MATH 414 Lecture 20
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Last Time
"Structure Theorem": is a linear time-invariant filter if and only if , where is the impulse response function, and is the system function.
We also discussed an acausal filter .
For , we have 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 * h(t) = \frac{1}{\pi} \int_{\lambda_c \, (t-t_c)}^{\lambda_c \, t} \frac{\sin{u}}{u} \, \mathrm{d}u }
Plotting this filtered function gives a function that seems to give an output before the signal actually arrives (hence acausal), violating the laws of physics!
Causal Filters
Don't violate physical laws
A linear time-invariant 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 said to be causal if and only if the output 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 0} until the input function arrives.
Let our input be 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) = 0} 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 < 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) = f * h(t) = 0} 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 < t_0} as well.
The Butterworth filter is causal
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 causal 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 h(t) = 0} 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 < 0} .
Partial Proof. (⇐) Assume 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) = 0} 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 < t_0} . We write
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 t < t_0} . Then for 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_0 \le \tau < \infty} , 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 - \tau < t_0 - \tau < 0} , so 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 h(t-\tau) = 0} , so
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_{t_0}^\infty f(\tau) \cdot 0 \,\mathrm{d}\tau = 0} .
Corollary. 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 causal 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 \hat{h}(\lambda) = \frac{1}{\sqrt{2\pi}} \mathcal{L} \left\{ h \right\} (i \, \lambda)}
Proof. 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 h}
is causal, 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 \mathcal{L} \left\{ h \right\} = \int_{-\infty}^{\infty} h(t) \, \mathrm{e}^{-s\,t} \,\mathrm{d}t = \int_{0}^{\infty} h(t) \, \mathrm{e}^{-s\,t}}
. If we 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 s = i \, \lambda}
, we get 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{L} \left\{ h \right\}(i \, \lambda) = \int_{0}^{\infty} h(t) \mathrm{e}^{-i\,\lambda\,t} \,\mathrm{d}t = \sqrt{2\pi} \, \hat{h}(\lambda)}
Suppose 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 \hat{h}(\lambda)} is the system 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 \hat{h}(\lambda) = \frac{1}{\sqrt{2\pi}} \, \int_{0}^{\infty} H(t) \, \mathrm{e}^{i \, \lambda \, t} \,\mathrm{d}t} , hence 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) = \frac{1}{\sqrt{2\pi}} \, \mathcal{L} \left\{ H \right\}\left( s = i \, \lambda \right)}
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 h(t) := \begin{cases} H(t) & 0 \le t \\ 0 & t < 0 \end{cases}} . 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{h}(\lambda) = \frac{1}{\sqrt{2\pi}} \, \mathcal{L} \left\{ H \right\}(i\,\lambda) = \frac{1}{\sqrt{2\pi}} \, \int_{0}^{\infty} H(t) \mathrm{e}^{-i\,\lambda\,t} \,\mathrm{d}t = \mathcal{F}[h]}
Butterworth
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) = \frac{A}{\sqrt{2\pi} \, (\alpha + i\,\lambda)} = \frac{A}{\sqrt{2\pi}} \, \frac{1}{\alpha + s}} , 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 \frac{1}{\alpha + s} = \int_{0}^{\infty} \mathrm{e}^{-\alpha \, t} \cdot \mathrm{e}^{-s\,t} \,\mathrm{d}t = \mathcal{L} \left\{ \mathrm{e}^{-\alpha \,t} \right\}}
Look up Papoulis in regard to signal processing
Example
Consider the simple circuit containing an input voltage 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 v} , a resistor of resistance 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 R} , and a capacitor with capacitance 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 R} connected in series with a current 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 I} running through it.
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 v(t) = I \, R + \frac{Q}{C}}
We also have 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 I = \frac{\mathrm{d}Q}{\mathrm{d}t}} , hence
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} v(t) &= R \, \frac{\mathrm{d}Q}{\mathrm{d}t} + \frac{Q}{C} \\ \frac{v(t)}{R} = \frac{\mathrm{d}Q}{\mathrm{d}t} + \frac{Q}{R \, C} \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 R\,C} has units of time.
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 \alpha = \frac{1}{R \, C}} . 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 \frac{v(t)}{R} = \frac{\mathrm{d}Q}{\mathrm{d}t} + \alpha \, Q = f(t)} . Suppose that at 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=0} , 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 Q(0) = 0} , 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(t)=0} 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 \le 0} .
Nothing can stop us from taking the Laplace transform of the 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} \mathcal{L} \left\{ Q = \alpha \, Q \right\}(s) &= s \, \mathcal{L} \left\{ Q \right\} + \alpha \mathcal{L} \left\{ Q \right\} \\ &= F(s) \\ \mathcal{L} \left\{ Q \right\} &= \frac{1}{s+\alpha} \, F(s) \\ F(s) &= \int_{0}^{\infty} f(t) \,\mathrm{e}^{-s\,t}\,\mathrm{d}t \\ F(i\,\lambda) &= \frac{0}{\infty} f(t) \, \mathrm{e}^{-i \, \lambda \, t} \,\mathrm{d}t = \sqrt{2\pi} \, \mathcal{L} \left\{ f \right\}(i\,\lambda) \\ \mathcal{L} \left\{ Q \right\} (i\,\lambda) &= \frac{1}{\alpha + i\,\lambda} \, \sqrt{2\pi} \, \hat{f}(\lambda) \\ \frac{1}{\alpha + i\,\lambda} &= \sqrt{2\pi} \cdot \frac{1}{\sqrt{2\pi}} \, \mathcal{F} \left[ A \, \mathrm{e}^{-\alpha \, t} \, H(t) \right] \\ \frac{\sqrt{2\pi}}{\alpha + i \, h} &= \mathcal{F} \left[ A \, \mathrm{e}^{-\alpha \, t} \right] (i \, \lambda) \\ F(i\,\lambda) = \sqrt{2\pi} \, \hat{f}(\lambda) \\ \frac{1}{\sqrt{2\pi}} \, \mathcal{L}[Q](i\,\lambda) &= \frac{1}{\alpha + i \, \lambda} \, \cancel{\sqrt{2\pi}} \, \hat{f}(\lambda) \\ &= \hat{Q}(\lambda) = \frac{1}{\alpha + i\,\lambda} \, \hat{f}(\lambda) \\ \ldots \end{align}}
Signals can be filtered by running them through a bunch of circuits
Sampling Theorem
We say 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)} is band-limited 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 \hat{f}(\lambda) = 0} for 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| \lambda \right| > \Omega} for some finite 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 \Omega}
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 \Omega} is a circular or angular frequency, and the frequency we would work with 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 \nu_{sig} = \frac{\Omega}{2\pi}}
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(t)} be band-limited
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) = 0} outside of the band 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[ -\Omega, \Omega \right]} . 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 f(t) = \sum_{j=-\infty}^\infty f \left( \frac{j\,\pi}{\Omega} \right) \, \mathrm{sinc} \left( \Omega \,t - j\, \pi \right)}
Hence, we only need a finite number of discrete samples of the signal 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(x)} to reconstruct the entire signal