**Introduction**

Signal modulation plays a pivotal role in telecommunications, making it possible to transmit information across significant distances. To gain a practical understanding of modulation principles, this exercise utilizes the Red Pitaya, a versatile, open-source, software-defined radio (SDR) platform. The Red Pitaya enables us to generate and modify signals, facilitating the exploration of key concepts such as mixing, Amplitude Modulation (AM), and the impact of frequency normalization. This hands-on approach bridges the gap between theory and practice, allowing users to delve into the intricacies of signal modulation in a tangible way.

**Mixing**

The operation of mixing involves combining two signals together, usually through a multiplication operation. The result of multiplying two signals is a process that creates sum and difference frequencies from the original frequencies.

In the analysis of mixing, two sinusoids were used because sinusoids are the basis functions for Fourier Analysis, and any periodic signal can be represented as a combination of sinusoids (according to the Fourier Series). As such, it provides a straightforward way to visualize and analyze the principles of mixing and modulation.

To achieve this modulation, we can view the mathematical result of multiplying two periodic waveforms. For this analysis case, we will employ two sinusoids.

$\begin{split}\begin{matrix} x_{1}(t) = A_{1}\cos\left( \omega_{1}t + \phi_{1} \right)x_{2}(t) = A_{2}\cos\left( \omega_{2}t + \phi_{2} \right)\ \\ \end{matrix}\end{split}$Multiplying these two is made simple by invoking the well known trig identity:

$\begin{split}\begin{matrix} \cos(x)\cos(y) = \frac{1}{2}\left\lbrack \cos(x - y) + cos(x + y) \right\rbrack\ \end{matrix}\end{split}$thus allowing:

$\begin{split}\begin{matrix} x_{1}(t)x_{2}(t) = \frac{A_{1}A_{2}}{2}\left\lbrack \cos\left( \omega_{1}t + \phi_{1} - \left\lbrack \omega_{2}t + \phi_{2} \right\rbrack \right) + cos\left( \omega_{1}t + \phi_{1} + \left\lbrack \omega_{2}t + \phi_{2} \right\rbrack \right) \right\rbrack\ \\ \end{matrix}\end{split}$This shows that:

- We generate data at the sum and difference frequencies
- Phases add directly
- The amplitude/energy of the resulting data is shared from both waveforms

In most mixing applications, the results of a mixer are then filtered to only include either the sum or the difference frequency.

**AM modulation**

Amplitude modulation (AM) is a simple yet effective modulation scheme where the message signal (information to be sent) changes the amplitude of a faster “carrier” signal. The carrier signal is the base waveform that is modified to carry the information. It is typically a high-frequency sinusoid, chosen because it can easily propagate through space as a radio wave.

Amplitude modulation involves using a message signal (bandlimited to $f_m$) to modulate the amplitude of a faster “carrier” signal at $f_c$. For simplicity, we will take the carrier to have no phase (this is justified by (3), where we can see that the mixed results simple add phase, and we can simply define $\phi_1$=0), and give the modulated signal some phase $\phi$.

$\begin{split}\begin{matrix} \begin{matrix} x_{c}(t) = A_{c}\cos\left( \omega_{c}t \right) \\ x_{m}(t) = A_{m}\cos\left( \omega_{m}t + \phi \right) \\ \end{matrix}\ \#(4) \\ \end{matrix}\end{split}$**Dual Side Band (DSB) Modulation**

In Dual Side Band modulation, both the upper and lower sidebands are transmitted. This effectively doubles the bandwidth required for transmission. DSB modulation is relatively straightforward, but is not the most efficient in terms of bandwidth utilization.

Employing the simplest mixing, we simply multiple the carrier (denoted $x_c(t)$) by the message signal (denoted $x_m(t)$). This makes the resulting signal $x_{DSB}(t)$

$\begin{split}\begin{matrix} x_{DSB}(t) = \ x_{c}(t)x_{m}(t) = \frac{1}{K}\frac{A_{c}A_{m}}{2}\left\lbrack \cos\left( \omega_{c}t - \left\lbrack \omega_{m}t + \phi \right\rbrack \right) + cos\left( \omega_{c}t + \left\lbrack \omega_{m}t + \phi \right\rbrack \right) \right\rbrack\ \#(5) \\ \end{matrix}\end{split}$This is what is known as Dual Side Band (DSB) or suppressed carrier modulation.

**Amplitude Modulation**

The traditional AM scheme includes a DC offset which allows the receiver to easily detect and remove the carrier signal and recover the original message. It is used because it is simple and the receiver circuitry is also simple. However, it is not power efficient as it needs to transmit the carrier along with the signal.

