Fix modulation index, add tri

README.md

@@ -82,13 +82,13 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
 ## Fourier transform identities and properties
 
 | Time domain $x(t)$                                                           | Frequency domain $X(f)$                                                                                                                   |
-| --------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- |
+| ---------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------- |
 | $\text{rect}\left(\frac{t}{T}\right)\quad\Pi\left(\frac{t}{T}\right)$        | $T \text{sinc}(fT)$                                                                                                                       |
 | $\text{sinc}(2Wt)$                                                           | $\frac{1}{2W}\text{rect}\left(\frac{f}{2W}\right)\quad\frac{1}{2W}\Pi\left(\frac{f}{2W}\right)$                                           |
 | $\exp(-at)u(t),\quad a>0$                                                    | $\frac{1}{a + j2\pi f}$                                                                                                                   |
 | $\exp(-a\lvert t \rvert),\quad a>0$                                          | $\frac{2a}{a^2 + (2\pi f)^2}$                                                                                                             |
 | $\exp(-\pi t^2)$                                                             | $\exp(-\pi f^2)$                                                                                                                          |
-| $1 - \frac{\lvert t \rvert}{T},\quad\lvert t \rvert < T$              | $T \text{sinc}^2(fT)$                                                                                                                     |
+| $1 - \frac{\lvert t \rvert}{T},\quad\lvert t \rvert < T\quad\text{tri}(t/T)$ | $T \text{sinc}^2(fT)$                                                                                                                     |
 | $\delta(t)$                                                                  | $1$                                                                                                                                       |
 | $1$                                                                          | $\delta(f)$                                                                                                                               |
 | $\delta(t - t_0)$                                                            | $\exp(-j2\pi f t_0)$                                                                                                                      |
@@ -130,6 +130,7 @@ along with this program. If not, see <http://www.gnu.org/licenses/>.
     \text{sgn}(t) &= \begin{cases} +1, & t > 0 \\ 0, & t = 0 \\ -1, & t < 0 \end{cases}&\text{Signum Function}\\
     \text{sinc}(2Wt) &= \frac{\sin(2\pi W t)}{2\pi W t}&\text{sinc Function}\\
     \text{rect}(t) = \Pi(t) &= \begin{cases} 1, & -0.5 < t < 0.5 \\ 0, & \lvert t \rvert > 0.5 \end{cases}&\text{Rectangular/Gate Function}\\
+    \text{tri}(t/T) &= \begin{cases} 1 - \frac{|t|}{T}, & \lvert t\rvert < T \\ 0, & \lvert t \rvert > T \end{cases}=\Pi(t/T)*\Pi(t/T)&\text{Triangle Function}\\
     g(t)*h(t)=(g*h)(t)&=\int_\infty^\infty g(\tau)h(t-\tau)d\tau&\text{Convolution}\\
 \end{align*}
 ```
@@ -156,16 +157,19 @@ Calculate $C_n$ coefficient as follows from $x_p(t)$:
 
 ```math
 \begin{align*}
-    C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt\\
-       &=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$}
+    C_n&=\frac{1}{T_s} \int_{T_s} x_p(t)\exp(-j2\pi f_s t)dt=\frac{1}{T_s} X(nf_s)\quad\color{red}\text{(TODO: Check)}\quad\color{white}\text{$x(t-nT_s)$ is contained in the interval $T_s$}
 \end{align*}
 ```
 
 <!-- MATH END -->
 
-### $\text{rect}$ function
+### Shape functions
 
-![rect](images/rect.drawio.svg)
+| $\text{rect}$ function          | $\text{tri}$ function |
+| ------------------------------- | --------------------- |
+| ![rect](images/rect.drawio.svg) | TODO: Add graphic.    |
+
+Tri placeholder: For $\text{tri}(t/T)=1-\|t\|/T$, Intersects $x$ axis at $-T$ and $T$ and $y$ axis at $1$.
 
 ### Bessel function
 
@@ -319,7 +323,7 @@ h(t)&=h_I(t)\cos(2\pi f_c t)-h_Q(t)\sin(2\pi f_c t)\\
 \begin{align*}
     x(t)&=A_c\cos(2\pi f_c t)\left[1+k_a m(t)\right]=A_c\cos(2\pi f_c t)\left[1+m_a m(t)/A_c\right], \\
     &\text{where $m(t)=A_m\hat m(t)$ and $\hat m(t)$ is the normalized modulating signal}\\
-    m_a &= \frac{\min_t|k_a m(t)|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\
+    m_a &= \frac{|\min_t(k_a m(t))|}{A_c} \quad\text{$k_a$ is the amplitude sensitivity ($\text{volt}^{-1}$), $m_a$ is the modulation index.}\\
     m_a &= \frac{A_\text{max}-A_\text{min}}{A_\text{max}+A_\text{min}}\quad\text{ (Symmetrical $m(t)$)}\\
     m_a&=k_a A_m \quad\text{ (Symmetrical $m(t)$)}\\
     P_c &=\frac{ {A_c}^2}{2}\quad\text{Carrier power}\\