Chapter 2
Classical Vector Transform Techniques
From the frequency domain revelations of the Fourier transform to the compact data representations enabled by cosine and sine transforms, classical vector transform techniques have long served as the linchpins of modern data and signal processing. This chapter delves into these foundational methods with a rigorous lens, unpacking not just their algorithms, but also the profound insights they provide into the structure, fidelity, and limitations of transformed data.
2.1 Fourier Transform: Theory and Algorithmic Implementation
The Fourier transform constitutes a cornerstone in signal processing and analysis by enabling the decomposition of functions or signals into their constituent frequencies. Central to this theory are the continuous Fourier transform (CFT) and its discrete counterpart, the discrete Fourier transform (DFT), each serving specific roles depending on the signal domain and sampling context.
The continuous Fourier transform of a function x(t) ? L1(R) is defined as
where j = and f denotes frequency in Hertz. This integral transform maps the time-domain signal x(t) to its frequency-domain representation X(f), capturing the amplitude and phase distribution of spectral components. The inverse transform reconstructs x(t) from X(f) via
The CFT assumes signals defined on a continuous time axis and infinite duration, often idealized in theory but foundational for understanding frequency content and linear time-invariant system responses.
For digital signal processing, signals are sampled and represented as finite-length sequences. The discrete Fourier transform is applicable here and is defined for a sequence x[n], n = 0,1,.,N - 1, as
where N is the sequence length. The corresponding inverse operation is
The DFT maps N time-domain samples into N frequency bins, discretely sampling the frequency spectrum and inherently assuming periodicity of the sequence with period N. This periodicity leads to considerations around spectral leakage and the choice of windowing functions to manage discontinuities at the boundaries.
Efficient computation of the DFT is critical for real-time and large-scale spectral analysis. The naive computation requires O(N2) operations due to the double summation. This complexity was dramatically reduced by the development of the Fast Fourier Transform (FFT) algorithm, which exploits symmetries and periodicities in the DFT matrix to reduce computational cost to O(N log N).
The Cooley-Tukey algorithm, the most common FFT approach, recursively decomposes an N-point DFT into smaller DFTs by factorizing N, typically into powers of two. This is expressed by splitting the input sequence into even and odd terms:
Recognizing that e-jk2n = e-jkn, the problem reduces to two N/2-point DFTs plus N twiddle factor multiplications
This recursive divide-and-conquer structure continues until reaching trivial DFT sizes (usually 2-point transforms).
Pseudocode for a radix-2 decimation-in-time FFT is as follows:
def fft(x): N = len(x) if N <= 1: return x even = fft(x[0::2]) odd = fft(x[1::2]) T = [exp(-2j * pi * k / N) * odd[k] for k in range(N//2)] return [even[k] + T[k] for k in range(N//2)] + \ [even[k] - T[k] for k in range(N//2)] The FFT's improved efficiency enables practical exploitation of spectral analysis in myriad applications: audio and image processing, telecommunications, and control systems. Given an adequately sampled signal, the FFT reveals dominant frequency...
