FM Demodulation on the DE1-SoC¶
Project Introduction¶
I built a system that demodulates live FM (frequency modulated) radio signals on the DE1-SoC board using a RTL-SDR module and Raspberry Pi, enabling real-time signal processing for clear, immediate audio output. Using the RTL-SDR as a receiver, I connected it to a Raspberry Pi to extract the I and Q values. These values were transmitted to the DE1-SoC's Hard Processor System (HPS) via a TCP socket. Once received, the I and Q values were demodulated, filtered, and sent to the DAC for audio playback of WVBR, Ithaca’s Alternative radio station.
High Level Design¶
Rationale and Sources of Idea¶
Since this class focuses on using FPGAs (field programmable gate arrays) for hardware acceleration, I anticipated that the FPGA would excel at handling complex, real-time computations. I also knew that radios receive and demodulate signals in real-time so I figured that the FPGA would be perfect for exploring digital signal processing for FM demodulation. Initially, I aimed to implement a complete software defined radio (SDR) system on the FPGA. However, given that I worked on the final project independently and had limited time to complete it, I decided to scale the scope down to focus on demodulation on the HPS, as the topic was new to me.
Background Math¶
IQ Modulation
In Software-Defined Radios (SDR), IQ modulation is used to represent RF signals as two orthogonal components: I (In-phase) and Q (Quadrature). The I component corresponds to the real part of the signal, aligned with a cosine wave, while the Q component corresponds to the imaginary part, aligned with a sine wave (90° out of phase). Together, the complex signal I+jQ allows full representation of the signal's amplitude and phase. This orthogonal decomposition makes it possible to encode and process complex modulated signals.
In the context of FM signals, the transmitted signal can be expressed as:
$$
s(t) = A \cdot \cos(2\pi f_c t + \theta(t))
$$
where:
$ f_c $ is the carrier frequency, and
$ \theta(t)$ is the instantaneous phase that encodes the modulating signal.
The IQ representation of this signal is:
$$
I(t) = A \cdot \cos(2\pi f_c t + \theta(t))
$$
and
$$
Q(t) = A \cdot \sin(2\pi f_c t + \theta(t))
$$
IQ Demodulation
IQ demodulation extracts the original baseband information from a modulated RF signal by mixing it with two local oscillator signals: a cosine wave for the I component and a sine wave for the Q component. These operations shift the signal to baseband while preserving the amplitude and phase. The orthogonality of the cosine and sine ensures no interference between I and Q, enabling precise reconstruction of the transmitted signal. By analyzing the resulting IQ components, both amplitude and phase information can be extracted for further processing.
FM Demodulation Using IQ Values
FM demodulation using IQ values relies on detecting changes in the instantaneous phase of the signal. Since FM signals encode information in phase variations rather than amplitude, the phase angle $\theta$ can be computed as:
$$
\theta(t) = \arctan\left(\frac{Q(t)}{I(t)}\right)
$$
To extract the frequency modulation, we take the derivative of the phase with respect to time. The instantaneous frequency $f(t)$ is given by the derivative of $\theta(t)$:
$$
f(t) = \frac{d\theta(t)}{dt} = \frac{d}{dt} \left( \arctan\left(\frac{Q(t)}{I(t)}\right) \right)
$$
This derivative gives the rate of change of phase, which is proportional to the instantaneous frequency deviation of the signal. The instantaneous frequency deviation corresponds to the original modulating signal. This FM demodulation technique is also called phase discrimination.
Derivative Approximation
In my code, I approximate this computation by taking an approximated derivative of consecutive I/Q values first. Then I take the arctangent of the real and imaginary components of this approximated derivataive to get the derivative of phase. Although this approach was less accurate, it simplified the C code.
$$
\text{real} = I(t) \cdot I(t + \Delta t) - Q(t) \cdot Q(t + \Delta t)
$$
$$
\text{imag} = I(t) \cdot Q(t + \Delta t) + Q(t) \cdot I(t + \Delta t)
$$
These two expressions approximate the real and imaginary parts of the derivative that reflects the changes in the in-phase and quadrature components, respectively, from one time sample to the next.
Next, the arctangent of the imaginary and real part are taken to compute the derivative of the phase angle:
$$
f(t) \approx \frac{d\theta(t)}{dt} \approx \arctan\left( \frac{\text{imag}}{\text{real}} \right)
$$
