Properties of Fourier Transform | Signals and Systems - Electrical Engineering (EE) PDF Download

Properties of Fourier Transform

Differentiation/Integration

Hence if 

then

Now,

Hence if,

then,

The inverse operation of taking the derivative is running the integral : 

eg : 

let 

This causes problem when 

 impulse in frequency. 

Example:

Scaling of the independent variable by a real constant a 
When a > 0 or a < 0

Hence the scaling of the independent variable is a self-dual operation.
Consider

Hence, x(t) and |a|^1/2 x(at) have the same energy. Therefore such scaling is called energy normalized scaling of the independent variable.

Properties of Fourier Series.

Using the properties we just proved for the Fourier Transform, we state now the corresponding properties for the Fourier series.

Time-shift

Recall, that if x(t) is periodic then X(f) is a train of impulses.

We know: 

Thus if x(t) is periodic with period T , x( t - t₀) has Fourier series coefficients  

Differentiation 

If the periodic signal is differentiable then

Thus if x(t) is periodic with period T , x'(t) has Fourier Series coeffici 

Scaling of the independent variable

 
If a > 0, x(at) is periodic with period ( T / a ) and now c k becomes Fourier coefficient corresponding to frequency .
If a < 0, x(at) is periodic with period ( T / -a) and now ck becomes Fourier coefficient corresponding to frequency

Multiplication by t 

Multiplication by t of-course will not leave a periodic signal periodic. But what we can do is, multiply by t in one period, and then consider a periodic extension. i.e: x(t) is periodic with period T, we see what the Fourier series coefficients of y(t), defined as follows is:

Note the k^th Fourier series co-efficient of x(t) is 

Similarly, let 

Therefore, k^th Fourier series coefficient of  

This idea is not of much use without knowledge of  

Conclusion:

In this lecture you have learnt:

• Properties of the Fourier Transform w.r.t. differentiation and integration
• Properties of the Fourier Transform w.r.t. scaling of the independent variable by a real constant a.
• Properties of the Fourier Series w.r.t. time shifting
• Properties of the Fourier Series w.r.t. differentiation
• Properties of the Fourier Series w.r.t. scaling of the independent variable
• Properties of the Fourier Series w.r.t. multiplication by t