#DiscreteFourierTransform

[govindhtech]

What is the Fast Fourier Transform A Modern Computing Pillar

The Fast Fourier Transform

Honouring the FFT and Computing's Future: Representation to Revolution

For the first Fast Fourier Transform deployment, IBM received an IEEE Milestone award on June 11, 2025. IBM researchers created this method in 1965, transforming computers.

Since it supports JPEG and MPEG standards and reconstructs MRI and CT scan images, the FFT has a wide impact. Also needed for scientific computing (spectrum approaches for solving PDEs), music and video compression (MP3, JPEG), and telecommunications (4G/5G, WiFi). Richard Hamming called the Cooley-Tukey FFT “the most important numerical algorithm of the lifetime”.

FFT's Key Innovation

Fundamental FFT Innovation James Cooley and John Tukey introduced the FFT in 1965 as a “better way to represent information” rather than a new scientific discovery. The Fourier transform can split a time-domain signal like a wave into smaller waves with various frequencies. Before the 1960s, Fourier transform computing was too slow for real-time applications. Cooley and Tukey's technique accelerated real-time signal processing by reducing the computing cost of the Discrete Fourier Transform (DFT) from O(N^2) to O(NlogN). The revelation was that changing a computing problem's mathematical representation can change it.

Quantum Computing Lessons:

The FFT's ongoing development aids quantum algorithm development. When building new  quantum algorithms, “choosing the right representation can make the impossible possible” is crucial.

Quantum Computing: Emerging Idea Beyond improving existing methods, quantum computing changes how information is represented and abstracted. Classical computing employs bits with deterministic binary values (0s and 1s) and Boolean operations, while quantum computing uses qubits. In complex vector spaces, qubits store information as probability amplitudes (α|0⟩ + β|1⟩), where α and β are complex numbers. Quantum computing uses unitary evolution of qubit states through matrix operations instead of classical logic, generating probabilistic results.

This new computational paradigm enables Grover's method, which quadraticly speeds up unstructured search, and Shor's algorithm, which uses the Quantum Fourier Transform to exponentially speed integer factorisation. Additionally, quantum simulation can mimic quantum systems that conventional machines cannot handle.

Future Quantum-Classical Synergy. The most innovative computer future may be a mix of quantum and classical. Traditional computers are fast at control logic, data storage, and predictable computations. However, quantum systems thrive in mimicking quantum phenomena, high-dimensional linear algebra, probabilistic sampling, and landscape optimisation, where classical information representation fails.

Together, these paradigms can solve problems neither system can. VQE and QAOA are two novel hybrid classical-quantum algorithms in development. Quantum advantage, where a quantum-classical combination outperforms classical computation, is nearing, and SQD and SKQD are being developed. Supply chain optimisation, material science, finance, and drug development may use these methodologies.

Quantum technologies are expected to boost traditional computing as “coprocessors with radically different capabilities” like GPUs on CPUs. As the computing bottleneck shifts from hardware limits to algorithmic innovation, new abstractions, representations, and algorithms are needed to balance workloads among complementary architectures. The current age may be the start of a more significant algorithmic era than the FFT.

Anticipating The FFT's anniversary reminds us that innovations often come from better questions, smarter representations, or new viewpoints, not more authority. Fusion of classical and quantum domains will release new processing capability, requiring daring abstractions, inventive representations, and innovative algorithms.