The computation of mathematical techniques used to manipulate signal data is known as digital signal processing (DSP). The Discrete Fourier Transform (DFT) is one of the most crucial tools in digital signal processing. It is typically used to create a representation of a signal’s frequency-domain (or spectral) properties.

We will go over the operation of DFT and how to use it to output the signal spectrum in this post.

Discrete Fourier Transform (DFT)

The fundamental mathematical concept of the DFT and the fundamental idea behind Spectral Decomposition, which comes to the conclusion that a signal is nothing more than the sum of sinusoids with various frequency components, is the Fourier Transform. A signal is a collection of time-domain samples because the signal data we are working with is all digital. DFT, which can be used to alternate between the time and frequency domains, can be used to perform the Fourier transform on such discrete signals. While the frequency domain depicts the spectrum of the sinusoids that make up the signal, the time domain holds the signal samples. The relationship between the time and frequency domains using DFT and IDFT is depicted in the image below.

Mathematically speaking, if we have a signal (xn) with N samples, the DFT of this signal is defined as

Where:

• N: Number of samples
• n: Current sample
• k: Current frequency where k ∈ [0, N−1]
• xn: The sine value at sample n
• Xk: The DFT which includes information on both amplitude and phase

An array of complex numbers that contain the data on the frequencies, amplitudes, and phases of the sinusoids that make up the input signal make up the DFT’s (Xk) output. The positive frequency terms are found in the first half of the DFT array (Xk),. While the negative frequency terms are found in the second half. Additionally, the spectrum is symmetric and the first half of the frequency terms is the conjugate of the second half of the frequency terms when the input signal is only a real-valued signal. Therefore, in the case of signals with real values,. We only pay attention to the first half (the positive frequency terms). In the event that the number of input samples (N) is odd or even,. The positive and negative frequency terms are each represented in the figure below.

The complex array (Xk) can be used to calculate the amplitude and phase of each sinusoid that contributes to the signal. (Im and Re stand for the Imagery and Real parts of a complex number, respectively.)

Inverse Discrete Fourier Transform (IDFT)

The Inverse Discrete Fourier Transform can be used to return from the frequency domain to the time domain in a manner similar to how you can go from the time domain to the frequency domain. When you want to use DFT to filter out specific frequency components from the signal. And IDFT to retrieve the signal back to its time domain, this process is very helpful in signal processing. So The Xk sequence that comes after the equation can be used to calculate the IDFT:

Conclusion

• We have discussed the discrete Fourier transform’s mathematical foundation and how to compute it for a discrete signal.
• We created a function that, using the mathematical equation, calculates the DFT,. And we used it to process a signal that was generated using the class Signal that we created in the previous post.
• So We now know that the DFT produces an array of complex numbers with N elements, the same as the input signal’s sample count. The amplitude and phase of the spectrum are described by these complex numbers.
• As we’ve seen, if the input signal is a real-value signal, the DFT’s output will be symmetrical to half of the sampling rate. And for this reason, we only pay attention to the frequencies that are positive.
• The Inverse Discrete Fourier Transform, or IDFT, calculation equation has been identified. Additionally, in order to extract the original signal from the spectrum, we developed a function to compute the IDFT.
• So We’ve talked about the functions we’ve built’s limitations,. And the Fast Fourier Transform algorithm is a quick way to compute Fourier transforms.