1. Introduction
Digital filters are discrete-time maps that perform mathematical operations on a sampled signal [
1]. Frequency response is usually applied to characterize filters [
2,
3]. Two main classes of digital filters are generally used. When an impulse response is not zero for a finite number of samples, then we have the finite impulse response (FIR) filters. In the case where the impulse response produces an infinite number of non-zero samples, then we have the infinite impulse response (IIR) [
4,
5]. The great performance of digital filters is believed to be one of the reasons explaining the popularity of DSP devices [
6].
The process of digital filtering is extensively used in many applications in communications, signal processing, electrical and biomedical engineering, and control [
7,
8,
9,
10,
11,
12,
13,
14,
15]; for example, coding and compression, signal augmentation, denoising, amplitude and frequency demodulation, analog-to-digital conversions, shape detection, and extraction [
16,
17,
18,
19,
20,
21,
22,
23,
24,
25]. For some applications, nonlinearity is tailored to a specific purpose [
26]. Recently, the authors of [
27] designed a digital sigma–delta truncated infinite impulse response filter, which furnishes adequate rejection with a digital-to-analog converter of no more than 8 bits. The application in [
27] is related to human body communication, which for many researchers is a promising research topic as it plays an important role in wireless body area networks because of its low power and hardware cost. In this area, it seems that digital filters of medium to low word length has again attracted the attention of researchers.
When digital filters are employed under fixed-point arithmetic platforms, e.g., microcontrollers, DSP, and FPGA, or with very demanding performance specifications, the importance of filter coefficient accuracy increases, because the signal may be distorted [
28,
29,
30]. Thus, a common goal in the finite precision analysis is to choose a word length such that the digital system presents sufficiently accurate realization. This design should consider the complexity and cost of hardware and software [
31].
In digital signal processing, the issues of finite word length are some of the most significant components when the discrete poles are very close to the unit circle. Mullis and Roberts [
32] and Hwang [
33] have demonstrated that the influence of quantization errors on the digital filter performance depends on the filter implementation. In addition, Rader and Gold [
34] have shown that for a given filter implementation it is possible that small errors in the denominator or numerator coefficients may cause large pole or zero offset. Moreover, Goodall and Donoghue [
35] and Jones et al. [
36] have observed a significant sensitivity of coefficient word lengths. This fact relates to the inability of computers to represent the infinite nature of real sets [
37]. The influence of computer limitations opens a new perspective for computer environment simulation. For example, Nepomuceno [
38] presents a theorem that identifies the reliability of calculations performed at fixed points; in [
39,
40], a technique has been developed to decline a simulation if a mandatory accuracy is greater than the lower bound error, growing numerical reliability in simulation, and still in [
41], the authors show how sensitive a simulated system is in different processors.
It seems clear that much research has been devoted to investigating the influence of finite precision on digital filters [
32,
34,
36,
42,
43]. In those investigations, there are many cases where the quality of filter is measured using the filter response or signal-to-noise ratio (SNR) [
43]. Despite the fact that the effect of filters on entropy has been pointed out since the work of Shannon [
44], there is much less attention given to the entropy effects due to finite precision digital filters on the filtered signal. One work in this direction has been undertaken by Badii et al. [
45], who show the influence of an infinite-impulse response in the fractal dimension of the attractor reconstructed from a filtered chaotic signal. Other works have employed entropy to the design of digital filters. For instance, Madan [
46] has introduced the use of the maximum entropy method for the design of linear phase FIR digital filters. In [
47], another attempt to use entropy in the design of digital FIR filters has been observed. However, no work has been found investigating the effects on entropy on a filtered signal by an IIR filter. This paper seeks to relate the computational limitations and the variation of the main parameters of a filter in the measured entropy. As entropy is a good index to detect increasing of noise in a signal, we have used a boundary technique to observe the effects of finite precision on the parameters of the filters according to the word length of 16 or 32 bits. We noticed that entropy is more sensitive than SNR. It was important to show that despite the ideal linear filter do not increase entropy, numerical experiments using the elliptic, Butterworth and Chebyshev filters have shown an increasing of entropy. Additionally, a positive correlation between order and entropy has been observed in the elliptic filter. This information can be useful to design or to evaluated digital filters in situations where the growth noise should be mitigated.
The remainder of this paper is organized as follows. The definitions of IIR, FIR filters, quantization, and entropy are given in
Section 2 as well as three scenarios of the simulation.
Section 3 presents the results, where three filter types are investigated: Butterworth, Chebyshev, and elliptic. The remaining section is devoted to summarizing our results.
4. Discussion and Conclusions
This work has investigated the effects of finite precision in the entropy of digital filtered signals. This allows us to quantify the introduction of noise due the action of such filters. We have shown that entropy is a good alternative to identify the presence of noise. It has presented a better result than the signal-to-noise ratio for small amount of variance. To observe the effects of entropy in filtered signals, we have designed three numerical experiments. In Numerical Experiment 1, we have evidenced the increasing of the entropy in all types of filters investigated (Butterworth, Chebyshev, and elliptic) for 16 and 32 bits. The entropy of the input signal is
$H=4.9255$, whereas in all the filtered signal the entropy is
$H>5.32$. This is not what is expected for an ideal linear filter (see [
44]). We should notice, according to
Table 2, that elliptic has been set up with the lowest order. Even in such circumstances, this type of filter has shown practically the same level of entropy in the filtered signal.
The results of Numerical Experiment 2 are shown in
Table 5. In this case, an ideally filter is simulated by taking out some of the frequency components of the signal. The entropy of filtered signal has been significantly increased varying from 6.5 to almost 8.
In Numerical Experiment 3, we have noticed another feature as described in
Table 6. This experiment shows a significant positive correlation at the 0.05 level (2-tailed) for elliptic with
p-value equals to 0.030. From these experiments, it seems clear that the elliptic filter introduces more uncertainty, that is, entropy, to the filtered signal when compared to Butterworth and Chebyshev filters.
Figure 4 shows the FFT of the signals. It is possible to notice a slight difference between subfigures (b) and (c).
The remarks made in this manuscript is coherent to what have been presented by DeBrunner et al. [
47]. As we are focusing our attention to the source noise furnished by arithmetical operations (see
Figure 1), design strategies that look for more efficient ways to implement mathematical expressions can be useful to reduce entropy. In our future work, we intend to test different topologies of filter (direct or cascade, for instance) to verify its influence in the increasing of entropy as done in this manuscript. This seems a quite reasonable pathway as the order is related to the increasing of the number of mathematical operations, which is a well-known source of the noise. We also intend to investigate the influence of sample rate and the number of samples in the computation of entropy.