Baseband Input Waveform and Its Power Spectrum, AWGN Channel Output and Its Power Spectrum, Signal Constellation Before and After Rayleigh Channel

In this document, we will explore the following topics:

1. Baseband input waveform and its power spectral density;

2. QPSK modulated signal and its power spectrum;

3. AWGN channel output and its power spectrum;

4. Signal constellation diagrams before and after transmission through Rayleigh channel;

5. Bit error rate performance under Additive White Gaussian Noise (AWGN) and Rayleigh fading channel conditions, including theoretical curves for AWGN. For comparative analysis, all BER performance curves will be plotted using identical coordinate scales.

First, we will provide a detailed examination of the baseband input waveform and its power spectral density. Here, we will explain the definition of baseband waveforms and their impact on system performance, including implementation aspects such as waveform generation using raised-cosine filtering with adjustable roll-off factors.

Next, we will investigate QPSK modulation signals and their spectral characteristics, discussing QPSK's application in communication systems with code examples showing modulation using I/Q components and carrier multiplication.

In the section covering AWGN channel output and power spectrum, we will analyze AWGN channel properties and methods for power spectral density calculation, demonstrating how to simulate AWGN using random number generation with proper variance adjustment based on signal-to-noise ratio requirements.

We will also present signal constellation diagrams before and after Rayleigh channel transmission, explaining how these visual tools can optimize communication system performance through techniques like channel equalization and diversity combining algorithms.

Finally, we will examine bit error rate performance under both AWGN and Rayleigh fading channel conditions, comparing their performance characteristics. We will detail BER definition and computational methods, including Monte Carlo simulation approaches and theoretical error probability calculations for different modulation schemes. The theoretical AWGN curve will be derived using Q-function approximations. All BER performance curves will maintain consistent coordinate scaling for effective comparison.

Through this documentation, readers will gain comprehensive understanding of key communication system concepts and technologies, learning how these elements interact to optimize overall system performance through proper implementation and algorithmic choices.