The Fourier Transform

Fourier Transform (Analysis Equation)

The Fourier transform will Uniquely define

For a non-periodic . Periodic Signals will use a Fourier Series instead.

Inverse Fourier Transform (Synthesis Equation)

Uniquely defines

Fourier Transform is Complex

and are uniquely defined for every

Dirichlet’s Conditions

If these are satisfied , then it must have Fourier Transform; if not, it may still have one.

is:

1. Single-valued with finite extrema in any finite interval
2. Finite discontinuities in any finite interval
3. Absolutely integrable:

All physically realizable functions have a Fourier Transform

Physically realizable functions are all energy functions defined as:

Fourier Transform is Conjugate Symmetric for real time-domain Functions

if ( is real), then will have an even real part and an odd imaginary part:

• • is even
• • is odd

But if is both real and even (, ) then is also real and even. This is shown in the rectangular function example

Inverse Relationship of the Fourier Transform

A pulse narrow in time has a wide range of frequencies and a function defined over a lot of time has a narrow frequency range

Common Fourier Pairs

Rectangular Function and Sinc

Rectangular Function

More generally:

Sinc Function

So:

Truncated Decaying Exponential Pulse

Unit Step Function

Decaying Exponential Pulse

Rising Exponential Pulse

Fourier Transform

So: