port from mathematics-physics notes
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# Amplitude modulation
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> *Theorem*: a multiplication of two harmonic functions results in a sum of harmonics withh the sum and difference of the original frequencies. This is called *heterodyne*.
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??? note "*Proof*:"
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Will be added later.
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For example if we have a harmonic signal $m: \mathbb{R} \to \mathbb{R}$ with $\omega, A \in \mathbb{R}$ given by
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$$
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m(t) = A \cos \omega t,
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$$
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for all $t \in \mathbb{R}$ and a harmonic carrier signal $c: \mathbb{R} \to \mathbb{R}$ with $\omega_c \in \mathbb{R}$ given by
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$$
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c(t) = \cos \omega_c t.
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$$
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for all $t \in \mathbb{R}$. Then the multiplication of both is given by
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$$
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m(t)c(t) = A \cos (\omega t) \cos (\omega_c t) = \frac{A}{2} \bigg(\cos t(\omega + \omega)c + \cos t(\omega - \omega_c) \bigg),
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$$
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obtaining heterodyne.
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> *Definition*: amplitude modulation makes use of a harmonic carrier signal $c: \mathbb{R} \to \mathbb{R}$ with a reasonable angular frequency $\omega_c \in \mathbb{R}$ given by
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>
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> $$
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> c(t) = \cos \omega_c t
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> $$
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>
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> for all $t \in \mathbb{R}$ to modulate a signal $m: \mathbb{R} \to \mathbb{R}$.
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<br>
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> *Theorem*: For the case that the carrier signal is not additionaly transmitted we obtain
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>
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> $$
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> m(t) c(t) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2} \big(M(\omega + \omega_c) + M(\omega - \omega_c) \big),
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> $$
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>
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> for all $t, \omega \in \mathbb{R}$.
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>
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> For the case that the carrier signal is additionaly transmitted we obtain
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>
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> $$
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> m(t) (1 + c(t)) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2} \Big(M(\omega + \omega_c) + M(\omega - \omega_c) + \pi \big(\delta(\omega + \omega_c) + \delta(\omega - \omega_c) \big) \Big)
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> $$
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>
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> for all $t, \omega \in \mathbb{R}$.
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>
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> Therefore multiple bandlimited signals can be transmitted simultaneously in frequency bands.
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??? note "*Proof*:"
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Will be added later.
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# The discrete Fourier transform
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> *Theorem*: sampling a signal with the impulse train makes the spectrum of the signal periodic.
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??? note "*Proof*:"
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Will be added later.
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A bandlimited signal implies that its frequency components are zero outside the bandwidth frequency interval.
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> *Theorem*: if a signal has a bandwidth $\omega_b \in \mathbb{R}$ then it can be completely determined from its samples at a sampling frequency $\omega_s \in \mathbb{R}$ given by
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>
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> $$
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> \omega_s > 2 \omega_b.
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> $$
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??? note "*Proof*:"
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Will be added later.
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When the sampling frequency does not comply to this statement, the reconstruction of the spectrum will exhibit imperfections known as aliasing. The critical value of the sampling frequency is known as the *Nyquist* frequency.
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## The discrete time Fourier transform
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> *Theorem*: let $f: \mathbb{R} \to \mathbb{C}$ be a signal with its sampled signal $f_s(t) = f(t) \delta_{T_s}(t)$ for all $t \in \mathbb{R}$ with sampling period $T_s \in \mathbb{R}$. Then the discrete time Fourier transform $F: \mathbb{R} \to \mathbb{C}$ of $f_s$ is given by
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>
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> $$
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> F(\Omega) = \sum_{m = -\infty}^\infty f[m] e^{-im\Omega},
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> $$
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>
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> for all $\Omega \in \mathbb{R}$. With $\Omega = \omega T_s$ the dimensionless frequency and $F_s(\omega) := F(\Omega)$.
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??? note "*Proof*:"
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Will be added later.
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## The discrete Fourier transform
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> *Theorem*: let $f: \mathbb{R} \to \mathbb{C}$ be a signal and $f_N: \mathbb{R} \to \mathbb{C}$ the truncated signal of $f$ by $N \in \mathbb{N}$ given by
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>
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> $$
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> f_N[m] = \begin{cases} f[m] &\text{ if } m \in \{0, \dots, N - 1\}, \\ 0 &\text{ if } m \notin \{0, \dots, N - 1\}, \end{cases}
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> $$
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>
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> sampled by $T_s \in \mathbb{R}$. Its discrete Fourier transform $F_N: \mathbb{R} \to \mathbb{C}$ is given by
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>
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> $$
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> F_N[k] = \sum_{m=0}^{N-1} f[m] \exp \bigg(-2\pi i \frac{km}{N} \bigg)
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> $$
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>
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> for all $k \in \{0, \dots, N-1\}$.
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??? note "*Proof*:"
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Will be added later.
