Procedure

Signal Plot

This section requires selection of signal and tweaking of some parameters of the signal to visualize how the signal changes with the change in parameters. The objective of this section is to visualize the signal and identify the parameters of a signal. Click on the "Plot" button to visualize the plots. Steps to be done are as follows

1. Select the signal in the drop down provided
2. Enter Parameters as necessary
3. Click on Plot button to visualize the plot in the figure

The parameters mentioned as "Parameter 1" and "Parameter 2" for the signals correspond to the parameters as listed below

Signal	Parameter 1	Parameter 2	Signal
Sine	Frequency	Amplitude	x(t) = $A \sin(2\pi f t)$
Cosine	Frequency	Amplitude	x(t) = $A \cos(2\pi f t)$
Ramp	-	Amplitude	x(t) = At
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) = $A e^{2\pi f t}$

The plot is obtained and it represents the selected signal with the specified parameters.

Signal Product

This section requires selection of two signals and tweaking of some parameters of the signals to visualize how the signals change with the change in parameters. The objective of this section is to visualize the product of two signals. Click on the "Plot" button to visualize the plots. Steps to be done are as follows

1. Select the first signal in the drop down provided
2. Enter Parameters as necessary
3. Select the second signal in the drop down provided
4. Enter Parameters as necessary
5. Click on Plot button to visualize the plot in the figure

The parameters mentioned as "Parameter 1" and "Parameter 2" for the signals correspond to the parameters as listed below

Signal	Parameter 1	Parameter 2
Sine	Frequency	Amplitude
Cosine	Frequency	Amplitude
Ramp	-	Amplitude
Pulse	-	Amplitude
Haar	Scale Parameter	Amplitude
Complex Exponential	Frequency	Amplitude

The plot is obtained and it represents the two selected signals with the specified parameters, and their product in the same figure.

Real Sinusoids

This section requires selection of two real sinusoidal signals (sine or cosine) and tweaking of a scale parameter of the signals to visualize how their product and orthogonality changes with the change in parameters. The objective of this section is to visualize the product of two real sinusoids and comment on their orthogonality. Click on the "Plot" button to visualize the plots. Steps to be done are as follows

1. Select the first signal in the drop down provided
2. Enter Parameter k
3. Select the second signal in the drop down provided
4. Enter Parameter k
5. Click on Plot button to visualize the plot in the figure

The plot is obtained and it represents the product of the two selected signals with the specified parameters. The areas are shaded using different colors. The areas above the x-axis (positive integrals) are shaded with one color (here, orange) and the areas below the x-axis (negative integrals) are shaded with another color (here, blue).

This coloring helps visualize the areas (integrals) that cancel each other out during a complete integration, which may result in a 0 integral or not, which determines the orthogonality between the signals.

The observartions tab at the bottom shows the integral of the product of the signals. The value present in this observation tab shows the value for the integral. Recalling from theory, this value is 0 for orthogonal signals and non-zero for signals which are not orthogonal to each other.

Complex Sinusoids

This section comprises of two complex sinusoidal signals (complex exponential) and tweaking of a scale parameter of the signals to visualize how their product and orthogonality changes with the change in parameters. The objective of this section is to visualize the product of two complex sinusoids and comment on their orthogonality. Click on the "Plot" button to visualize the plots. Steps to be done are as follows

1. Enter Parameter k for the first signal
2. Enter Parameter k for the second signal
3. Click on Plot button to visualize the plot in the figure

The plot is obtained and it represents the product of two complex exponentials with the specified parameters. The figure is divided into 2 subplots. The first one represents the real part of the signal and the second part represents the imaginary part of the signal. The areas are shaded using different colors. The areas above the x-axis (positive integrals) are shaded with one color and the areas below the x-axis (negative integrals) are shaded with another color.

This coloring helps visualize the areas (integrals) that cancel each other out during a complete integration, which may result in a 0 integral or not, which determines the orthogonality between the signals.

The observartions tab at the bottom shows the integral of the product of the signals. The value present in this observation tab shows the value for the integral. Recalling from theory, this value is 0 for orthogonal signals and non-zero for signals which are not orthogonal to each other.

Haar Wavelet

This section comprises of two Haar wavelets and tweaking of a scale parameter of the wavelets to visualize how their product and orthogonality changes with the change in parameters. The objective of this section is to visualize the product of two Haar wavelets and comment on their orthogonality. Click on the "Plot" button to visualize the plots. Steps to be done are as follows

1. Select the Parameter k for the first signal from the drop down
2. Select the Parameter k for the second signal from the drop down
3. Click on Plot button to visualize the plot in the figure

The plot is obtained and it represents the product of two Haar wavelets with the specified parameters. The areas are shaded using different colors. The areas above the x-axis (positive integrals) are shaded with one color (here, orange) and the areas below the x-axis (negative integrals) are shaded with another color (here, blue).

This coloring helps visualize the areas (integrals) that cancel each other out during a complete integration, which may result in a 0 integral or not, which determines the orthogonality between the signals.

The observartions tab at the bottom shows the integral of the product of the signals. The value present in this observation tab shows the value for the integral. Recalling from theory, this value is 0 for orthogonal signals and non-zero for signals which are not orthogonal to each other.

Orthogonality Test

This section comprises of two signals to be selected and tweaking of a scale parameter of the wavelets to visualize how their product and orthogonality changes with the change in parameters. The objective of this section is to visualize the product of any two selected signals and comment on their orthogonality. Click on the "Plot" button to visualize the plots. Steps to be done are as follows

1. Select the first signal in the drop down provided
2. Enter Parameters as necessary
3. Select the second signal in the drop down provided
4. Enter Parameters as necessary
5. Click on Plot button to visualize the plot in the figure

The plot is obtained and it represents the product of the two selected signals with the specified parameters. The areas are shaded using different colors. The areas above the x-axis (positive integrals) are shaded with one color and the areas below the x-axis (negative integrals) are shaded with another color.

This coloring helps visualize the areas (integrals) that cancel each other out during a complete integration, which may result in a 0 integral or not, which determines the orthogonality between the signals.

The observartions tab at the bottom shows the integral of the product of the signals. The value present in this observation tab shows the value for the integral. Recalling from theory, this value is 0 for orthogonal signals and non-zero for signals which are not orthogonal to each other.