The Digital Transmission Chain: From Bits to RF

Date: 2025-11-24 Tags: RF, DSP, SDR, Modulation, Engineering Author: Wissam Ztaoui

Introduction

A Software Defined Radio (SDR) system abstracts hardware complexity into software blocks. This guide dissects the complete digital transmission chain, explaining the mathematical and engineering principles behind transforming a digital bitstream into an electromagnetic wave.

1. Source Coding (Compression)

Before transmission, data must be minimized.

• Entropy: Shannon’s Source Coding Theorem states that $H(X)$ is the limit of lossless compression.
• Algorithm: Huffman coding or Lempel-Ziv-Welch (LZW).
• Goal: Remove redundancy.

2. Channel Coding (Forward Error Correction)

The wireless channel is noisy (AWGN, fading). We must add redundancy to detect and correct errors.

• Block Codes: Hamming, Reed-Solomon.
• Convolutional Codes: Viterbi algorithm for decoding.
• Turbo/LDPC Codes: Used in 4G/5G/Wi-Fi to approach the Shannon Limit ($C = B \log_2(1 + S/N)$).

Example (Hamming 7,4): Adds 3 parity bits to 4 data bits. Can correct 1 bit flip.

3. Modulation Mapping (Constellations)

We map bits to complex symbols ($I + jQ$).

Phase Shift Keying (PSK)

• BPSK: 1 bit/symbol. Phases: $0, \pi$. Robust but slow.
• QPSK: 2 bits/symbol. Phases: $\pi/4, 3\pi/4, 5\pi/4, 7\pi/4$.

Quadrature Amplitude Modulation (QAM)

Varies both amplitude and phase.

• 16-QAM: 4 bits/symbol.
• 64-QAM: 6 bits/symbol. Higher throughput, requires higher SNR.

IQ Representation: $$ s(t) = I(t)\cos(2\pi f_c t) - Q(t)\sin(2\pi f_c t) $$

4. Pulse Shaping (Filtering)

Rectangular pulses have infinite bandwidth (sinc function in frequency domain). To prevent Inter-Symbol Interference (ISI) and limit bandwidth, we use a Root Raised Cosine (RRC) filter.

• Roll-off factor ($\alpha$): Determines excess bandwidth.

5. Digital Up-Conversion (DUC)

Inside the FPGA/DSP:

1. Interpolation: Increase sample rate.
2. NCO (Numerically Controlled Oscillator): Generates digital sine/cosine.
3. Digital Mixer: Multiplies baseband signal by NCO output to shift to Intermediate Frequency (IF).

6. Analog Front-End

1. DAC: Converts digital samples to analog voltage.
2. Mixer: Up-converts IF to RF (e.g., 2.4 GHz).
3. Power Amplifier (PA): Boosts signal power. Non-linearity here causes spectral regrowth (distortion).
4. Antenna: Radiates the signal.

7. GNU Radio Implementation

In GNU Radio Companion (GRC), this chain is built visually:

1. File Source (Bits)
2. Constellation Modulator (Maps bits to symbols + RRC Filter)
3. USRP Sink (Sends IQ data to hardware)

Conclusion

Understanding the transmission chain is fundamental for any RF engineer. It bridges the gap between abstract information theory and physical reality.