Fourier transformation

The convolution integral

• The output y(t) to an input x(t) is seen as a weighted superposition of impulse response time-shifted by \(\tau\).
• The expression is called the convolution integral and denoted by the same symbol * as in the discrete-time case\[y(t) = \int_{-\infty}^\infty x(\tau)h(t-\tau)d\tau = x(t)*h(t)\]

Harmonic signals (complex exponential) are eigenfunctions for LTI systems. That is,
\[\mathcal{H}f = \lambda f\].

Suppose the input is \(x(t) = Ae^{st}\). The output of the system with impulse repose h(t) is then
\begin{align}\int_{-\infty}^\infty h(t-\tau)Ae^{s\tau}d\tau & = \int_{-\infty}^{\infty}h(\tau)Ae^{s(t-\tau)}d\tau = Ae^{st}\int_{-\infty}^{\infty}h(\tau)e^{-s\tau}d\tau \\ & =Ae^{st}H(s)\end{align}
where H(s) is a scalar dependent only on the parameter s.

Fourier series
Continuous time
\[X[k] = \frac{1}{T}\int_0^Tx(t)e^{-jk\omega_0t}dt\]

Discrete time
\[X[k] = \frac{1}{N}\sum_{n=<N>}x[n]e^{-jk\Omega_0n}\]