MATH 414 Lecture 18

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Convolution Theorem

By definition, the convolution of two functions Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(t)} is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( f * g \right)(t) = \int_{-\infty}^{\infty} f(\tau) \, g(t - \tau) \,\mathrm{d}\tau}

Theorem. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{F} \left[ f * g \right] (\lambda) = \sqrt{2\pi} \, \hat{f}(\lambda) \, \hat{g}(\lambda)}

Proof. (omitted)

quod erat demonstrandum

Plancheral (Parseval) Theorem

Given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{g}} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle f, g \right\rangle_{L^2} = \int_{-\infty}^{\infty} f(t) \, g(t) \,\mathrm{d}t}

Show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle f,g \right\rangle_{L^2} = \left\langle \hat{f}, \hat{g} \right\rangle_{L^2}}

What does this mean?

let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=g}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} {\left\| f \right\|_{L^2}}^2 &= \int_{-\infty}^{\infty} \left| f(t) \right|^2 \,\mathrm{d}t \\ {\left\| \hat{f} \right\|_{L^2}}^2 &= \int_{-\infty}^{\infty} \left| \hat{f}(t) \right|^2 \,\mathrm{d}t \\ \hat{f}(\lambda) &= \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^{\infty} f(t) \, \mathrm{e}^{-i\,\lambda\,t}\,\mathrm{d}t \\ f(t) &= \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^{\infty} \hat{f}(\lambda) \, \mathrm{e}^{+i\,\lambda\,t}\,\mathrm{d}\lambda \\ \end{align}}

Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{e}^{i \, \lambda \, t}} is a "pure tone frequency" for fixed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} .

Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{f}(\lambda)} represents the amplitude and phase.

When we compute the synthesis

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t) = \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^{\infty} \hat{f} (\lambda) \, \mathrm{e}^{i \, \lambda \, t} \,\mathrm{d}\lambda}

We are expressing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)} as a "linear combination" of "pure" tones. In Fourier Series, we do the same thing (synthesis of "pure" tones)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t) = \sum_{n = -\infty}^\infty c_n \, \mathrm{e}^{i \, n \, t}}

Parseval States that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{-\infty}^\infty \left| f(t) \right|^2 \,\mathrm{d}t} is the total energy in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} .

Theorem. [Plancheral / Parseval].

Proof.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int_{-\infty}^{\infty} f(t) \, \overline{g(t)} \,\mathrm{d}t &= \int_{-\infty}^{\infty} f(t) \, \overline{ \left( \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^\infty \hat{g}(\lambda) \, \mathrm{e}^{+i \, \lambda \, t} \,\mathrm{d}\lambda \right)} \,\mathrm{d}t \\ &= \int_{-\infty}^{\infty} f(t) \, \left( \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^\infty \overline{\hat{g}(\lambda)} \, \mathrm{e}^{-i \, \lambda \, t} \,\mathrm{d}\lambda \right) \,\mathrm{d}t \\ &\underset{=}{fubini} \int_{-\infty}^{\infty} \overline{\hat{g}(\lambda)} \, \left( \frac{1}{\sqrt{2\pi}} \,\int_{-\infty}^\infty f(t) \, \mathrm{e}^{-i \, \lambda \, t} \, \mathrm{d}t \right) \,\mathrm{d}\lambda \\ &= \int_{-\infty}^\infty \hat{f}(\lambda) \, \overline{\hat{g}(\lambda)} \,\mathrm{d}\lambda \\ &= \left\langle \hat{f} , \hat{g} \right\rangle_{L^2} \end{align}}

quod erat demonstrandum

Linear Filters

space of signals: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2} (work with piecewise continuous)

A Linear Filter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} is a linear transformation from the space of signals into the space of signals (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2} ). This means Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L \left[ \alpha \, f + \beta \, g \right] = \alpha \, L[f] + \beta \, L[g]} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} are constants, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} are signals.

We shall use the notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L[f](t) = \tilde{f}(t)}

Time invariance

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)} , start at 12 noon, but we perform a shift (time translation) by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and apply again, we'll get the same signal back.

Define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_a(t) := f(t-a)}

When Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a>0} , then the signal shifts to the right; and when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a<0} , the signal shifts to the left.

A linear filter is said to be time-invariant if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L[f_a] = (L[f])_a = \tilde{f}(t-a)} Intuitively, when we shift a function and filter, we should get the same thing as if we would have applied the filter and then shifted.

All time-invariant filters look like convolutions

Example: Running Filter

Take some time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d > 0} and a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , we define

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L[f](t) = \frac{1}{d} \, \int_{t-d}^{t} f(\tau) \,\mathrm{d}\tau}

This is the average of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} over the interval from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t-d} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} . This has the effect of smoothing a "rough" function.