Discrete Fourier Transform (DFT) and its Applications

Sura Fahmy Yousif

Chemical Engineering Department

Fourier transform (FT) is one of the most commonly used techniques in (linear) signal processing and control theory. It provides one-to-one transform of signals from a time-domain representation to a frequency domain representation. It allows a frequency content (spectral) analysis of a signal. Frequency spectrum means that you can see which frequencies are inside a signal where every signal can be described as a superposition of sine and cosine signals. FT can be categorized to two main types: Continuous Fourier Transform (CFT) and Discrete Fourier Transform (DFT). Although the CFT is practically great, it is difficult to use it in real life. This is because, in real life, we are usually dealing with discrete data that was sampled using some kind of sensors. If we think of any time series data like weather or traffic, we get the discrete values at each time point (e.g. 1 second, 2 second, 3 second, etc.) and we don’t know what’s in between those samples. So, we need accurate approximation of the real data. For this case, the discrete equivalent version of the Fourier Transform is required. This is where Discrete Fourier Transform comes into play. CFT deals with aperiodic signal, while DFT deals with periodic signals and this is the method that we usually use for real world data. In many practical applications, the DFT is the most important discrete transform used to perform Fourier analysis. Also, it is one of the most significant tools in Digital Signal Processing.

DFT can calculate a signal's frequency spectrum. This is a direct examination of information encoded in the frequency, phase, and amplitude of the component sinusoids. For example, human speech and hearing use signals with this type of encoding. Second, the DFT can find a system's frequency response from the system's impulse response, and vice versa. This allows systems to be analyzed in the frequency domain, just as convolution allows systems to be analyzed in the time domain. Third, the DFT can be used as an intermediate step in more elaborate signal processing techniques. The classic example of this is Fast Fourier Transform (FFT) convolution, an algorithm for convolving signals that is hundreds of times faster than conventional methods. The DFT is also used to efficiently solve partial differential equations, and to perform other operations such as multiplying large integers. Additionally, it can be implemented in computers by numerical algorithms or even dedicated hardware.