In this section, you can select a signal and adjust its parameters to see how the signal changes. The goal is to visualize the signals and comment on their orthogonality.

Follow these steps:

1. Choose a signal from the drop-down menu
2. Enter any necessary parameter(s)
3. Click the "Plot" button to view the plot in the figure

The "Parameter 1" and "Parameter 2" for the signals correspond to the following listed parameters:

Signal	Parameter 1	Parameter 2	Formula
Sine	Frequency	Amplitude	\( x(t) = A\ \texttt{sin}(\frac{2\pi kt}{N}) \)
Cosine	Frequency	Amplitude	\( x(t) = A\ \texttt{cos}(\frac{2\pi kt}{N}) \)
Ramp	-	Amplitude	\( x(t) = A\ t \)
Pulse	-	Amplitude	\( x(t) = A\ \mathcal{I}_{[\frac{-2}{3},\frac{2}{3}]} \)
Haar	Scale Parameter	Amplitude	\( x(t) = A\ \mathcal{I}_{[-2,-2+\frac{1}{2^{s-1}}]} - \mathcal{I}_{[-2+\frac{1}{2^{s-1}},-2+\frac{1}{2^{s-2}}]} \)
Complex Exponential	Frequency	Amplitude	\( x(t) = A\ e^{jft} \)

In this section, you can select two signals and adjust their parameters to visualize how their product changes. The goal is to visualize the product of the two signals.

Follow these steps:

1. Choose the first signal from the dropdown menu
2. Enter the necessary parameter(s) for the first signal
3. Choose the second signal from the dropdown menu
4. Enter the necessary parameter(s) for the second signal
5. Click the "Plot" button to view the plot in the figure

The "Parameter 1" and "Parameter 2" for the signals correspond to the following listed parameters:

Signal	Parameter 1	Parameter 2	Formula
Sine	Frequency	Amplitude	\( x(t) = A\ \texttt{sin}(\frac{2\pi kt}{N}) \)
Cosine	Frequency	Amplitude	\( x(t) = A\ \texttt{cos}(\frac{2\pi kt}{N}) \)
Ramp	-	Amplitude	\( x(t) = A\ t \)
Pulse	-	Amplitude	\( x(t) = A\ \mathcal{I}_{[\frac{-2}{3},\frac{2}{3}]} \)
Haar	Scale Parameter	Amplitude	\( x(t) = A\ \mathcal{I}_{[-2,-2+\frac{1}{2^{s-1}}]} - \mathcal{I}_{[-2+\frac{1}{2^{s-1}},-2+\frac{1}{2^{s-2}}]} \)
Complex Exponential	Frequency	Amplitude	\( x(t) = A\ e^{jft} \)

In this section, you can select two real sinusoidal signals (sine or cosine) and adjust a scale parameter to observe how their product and orthogonality change with different parameters. The goal is to visualize the product of the two signals and determine their orthogonality.

The signal plotted is x(t) = \( A\ \texttt{cos}(\frac{2\pi kt}{N}) \) or x(t) = \( A\ \texttt{sin}(\frac{2\pi kt}{N}) \) with A = 1 and N = 8.

Follow these steps:

1. Choose the first signal from the dropdown menu
2. Choose harmonic k for the first signal
3. Choose the second signal from the dropdown menu
4. Choose harmonic k for the second signal
5. Click the "Plot" button to view the plot in the figure

The resulting plot shows the product of the two signals with the selected parameters. The shaded areas above the x-axis (positive integrals) and below the x-axis (negative integrals) use different colors to indicate which areas cancel out during a complete integration, determining the orthogonality between the signals.

The observations tab at the bottom displays the integral of the product of the signals. If the value is zero, the signals are orthogonal. If the value is non-zero, the signals are not orthogonal.

This section shows two complex sinusoidal signals and their product.

The signal plotted is x(t) = \( A e^{\frac{2\pi kt}{N}} \) with A = 1 and N = 8.

To visualize the product and comment on its orthogonality, follow these steps:

1. Enter the harmonic "k" for the first signal (between 0 and N-1)
2. Enter the harmonic "k" for the second signal (between 0 and N-1)
3. Click the "Plot" button to view the plot in the figure

The figure has two subplots, one for the real part and the other for the imaginary part of the product of the signals. Shading is used to indicate positive and negative integrals.

The observations tab at the bottom displays the integral of the product of the signals, which is zero for orthogonal signals and non-zero otherwise.

To visualize the product and orthogonality of two Haar wavelets, follow these steps:

1. Enter the scale parameter "k" for the first signal
2. Enter the scale parameter "k" for the second signal
3. Click the "Plot" button to view the plot in the figure

Shading is used to indicate positive and negative integrals.

The observations tab at the bottom displays the integral of the product of the signals, which is zero for orthogonal signals and non-zero otherwise.

In this section, you can select a signal and adjust its parameters to check if they are orthogonal. The goal is to visualize the signals and comment on their orthogonality.

Follow these steps:

1. Choose a signal from the drop-down menu
2. Enter any necessary parameter(s)
3. Click the "Plot" button to view the plot in the figure

The "Parameter 1" and "Parameter 2" for the signals correspond to the following listed parameters:

Signal	Parameter 1	Parameter 2	Formula
Sine	Frequency	Amplitude	\( x(t) = A\ \texttt{sin}(\frac{2\pi kt}{N}) \)
Cosine	Frequency	Amplitude	\( x(t) = A\ \texttt{cos}(\frac{2\pi kt}{N}) \)
Ramp	-	Amplitude	\( x(t) = A\ t \)
Pulse	-	Amplitude	\( x(t) = A\ \mathcal{I}_{[\frac{-2}{3},\frac{2}{3}]} \)
Haar	Scale Parameter	Amplitude	\( x(t) = A\ \mathcal{I}_{[-2,-2+\frac{1}{2^{s-1}}]} - \mathcal{I}_{[-2+\frac{1}{2^{s-1}},-2+\frac{1}{2^{s-2}}]} \)
Complex Exponential	Frequency	Amplitude	\( x(t) = A\ e^{jft} \)