MATH 414 Lecture 19

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Linear Time-Invariant Filters

Recall 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 f_a(t) = f(t-a)} , which 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 f} shifted to the right 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} units.

A filter is called 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}

At the end of last lecture, we discused how if $L(t)=f*h$ 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 h} , 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 L} is linear and time-invariant.

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 L} is a linear time-invariant filter 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) = f * h} , 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 h} is the impulse response function.

Proof. We've already shown that (⇒) is true, this will only show (⇐).

Suppose 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 linear and time-invariant. Let's apply 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} 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 \mathrm{e}^{i \, \lambda \, t}} :

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( \mathrm{e}^{i \, \lambda \, t} \right) = h^\lambda(t)}

Applying the filter on a shifted function gives

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( \mathrm{e}^{i \, \lambda \, (t-a)} \right) = h^\lambda(t-a)}

Observe 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-a)} = \mathrm{e}^{-i \, \lambda \, a} \cdot \mathrm{e}^{i \, \lambda \, t}}

The first term is just a scalar (with respect 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 \lambda} , but not with respect 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 t} ). Therefore

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} h^\lambda (t-a) &= L \left( \mathrm{e}^{-i \, \lambda \, a} \cdot \mathrm{e}^{i \, \lambda \, t} \right) \\ &= \mathrm{e}^{-i \, \lambda \, a} \, L \left( \mathrm{e}^{i \, \lambda \, t} \right) \\ &= \mathrm{e}^{-i \, \lambda \, a} \, h^\lambda(t) \end{align}}

holds for all 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 for all 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} ; so we can set 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 = t} .

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} h^{\lambda} (t-t) &= h^\lambda(0) = \mathrm{e}^{-i\,\lambda\,t} \, h^{\lambda}(t) \\ h^\lambda(t) &= \mathrm{e}^{i\,\lambda\,t} \, h^{\lambda}(0) \\ L \left( \mathrm{e}^{i \, \lambda \, t} \right) &= \mathrm{e}^{i\,\lambda\,t} \, h^{\lambda}(0) \end{align}}

Let's apply 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} to the inverse fourier transform of 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 \begin{align} L(f) &= L \left( \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty \hat{f} \, \mathrm{e}^{+i\,\lambda\,t} \,\mathrm{d}\lambda \right) \\ &= \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^{\infty} \hat{f} \, L \left( \mathrm{e}^{i \, \lambda \, t} \right) \,\mathrm{d}\lambda \\ &= \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^{\infty} \hat{f} \, h^\lambda (0) \, \mathrm{e}^{i \,\lambda \,t} \,\mathrm{d}\lambda \end{align}}

Let's define ${\hat {h}}(\lambda ):={\frac {1}{\sqrt {2\pi }}}\,h^{\lambda }(0)$. 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 \begin{align} L(f) &= \sqrt{2\pi} \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi}} \, \hat{f}(\lambda) \, \hat{h}(\lambda) \mathrm{e}^{i\,\lambda\,t} \,\mathrm{d}\lambda &= \mathcal{F}^{-1} \left( \hat{f}(\lambda) \, \hat{h}(\lambda) \right) \, \sqrt{2\pi} \\ &= f * h \end{align}}

quod erat demonstrandum

Impulse Response

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_{\delta}(t) = \theta(x+\delta) \, \frac{1}{2\delta} - \theta(x-\delta) \frac{1}{2\delta}}

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 \lim_{\delta \to 0^+} f_\delta(t)} is a spike such 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 f(t) = \begin{cases} \infty & t = 0 \\ 0 & \mbox{otherwise}\end{cases}}

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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} L(f_\delta(t)) &= \int_{-\infty}^\infty f(\tau) \, h(\tau-t) \,\mathrm{d}\tau \\ &= \ldots \end{align}}

Constructing Filters

${\begin{aligned}L(f)&=f*h\\\left(L(f)\right)&={\sqrt {2\pi }}{\hat {f}}(\lambda )\,{\hat {h}}(\lambda )\end{aligned}}$

Low pass filter

Reduces amplitudes of high magnitude frequencies

High pass filter

Reverse of low pass filter

Notch filter

Removes selection of frequencies.

"Ideal" Low Pass Filter

Take 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{h}(\lambda) = \begin{cases} \frac{1}{\sqrt{2\pi}} & \left| \lambda \right| \le \lambda_0 \\ 0 & \mbox{otherwise} \end{cases}}

Taking the convolution 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) = f * h} , 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 \begin{align} h &= \mathcal{F}^{-1} \left( \hat{h}(\lambda) \right) \\ &= \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^{\infty} \hat{h}(\lambda) \, \mathrm{e}^{+i \, \lambda \, t} \,\mathrm{d}\lambda \\ &= \frac{1}{2\pi} \int_{-\lambda_0}^{\lambda_0} \mathrm{e}^{i\,\lambda t} \,\mathrm{d}\lambda &= \frac{\sin{(\lambda_0 \, t)}}{\pi \, t} \end{align}}

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) = \int_{-\infty}^{\infty} f(\tau) \, \frac{\sin{(\lambda_0 \, (t-\tau))}}{\pi \, (t-\tau)} \,\mathrm{d}\tau}

Put 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(t) = \theta(t) - \theta(t-t_0)} . 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 L(f) = \int_{0}^{b} \frac{\sin{(\lambda_0 \, t)}}{\pi \, t}}

Perform 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 u} -substitution with 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 u = \lambda_0 \, (t-\tau)}

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} L(f) &= \int_{\lambda_0 \, t}^{\lambda_0 \, (t-t_0)} \frac{\sin{u}}{\pi \frac{u}{\lambda_0}} \, \frac{-\mathrm{d}u}{\lambda_0} \\ &= \int_{\lambda_0 \, (t-t_0)}^{\lambda_0 \, t} \frac{\sin{u}}{\pi\,u} \,\mathrm{d}u \\ &= \frac{1}{\pi} \left( \mathrm{Si}(\lambda_0 \, t) - \mathrm{Si}(\lambda_0 \, (t-t_0)) \right) \end{align}}

See Definition 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 \mathrm{Si}(x)} below.

Butterworth 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 h(t) = \begin{cases} a \, \mathrm{e}^{-\alpha \, t} & t \ge 0 \\ 0 & t < 0 \end{cases}}

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} \hat{h}(\lambda) &= \frac{1}{\sqrt{2\pi}} \, \int_{-\infty}^{\infty} h(t) \, \mathrm{e}^{-i\,\lambda\,t} \,\mathrm{d}t \\ &= \frac{A}{\sqrt{2\pi}} \, \int_{0}^{\infty} \mathrm{e}^{-\alpha \,t - i \,\lambda\,t} \,\mathrm{d}t \\ &= \frac{A}{\sqrt{2\pi} \, \left( \alpha + i \, \lambda \right)} \end{align}}

${\hat {L}}(f)={\cancel {\frac {\sqrt {2\pi }}{\sqrt {2\pi }}}}\,{\frac {A\,{\hat {f}}(\lambda )}{\alpha +i\,\lambda }}={\frac {A}{a+i\,\lambda }}\,{\hat {f}}(\lambda )$

Now 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| \hat{L}(f) \right|^2 = \frac{A^2}{\alpha^2 + \lambda^2} \, \left| \hat{f}(\lambda) \right|^2} . As 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} gets big, then the frequencies returned 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 \hat{f}} are reduced. Therefore this is essentially a low-pass filter

The Sine Integral

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{Si}(x) := \int_{0}^{x} \frac{\sin{u}}{u} \,\mathrm{d}u}