From 896ccfe17c05a8b1f47ae1622e44c50a095108bc Mon Sep 17 00:00:00 2001 From: Luc Date: Sun, 21 Jan 2024 19:56:36 +0100 Subject: [PATCH] Added various sections to signal analysis. --- config/en/mkdocs.yaml | 5 +- .../signal-analysis/amplitude-modulation.md | 59 +++++++++++++ .../discrete-fourier-transform.md | 84 +++++++++++++++++++ .../signal-analysis/fourier-transform.md | 4 +- .../signal-analysis/signal-filters.md | 2 + .../signal-analysis/signals.md | 2 +- .../signal-analysis/systems.md | 0 7 files changed, 152 insertions(+), 4 deletions(-) create mode 100644 docs/en/physics/mathematical-physics/signal-analysis/amplitude-modulation.md create mode 100644 docs/en/physics/mathematical-physics/signal-analysis/discrete-fourier-transform.md create mode 100644 docs/en/physics/mathematical-physics/signal-analysis/signal-filters.md create mode 100644 docs/en/physics/mathematical-physics/signal-analysis/systems.md diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index 1f5cdca..8a4d7f3 100755 --- a/config/en/mkdocs.yaml +++ b/config/en/mkdocs.yaml @@ -120,7 +120,10 @@ nav: - 'Signals': physics/mathematical-physics/signal-analysis/signals.md - 'Fourier series': physics/mathematical-physics/signal-analysis/fourier-series.md - 'Fourier transform': physics/mathematical-physics/signal-analysis/fourier-transform.md -# - 'Discrete Fourier transform': + - 'Discrete Fourier transform': physics/mathematical-physics/signal-analysis/discrete-fourier-transform.md + - 'Amplitude modulation': physics/mathematical-physics/signal-analysis/amplitude-modulation.md + - 'Signal filters': physics/mathematical-physics/signal-analysis/signal-filters.md + - 'Systems': physics/mathematical-physics/signal-analysis/systems.md - 'Electromagnetism': # - 'Electrostatics': # - 'Magnetostatics': diff --git a/docs/en/physics/mathematical-physics/signal-analysis/amplitude-modulation.md b/docs/en/physics/mathematical-physics/signal-analysis/amplitude-modulation.md new file mode 100644 index 0000000..0a56812 --- /dev/null +++ b/docs/en/physics/mathematical-physics/signal-analysis/amplitude-modulation.md @@ -0,0 +1,59 @@ +# Amplitude modulation + +> *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*. + +??? note "*Proof*:" + + Will be added later. + +For example if we have a harmonic signal $m: \mathbb{R} \to \mathbb{R}$ with $\omega, A \in \mathbb{R}$ given by + +$$ + m(t) = A \cos \omega t, +$$ + +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 + +$$ + c(t) = \cos \omega_c t. +$$ + +for all $t \in \mathbb{R}$. Then the multiplication of both is given by + +$$ + m(t)c(t) = A \cos (\omega t) \cos (\omega_c t) = \frac{A}{2} \bigg(\cos t(\omega + \omega)c + \cos t(\omega - \omega_c) \bigg), +$$ + +obtaining heterodyne. + +> *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 +> +> $$ +> c(t) = \cos \omega_c t +> $$ +> +> for all $t \in \mathbb{R}$ to modulate a signal $m: \mathbb{R} \to \mathbb{R}$. + +
+ +> *Theorem*: For the case that the carrier signal is not additionaly transmitted we obtain +> +> $$ +> m(t) c(t) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2} \big(M(\omega + \omega_c) + M(\omega - \omega_c) \big), +> $$ +> +> for all $t, \omega \in \mathbb{R}$. +> +> For the case that the carrier signal is additionaly transmitted we obtain +> +> $$ +> m(t) (1 + c(t)) \overset{\mathcal{F}}\longleftrightarrow \frac{1}{2} \Big(M(\omega + \omega_c) + M(\omega - \omega_c) + \pi \big(\delta(\omega + \omega_c) + \delta(\omega - \omega_c) \big) \Big) +> $$ +> +> for all $t, \omega \in \mathbb{R}$. +> +> Therefore multiple bandlimited signals can be transmitted simultaneously in frequency bands. + +??? note "*Proof*:" + + Will be added later. \ No newline at end of file diff --git a/docs/en/physics/mathematical-physics/signal-analysis/discrete-fourier-transform.md b/docs/en/physics/mathematical-physics/signal-analysis/discrete-fourier-transform.md new file mode 100644 index 0000000..c6ef721 --- /dev/null +++ b/docs/en/physics/mathematical-physics/signal-analysis/discrete-fourier-transform.md @@ -0,0 +1,84 @@ +# The discrete Fourier transform + +> *Theorem*: sampling a signal with the impulse train makes the spectrum of the signal periodic. + +??? note "*Proof*:" + + Will be added later. + +A bandlimited signal implies that its frequency