State convolution theorem for 1D
The convolution theorem for 1D signals relates the convolution operation in the time (or spatial) domain to multiplication in the frequency domain. It states:
Convolution Theorem (1D): Let f(x)f(x)f(x) and g(x)g(x)g(x) be two functions (signals) defined on the real line R\mathbb{R}R. Denote the Fourier transforms of fff and ggg as F{f(x)}=F(ω)\mathcal{F}\{f(x)\} = F(\omega)F{f(x)}=F(ω) and F{g(x)}=G(ω)\mathcal{F}\{g(x)\} = G(\omega)F{g(x)}=G(ω), respectively. Then, the Fourier transform of their convolution h(x)=(f∗g)(x)h(x) = (f * g)(x)h(x)=(f∗g)(x) is given by the pointwise multiplication of their Fourier transforms: F{(f∗g)(x)}=H(ω)=F(ω)⋅G(ω).\mathcal{F}\{(f * g)(x)\} = H(\omega) = F(\omega) \cdot G(\omega).F{(f∗g)(x)}=H(ω)=F(ω)⋅G(ω).
In mathematical terms: F{(f∗g)(x)}=F(ω)⋅G(ω),\mathcal{F}\{(f * g)(x)\} = F(\omega) \cdot G(\omega),F{(f∗g)(x)}=F(ω)⋅G(ω), where ⋅\cdot⋅ denotes pointwise multiplication.
Explanation:
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Convolution in Time Domain: The convolution (f∗g)(x)(f * g)(x)(f∗g)(x) of two functions f(x)f(x)f(x) and g(x)g(x)g(x) is defined as: (f∗g)(x)=∫−∞∞f(τ)g(x−τ) dτ.(f * g)(x) = \int_{-\infty}^{\infty} f(\tau) g(x - \tau) \, d\tau.(f∗g)(x)=∫−∞∞?f(τ)g(x−τ)dτ. This operation combines the effects of fff and ggg over all possible shifts of ggg across fff.
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Multiplication in Frequency Domain: The Fourier transform F(ω)F(\omega)F(ω) of f(x)f(x)f(x) and G(ω)G(\omega)G(ω) of g(x)g(x)g(x) are complex functions that represent the frequency content (spectrum) of fff and ggg, respectively. According to the convolution theorem, multiplying F(ω)F(\omega)F(ω) and G(ω)G(\omega)G(ω) gives the Fourier transform H(ω)H(\omega)H(ω) of the convolution h(x)=(f∗g)(x)h(x) = (f * g)(x)h(x)=(f∗g)(x).
Importance:
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Computational Efficiency: The convolution theorem allows us to compute convolutions efficiently using the Fast Fourier Transform (FFT) algorithm, which reduces the computational complexity from O(n2)O(n^2)O(n2) to O(nlog?n)O(n \log n)O(nlogn) for nnn samples.
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Signal Processing Applications: In signal processing, this theorem is fundamental for designing filters, analyzing systems, and understanding the frequency response of linear time-invariant systems.
Understanding the convolution theorem provides insights into how signals can be transformed between time and frequency domains, enabling efficient processing and analysis in various scientific and engineering disciplines.
