ON FINITE-PRECISION EFFECTS IN LATTICE WAVE-DIGITAL FILTERS

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1 ON FINITE-PRECISION EFFECTS IN LATTICE WAVE-DIGITAL FILTERS Isabel M. Álvarez-Gómez, J.D. Osés and Antonio Álvarez-Vellisco EUIT de Telecomunicación, Camino de la Arboleda s/n, Universidad Politécnica de Madrid, 2831 Madrid (Spain) Tel:[+34] Fax[+34] ; ; ABSTRACT Digital filters are a basic building block in many Soft Computing applications, specifically in those that require Signal Processing Techniques to obtain feature vectors representative of the real-world problem. Practical implementation of digital filters implies using finite-length registers for filter coefficients storing. It is known that if we wanted to achieve ideal filtering performances, infinite-length filters would be needed. But, such filters are impossible to construct by using infinite precision models. In this paper, in order to minimize the above mentioned errors, an application for the design and analysis of LWDF (Lattice Wave Digital Filters) has been developed in MATLAB. The simulation results are satisfactory and show the importance of using the above mentioned application when analyzing the quantization effects in digital filters, before carrying out their practical implementation. KEYWORDS: Lattice Wave Digital Filter, Limit cycles, Iterations, Samples, Noise, Word-length, Ceil, Floor, Saturate, Wrap. 1. INTRODUCTION Minimizing the effects of quantization errors that arise when implementing digital filters has been one of the greatest challenges that filter designers have faced for several decades. In order to achieve the above mentioned objective, many algorithms have been built by using advanced filtering techniques. In this paper, a novel application based on lattice wave-digital filter (LWDF) techniques is presented [1]. This algorithm allows us to carry out as much simulations as we need to obtain the desired filter characteristic response. In short, when designing the digital filter by using the proposed algorithm the designer can choose not only the filter characteristics, but also its quantization parameters. In addition, the filter characteristics that the designer can choose are the following: Type of filter: low-pass filter, band-pass filter, high-pass filter or band-stop filter. Passband attenuation and stopband attenuation. Passband frequency and stopband frequency. Sampling frequency. Filter design: Butterworth filter, Chebyshev filter, Inverse Chebyshev filter, Cauer (elliptic) filter. Furthermore, the following filter quantization parameters can be chosen: Round off: floor, ceil, fix, round and convergence.

Such an algorithm is easy to use and it gives the designer the following information: Frequency response of the filter, impulse response of the filter and the polezero diagram for both the ideal

2 Overflow: saturate and warp. Number of bits. 2. THE ALGORITHM The algorithm for the digital filter implementation with minimized quantization errors was built by using the explicit formulas for designing LWDF [2]. Such an algorithm is easy to use and it gives the designer the following information: Frequency response of the filter, impulse response of the filter and the polezero diagram for both the ideal filter (infinite precision) and the finite precision Quantization error. Quantization noise. Limit cycles. Furthermore, one of the most important parts of this algorithm is the one related to the quantization noise. This part was carried out by applying the Method for measuring the performance of weakly nonlinear system in [3]. Figure 1: Filter design Figure 2: Limit cycles 2 b) 2 iterations 4 iterations -2 2 iterations 4 iterations rad Figure 3: Results with 2 and 4 iterations: Response in Frequency; b) Noise spectral density

