## 2 FM-ACSK Communication System
### Frequency Modulation  
The ACSK-modulated signal is frequency-modulated using a digital modulator based on a numerically controlled oscillator (NCO) (see Figure 6). A Simulink-assisted design is used for fixed-point VHDL implementation using an HDL Coder toolbox. The passband frequency is 10.7 MHz, which is a typical FM intermediate frequency, and is sufficient for the prototype system's testing purposes without the introduction of an actual wireless communication part. The chosen frequency can be easily combined with existing broadband FM radio modules and designs, which can be used for further tests in wireless connectivity. Also, this frequency is in a range that is well suited for generation on an FPGA with a  
Figure 4: General structure of fourth-order modified Chua chaos generator implementation.  
Figure 5: Linear error feedback chaotic synchronization scheme between master and slave chaos generators.  
50 MHz clock, as it is close to half of the Nyquist frequency. With a frequency modulation index of 3 and an ACSK signal maximum noteworthy frequency of 215 kHz, the bandwidth of the FM-ACSK signal (according to Carson's Rule) is 1720 kHz.  
This digital FM system was specifically designed for working with a 50 MHz sampling clock, since the frequencies of all system signals are relative to the sampling frequency. The NCO uses 26 bits for the accumulator, 13 bits for phase quantization, and 13 bits for phase dithering. It ensures an output frequency resolution of about 1 Hz and a spurious free dynamic range (SFDR) of approximately 90 dB. The carrier increment sets the 10.7 MHz carrier output, while the deviation increment is used for scaling the ACSK output signal to achieve the desired deviation of 645 kHz.  
The digital FM demodulator (see Figure 7) uses a 30-tap FIR Hilbert filter and a delay of corresponding length (half of the Hilber filter's length plus its latency) to obtain the analytic signal from the noisy FM-ACSK input. Then, it is multiplied by a carrier generated by a similar NCO as the one used in the FM modulator. As a result, one part of the signal is shifted to the baseband, while the other is shifted to the double carrier frequency. A lowpass 100-tap FIR filter with an 860 kHz cutoff frequency removes the signal copy at double the carrier frequency, as well as part of the channel noise. The retrieval of the angle of the complex baseband analytic signal recovers the instantaneous phase of the modulated signal. The recovery of the modulating ACSK signal (instantaneous frequency) is performed using a simple two-tap difference filter, which additionally removes any DC bias (which depends on the carrier phase offset). Right after that, phase unwrapping is performed (when necessary) by adding or subtracting 2\(\pi\) if the signal value is below \(-\pi\) or above \(\pi\), respectively. Hard clipping for signal amplitudes above the expected range (defined by the frequency deviation) is performed to protect the data from overflows in the presence of strong noise. At the end, the signal is amplified to the power level, which is close to the ACSK output power in the transmitter. Before passing the recovered ACSK signal to the ACSK demodulator, additional filtering is performed with a 215 kHz lowpass FIR filter to further reduce the impact of the channel noise.  
Figure 6: Simulink schematic of digital FM modulator.  
Figure 7: Simulink schematic of digital FM demodulator.