Computational Complexity of Radix-2, Radix-4 and Bluestein Algorithms Implementation of the Discrete Fourier Transform (DFT)

<p dir="ltr">The computational complexity of Discrete Fourier Transform (DFT) algorithms plays a pivotal role in signal processing, influencing their applicability in various domains. This paper investigates three prominent Fast Fourier Transform (FFT) algorithms: Radix-2, Radix-4, and Bluestein, with a focus on their computational efficiency and suitability for different sequence lengths. MATLAB implementations were developed to optimize these algorithms, reducing the number of multiplications and additions required during runtime. A comparative analysis reveals that Radix-2 and Radix-4 algorithms are highly efficient for power-of-two and power-of-four data lengths, respectively, while the Bluestein algorithm provides unparalleled flexibility for arbitrary sequence lengths, including primes. The study demonstrates the trade-offs associated with each algorithm, highlighting their strengths and limitations. Radix-4 achieves greater efficiency over Radix-2 for longer sequences, while Bluestein eliminates the need for zero-padding at the cost of increased computational complexity. This research offers valuable insights into the selection of FFT algorithms based on application-specific requirements and data characteristics, laying the groundwork for further optimization and hybrid algorithm development. The findings underscore the enduring importance of FFTs in addressing the computational demands of modern signal processing tasks.</p>