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We have that $F_N[k] = F_N(k\Delta \omega)$ with $\Delta \omega = \frac{2\pi}{N T_s}$ the angular frequency resolution.
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> *Theorem*: let $F_N: \mathbb{R} \to \mathbb{C}$ be a spectrum of a signal truncated by $N \in \mathbb{N}$ then its inverse discrete Fourier transform $f_N: \mathbb{R} \to \mathbb{C}$ is given by
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>
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> $$
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> f[m] = \frac{1}{N} \sum_{k=0}^{N-1} F_N[k] \exp \bigg(2\pi i \frac{km}{N} \bigg)
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> $$
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>
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> for all $m \in \{0, \dots, N - 1\}$.
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??? note "*Proof*:"
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Will be added later.
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> *Definition*: therefore $f_N$ and $F_N$ with $N \in \mathbb{N}$ form a discrete Fourier transform pair denoted by
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>
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> $$
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> f_N \overset{\mathcal{DF}}\longleftrightarrow F_N,
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> $$
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>
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> therefore we have
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>
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> $$
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> \begin{align*}
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> &f_N[m] = \mathcal{DF}^{-1}[F_N[k]], \quad &\forall m \in \{0, \dots, N - 1\}, \\
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> &F_N[k] = \mathcal{DF}[f[m]], \quad &\forall k \in \{0, \dots, N - 1\}.
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> \end{align*}
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> $$
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# Fourier series
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> *Theorem*: the "Fourier" inner product of two functions $g, f: \mathbb{C} \to \mathbb{C}$ is defined as
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>
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> $$
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> \langle f, g \rangle = \int_a^b f(t) \overline g(t) dt
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> $$
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>
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> with $f, g$ members of the square integrable functions $L^2[a,b]$ with $a,b \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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The overline generally implies the complex conjugate.
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> *Corollary*: the "Fourier" norm of a square integrable function $f: \mathbb{C} \to \mathbb{C}$ is defined as
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>
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> $$
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> \|f\| = \sqrt{\langle f, f \rangle}.
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> $$
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f: \mathbb{R} \to \mathbb{R}$ be a periodic function with period $T_0 \in \mathbb{R}$ then the autocorrelation of $f$ will create peaks for $t = zT_0$ for all $t \in \mathbb{R}$ and $z \in \mathbb{Z}$.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Definition*: two functions $f, g: \mathbb{C} \to \mathbb{C}$ are orthogonal if and only if
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>
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> $$
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> \langle f, g \rangle = 0
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> $$
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## Approximating functions
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> *Lemma*: a function $f: \mathbb{R} \to \mathbb{C}$ can be approximated with a linear combination of orthogonal functions $b_k: \mathbb{R} \to \mathbb{C}$ given by
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>
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> $$
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> \phi_n(t) = \sum_{k=0}^n c_k b_k(t),
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> $$
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>
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> for all $t \in \mathbb{R}$ with $n \in \mathbb{N}$ the order. The coefficients $c_k \in \mathbb{C}$ that minimise $\|f - \phi_n\|$ may be determined by
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>
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> $$
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> c_k = \frac{\langle f, b_k \rangle}{\langle b_k, b_k \rangle}.
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> $$
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??? note "*Proof*:"
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Will be added later.
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The orthogonal functions $b_k: \mathbb{R} \to \mathbb{C}$ have not yet been specified. There are many possible choices (Legendre polynomials, Bessel functions, spherical harmonics etc.) for these functions, for the Fourier series specifically we make use trigonometric or more generally imaginary exponential functions.
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> *Lemma*: in the special case that $b_k: \mathbb{R} \to \mathbb{C}$ is given by
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>
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> $$
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> b_k(t) = \exp(i k \omega_0 t),
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> $$
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>
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> for all $t \in \mathbb{R}$ with $k \in \mathbb{Z}$ and $\omega_0 \in \mathbb{R}$ the angular frequency. A periodic function $f: \mathbb{R} \to \mathbb{C}$ with period $T_0 = \frac{2\pi}{\omega_0}$ may be approximated by
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>
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> $$
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> \phi_n(t) = \sum_{k = 0}^n c_k e^{i k \omega_0 t},
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> $$
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>
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> for all $t \in \mathbb{R}$. With the coefficients $c_k \in \mathbb{C}$ given by
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>
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> $$
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> c_k = \frac{1}{T_0} \int_0^{T_0} f(t) e^{-i k \omega_0 t}dt.
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> $$
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Lemma*: For a periodic function $f: \mathbb{R} \to \mathbb{C}$ and its approximation $\phi_n$ given in the above lemma we have
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>
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> $$
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> \lim_{n \to \infty} \|f - \phi_n \| = 0,
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> $$
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>
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> implies that the resulting series approximation converges to $f$. Similarly the series approximation converges also pointwise
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>
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> $$
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> \lim_{n \to \infty} |f(t) - \phi_n(t)| = 0,
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> $$
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>
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> for all $t \in D$ with $D \subseteq \mathbb{R}$ the interval where $f$ is continuous.
