Fourier Series and Transform

What is Transform

Changing Point of View

 

Transforming a function to another

 

Changing independent variable of a function

 

Usually denoted by $x(.) \leftrightarrow X(.)$ or $x(.) \leftrightarrow \cal{T}(x(.))$

 

In case of Fourier Transform, it's:

$x(t) \leftrightarrow X(\jmath \omega)$ or $x(t) \leftrightarrow \cal{F}(x(t))$

Why Transform?

Time domain : Complex Mathematics of Systems

 

Complex Equation to simple algebric form

 

Solve algebric equation

 

Go back to Time Domain

What is Frequency Domain?

A talk in terms of Frequency

 

In Time Domain, $x(t)$ will give value/energy of signal at $t$

 

In Frequency Domain, $X(\jmath \omega)$ will give value/energy of signal at $\omega$.

Why Frequency Domain?

Dish

Cook

Engineer

Plot

Ingredients ($\omega$)

Quantity($X(\jmath \omega)$) of each ingredients

Change in Quantity for better results

New Dish

New Plot

Why Frequency Domain (Contd.)

We want to know:

 

Ingredients of Plot

 

Value/Quantity of ingredients

 

Effect on Plot if we decrease quantity of ingredients

Fourier Series

Periodic Signal $ \longleftrightarrow $ Sum of Sinusoidal waves with frequency $0$, $\omega_0$, $2\omega_0 \ldots k \omega_0$

If $x(t) = x(t+T) $

$x(t) = \Sigma_{k=-\infty}^{\infty} a_k e^{jk\omega_0 t}$

Why only these frequencies?

What is $a_k$

Amplitude of wave having frequency $ k \omega_0 $

Corresponds to Energy of component of Signal with frequency $ k \omega_0 $

Periodic wave in terms of Fourier Series

Fourier Transform

For $ T_0 \rightarrow \infty $ i.e. Non-periodic signal,

$$ 2\omega_0 - \omega_0 = d\omega_0 $$

$$ \Rightarrow a_k \rightarrow X(\jmath \omega) $$

Fourier Transform

Suppose $x(t)$ is a non-periodic signal

$$X(\jmath \omega) = \int^{\infty}_{-\infty} x(t) e^{-\jmath \omega t} dt $$

$$x(t) = \frac{1}{2\pi}\int^{+\infty}_{-\infty}X(\jmath \omega) e^{\jmath \omega t} d\omega $$

Why not only specific frequencies?

Summation of sinusoidal waves with all frequencies. i.e. $-\infty$ to $\infty$

What is $X(\jmath \omega)$

Fourier Transform of Periodic Signal

  • Periodic Signal must be addition of sinusoidal waves of particular frequencies.
  • This means no influence of all other frequencies.

Impulse in Frequency Domain

 

Sampling of $X(\jmath \omega)$ at every $\omega = k\omega_0$

$\Rightarrow X(\jmath \omega) = 0$ for $\forall \omega \ne k\omega_0$

Laplace and Fourier Transform

Fourier Transform $X(\jmath \omega) \subset$ Laplace Transform $\cal{L}(s) $


$ s = \sigma+\jmath \omega \Rightarrow $ Real + Imaginary Analysis $\Rightarrow$ Laplace Transform


$\sigma > 0 \Rightarrow $ Unstable System


$\sigma ≤ 0 \Rightarrow $ Stable System


$ s= \jmath \omega \Rightarrow$ Imaginary Analysis $\Rightarrow$ Fourier Transform


Laplace is used for Transient Analysis and Stability Analysis of System


Fourier is used for Frequency Analysis of Stable System

Thank You!