The proposed approach is based on the design of digital signals in which the duty-cycle (δ) can be programmed. This fact is used for the high harmonics exact cancellation. Signal processing is performed in the frequency domain, whereas signal manipulation is done in the time domain. For a given digital signal f_d(t), with period T, amplitude A and interval d at high state, the C_n Fourier coefficients at the harmonics ω_n = nω_o, are given by

The sin(x) function has zeros when πnδ = mπ (n, m belong to N), being δ = d/T. A method to derive a sinusoidal voltage signal based on δ selection is proposed.

The mathematical basis of the proposed method is the time convolution of two signals. For two given functions, f_d1(t) and f_d2(t), the time convolution has a Fourier spectrum defined by the product of independent spectra of the functions F_d1(ω) and F_d2(ω) respectively, expressed as

In particular, when the signals have a rectangular form with the same period, T, but different duty cycle δ, the frequency spectrum of each signal will have zero components at frequencies in which sin(πnδ) is cancelled. In this work, we assume that all digital outputs have the same amplitude. The basic block pro- cessing scheme (Figure 1) considers the FFT’s of M rectangular signals with the same period T, and different δ_n(n = 1, 2, ∙∙∙, M).

For f_d1(t) rectangular signal, with δ = 0.5, all even C_n coefficients are cancelled. The spectrum of f_d2(t) must be designed to null the first useless odd harmonic of f_d1(t) that is 3ω_o. By selecting its duty-cycle as one third of T, the first non-desired odd harmonic (3ω_o) is cancelled since sin(πn/3) is zero when 3|n (C₃, C₆, ∙∙∙). It can be easily deduced that considering only two functions f_d1(t) and f_d2(t), with δ = 0.5 and δ = 0.33, all non-eliminated harmonics are co-primes with 2 and 3. This is shown more clearly in Figure 2. For this reason, the Fourier coefficients after multiplying these functions are given by

where V_n,i represents the nth coefficient of the complex Fourier series for the convolution of the i signals, and P_i, the ith element of the ordered set composed by the prime numbers, P, i.e. P = {2, 3, 5, ∙∙∙}. In the case i = 2, Equation (4) is obtained. This process can be extended to three or more digital signals, increasing the lower element in P set, which means to increase the sinusoidal voltage signal quality.

The proposed system has been tested on the 8 bits microcontroller (µC) Ar- duino™ Uno with ATmega328 version. Arduino™ is an open-source electronics prototyping platform [8] . Besides the µC, we use a DAC 0808 to create an analog version, and a first order RC Smoothing Active Filter (SAF). The cutoff frequency of the SAF is programmed to the nearest higher frequency as the wanted signal. This paper will show the output to 100 Hz, 1 kHz and 10 kHz signals, and the SAF cutoff frequencies are 200 Hz, 2 kHz and 20 kHz respectively.

Matlab® R2014b simulations have been performed for several digital signals. Figure 3 shows the waveforms obtained for rectangular functions with 1, 2, 3 and 4 convoluted signals, with duty-cycles of 0.5, 0.33, 0.2 and 0.14 respectively. Waveforms are presented after Fourier coefficients have been multiplied in the frequency domain, and then, the inverse Fourier transform is calculated. To evaluate the quality of the sinusoidal signals obtained, the Total Harmonic Distortion (THD) is observed in Figure 4.

Taking only three convoluted signals it is possible to obtain a THD below 1% (about 0.5%), which is usually enough in ECIS technique. Total Harmonic distortion expected as a function of the number of convoluted signals is represented in Figure 4. To select the working frequency it must be set the µC clock signal, a difference to other µC based realizations [5] . It was considered the ARM Cortex-M7 µC as an example for further implementation.

Real-time signal generated by the 8 bits microcontroller is presented in the following figures, in which channel 1 is a sinusoidal wave produced by the Press 2 MHz Function Generator GF-232, and channel 2 shows the proposed generator output. In Figure 5, we can see that the approximate generator GF-232 THD is

0.88%, while the proposed generator is 0.85%, being these two results approximately 0.3% higher than the theoretical THD expected result, around 0.5%, shown in section 3.1.

In Figure 6, the outputs of the four selected frequencies (100 Hz, 1 KHz, 10 KHz and 18.5 KHz) are presented. The highest frequency that can be achieved

with the selected µC, with an acceptable rate of noise, is around 18.5 KHz. This is derived, mainly, by the clock drift and the high level programming language, which can be easily sorted out with another µC. With the proposed system, the frequency can be easily changed in real-time, what is of utmost importance in the monitorization of ECIS biological experiments.