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??? note "*Proof*:"
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Will be added later.
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## The Fourier series
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With the above lemmas we may state the following theorems.
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> *Theorem*: the classical Fourier series of a periodic function $f: \mathbb{R} \to \mathbb{C}$ with period $T_0 = \frac{2\pi}{\omega_0}$ may be given by
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>
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> $$
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> f(t) = \sum_{k = -\infty}^\infty c_k e^{i k \omega_0 t},
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> $$
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>
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> for all $t \in \mathbb{R}$. With the coefficients $c_k \in \mathbb{C}$ given by
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>
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> $$
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> c_k = \frac{1}{T_0} \int_0^{T_0} f(t) e^{-i k \omega_0 t}dt.
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> $$
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??? note "*Proof*:"
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Will be added later.
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Expanding the Fourier series such that it can also approximate aperiodic functions obtains.
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> *Theorem*: the Fourier series of an aperiodic function $f: \mathbb{R} \to \mathbb{C}$ may be given
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>
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> $$
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> f(t) = \frac{1}{2\pi} \int_{-\infty}^\infty F(\omega) e^{i \omega t} d\omega,
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> $$
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>
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> for all $t \in \mathbb{R}$. The expansion coefficient $F: \mathbb{R} \to \mathbb{C}$ is given by
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>
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> $$
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> F(\omega) = \int_{-\infty}^\infty f(t) e^{-i \omega t}dt
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> $$
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>
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> for all $\omega \in \mathbb{R}$. Is called the Fourier transform of $f$ and represents the continuous frequency spectrum of $f$.
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??? note "*Proof*:"
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Will be added later.
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# Fourier transformations
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## Definition of the Fourier transform
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> *Definition*: let $f, F: \mathbb{R} \to \mathbb{C}$, the Fourier transform of $f$ is given by
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>
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> $$
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> F(\omega) = \int_{-\infty}^\infty f(t) e^{-i \omega t}dt,
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> $$
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>
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> for all $\omega \in \mathbb{R}$. The inverse Fourier transform of $F$ is given by
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>
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> $$
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> f(t) = \frac{1}{2\pi} \int_{-\infty}^\infty F(\omega) e^{i \omega t} d\omega,
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> $$
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>
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> for all $t \in \mathbb{R}$. Therefore $f$ and $F$ form a Fourier transform pair denoted by
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>
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> $$
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> f \overset{\mathcal{F}}\longleftrightarrow F,
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> $$
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>
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> therefore we have
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>
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> $$
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> \begin{align*}
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> &f(t) = \mathcal{F}^{-1}[F(\omega)], \quad &\forall t \in \mathbb{R}, \\
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> &F(\omega) = \mathcal{F}[f(t)], \quad &\forall \omega \in \mathbb{R}.
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> \end{align*}
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> $$
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## Properties of the Fourier transform
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> *Proposition*: let $f, g, F, G: \mathbb{R} \to \mathbb{C}$, we have linearity given by
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>
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> $$
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> af(t) + bg(t) \overset{\mathcal{F}}\longleftrightarrow aF(\omega) + bG(\omega),
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> $$
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>
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> with $a,b \in \mathbb{C}$.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$, we have time shifting given by
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>
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> $$
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> f(t - t_0) \overset{\mathcal{F}}\longleftrightarrow F(\omega) e^{-i\omega t_0},
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> $$
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>
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> with $t_0 \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$, we have frequency shifting given by
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>
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> $$
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> e^{i \omega_0 t} f(t) \overset{\mathcal{F}}\longleftrightarrow F(\omega - \omega_0)
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> $$
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>
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> with $\omega_0 \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$, we have time or frequency scaling given by
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>
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> $$
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> f(t/a) \overset{\mathcal{F}}\longleftrightarrow |a| F(a\omega)
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> $$
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>
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> with $a \in \mathbb{R}$.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f, g, F, G: \mathbb{R} \to \mathbb{C}$, we have time convolution given by
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>
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> $$
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> f(t) * g(t) \overset{\mathcal{F}}\longleftrightarrow F(\omega) G(\omega).
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> $$
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f, g, F, G: \mathbb{R} \to \mathbb{C}$, we have frequency convolution given by
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>
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> $$
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> f(t) g(t) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2\pi} F(\omega) * G(\omega).
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> $$
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$ be differentiable, we have time differentation given by
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>
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> $$
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> f'(t) \overset{\mathcal{F}}\longleftrightarrow i \omega F(\omega).
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> $$
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: let $f,F: \mathbb{R} \to \mathbb{C}$ be differentiable, we have time integration given by
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>
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> $$
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> \int_{-\infty}^t f(u)du \overset{\mathcal{F}}\longleftrightarrow \frac{1}{i\omega} F(\omega) + \pi F(0)\delta(\omega).