A more traditional AM modulation method for AM modulation employs a slightly modified version of DSB modulation. This time, the modulating signal is 1) scaled by the carrier amplitude, and 2) given a DC offset. This makes the new modulation function $m(t)$ take the form

$\begin{split}\begin{matrix} m(t) = 1 + \frac{x_{m}(t)}{A_{c}}\ \#(6) \\ \end{matrix}\end{split}$This makes the modulated signal simply $x_{AM}(t) = x_{c}(t)m(t)$. This can be shown with some more clever trig identities to take the form

$\begin{split}\begin{matrix} x_{AM}(t) = A_{c}\cos(2\pi f) + \frac{A_{c}}{2}\left\lbrack \cos\left( \omega_{c}t - \left\lbrack \omega_{m}t + \phi \right\rbrack \right) + cos\left( \omega_{c}t + \left\lbrack \omega_{m}t + \phi \right\rbrack \right) \right\rbrack\ \#(7) \\ \end{matrix}\end{split}$**Sidenote: Modulation Index**

The modulation index is a key parameter of any AM signal. It defines the extent of the variation in a carrier signal according to the information being sent. A high modulation index will cause a large amount of variation in the carrier signal, making it more susceptible to distortion or noise, while a low modulation index will lead to a lower signal quality.

Since there are now two terms, a carrier and the encoded message signal, we can consider the case of analyzing the peak of the message signal compared to the peak of the carrier. This ratio is known as the modulation index $\mu$, and describes the modulation “strength” of the message onto the carrier.

$\begin{split}\begin{matrix} \mu = \frac{\left| m(t) \right|}{A_{c}}\ \#(8) \\ \end{matrix}\end{split}$Full strengths modulation corresponds to a 100% index, and means that potential peaks of the carrier can be suppressed into a null. This parameter is not so important for this lab, but will be of interest to the analysis of communication systems in a future course.

Figure : Modulation Index visualized. Credit: Wikipedia Modulation

**Normalized Frequency**

Frequency normalization is often employed in discrete systems to facilitate comparison between systems of different sizes or specifications. Normalized frequency is simply the frequency represented in terms of the Nyquist frequency or the sampling rate, thereby abstracting away the actual values and instead focusing on the underlying behavior of the system.

After the act of sampling, it becomes convenient to rescale (normalize) frequency. This is done by the relation

$\begin{split}\begin{matrix}\widehat{\omega} = \omega T_{s} = \frac{2\pi f}{f_{s}}\ \#(8) \\\end{matrix}\end{split}$Where $\omega=2\pi f \space and \space T_s=\frac{1}{f_s}$ is the sampling time. This representation is oftentimes used in discrete time systems as it allows for the consideration of systems in reference to the total bandwidth of the discrete system.

**Tasks/Questions**

**Theory**
1. Why in the analysis of mixing, were two sinusoids used? (Hint, sinusoids are what for the space of periodic functions?)
*Two sinusoids were used because sinusoids form the basis of the space of periodic functions (Fourier series). In other words, any periodic function can be represented as a sum of sinusoids of various frequencies, amplitudes, and phases.*
2. Why is the carrier being a sinusoid preferrable from a transmission perspective?
*A sinusoid is preferable as a carrier signal because it is easy to generate, mathematically tractable, and well-suited for transmission over a medium (such as air for radio). Its constant amplitude makes it less prone to distortion as it propagates.*
3. In both described AM schemes (DSB, AM w/modulation index), is there a way to reduce the total bandwidth of the system anymore? (Hint, do you need both sides of a spectrum to retrieve a signal if you know the signal is real valued?)
• In DSB and traditional AM, there’s redundancy because the information is contained in both the upper and lower sidebands. If the signal is real-valued (as is often the case), you can use single-sideband (SSB) modulation, which cuts the bandwidth requirement in half.*
1. It was stated in the theory, that for AM, usually $f_c$> $10x f_m$. Why would this be true, and why would one want $f_c$ to be even larger. For example, FM radio operates on a carrier of $f_c\approx88MHz-108MHz$, but the bandwidth of audio signals is only 20kHz (as was demonstrated last lab).
$f_c=10x f_m$* is often chosen to ensure the message signal doesn’t interfere with the carrier signal, to simplify the process of demodulation, and to adhere to regulations that prevent signals from occupying too much bandwidth. Also, a higher* $f_c$* allows for better propagation of the signal.*
2. Why is the carrier generally a very powerful signal in real systems? (Hint: how far are you from the radio tower when you listen to the radio? As all signals travel, they will spread out unless coerced otherwise)
*The carrier signal is powerful in real systems because as signals travel, they lose power due to various factors (like propagation loss). A stronger carrier signal ensures that the signal can be received at a greater distance from the transmitter.*