Convolution Theorem

Fourier Analysis · Axiom Academy

LESSON The Convolution Theorem Discovering the fundamental duality between convolution in time and multiplication in frequency Convolution is an operation that combines two functions to produce a third function. For continuous functions f and g, the convolution (f * g)(t) is defined as: Intuitively, convolution "slides" one function over another, multiplying and integrating at each position. The variable τ (tau) is the integration variable, while t is the output position. The Convolution Theorem states that the Fourier transform of a convolution is the product of the Fourier transforms: This remarkable result means that a complex convolution integral in the time domain becomes simple multiplication in the frequency domain. This is the foundation of frequency-domain filtering and system analysis. 3. The Dual: Multiplication Theorem The Convolution Theorem has a beautiful dual: the Fourier transform of a product is the convolution of the Fourier transforms (scaled by 1/2π): This duality is symmetric: just as time-domain convolution becomes frequency-domain multiplication, time-domain multiplication becomes frequency-domain convolution. This explains phenomena like amplitude modulation and windowing effects. The proof uses the definition of the Fourier transform and Fubini's theorem to interchange the order of integration:

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