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> $$
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??? note "*Proof*:"
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Will be added later.
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<br>
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@ -0,0 +1,116 @@
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# Signal filters
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## The impedance
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> *Proposition*: considering an ideal resistor $R \in \mathbb{R}$. When a voltage difference $v_i - v_o$ is applied to both ends a current $I: \mathbb{R} \to \mathbb{R}$ will be induced given by
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>
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> $$
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> v_i(t) - v_o(t) = R I(t),
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> $$
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>
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> for all $t \in \mathbb{R}$. By taking the Fourier transform of both sides of the expression we find
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>
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> $$
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> V_i(\omega) - V_o(\omega) = R I(\omega),
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> $$
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>
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> for all $\omega \in \mathbb{R}$ with $V_{i,o}: \mathbb{R} \to \mathbb{C}$ the fourier transforms of the voltage difference.
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??? note "*Proof*:"
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Will be added later.
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<br>
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> *Proposition*: considering a load coil with inductance $L \in \mathbb{R}$. When a voltage difference $v_i - v_o$ is applied to both ends a current $I: \mathbb{R} \to \mathbb{R}$ will be induced given by
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>
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> $$
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> v_i(t) - v_o(t) = L I'(t),
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> $$
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>
|
||||
> for all $t \in \mathbb{R}$. By taking the Fourier transform of both sides of the expression we find
|
||||
>
|
||||
> $$
|
||||
> V_i(\omega) - V_o(\omega) = i \omega L I(\omega),
|
||||
> $$
|
||||
>
|
||||
> for all $\omega \in \mathbb{R}$ with $V_{i,o}: \mathbb{R} \to \mathbb{C}$ the fourier transforms of the voltage difference.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
<br>
|
||||
|
||||
> *Proposition*: considering a capacitor with capacity $C \in \mathbb{R}$. When a voltage difference $v_i - v_o$ is applied to both ends a current $I: \mathbb{R} \to \mathbb{R}$ will be induced given by
|
||||
>
|
||||
> $$
|
||||
> v_i(t) - v_o(t) = \frac{1}{C} \int_{-\infty}^t I(t)dt,
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$. By taking the Fourier transform of both sides of the expression we find
|
||||
>
|
||||
> $$
|
||||
> V_i(\omega) - V_o(\omega) = \bigg(\frac{1}{i \omega C} + \frac{\pi \delta(\omega)}{C} \bigg) I(\omega),
|
||||
> $$
|
||||
>
|
||||
> for all $\omega \in \mathbb{R}$ with $V_{i,o}: \mathbb{R} \to \mathbb{C}$ the fourier transforms of the voltage difference.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition*: the complex impedance $Z: \mathbb{R} \to \mathbb{C}$ is defined as
|
||||
>
|
||||
> $$
|
||||
> V_i(\omega) - V_o(\omega) = Z(\omega) I(\omega)
|
||||
> $$
|
||||
>
|
||||
> for all $\omega \in \mathbb{R}$.
|
||||
|
||||
Therefore the complex impedance for the ideal resistor is given by $Z(\omega) = R$ and for the load coil $Z(\omega) = i \omega L$ for all $\omega \in \mathbb{R}$.
|
||||
|
||||
> *Proposition*: the impedance elements $Z_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in series can be summed to obtain $Z$
|
||||
>
|
||||
> $$
|
||||
> Z = Z_1 + \dots + Z_n.
|
||||
> $$
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
<br>
|
||||
|
||||
> *Proposition*: the impedance elements $Z_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in parallel can be inversely summed to obtain $Z$
|
||||
>
|
||||
> $$
|
||||
> \frac{1}{Z} = \frac{1}{Z_1} + \dots + \frac{1}{Z_n}.
|
||||
> $$
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
<br>
|
||||
|
||||
## The transfer function
|
||||
|
||||
> *Definition*: the relation between the input and output voltage in the frequency domain $V_{i,o}: \mathbb{R} \to \mathbb{C}$ can be written as
|
||||
>
|
||||
> $$
|
||||
> V_o(\omega) = H(\omega) V_i(\omega),
|
||||
> $$
|
||||
>
|
||||
> for all $\omega \in \mathbb{R}$ with $H: \mathbb{R} \to \mathbb{C}$ the transfer function.
|
||||
|
||||
The transfer function may be interpreted as a frequency filter of the signal.
|
||||
|
||||
Some ideal filters are given in the list below
|
||||
|
||||
* a *low-pass* filter removes all frequency components $\omega > \omega_c$ with $\omega_c \in \mathbb{R}$ the cut-off frequency,
|
||||
* a *high-pass* filter removes all frequency components $\omega < \omega_c$,
|
||||
* a *band-pass* filter removes all frequency componets outside a particular frequency range,
|
||||
* a *band-stop* filter removes all frequency compnents inside a particular frequency range.
