The Continuous Time Fourier Transform

Fourier Transform or Fourier Integral of an aperiodic signal x(t) is given by

$\displaystyle X(t) = \int_{-\infty}^{\infty} x(t) e^{-j \omega t} dt $

and Inverse Fourier transform equation is given by

$\displaystyle x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j \omega) e^{j \omega t} d\omega $

The Fourier Transform for Periodic Signals :

Let us consider a signal x(t) having Fourier transform X(jω) that is a single impulse of area 2π at ω = ω0 i.e.

X(jω) = 2π δ(ω – ω0)

Then , $\displaystyle x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j \omega) e^{j \omega t} d\omega $

$\displaystyle x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} 2 \pi \delta (\omega – \omega_0) e^{j \omega t} d\omega $

$\displaystyle x(t) = e^{j \omega_0 t} $

If x(jω) is of the form of a linear combination of impulses equally spaced in frequency i.e.

$\displaystyle X(j\omega) =\Sigma_{k=-\infty}^{\infty} 2 \pi a_k \delta ( \omega – k \omega_0 ) $

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

Hence , Fourier Transform of a periodic signal with Fourier series coefficients {ak} can be expressed as a train of impulses occurring at the harmonically related frequencies and for which the area of the impulse at the kth frequency kω0 is 2π times the kth Fourier series coefficients ak

Properties of the Continuous Time Fourier Transform :

For the sake of convenience , X(j ω) is referred with notation F{x(t)} and to x(t) with the notation F-1{X(j ω)} . Also X(t) and X(j ω) are referred as a Fourier Transform pair with the notation .

(i) Linearity : $x(t) \longleftrightarrow X(j\omega)$

If $x(t) \longleftrightarrow X(j\omega)$

and $y(t) \longleftrightarrow Y(j\omega)$

then $a x(t) + b y(t) \longleftrightarrow a X(j\omega) + b Y(j \omega)$

(ii) Time Shifting : If $x(t) \longleftrightarrow X(j\omega)$

then , $x(t-t_0) \longleftrightarrow e^{-j \omega t_0}X(j\omega)$

(iii) Conjugation and Conjugate Symmetry : The Conjugation Property states that ,

If $x(t) \longleftrightarrow X(j\omega)$

then $x^\ast(t) \longleftrightarrow X^\ast (-j\omega)$

This property also says that if x(t) is real , then X(jω) has Conjugate Symmetry , i.e.

$ X(-j\omega) = X^\ast (j\omega)$

(iv) Differentiation and Integration :

$\displaystyle \frac{d x(t)}{dt} \longleftrightarrow j\omega X(j\omega)$

$\displaystyle \int x(t) dt \longleftrightarrow \frac{1}{j\omega} X(j\omega)$

(v) Duality : Multiplication by jt in time domain corresponds roughly to differentiation in the frequency domain .

$\displaystyle – jt x(t) \longleftrightarrow \frac{d}{d \omega} X(j\omega) $

Similarly , the dual properties of time shifting property and integration property are ,

$\displaystyle e^{j \omega_0 t} x(t) \longleftrightarrow X(j(\omega – \omega_0)) $

and , $\displaystyle -\frac{1}{j t} x(t) + \pi x(0)\delta(t) \longleftrightarrow \int_{-\infty}^{\infty} x(\eta) d\eta $

(vi) Parseval’s Relation : If x(t) and X(jω) are the Fourier Transform pair , then

$\displaystyle \int_{-\infty}^{\infty}| x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(j \omega)|^2 d\omega $

The term on the LHS of above equation is the Total energy in the signal x(t) and Parseval’s relation says that it can be determined either by computing the energy per unit time |x(t)|2 and integrating over all time or by computing energy per unit time frequency ( |(X(jω)|2 /2π ) and integrating over frequencies .

(vii) The convolution property : The convolution property states that the convolution in tie domain corresponds to multiplication in frequency domain . For example , an LTI system has impulse response h(t) and input x(t) being applied then the output y(t) is the convolution of h(t) and x(t) , i.e.

$y(t) = x(t) \ast h(t) $

Then , $\displaystyle y(t) \longleftrightarrow Y(j(\omega) = X(j(\omega) . H(j(\omega)$

Where , X(jω) = Z[x(t)]

H(jω) = Z[h(t)]

(viii) The Multiplication Property : Because of duality between time and frequency domains , the dual of convolution property also exists and referred to as the convolution property which states that multiplication in time domain corresponds to convolution in the frequency domain , i.e.

$ r(t) = s(t) p(t) \longleftrightarrow R(j(\omega) = \frac{1}{2\pi}[s(j\omega \ast \phi(j\omega))] $