import marimo

__generated_with = "0.25.1"
app = marimo.App()


@app.cell
def _():
    import marimo as mo

    return (mo,)


@app.cell(hide_code=True)
def _(mo):
    mo.md(r"""
    # Input-Output Relation of a First-Order System

    """)
    return


@app.cell(hide_code=True)
def _(mo):
    mo.md(r"""
    A periodic function can be characterized by the properties: amplitude, frequency, period, and phase. Let's exemplify these properties for a pediodic function composed by a single frequency, the sine wave or sinusoid [trigonometric function](http://en.wikipedia.org/wiki/Trigonometric_functions):

    $$ x(t) = Asin(2 \pi f t + \phi) $$

    Where $A$ is the amplitude, $f$ the frequency, $\phi$ the phase, and $T=1/f$ the period of the function $x(t)$.

    We can define $\omega=2\pi f = 2\pi/T$ as the angular frequency, then:

    $$ x(t) = Asin(\omega t + \phi) $$

    Let's visualize this function:
    """)
    return


@app.cell
def _():
    from __future__ import division, print_function  # for version compatibility
    import numpy as np
    import matplotlib.pyplot as plt
    from scipy import signal
    from math import pi, sin, cos, sqrt, atan2
    import warnings
    warnings.filterwarnings('ignore')
    return np, plt


@app.cell
def _(np):
    t = np.linspace(-2, 2, 101)                    # time vector
    qi = 1 * np.sin(2*t) + 0.3*np.sin(20*t)
    qo1 = 0.93 * np.sin(2*t - 21.8/180*np.pi) + 0.072 * np.sin(20*t - 76./180*np.pi)
    qo2 = 1 * np.sin(2*t - 0.23/180*np.pi) + 0.3 * np.sin(20*t - 2.3/180*np.pi)
    return qi, qo1, qo2, t


@app.cell
def _(plt, qi, qo1, qo2, t):
    plt.figure(figsize=(10,8))
    plt.plot(t,qi,'bo',t,qo1,'r-.',t,qo2,'g-',lw=2)
    plt.grid('on')
    plt.xlabel('$t$',fontsize=18)
    plt.ylabel('$q_i,q_o$',fontsize=18)
    plt.show()
    return


if __name__ == "__main__":
    app.run()