Phase-sensitive demodulation (PSD) is a simple and reliable demodulation method ... For that, a dedicated multiplier of a XtremeDSP Slice at the FPGA is used.
Sensors 2016, 16, 1158; doi:10.3390/s16081158
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Supplementary Materials: System Description and First Application of an FPGA-Based Simultaneous Multi-Frequency Electrical Impedance Tomography Susana Aguiar Santos, Anne Robens, Anna Boehm, Steffen Leonhardt and Daniel Teichmann 1. Direct Digital Synthesis Direct Digital Synthesis (DDS) was used to create the sine and cosine waveforms. The desired output frequency f out is set by changing the value of the phase accumulator and can be calculated from
f out =
f clk · ∆θ 2Bθ (n)
⇒ f out = ∆θ · a
(1)
where f clk is the system clock frequency, ∆θ is the phase increment value and Bθ (n) is the number of bits of the phase accumulator. The constant a gives the frequency resolution. In the presented EIT system, f clk = 245.76 MHz and Bθ (n) = 16 bit were chosen, yielding a frequency resolution of 3750 Hz. If another resolution is required, the number of bits of the phase accumulator can be adapted accordingly. Thus, in order to generate a sine wave of 120 kHz, a ∆θ = 32 has to be given ∆θ =
f out 120000 Hz = = 32 a 3750 Hz
(2)
2. Phase-Sensitive (I/Q) Demodulation Phase-sensitive demodulation (PSD) is a simple and reliable demodulation method to calculate the in-phase (I) and quadrature (Q) components of the measured voltages. Considering a number of frequencies n, the measured voltages (Vmeas ) modulated by the body are represented by: n
Vmeas =
∑ Ai · sin(ωi t + ϕi )
(3)
i =1
where Ai is the modulated amplitude and ϕi is the phase induced by the tissues impedances, for each frequency wi . The demodulation of the I and Q components for each frequency is performed separately by multiplying the measured signal Vmeas with the original sine and cosine of each frequency, respectively. For that, a dedicated multiplier of a XtremeDSP Slice at the FPGA is used. Considering the number of frequencies n = 2, the I and Q components for the measured voltage at frequency ω1 (V1 ) are obtained by: V1,I
= ( A1 · sin(ω1 t + ϕ1 ) + A2 · sin(ω2 t + ϕ2 )) · ( Aref · sin(ω1 t))
(4)
V1,Q
= ( A1 · sin(ω1 t + ϕ1 ) + A2 · sin(ω2 t + ϕ2 )) · ( Aref · cos(ω1 t))
(5)
which is equivalent to: 1 V1,I = ( A1 Aref · cos( ϕ1 ) − A1 Aref · cos(2ω1 t + ϕ1 )+ 2 + A2 Aref · cos(ω2 t + ϕ2 − ω1 t) − A2 Aref · cos(ω2 t + ϕ2 + ω1 t))
(6)
Sensors 2016, 16, 1158; doi:10.3390/s16081158
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1 V1,Q = ( A1 Aref · sin( ϕ1 ) + A1 Aref · sin(2ω1 t + ϕ1 )+ 2 + A2 Aref · sin(ω2 t + ϕ2 − ω1 t) + A2 Aref · sin(ω2 t + ϕ2 + ω1 t))
(7)
When low-pass filtering or integrating these signals for a complete period of the respective frequency, the alternating components of the signal vanish and only constant components remain. The following equations show the generalized Ii and Qi component of a frequency wi . Vi,I
=
Vi,Q
=
Ai Aref · cos( ϕi ) = Ii 2 Ai Aref · sin( ϕi ) = Qi 2
mfDummy Amplitude 3. Results of Resistive/Capacitive Test Unit and
(8) (9)
Phase 17 of 21
Version February 23, 2016 submitted to Sensors
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Sensors 2016, 16, 1158; doi:10.3390/s16081158
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