|
||||
220
docs/physics/mathematical-physics/signal-analysis/signals.md
Normal file
220
docs/physics/mathematical-physics/signal-analysis/signals.md
Normal file
|
|
@ -0,0 +1,220 @@
|
|||
# Signals
|
||||
|
||||
## Definitions
|
||||
|
||||
> *Definition*: a signal is a function of space and time.
|
||||
>
|
||||
> * Output can be analog or quantised.
|
||||
> * Input can be continuous or discrete.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition*: a signal can be sampled at particular moments $k T_s$ in time, with $k \in \mathbb{Z}$ and $T_s \in \mathbb{R}$ the sampling period. For a signal $f: \mathbb{R} \to \mathbb{R}$ sampled with a sampling period $T_s$ may be denoted by
|
||||
>
|
||||
> $$
|
||||
> f[k] = f(kT_s), \qquad \forall k \in \mathbb{Z}.
|
||||
> $$
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition*: signal transformations on a function $x: \mathbb{R} \to \mathbb{R}$ obtaining the function $y: \mathbb{R} \to \mathbb{R}$ are given by
|
||||
>
|
||||
> | Signal transformation | Time | Amplitude |
|
||||
> | :-: | :-: | :-: |
|
||||
> | Reversal | $y(t) = x(-t)$ | $y(t) = -x(t)$ |
|
||||
> | Scaling | $y(t) = x(at)$ | $y(t) = ax(t)$ |
|
||||
> | Shifting | $y(t) = x(t - b)$ | $y(t) = x(t) + b$ |
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
For sampled signals similar definitions hold.
|
||||
|
||||
### Symmetry
|
||||
|
||||
> *Definition*: consider a signal $f: \mathbb{R} \to \mathbb{R}$ which is defined in an interval which is symmetric around $t = 0$, we define.
|
||||
>
|
||||
> * $f$ is *even* if $f(t) = f(-t)$, $\forall t \in \mathbb{R}$.
|
||||
> * $f$ is *odd* if $f(t) = -f(-t)$, $\forall t \in \mathbb{R}$.
|
||||
|
||||
For sampled signals similar definitions hold.
|
||||
|
||||
> *Theorem*: every signal can be decomposed into symmetric parts.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
### Periodicity
|
||||
|
||||
> *Definition*: a signal $f: \mathbb{R} \to \mathbb{R}$ is defined to be periodic in $T$ if and only if
|
||||
>
|
||||
> $$
|
||||
> f(t + T) = f(t), \qquad \forall t \in \mathbb{R}.
|
||||
> $$
|
||||
|
||||
For sampled signals similar definitions hold.
|
||||
|
||||
> *Theorem*: a summation of two periodic signals with periods $T_1, T_2 \in \mathbb{R}$ respectively is periodic if and only if
|
||||
>
|
||||
> $$
|
||||
> \frac{T_1}{T_2} \in \mathbb{Q}.
|
||||
> $$
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
### Signals
|
||||
|
||||
> *Definition*: the Heaviside step signal $u: \mathbb{R} \to \mathbb{R}$ is defined by
|
||||
>
|
||||
> $$
|
||||
> u(t) = \begin{cases} 1 &\text{ if } t > 0,\\ 0 &\text{ if } t < 0,\end{cases}
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
For a sampled function the Heaviside step signal is given by
|
||||
|
||||
$$
|
||||
u[k] = \begin{cases} 1 \text{ if } k \geq 0, \\ 0 \text{ if } k < 0, \end{cases}
|
||||
$$
|
||||
|
||||
for all $k \in \mathbb{Z}$.
|
||||
|
||||
> *Definition*: the rectangular signal $\text{rect}: \mathbb{R} \to \mathbb{R}$ is defined by
|
||||
>
|
||||
> $$
|
||||
> \text{rect} (t) = \begin{cases} 1 &\text{ if } |t| < \frac{1}{2}, \\ 0 &\text{ if } |t| > \frac{1}{2},\end{cases}
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
The rect signal can be normalised obtaining the scaled rectangular signal $D: \mathbb{R} \to \mathbb{R}$ defined by
|
||||
|
||||
$$
|
||||
D(t, \varepsilon) = \begin{cases} \frac{1}{\varepsilon} &\text{ if } |t| < \frac{\varepsilon}{2},\\ 0 &\text{ if } |t| > \frac{\varepsilon}{2},\end{cases}
|
||||
$$
|
||||
|
||||
for all $t \in \mathbb{R}$.
|
||||
|
||||
The following signal has been derived from the scaled rectangular signal $D: \mathbb{R} \to \mathbb{R}$ used on a signal $f: \mathbb{R} \to \mathbb{R}$ for
|
||||
|
||||
$$
|
||||
\lim_{\varepsilon \;\downarrow\; 0} \int_{-\infty}^{\infty} f(t) D(t, \varepsilon)dt = \lim_{\varepsilon \;\downarrow\; 0} \frac{1}{\varepsilon} \int_{-\frac{\varepsilon}{2}}^{\frac{\varepsilon}{2}} f(t) dt = f(0),
|
||||
$$
|
||||
|
||||
using the mean [value theorem for integrals](../../../mathematics/calculus/integration.md#the-mean-value-theorem-for-integrals).
