

Transform of

Convolution

• A mathematical operation that combines two functions to produce a third function.

• It describes how one function "overlaps" or "interacts" with another as one of them is shifted.

Laplace transform of Convolution

Laplace of Derivatives

FOURIER SERIES

• A technique for expressing a periodic function in terms of sinusoids.

• It was named after Jean Baptiste Joseph Fourier(1768-1830).

a0 term in Fourier Series

an term in Fourier Series

bn term in Fourier Series

The mean / DC value in the equation of Fourier Series

Period used in Fourier Series

FOURIER TRANSFORM

• allows us to extend the concept of a frequency spectrum to non-periodic functions.

• The transform assumes that a non-periodic function is a periodic function with an infinite period.

Fourier-transform of 1

Fourier-transform of

Fourier-transform of S

Fourier-transform of

Fourier-transform of

Fourier-transform of

Z-TRANSFORM

It transforms the time-domain signal x[n] into its complex-plane representation X[z].

Z-transform of

Z-transform of
**ROC lzl > a

Z-transform of
**ROC lzl < a

REGION OF CONVERGENCE

Values of z that does not make it undefined

diverging

Values of z that does make it undefined