[DSP] W07 - Continuous-Time
contents
continuous-time
- the physical world is assumed to be continuous-time
- analog is continuous-time
- the computer world is discrete-time
fig: continuous-time world vs. discrete-time world metaphor
analog world
- calculus
- distributions
- systems theory
-
electronics
- mathematical parallel:
- real-values time: \(t\) (sec)
- functions: \(x(t) \in L_2(\mathbb{R}) \)
- frequency: \( \Omega \in \mathbb{R} (\frac{rad}{sec}) \)
- unlimited, not bound
- fourier-transform: \( L_2(\mathbb{R}) \mapsto L_2(\mathbb{R}) \)
discrete world
- arithmetic
- combinatorics
- computer science
-
dsp
- mathematical parallel:
- countable integer index: \(n\)
- sequences: \( x[n] \in \ell_2(\mathbb{Z}) \)
- frequency: \( \omega \in [-\pi,\pi] \)
- bounded
- dtft: \( \ell_2(\mathbb{Z}) \mapsto L_2([-\pi,\pi]) \)
analog-analog conversions
- dsp is used for analog-analog transform with intermediate digital processing
- example:
- mp3
- digital photography
- process input \(x(t)\) and output \(y(t)\) are both continuous-time
- process intermediate processing in on discrete sequences \(x[n], y[n]\)
- usually a storage, transport and reproduction process
digital-analog conversions
- there can be conversion from discrete-time to analog-time done with dsp
- examples:
- computer graphics
- video games
- process input \(x[n] \) is discrete-time
- process output \(y(t)\) is continuous-time
-
intermediate processing in on discrete sequences \( x[n], y[n] \)
- usually a synthesis process
analog-digital conversions
- there can be conversion from discrete-time to analog-time done with dsp
- examples:
- control systems
- measurement systems
- surveillance applications
- process input \(x(t)\) is continuous-time
- process output \(y[n] \) is discrete-time
-
intermediate processing in on discrete sequences \( x[n], y[n] \)
- usually a monitoring and reactive process
conversion cycle
fig: \(x(t)\) - continuous-time; \(x[n]\) - discrete-time
continuous-time dsp
- time: real variable \(t\)
- signal \(x(t)\) - complex function of a real variable
- finite energy: \( x(t) \in L_2(\mathbb{R}) \)
- inner product in \(L_2(\mathbb{R})\) \[ \langle x(t), y(t) \rangle = \int_{-\infty}^{\infty} x^{*}(t)y(t)dt \]
- energy: \[ \vert\vert x(t) \vert\vert^2 = \langle x(t), x(t) \rangle \]
- fourier-transform: \( L_2(\mathbb{R}) \mapsto L_2(\mathbb{R}) \)
analog LTI filters
fig: block diagram of an analog LTI
-
these are the analog parallel of discrete LTI filters
-
here, \[ \begin{align} y(t) & = (x * h)(t) \\
& = \langle h^*(t - \tau) , x(\tau) \rangle \\
& = \int_{\infty}^{\infty} x(\tau) h(t- \tau) d \tau \\
\end{align} \]
continuous-time fourier transform (CTFT)
- in discrete-time max angular frequency is \(\pm \pi\)
- in continuous-time no upper bound to frequency \( \omega \in \mathbb{R} \)
- however, the concept of breaking a function down into component sines is still the same
\[ \begin{align}
& \text{forward fourier-transform} \\
X(j\Omega) & = \int_{\infty}^{\infty} x(t) e^{-j\Omega t} dt & \leftarrow \text{not periodic} \\
\\
& \text{inverse fourier-transform} \\
x(t) & = \frac{1}{2\pi} \int_{\infty}^{\infty} X(j\Omega) e^{j\Omega t} d\Omega & \\
\end{align} \]
real-world frequency
- \( \Omega \) is expressed in \( \frac{rad}{s} \)
- \( F = \frac{\Omega}{2\pi} \) expressed in Hertz \( \left( \frac{1}{s} \right) \)
- period \( T = \frac{1}{F} = \frac{2\pi}{\omega} \)
example
fig: gaussian signal (analog)
- the fourier transform of above gaussian signal as a bell shaped magnitude curve
- rescaled appropriately
fig: gaussian signal fourier transform magnitude
continuous-time convolution
- for following filter action
fig: block diagram of an analog LTI
- the filter output fourier transform is scaling the input fourier transform with the frequency response of the filter
- frequency response is nothing but the fourier transform of the filter impulse response
\[ Y(j\Omega) = X(j\Omega)H(j\Omega) \]
bandlimited functions
- \(\Omega_N\)-bandlimitedness \[ X( j\Omega ) = 0 \text{ for } \vert \Omega \vert > \Omega_N \]
example - continuous-time fourier transform
- a bandlimited rect function
fig: prototypical bandlimited function
fourier transfer
\[ \begin{align}
\Phi(j\Omega) & = G \text{ } rect \Bigg ( \frac{\Omega}{2\Omega_N} \Bigg ) \\
\
\varphi(t) & = G \text{ } \frac{1}{2\pi} \int_{-\infty}^{\infty} \phi(j\Omega)e^{j\Omega t} d\Omega \\
& = G \frac{\Omega_N}{\pi} sinc \Bigg ( \frac{\Omega_N}{\pi} t \Bigg ) \\
\end{align}
\]
- normalization: \( G = \frac{\pi}{\Omega_N} \)
- total bandwidth: \( \Omega_B = 2\pi\Omega_N \)
- define: \( T_s = \frac{2\pi}{\Omega_B} = \frac{\pi}{\Omega_N} \)
fig: prototypical bandlimited function
- with define substitutions
\[ \begin{align}
\Phi(j\Omega) & = \frac{\pi}{\Omega_N} \text{ } rect \Bigg ( \frac{\Omega}{2\Omega_N} \Bigg ) \\
\
\varphi(t) & = sinc \Bigg ( \frac{t}{T_s} \Bigg ) \\
\end{align} \]
fig: prototypical bandlimited function with substitution
fig: fourier transform of prototypical bandlimited function