In this topic, you study the Fourier Transform Pairs of Basic Signals as Decaying Exponential, Impulse function, DC, Cosine function, Sine function, Unit step function, Signum function, Complex Exponential, and, Exponential Pulse.
Sine Function
\[\sin {\omega _0}t \longleftrightarrow j\pi \left[ {\delta (\omega + {\omega _0}) {}- \delta ({}\omega {}- {\omega _0})} \right]\]
Cosine Function
\[\cos {\omega _0}t \longleftrightarrow \pi \left[ {\delta (\omega + {\omega _0}) + \delta (\omega{}{} – {\omega _0})} \right]\]
Unit step Function
\[u(t) \longleftrightarrow \pi \delta (\omega ) + \frac{1}{{j\omega }}\]
Impulse Function
\[\delta (t) \longleftrightarrow 1\]
Signum Function
\[\operatorname{sgn} (t) \longleftrightarrow \frac{2}{{j\omega }}\]
1
\[1 \longleftrightarrow 2\pi \delta (\omega )\]
Complex Exponential
\[{e^{j{\omega _0}t}} \longleftrightarrow 2\pi \delta (\omega – {\omega _0})\]
Exponential Decaying
\[{e^{ – at}}u(t) \longleftrightarrow \frac{1}{{a + j\omega }}\quad \quad a > 0\]
Exponential Pulse
\[{e^{ – a\left| t \right|}} \longleftrightarrow \frac{{2a}}{{{a^2} + {\omega ^2}}}{\kern 1pt} \quad a > 0\]