# What is Transform

Changing Point of View

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TransformingÂ a function to another

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Changing independentÂ variable of a function

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

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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

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Complex Equation to simple algebric form

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Solve algebric equation

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Go back to Time Domain

# What is Frequency Domain?

A talk in terms of Frequency

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In Time Domain, $x(t)$ will give value/energyÂ of signalÂ at $t$

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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:

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Ingredients of Plot

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Value/QuantityÂ of ingredients

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Effect on Plot if we decreaseÂ 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 $k \omega_0$

# Fourier Transform

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

$$2\omega_0 -Â \omega_0 = d\omega_0$$

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

# 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 $\infty$

# Fourier Transform of Periodic Signal

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

Impulse in Frequency Domain

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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 $\cal{L}(s)$

$s = \sigma+\jmath \omegaÂ \Rightarrow$ 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 TransientÂ Analysis and Stability Analysis of System

Fourier is used for Frequency Analysis of Stable System