3 The information given in Fig.1 is used to carry out the filter design, introduce information about a new one to be designed, or simulate the limit cycles in a third order filter (see Fig. 2). Also, Fig. 3 shows the response of the filter. 3. CHARACTERISTICS OF THE METHOD FOR MEASURING THE PERFORMANCE OF WEAKLY NONLINEAR SYSTEM This method [3] allows us to estimate the frequency response of the quantized j filter, [ H ( e )] ω jω Q, and the power spectral density of the noise Φ ee ( e ) corrupting the information, from the output sequence consisting of the points ω k = k 2π / N. The input to the filter is a random signal v ~ λ [ n], λ = 1: L, where L is for the number of independent trials. L should be chosen in such a way that the desired precision be achieved. Thus, the more samples we take, the better the filter. An study of the influence of the number of iterations and simples on this method follows. Study of the influence of the number of iterations First, a filter with the characteristics given in Table 1 is chosen. LWDF LOWPASS FILTER (Chebyshev Approx.) a p () a s () f p () f s () Order Table 1 Figure 4 shows that one of the best properties of this method, is that it allows the designer to achieve a response as exact as the one needed. Also, Fig. 3 shows no clear difference between using 2 iterations and 4 iterations, but the time consumption between both results is huge. Study of the influence of the number of samples LWDF LOWPASS FILTER (Elliptic Approx.). a p () a s () f p () f s () Order Table 2 1 b) samples 124 samples samples 124 samples muestras: Respuesta enespectral uido Ree -8 Fig 4: Results with 512 and 124 sa rens rad Figure 4: Results with 512 and 124 samples Response in Frequency b) Noise spectral density

4 On the one hand, the higher number of samples, the better the filter. On the other hand, the higher the number of simples, the higher the time consumption. So, a trade-off should be established between the desired response and the time consumption in designing the filter. Thus, Fig. 4 shows that using a huge number of samples is not necessary; the results are very similar for the case under study using 512 samples and 124 samples. 4. RESULTS OF THE EXPERIMENT This section is devoted to show the influence of the parameters of the quantization, studied in [3]. Study of the influence of the round-off errors on the filter design As an example, experimental results of the application of the proposed algorithm to the filter with the characteristics given in Table 2 are shown. 1 b) Figure 5: Response in Frequency using ceil ( y floor (b) floor ceil ra d Figure 6: Noise spectral density Figure 5 shows the frequency response of the filter depending on the round-off method. That is to say, depending on the quantization parameter used, either ceil or

5 floor. Also, it can be seen that there is a clear difference in the attenuated band. Furthermore, the noise power spectral densities for both cases are very similar to each other (see Fig. 6). Study of the influence of the overflow on the filter design In order to study the difference between carrying out the quantization by using saturate and wrap, a filter with the characteristics given in Table 3 analyzed. LWDF LOWPASS FILTER (Butterworth Approx.) a p () a s () f p () () f s Order Table 3 1 b) Figure 7: Response in Frequency using saturate ( y wrap (b) The passband does not change practically but the attenuated band has undesirable peaks. Study of the influence of the number of bits on the filter design The higher the number of bits of the filter, the more the response of the filter is similar to the ideal one. Finite precision models introduce distortion in the system. In addition, the quality of the filter increases (see Fig. 9). 5 8 B ITS 5 12 B ITS B ITS B ITS Figure 8: Response in frequency whit several word-lengths

6 bits 16 bits ra d Figure 9: Noise spectral density with 12 and16 bits of quantization 5. CONCLUSIONS The priori estimation of the quantization parameters is not an easy task. Such estimation depends on the chosen filter structure. However, generally speaking, it can be said that 1.- Regarding the quantization parameters: No filter structure can be chosen to carry out the treatment of the round-off errors and the overflow. Thus, in order to obtain a satisfactory response of the filter, it is necessary to choose a high number of iterations. In order to achieve a satisfactory response of the filter, a filter with 16 or more bits can be chosen. What is more, with such a number of bits the filter response exhibits low distortion levels. However, it must not be forgotten that the higher the number of bits, the more expensive the filter. 2.- Regarding the parameters used in The Method for measuring the performance of weakly nonlinear system In order to achieve good levels of approximation, a number of 2 iterations or higher yields satisfactory results. However, it should not be forgotten that the higher the number of iterations, the better the design. Moreover, reliable results can be achieved by using 512 samples or more. 6. REFERENCES [1] A. Fettweis, H. Levin and A. Sedlmeyer, 'Wave Digital Lattice Filters', Int. Journal of Circuit Theory and applications, Vol.2, pag , [2] A. Fettweis, 'Wave Digital Filters: Theory and Practice', Proc. IEEE, Vol. 74, pag , [3] H. W. Schüßler and Y. Dong, A New Method For Measuring The Performance Of Weakly Nonlinear System, Proc. ICASSP, pag , 1989.

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