|
||||
|
||||
> *Definition*: the Dirac signal $\delta$ is a generalized signal defined by the properties
|
||||
>
|
||||
> $$
|
||||
> \begin{align*}
|
||||
> \delta(t - t_0) = 0 \quad \text{ for } t \neq t_0,& \\
|
||||
> \int_{-\infty}^\infty f(t) \delta(t - t_0) dt = f(t_0),&
|
||||
> \end{align*}
|
||||
> $$
|
||||
>
|
||||
> for a signal $f: \mathbb{R} \to \mathbb{R}$ continuous in $t_0$.
|
||||
|
||||
For sampled signals the $\delta$ signal is given by
|
||||
|
||||
$$
|
||||
\delta[k] = \begin{cases} 1 &\text{ if } k = 0, \\ 0 &\text{ if } k \neq 0.\end{cases}
|
||||
$$
|
||||
|
||||
## Signal sampling
|
||||
|
||||
We already established that a signal $f: \mathbb{R} \to \mathbb{R}$ can be sampled with a sampling period $T_s \in \mathbb{R}$ obtaining $f[k] = f(kT_s)$ for all $k \in \mathbb{Z}$. We can also define a *time-continuous* signal $f_s: \mathbb{R} \to \mathbb{R}$ that represents the sampled signal using the Dirac signal, obtaining
|
||||
|
||||
$$
|
||||
f_s(t) = f(t) \sum_{k = - \infty}^\infty \delta(t - k T_s), \qquad \forall t \in \mathbb{R}.
|
||||
$$
|
||||
|
||||
> *Definition*: the sampling signal or impulse train $\delta_{T_s}: \mathbb{R} \to \mathbb{R}$ is defined as
|
||||
>
|
||||
> $$
|
||||
> \delta_{T_s}(t) = \sum_{k = - \infty}^\infty \delta(t - k T_s)
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$ with a sampling period $T_s \in \mathbb{R}$.
|
||||
|
||||
Then integration works out since we have
|
||||
|
||||
$$
|
||||
\int_{-\infty}^\infty f(t) \delta_{T_s}(t) dt = \sum_{k = -\infty}^\infty \int_{-\infty}^\infty f(t) \delta(t - k T_s) dt = \sum_{k = -\infty}^\infty f [k],
|
||||
$$
|
||||
|
||||
by definition.
|
||||
|
||||
## Convolutions
|
||||
|
||||
> *Definition*: let $f,g: \mathbb{R} \to \mathbb{R}$ be two continuous signals, the convolution product is defined as
|
||||
>
|
||||
> $$
|
||||
> f(t) * g(t) = \int_{-\infty}^\infty f(u)g(t-u)du
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Proposition*: the convolution product is commutative, distributive and associative.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
> *Theorem*: let $f: \mathbb{R} \to \mathbb{R}$ be a signal then we have for the convolution product between $f$ and the Dirac signal $\delta$ and some $t_0 \in \mathbb{R}$
|
||||
>
|
||||
> $$
|
||||
> f(t) * \delta(t - t_0) = f(t - t_0)
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
let $f: \mathbb{R} \to \mathbb{R}$ be a signal and $t_0 \in \mathbb{R}$, using the definition of the Dirac signal
|
||||
|
||||
$$
|
||||
f(t) * \delta(t - t_0) = \int_{-\infty}^\infty f(u) \delta(t - t_0 - u)du = f(t - t_0),
|
||||
$$
|
||||
|
||||
for all $t \in \mathbb{R}$.
|
||||
|
||||
In particular $f(t) * \delta(t) = f(t)$ for all $t \in \mathbb{R}$; $\delta$ is the unity of the convolution.
|
||||
|
||||
The average value of a signal $f: \mathbb{R} \to \mathbb{R}$ for an interval $\varepsilon \in \mathbb{R}$ may be given by
|
||||
|
||||
$$
|
||||
f(t) * D(t, \varepsilon) = \frac{1}{\varepsilon} \int_{t - \frac{\varepsilon}{2}}^{t + \frac{\varepsilon}{2}} f(u)du.
|
||||
$$
|
||||
|
||||
For sampled/discrete signals we have a similar definition for the convolution product, given by
|
||||
|
||||
$$
|
||||
f[k] * g[k] = \sum_{m = -\infty}^{\infty} f[m]g[k - m],
|
||||
$$
|
||||
|
||||
for all $k \in \mathbb{Z}$.
