• Set the frequency of the message signal to 0.1. Show a plot of the acquired waveform. What does a normalized frequency $\widehat{\omega} < \frac{1}{2\pi}$ mean, and why does it introduce odd behavior into the observed waveforms?

In this scenario, you would need to configure your Red Pitaya setup to generate a message signal with the specified frequency and observe the waveform. The acquired waveform can then be plotted using a suitable software tool (e.g., MATLAB or Python). A normalized frequency of  $\widehat{\omega} < \frac{1}{2\pi}$ would suggest that the frequency of the signal is less than half the sampling frequency. This introduces odd behavior into the observed waveforms due to the phenomenon known as “aliasing,” where the signal is under sampled, causing it to appear as a lower frequency signal.

1. What happens when the message signal frequency is the same size or greater than the carrier frequency?

When the message signal frequency is the same size or greater than the carrier frequency, there could be issues with effective modulation. The carrier signal, as the name suggests, is supposed to “carry” the message signal, and it is typically of higher frequency. If the message frequency equals or surpasses the carrier frequency, the information might not be effectively encoded into the carrier signal, leading to poor reception or loss of data.

2. Use a message signal that is not a pure sinusoid (e.g. use anything that is a superposition of sinusoids), show the resulting spectrum, and comment as to the bandwidth of the modulated signal.

When a message signal that is not a pure sinusoid, e.g., a signal that is a superposition of sinusoids, is used, the resulting spectrum shows peaks at the frequencies of the individual sinusoids. The bandwidth of the modulated signal, in this case, will be broader. This is because the modulated signal now carries the information of multiple sinusoids, each with its own frequency, thus widening the total range of frequencies (bandwidth) in the signal.

3. Use a carrier signal that is not a pure sinusoid (e.g. use the square function), show the resulting spectrum, and comment as to the resulting signal strength in any one peak when compared to a pure sinusoidal carrier.

When a carrier signal is not a pure sinusoid, such as a square wave, the resulting spectrum of the modulated signal would contain additional harmonics due to the rich harmonic content of the square wave. The strength of the signal at any one peak could potentially be less than that of a pure sinusoidal carrier. This is because the energy of the square wave is distributed across several harmonics, while a pure sinusoidal carrier concentrates all its power at a single frequency.

4. Demonstrate aliasing with the modulated signal. This will involve you setting the message signal to have frequency content that passes the sampling frequency when modulated by the carrier. Show a plot of the aliased content in the time domain, and the frequency domain.

Aliasing with the modulated signal can be demonstrated by choosing a message signal with frequency content that, when modulated by the carrier, exceeds the sampling frequency. In such a case, the sampling theorem is violated and aliasing occurs. Aliasing is a form of distortion where higher frequency components get mapped onto lower frequencies. A plot of the time-domain signal will show this as distortions or anomalies in the signal, while in the frequency domain, you would see mirrored content about the Nyquist frequency. Remember, the exact appearance of aliasing will depend on the specific frequencies of your message and carrier signals, as well as your sampling frequency.

Conclusion

In conclusion, the Red Pitaya platform provides an effective, hands-on method for exploring signal modulation techniques, such as mixing and amplitude modulation. It allows users to directly observe the effects of varying signal frequencies and shapes, revealing the impacts of phenomena like aliasing. Despite some potential deviations from ideal scenarios due to hardware limitations or environmental noise, the Red Pitaya serves as a valuable learning tool for bridging theoretical principles with practical applications in signal processing and modulation.