MATLAB Implementation of 1D and 2D Discrete Fourier Transform (DFT)

The Discrete Fourier Transform (DFT) serves as a fundamental tool in signal processing. While MATLAB's built-in `fft` function provides efficient computation, its default output places high-frequency components at the spectrum edges, which may not suit certain visualization requirements. Implementing DFT manually offers flexibility in adjusting spectrum layout to center low-frequency components.

1D DFT Implementation Key Points Core Algorithm: Following the DFT definition formula, iterate through each frequency point to compute the inner product between the signal and complex exponential basis functions. Frequency Shifting: After computation, apply `fftshift` to relocate the zero-frequency component to the spectrum center. Performance Consideration: Compared to MATLAB's native `fft`, manual DFT implementation better illustrates mathematical principles but requires attention to computational efficiency with large datasets due to loop operations.

2D DFT Extension Separable Computation: Perform 1D DFT on each image row first, then apply 1D DFT to each column of the intermediate result, leveraging the separable property of 2D transformation. Bidimensional Frequency Shift: Apply spectrum shifting separately along row and column directions to ensure low-frequency concentration at the matrix center. Visualization Advantage: Centered spectrum presentation aligns better with human observation habits, facilitating analysis of dominant frequency components in images.

Application Scenarios Educational Demonstration: Helps students grasp the mathematical essence of DFT. Image Processing: Suitable for tasks requiring low-frequency focus (e.g., filter design). Custom Requirements: Adapting spectrum display logic to meet specific algorithm input specifications.

This implementation balances theoretical clarity with practical value, particularly useful for analysis scenarios requiring frequency-domain centered output. However, engineering applications should consider hybrid approaches (e.g., manual frequency shifting after calling `fft`) based on efficiency requirements.