|
||||
|
||||
## Correlations
|
||||
|
||||
> *Definition*: let $f,g: \mathbb{R} \to \mathbb{R}$ be two continuous signals, the cross-correlation is defined as
|
||||
>
|
||||
> $$
|
||||
> f(t) \star g(t) = \int_{-\infty}^\infty f(u) g(t + u)du
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
Especially the auto-correlation of a continuous signal $f: \mathbb{R} \to \mathbb{R}$ given by $f(t) \star f(t)$ for all $t \in \mathbb{R}$ is useful, as it can detect periodicity. This is proved in the section [Fourier series](fourier-series.md).
|
||||
|
||||
For sampled/discrete signals a similar definition exists given by
|
||||
|
||||
$$
|
||||
f[k] \star g[k] = \sum_{m = -\infty}^{\infty} f[m]g[k + m],
|
||||
$$
|
||||
|
||||
for all $k \in \mathbb{Z}$.
|
||||
213
docs/physics/mathematical-physics/signal-analysis/systems.md
Normal file
213
docs/physics/mathematical-physics/signal-analysis/systems.md
Normal file
|
|
@ -0,0 +1,213 @@
|
|||
# Systems
|
||||
|
||||
> *Definition*: a system transforms signals.
|
||||
|
||||
## Operators
|
||||
|
||||
> *Definition*: let $x,y: \mathbb{R} \to \mathbb{R}$ be the input and output signal related to an operator $T$ by
|
||||
>
|
||||
> $$
|
||||
> y(t) = T[x(t)]
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
For example for a time shift of the signal $S_{t_0}: y(t) = x(t - t_0)$ we have $y(t) = S_{t_0}[x(t)]$ for all $t \in \mathbb{R}$. For an amplifier of the signal $P: y(t) = k(t) x(t)$ we have $y(t) = P[x(t)]$ for all $t \in \mathbb{R}$.
|
||||
|
||||
> *Definition*: for systems $T_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in parallel we define operator addition by
|
||||
>
|
||||
> $$
|
||||
> T = T_1 + \dots + T_n,
|
||||
> $$
|
||||
>
|
||||
> such that for $x,y: \mathbb{R} \to \mathbb{R}$ the input and output signal obtains
|
||||
>
|
||||
> $$
|
||||
> y(t) = T[x(t)] = (T_1 + \dots + T_n)[x(t)] = T_1[x(t)] + \dots + T_n[x(t)],
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition*: for systems $T_i$ for $i \in \{1, \dots, n\}$ with $n \in \mathbb{N}$ in series we define operator multiplication by
|
||||
>
|
||||
> $$
|
||||
> T = T_n \cdots T_1,
|
||||
> $$
|
||||
>
|
||||
> such that for $x,y: \mathbb{R} \to \mathbb{R}$ the input and output signal obtains
|
||||
>
|
||||
> $$
|
||||
> y(t) = T[x(t)] =T_n \cdots T_1 [x(t)] = T_n[T_{n-1}\cdots T_1[x(t)]],
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
It may be observed that the operator product is not commutative.
|
||||
|
||||
## Properties of systems.
|
||||
|
||||
> *Definition*: a system $T$ with inputs $x_{1,2}: \mathbb{R} \to \mathbb{R}$ is linear if and only if
|
||||
>
|
||||
> $$
|
||||
> T[a x_1(t) + b x_2(t)] = a T_1[x_1(t)] + b T_2[x_2(t)]
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$ with $a,b \in \mathbb{C}$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition*: a system $T$ is time invariant if and only if for all $t \in \mathbb{R}$ a shift in the input $x: \mathbb{R} \to \mathbb{R}$ results only in a shift in the output $y: \mathbb{R} \to \mathbb{R}$
|
||||
>
|
||||
> $$
|
||||
> y(t) = T[x(t)] \iff y(t - t_0) = T[x(t - t_0)],
|
||||
> $$
|
||||
>
|
||||
> for all $t_0 \in \mathbb{R}$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition*: a system $T$ is invertible if distinct input $x: \mathbb{R} \to \mathbb{R}$ results in distinct output $y: \mathbb{R} \to \mathbb{R}$; the system is injective. The inverse of $T$ is defined such that
|
||||
>
|
||||
> $$
|
||||
> T^{-1}[y(t)] = T^{-1}[T[x(t)]] = x(t)
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition*: a system $T$ is memoryless if the image of the output $y(t_0)$ with $y: \mathbb{R} \to \mathbb{R}$ depends only on the input $x(t_0)$ with $x: \mathbb{R} \to \mathbb{R}$ for all $t_0 \in \mathbb{R}$.
|
||||
|
||||
<br>
|
||||
|
||||
> *Definition*: a system $T$ is causal if the image of the output $y(t_0)$ with $y: \mathbb{R} \to \mathbb{R}$ depends only on images of the input $x(t)$ for $t \leq t_0$ with $x: \mathbb{R} \to \mathbb{R}$ for all $t_0 \in \mathbb{R}$.
|
||||
|
||||
It is commenly accepted that all physical systems are causal since by definition, a cause precedes its effect. But do not be fooled.
