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-frequency data. The DTFT is also useful because it provides a theoretical basis for the Z transform.
The resulting function, 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 mulitplication in the discrete frequency domain.
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:
Energy in Frequency
Likewise, we can make a formula that represents the power in the continuous-frequency output of the DTFT:
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:
Power Density Spectrum
We can define the power density spectrum of the continuous-time frequency output of the DTFT as follows:
The area under the power density spectrum curve is the total energy of the signal.