components are zero outside the bandwidth frequency interval. + +> *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 +> +> $$ +> \omega_s > 2 \omega_b. +> $$ + +??? note "*Proof*:" + + Will be added later. + +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. + +## The discrete time Fourier transform + +> *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 +> +> $$ +> F(\Omega) = \sum_{m = -\infty}^\infty f[m] e^{-im\Omega}, +> $$ +> +> for all $\Omega \in \mathbb{R}$. With $\Omega = \omega T_s$ the dimensionless frequency and $F_s(\omega) := F(\Omega)$. + +??? note "*Proof*:" + + Will be added later. + +## The discrete Fourier transform + +> *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 +> +> $$ +> f_N[m] = \begin{cases} f[m] &\text{ if } m \in \{0, \dots, N - 1\}, \\ 0 &\text{ if } m \notin \{0, \dots, N - 1\}, \end{cases} +> $$ +> +> sampled by $T_s \in \mathbb{R}$. Its discrete Fourier transform $F_N: \mathbb{R} \to \mathbb{C}$ is given by +> +> $$ +> F_N[k] = \sum_{m=0}^{N-1} f[m] \exp \bigg(-2\pi i \frac{km}{N} \bigg) +> $$ +> +> for all $k \in \{0, \dots, N-1\}$. + +??? note "*Proof*:" + + Will be added later. + +We have that $F_N[k] = F_N(k\Delta \omega)$ with $\Delta \omega = \frac{2\pi}{N T_s}$ the angular frequency resolution. + +> *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 +> +> $$ +> f[m] = \frac{1}{N} \sum_{k=0}^{N-1} F_N[k] \exp \bigg(2\pi i \frac{km}{N} \bigg) +> $$ +> +> for all $m \in \{0, \dots, N - 1\}$. + +??? note "*Proof*:" + + Will be added later. + +> *Definition*: therefore $f_N$ and $F_N$ with $N \in \mathbb{N}$ form a discrete Fourier transform pair denoted by +> +> $$ +> f_N \overset{\mathcal{DF}}\longleftrightarrow F_N, +> $$ +> +> therefore we have +> +> $$ +> \begin{align*} +> &f_N[m] = \mathcal{DF}^{-1}[F_N[k]], \quad &\forall m \in \{0, \dots, N - 1\}, \\ +> &F_N[k] = \mathcal{DF}[f[m]], \quad &\forall k \in \{0, \dots, N - 1\}. +> \end{align*} +> $$ \ No newline at end of file diff --git a/docs/en/physics/mathematical-physics/signal-analysis/fourier-transform.md b/docs/en/physics/mathematical-physics/signal-analysis/fourier-transform.md index 15cb4e3..0a07fb3 100644 --- a/docs/en/physics/mathematical-physics/signal-analysis/fourier-transform.md +++ b/docs/en/physics/mathematical-physics/signal-analysis/fourier-transform.md @@ -24,8 +24,8 @@ > > $$ > \begin{align*} -> f(t) = \mathcal{F}^{-1}[F(\omega)], \quad \forall t \in \mathbb{R}&, \\ -> F(\omega) = \mathcal{F}[f(t)], \quad \forall \omega \in \mathbb{R}&. +> &f(t) = \mathcal{F}^{-1}[F(\omega)], \quad &\forall t \in \mathbb{R}, \\ +> &F(\omega) = \mathcal{F}[f(t)], \quad &\forall \omega \in \mathbb{R}. > \end{align*} > $$ diff --git a/docs/en/physics/mathematical-physics/signal-analysis/signal-filters.md b/docs/en/physics/mathematical-physics/signal-analysis/signal-filters.md new file mode 100644 index 0000000..b23ee5e --- /dev/null +++ b/docs/en/physics/mathematical-physics/signal-analysis/signal-filters.md @@ -0,0 +1,2 @@ +# Signal filters + diff --git a/docs/en/physics/mathematical-physics/signal-analysis/signals.md b/docs/en/physics/mathematical-physics/signal-analysis/signals.md index 6a9fbae..40aabc7 100644 --- a/docs/en/physics/mathematical-physics/signal-analysis/signals.md +++ b/docs/en/physics/mathematical-physics/signal-analysis/signals.md @@ -125,7 +125,7 @@ $$ ## Signal sampling -We already established that a signal $f: \mathbb{R} \to \mathbb{R}$ can be sampled with a sampling period $T_s \in \mathbb{R}$ obtaining $f[k] = f(kT_s)$ for all $k \in \mathbb{Z}$. We can also define a *time-continuous* signal $f_s(t)$ that represents the sampled signal using the Dirac signal, obtaining +We already established that a signal $f: \mathbb{R} \to \mathbb{R}$ can be sampled with a sampling period $T_s \in \mathbb{R}$ obtaining $f[k] = f(kT_s)$ for all $k \in \mathbb{Z}$. We can also define a *time-continuous* signal $f_s: \mathbb{R} \to \mathbb{R}$ that represents the sampled signal using the Dirac signal, obtaining $$ f_s(t) = f(t) \sum_{k = - \infty}^\infty \delta(t - k T_s), \qquad \forall t \in \mathbb{R}. diff --git a/docs/en/physics/mathematical-physics/signal-analysis/systems.md b/docs/en/physics/mathematical-physics/signal-analysis/systems.md new file mode 100644 index 0000000..e69de29