There are two views of the world:

• the continuous-time world, where phenomena are described functions of a real variable $$t \in \mathbb{R}$$, and where frequency $$f \in \mathbb{R}$$ is expressed in Hz. The continuous-time Fourier transform (FT) is the type of Fourier transform for these functions. The FT is not periodic.

• the discrete-time world, i.e. a model of reality that lives inside a digital computer. In discrete-time, phenomena are sequences indexed an integer variable $$n \in \mathbb{Z}$$ and frequency $$\omega \in [ - \pi, \pi]$$ is dimensionless. The DTFT is the type of Fourier transform for these sequences. The DTFT is $$2\pi-$$periodic.

In continuous-time, many of the signal processing concepts that we have studied in discrete-time still hold their meaning:

• the inner product between two functions is defined as $$\langle x(t), y(t) \rangle = \int_{-\infty}^{+\infty} x^*(t)y(t)dt$$.

• the FT maps time to frequency (and the inverse transform goes back to time). A notable class of functions is represented by the set of bandlimited functions. The prototypical bandlimited function is the sinc(). Bandlimited functions will play a fundamental role in sampling and interpolation.

• we can also define continuous-time filters and derive an equivalent convolution theorem. The FT of the output of the filter is the product of the FT of the transfer function and the FT of the input.