Digital Signal Processing/Discrete-Time Fourier Transform

The Discrete-Time Fourier Transform is a version of the fourier transform that is used to convert a discrete data set into a continuous-frequency representation. The DTFT is used mostly in theory, and less in practice, because computers are not usually capable of handling continuous-time frequency data. The DTFT is also useful because it provides a theoretical basis for the Z transform.

DTFT

$X(e^{j\omega })=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}x[n]e^{-j\omega n}$

The resulting function, $X(e^{j\omega })$ is a continuous function that is interesting for analysis. It can be used in programs, such as Matlab, to design filters and obtain the corresponding time-domain filter values.

DTFT Convolution Theorem

Like the CTFT, the DTFT has a convolution theorem associated with it. However, since the DTFT results in discrete-frequency values, the convolution theorem needs to be modified as such:

Energy in Time

It is sometimes helpful to calculate the amount of energy that exists in a certain set. These calculations are based off the assumption that the different values in a set are voltage values, however this doesn't necessarily need to be the case to employ these operations.

We can show that the energy of a given set can be given by the following equation:

${ℰ}_x=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{|x[n]|}^2$

Energy in Frequency

Likewise, we can make a formula that represents the power in the continuous-frequency output of the DTFT:

${ℰ}_x=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{|X(e^{j\omega })|}^{2}d\omega$

Parseval's Theorem

Parseval's theorem states that the energy amounts found in the time domain must be equal to the energy amounts found in the frequency domain:

$\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{|x[n]|}^2=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{|X(e^{j\omega })|}^{2}d\omega$

Energy Density Spectrum

We can define the energy density spectrum of the continuous-time frequency output of the DTFT as follows:

$S_{xx}(e^{j\omega })={|X(e^{j\omega })|}^2$

The area under the energy density specrtum curve is the total energy of the signal.