# DESIGN OF NON-RECURSIVE AND RECURSIVE DIGITAL BAND PASS FILTERS FOR GENERAL PURPOSE APPLICATIONS

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1 DESIGN OF NON-RECURSIVE AND RECURSIVE DIGITAL BAND PASS FILTERS FOR GENERAL PURPOSE APPLICATIONS Barreto Guerra, Jean Paul Professor: Penny Cabrera, Oscar Course: CE 0907 Digital Signal Processing 2011-I Professional School of Electronic Engineering Ricardo Palma University SUMMARY: This work is based on the study of non-recursive and recursive digital band pass filters for an audio equalizer. The design of FIR (nonrecursive) and IIR (recursive) filters were made following the design specifications for this application. To find the solution of this problem, first we will define the filter characteristics to find the respective coefficients for both types of filter. Then, we will deduce the expression of H(z) and the structure diagram of the filters with their correspondent coefficients. Next, we will deduce the expression of the difference equations that can be used for programing the algorithms for the simulation of the filters in a DSP microprocessor. Finally we will present the response of magnitude and phase of the designed filters complying with the required design characteristics. Whole design was simulated using the software Mathcad. RESUMEN: Este trabajo está basado en el estudio de filtros digitales pasa banda no-recursivos y recursivos para un ecualizador de audio. El diseño de filtros FIR (no-recursivo) e IIR (recursivo) fueron hechos siguiendo las especificaciones de diseño para esta aplicación. Para hallar la solución de este problema, primero vamos a definir las características del filtro para hallar los coeficientes respectivos para ambos tipos de filtro. Luego, deduciremos la expresión H(z) y el diagrama de estructura de los filtros con sus coeficientes correspondientes. A continuación, deduciremos la expresión de la ecuación diferencial que se usará para programar los algoritmos para la simulación de los filtros en un microprocesador DSP. Finalmente se presentará la respuesta en magnitud y fase de los filtros diseñados cumpliendo con las características de diseño requeridas. Todo el diseño fue simulado usando el software Mathcad. 1 INTRODUCTION This report presents the study of recursive and non-recursive filter, which are used on many applications. The design methods for each of these two classes of filters are different because of their distinctly different properties. In non-recursive filter structures the output depends only on the input, where we have feed-forward paths. The FIR filter has a finite memory and can have excellent linear phase characteristics, but it requires a large number of terms, to obtain a relatively sharp cutoff frequency response. In recursive filter structures the output depends both on the input and on the previous outputs, where we have both feed-forward and feed-back paths. The IIR filter has an infinite memory and tends to have fewer terms, but its phase characteristics are not as linear as FIR. Filter design can be implemented in the time-domain or frequency-domain. In this case, we shall be dealing with the design of filters specified in the frequency-domain because it is more suitable and precise for us. We will use

2 2 PRESENTATION OF THE PROBLEM The following problem requires 2 designs. First we require designing a non-recursive digital band pass filter with the following characteristics: Sampling Frequency 1000 KHz Number of coefficients 51 Band Width 100 KHz Centre Frequency 250 KHz (For Hamming, Von Hann and Kaiser Beta=4 and Beta=8 filters) Hanning Second we require designing a recursive digital band pass filter with the following characteristics: Sampling Frequency Range of the pass band (ripple free attenuation between 0 and 3 db) Attenuation must be at least 30 db for frequencies 1000 KHz From 200 to 300 KHz Less than 150 KHz and more than 350 KHz Kaiser β = 4 (For Butterworth and Chebyshev I filters) 3 DESCRIPTION OF THE SOLUTION FIR Filters Design (non-recursives filters) First of all, we define the filter characteristics: Kaiser β = 8 Then, we define the corresponding window to find the coefficients for each filter. For each filter, we have to convert the low pass to band pass. Hamming With these formulas, we find the coefficients (this result is shown in Part 4). IIR Filters Design (recursives filters) First of all, we define the parameters specified in the filter characteristics.

3 n=ceil(n) was chosen, because n have to be integer, leaving the expression: (the first comes from the second) So, we have to separate H(s), this means to find the roots of (-s^2/wc^2)-(-1)^(1/n): Then, using the analog models of the Butterworth and Chebyshev I filters and the bilinear transformation principles, we will deduce the correspondent poles of the filters. Butterworth Filter Taking the analog Butterworth filter model, that responds to the expression: S K shows the correspondent poles of the Butterworth Filter. Chebyshev I Filter Now, taking the analog Butterworth filter model, that responds to the expression: So, we have to find the value of n, so that from Fd2 Hz, the signal attenuate Att db. So, we have to find the value of n, so that from Fd2 Hz, the signal attenuate Att db.

