Digital Signal Processing/Discrete-Time Fourier Transform

The Discrete-Time Fourier Transform (DTFT) is the cornerstone of all DSP, because it tells us that from a discrete set of samples of a continuous function, we can create a periodic summation of that function's Fourier transform. At the very least, we can recreate an approximation of the actual transform and its inverse, the original continuous function. And under certain idealized conditions, we can recreate the original function with no distortion at all. That famous theorem is called the Nyquist-Shannon sampling theorem.

DTFT

[DTFT]

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

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:

DTFT Convolution Theorem

Multiplication in the continuous time domain becomes discrete convolution in the discrete frequency domain. Convolution in the continuous time domain becomes multiplication in the discrete frequency domain.

Energy

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:

${\mathcal {E}}_{x}=\sum _{n=-\infty }^{\infty }\left|x[n]\right|^{2}$

Energy in Frequency

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

${\mathcal {E}}_{x}={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\left|X(e^{j\omega })\right|^{2}d\omega$

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:

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

Power Density Spectrum

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

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

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