Class notes Mathematics 3

Module 03: Fourier Transform & Z-transforms



Contents
 Introduction
 Infinite Fourier transform and Inverse Fourier transform
 Properties of Fourier transform
 Fourier cosine and Fourier sine transform
 Inverse Fourier cosine and sine transform with associated properties.
 Convolution Theorem and its applications
 Z-transforms
 Definition and property
 Z-transform of some standard functions
 Properties of Z-transform
 Initial value theorem and Boundary Value theorem
 Inverse Z-transforms
 Solution of difference equations using Z-transform.

, Introduction
Many engineering problems lead to ordinary or partial differential equations which have
to be solved under various types of conditions formulated from the problem. We are already
familiar with the solution of higher order ordinary differential equations with initial conditions
(initial value problems) using Laplace transforms. Solution of some partial differential equations
with boundary conditions (boundary value problems) can be obtained with the help of Fourier
transforms.
Infinite Fourier transform and Inverse Fourier transform:
The infinite Fourier transform or simply the Fourier transform of a real valued function 𝑓(𝑥) is
defined by 𝐹[𝑓(𝑥)] = ∫ 𝑓(𝑥) 𝑒 𝑑𝑥 = 𝐹(𝑢) (1)
Provided the integral exists. On integration we obtain a function of 𝑢 which is usually denoted
by 𝐹[𝑢] 𝑜𝑟 𝑓 (𝑢). (2)
The inverse Fourier transform of 𝐹(𝑢) denoted by 𝐹 [𝐹(𝑢)] 𝑜𝑟 𝐹 [𝑓 (𝑢)] is defined by the

integral ∫ 𝐹(𝑢) 𝑒 𝑑𝑢 on integration we obtain a function of 𝑥.

i.e., 𝑓(𝑥) = ∫ 𝐹(𝑢) 𝑒 𝑑𝑢.

Properties of Fourier Transform:
1. Linearity Property:
If 𝑐 , 𝑐 , 𝑐 … 𝑐 are constants then
𝐹[𝑐 𝑓 (𝑥) + 𝑐 𝑓 (𝑥) + ⋯ + 𝑐 𝑓 (𝑥)] = 𝑐 𝐹[𝑓 (𝑥)] + 𝑐 𝐹[𝑓 (𝑥)] + ⋯ + 𝑐 𝐹[𝑓 (𝑥)]
2. Change of Scale property:

If 𝐹[𝑓(𝑥)] = 𝑓 (𝑢), then 𝐹[𝑓(𝑎𝑥)] = 𝑓 = 𝐹 .

3. Shifting property:
If 𝐹[𝑓(𝑥)] = 𝑓 (𝑢), then 𝐹[𝑓(𝑥 − 𝑎)] = 𝑒 𝑓 (𝑢) = 𝑒 𝐹(𝑢).
4. Modulation property:
If 𝐹[𝑓(𝑥)] = 𝑓 (𝑢), then

𝐹[𝑓(𝑥) cos 𝑎𝑥] = 𝑓 (𝑢 + 𝑎) + 𝑓 (𝑢 − 𝑎) = [𝐹(𝑢 + 𝑎) + 𝐹(𝑢 − 𝑎)]