Fourier Transforms

The main drawback of Fourier series is, it is only apppcable to periodic signals. There are some naturally produced signals such as nonperiodic or aperiodic, which we cannot represent using Fourier series. To overcome this shortcoming, Fourier developed a mathematical model to transform signals between time (or spatial) domain to frequency domain & vice versa, which is called Fourier transform .

Fourier transform has many apppcations in physics and engineering such as analysis of LTI systems, RADAR, astronomy, signal processing etc.

Deriving Fourier transform from Fourier series

Consider a periodic signal f(t) with period T. The complex Fourier series representation of f(t) is given as

$$ f(t) = sum_{k=-infty}^{infty} a_k e^{jkomega_0 t} $$

$$ quad quad quad quad quad = sum_{k=-infty}^{infty} a_k e^{j {2pi over T_0} kt} ... ... (1) $$

Let ${1 over T_0} = Delta f$, then equation 1 becomes

$f(t) = sum_{k=-infty}^{infty} a_k e^{j2pi k Delta ft} ... ... (2) $

but you know that

$a_k = {1over T_0} int_{t_0}^{t_0+T} f(t) e^{-j komega_0 t} dt$

Substitute in equation 2.

(2) $ Rightarrow f(t) = Sigma_{k=-infty}^{infty} {1 over T_0} int_{t_0}^{t_0+T} f(t) e^{-j komega_0 t} dt, e^{j2pi k Delta ft} $

Let $t_0={Tover2}$

$ = Sigma_{k=-infty}^{infty} [ int_{-Tover2}^{Tover2} f(t) e^{-j2 pi k Delta ft} dt ] , e^{j2 pi k Delta ft}.Delta f $

In the pmit as $T o infty, Delta f$ approaches differential $df, k Delta f$ becomes a continuous variable $f$, and summation becomes integration

$$ f(t) = pm_{T o infty} ⁡left{ Sigma_{k=-infty}^{infty} [ int_{-Tover2}^{Tover2} f(t) e^{-j2 pi k Delta ft} dt ] , e^{j2 pi k Delta ft}.Delta f ight} $$

$$ = int_{-infty}^{infty} [ int_{-infty}^{infty},f(t) e^{-j2pi ft} dt] e^{j2pi ft} df $$

$$f(t) = int_{-infty}^{infty}, F[omega] e^{jomega t} d omega$$

$ ext{Where},F[omega] = [ int_{-infty}^{infty}, f(t) e^{-j2 pi ft} dt]$

Fourier Transform of Basic Functions

Let us go through Fourier Transform of basic functions:

FT of GATE Function

$$F[omega] = AT Sa({omega T over 2})$$

FT of Impulse Function

$FT [omega(t) ] = [int_{- infty}^{infty} delta (t) e^{-jomega t} dt] $

$quad quad quad quad = e^{-jomega t}, |, t = 0 $

$quad quad quad quad = e^{0} = 1 $

$quad herefore delta (omega) = 1 $

FT of Unit Step Function:

$U(omega) = pi delta (omega)+1/jomega$

FT of Exponentials

$ e^{-at}u(t) stackrel{mathrm{F.T}}{longleftrightarrow} 1/(a+jω)$

$ e^{-at}u(t) stackrel{mathrm{F.T}}{longleftrightarrow} 1/(a+jomega )$

$ e^{-a,|,t,|} stackrel{mathrm{F.T}}{longleftrightarrow} {2a over {a^2+ω^2}}$

$ e^{j omega_0 t} stackrel{mathrm{F.T}}{longleftrightarrow} delta (omega - omega_0)$

FT of Signum Function

$ sgn(t) stackrel{mathrm{F.T}}{longleftrightarrow} {2 over j omega }$

Conditions for Existence of Fourier Transform

Any function f(t) can be represented by using Fourier transform only when the function satisfies Dirichlet’s conditions. i.e.

The function f(t) has finite number of maxima and minima.

There must be finite number of discontinuities in the signal f(t),in the given interval of time.

It must be absolutely integrable in the given interval of time i.e.

$ int_{-infty}^{infty}, |, f(t) | , dt < infty $

Discrete Time Fourier Transforms (DTFT)

The discrete-time Fourier transform (DTFT) or the Fourier transform of a discrete–time sequence x[n] is a representation of the sequence in terms of the complex exponential sequence $e^{jomega n}$.

The DTFT sequence x[n] is given by

$$ X(omega) = Sigma_{n= -infty}^{infty} x(n)e^{-j omega n} ,, ...,... (1) $$

Here, X(ω) is a complex function of real frequency variable ω and it can be written as

$$ X(omega) = X_{re}(omega) + jX_{img}(omega) $$

Where X_re(ω), X_img(ω) are real and imaginary parts of X(ω) respectively.

$$ X_{re}(omega) = |, X(omega) | cos heta(omega) $$

$$ X_{img}(omega) = |, X(omega) | sin heta(omega) $$

$$ |X(omega) |^2 = |, X_{re} (omega) |^2+ |,X_{im} (omega) |^2 $$

And X(ω) can also be represented as $ X(omega) = |,X(omega) | e^{j heta (ω)} $

Where $ heta(omega) = arg{X(omega) } $

$|,X(omega) |, heta(omega)$ are called magnitude and phase spectrums of X(ω).

Inverse Discrete-Time Fourier Transform

$$ x(n) = { 1 over 2pi} int_{-pi}^{pi} X(omega) e^{j omega n} domega ,, ...,... (2)$$

Convergence Condition:

The infinite series in equation 1 may be converges or may not. x(n) is absolutely summable.

$$ ext{when},, sum_{n=-infty}^{infty} |, x(n)|, < infty $$

An absolutely summable sequence has always a finite energy but a finite-energy sequence is not necessarily to be absolutely summable.