ECE438 - Digital Signal Processing

CTFT

\[X(f) = \int_{-\infty}^{\infty} x(t) e^{-2 \pi j f t} dt\]

\[x(t) = \int_{-\infty}^{\infty} X(f) e^{2 \pi j f t} df\]

As a convention, we require \(x(t)\) has the form \(u(t)f(t)\) so that infinite negative tail of \(e^{-2 \pi j f t}\) is eliminated.

An alternative notation:

\[X(\omega) = \int_{-\infty}^{\infty} x(t) e^{j f t} dt\]

\[x(t) = \frac{1}{2 \pi} \int_{-\infty}^{\infty} X(f) e^{j f t} df\]

Useful CTFT Pairs

\[\text{sinc}(t) = \frac{\sin (\pi t)}{\pi t} \xrightarrow[]{CTFT} \text{rect}(f)\] \[\text{comb}_T(x(t)) \xrightarrow[]{CTFT} \frac{1}{T} \text{rep}_{1/T}(X(f))\]

Duality of fourier transform

\[x(t) \xrightarrow[]{CTFT} X(f)\]

\[X(t) \xrightarrow[]{CTFT} x(-f)\]

Time Shifting

\[x(t - t_0) \xrightarrow[]{CTFT} X(f)e^{-j 2 \pi ft_0}\]

Time Scaling

\[x\left(\frac{t}{a}\right) \xrightarrow[]{CTFT} |a| X(af)\]

DTFT

\[X_d (\omega) = \sum_{n = - \infty}^\infty x[n] e^{-j \omega n}\]

\[x[n] = \frac{1}{2 \pi} \int_{-\infty}^{\infty} X_d(\omega) e^{j \omega n}\]

Reconstruction