|
||||
|
||||
> *Definition*: a system $T$ is bounded-input $\implies$ bounded-output (BIBO) -stable if and only if for all $t \in \mathbb{R}$ the output $y: \mathbb{R} \to \mathbb{R}$ is bounded for bounded input $x: \mathbb{R} \to \mathbb{R}$. Then
|
||||
>
|
||||
> $$
|
||||
> |x(t)| \leq M \implies |y(t)| \leq P,
|
||||
> $$
|
||||
>
|
||||
> for all $M, P \in \mathbb{R}$.
|
||||
|
||||
## Linear time invariant systems
|
||||
|
||||
Linear time invariant systems are described by linear operators whose action on a system does not expicitly depend on time; time invariance.
|
||||
|
||||
> *Definition*: consider a LTI-system $T$ given by
|
||||
>
|
||||
> $$
|
||||
> y(t) = T[x(t)],
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$. The impulse response $h: \mathbb{R} \to \mathbb{R}$ of this systems is defined as
|
||||
>
|
||||
> $$
|
||||
> h(t) = T[\delta(t)]
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$ with $\delta$ the Dirac delta function.
|
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|
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It may be literally interpreted as the effect of an impulse at $t = 0$ on the system.
|
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|
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> *Theorem*: for a LTI-system $T$ with $x,y,h: \mathbb{R} \to \mathbb{R}$ the input, output and impulse response of the system we have
|
||||
>
|
||||
> $$
|
||||
> y(t) = h(t) * x(t),
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
Therefore the system $T$ is completely characterized by the impulse response of $T$.
|
||||
|
||||
> *Theorem*: for two LTI-systems in parallel given by $T = T_1 + T_2$ with $x,y,h_1,h_2: \mathbb{R} \to \mathbb{R}$ the input, output and impulse response of both systems we have
|
||||
>
|
||||
> $$
|
||||
> y(t) = (h_1(t) + h_2(t)) * x(t),
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
<br>
|
||||
|
||||
> *Theorem*: for two LTI-systems in series given by $T = T_2 T_1$ with $x,y,h_1,h_2: \mathbb{R} \to \mathbb{R}$ the input, output and impulse response of both systems we have
|
||||
>
|
||||
> $$
|
||||
> y(t) = (h_2(t) * h_1(t)) * x(t),
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
From the definition of convolutions we have $h_2 * h_1 = h_1 * h_2$ therefore the product of LTI-systems *is* commutative.
|
||||
|
||||
For a causal system there is no effect before its cause, a causal LTI system therefore must have an impulse response $h: \mathbb{R} \to \mathbb{R}$ that must be zero for all $t \in \mathbb{R}^-$.
|
||||
|
||||
> *Theorem*: for a LTI-system and its impulse response $h: \mathbb{R} \to \mathbb{R}$ we have
|
||||
>
|
||||
> $$
|
||||
> h(t) \overset{\mathcal{F}}\longleftrightarrow H(\omega),
|
||||
> $$
|
||||
>
|
||||
> for all $t, \omega \in \mathbb{R}$ with $H: \mathbb{R} \to \mathbb{C}$ the transfer function.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
<br>
|
||||
|
||||
> *Theorem*: for a LTI system $T$ with $x,y,h: \mathbb{R} \to \mathbb{R}$ the input, output and its impulse if the inverse system $T^{-1}$ exists it has an impulse response $h^{-1}: \mathbb{R} \to \mathbb{R}$ such that
|
||||
>
|
||||
> $$
|
||||
> x(t) = h^{-1}(t) * y(t),
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$ if and only if
|
||||
>
|
||||
> $$
|
||||
> h^{-1} * h(t) = \delta(t),
|
||||
> $$
|
||||
>
|
||||
> for all $t \in \mathbb{R}$. The transfer function of $T^{-1}$ is then given by
|
||||
>
|
||||
> $$
|
||||
> H^{-1}(\omega) = \frac{1}{H(\omega)},
|
||||
> $$
|
||||
>
|
||||
> for all $\omega \in \mathbb{R}$.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
|
||||
<br>
|
||||
|
||||
Therefore a LTI-system is invertible if and only if $H(\omega) \neq 0$ for all $\omega \in \mathbb{R}$.
|
||||
|
||||
> *Theorem*: the low pass filter $H: \mathbb{R} \to \mathbb{C}$ given by the transfer function
|
||||
>
|
||||
> $$
|
||||
> H(\omega) = \text{rect} \frac{\omega}{2\omega_b},
|
||||
> $$
|
||||
>
|
||||
> for all $\omega \in \mathbb{R}$ with $\omega_b \in \mathbb{R}$ is not causal. Therefore assumed to be not physically realisable.
|
||||
|
||||
??? note "*Proof*:"
|
||||
|
||||
Will be added later.
|
||||
Loading…
Add table
Add a link
Reference in a new issue