4 Hamming The coeffcientes for this filter are: S K shows the correspondent poles of the Chebyshev I Filter. The expression of H(z) (that correspond to the filter), the structure diagram, and the algorithm for the simulation are: 4 RESULTS FIR Filters Results Applying the formulas listed above; there are 51 coefficients, from 0 to 50. We are going to show, the coefficients, the expression of H(z), the structure diagram and the algorithm for the simulation for each of the windows. Finally, we are going to show simultaneously, the magnitude and phase response for all these 4 filters.

5 Hanning The coeffcientes for this filter are: The expression of H(z) (that correspond to the filter), the structure diagram, and the algorithm for the simulation are:

6 Kaiser β = 4 Kaiser β = 8 The expression of H(z) (that correspond to the filter), the structure diagram, and the algorithm for the simulation are: The expression of H(z) (that correspond to the filter), the structure diagram, and the algorithm for the simulation are:

7 Hanning Kaiser β = 4 Finally, we will compare the magnitude and phase response of the 4 filters designed, but before that, we will show the correspondent formulas to get these responses for some frequencies. We define the variable F, shown as: Kaiser β = 8 The deduced expressions of amplitude and phase are: Now, we ll represent the response of magnitude and phase of all these filters: - The black graphic shows the hamming filter response. - The blue graphic shows the hanning filter response. - The red graphic shows the Kaiser β=4 filter response. Now calculating some attenuation for the different windows: Hamming - The pink graphic shows the Kaise β=8 filter response. From Figure 1, we can see that all the filters follow the characteristics specified for the design, we can see the cut frequencies and the stop frecuencies.

8 Then, we have to define many equations to find the coefficients: Figure 1. Magnitude and Phase response for FIR filters IIR Filters Results Applying the formulas listed above, we are going to show, the coefficients, the expression of H(z), the structure diagram and the algorithm for the simulation for each filter. Finally, we are going to show simultaneously, the magnitude and phase response for both designed filters. And making Butterworth Filter We will take the n values that belong to: And then, it will be necessary to accommodate the expression like this: Then, we have To find H(z), we will employ

9 The coefficients go from a 0 to a 10 and similarly from b 0 to b 10, following the sequence shown above. Then we have to compensate: Finally, we obtain the canonic function of the filter, with the corresponding coefficients: The next step is to define every coefficient:

10 To find H(z), we will employ The structure diagram, and the algorithm for the simulation are: Then, we have to define many equations to find the coefficients: Chebyshev I Filter And making We will take the n values that belong to: And then, it will be necessary to accommodate the expression like this:

11 The next step is to define every coefficient: Then, we have The coefficients go from a 0 to a 6 and similarly from b 0 to b 6, following the sequence shown above. Then we have to compensate: Finally, we obtain the canonic function of the filter, with the corresponding coefficients:

12 Now calculating some attenuation both filters: Butterworth Filter The structure diagram, and the algorithm for the simulation are: Finally, we will compare the magnitude and phase response of both designed filters, but before that, we will show the correspondent formulas to get these responses for some frequencies. The frequency response is defined by: Figure 2. Magnitude and Phase response for Butterworth filter (IIR filter) Chebyshev I Filter

13 - The red graphic shows the Butterworth filter response. - The blue graphic shows the Chebyshev I filter response. Figure 4. Magnitude and Phase response for IIR filters 5 CONCLUSIONS FIR Filters Results The response of all the filters, in terms of phase, is lineal and resembles each other. So, since these filters are FIR filters, they don t have phase distortion, that it s very important. Talking about the amplitude, the best filter of this four, is the Kaiser filter with β = 4, because this filter attenuate best the frequencies that are not in the band pass, also this filter is the least distorts the amplitude, so these two reasons make this filter the best in comparison with the other three. Figure 3. Magnitude and Phase response for Chebyshev I filter (IIR filter) From Figure 2 and Figure 3, we can see that Butterworth and Chebyshev I filter follow the characteristics specified for the design; we can see the cut frequencies and the stop frequencies required. Finally, we will compare the two filter responses, in magnitude and phase: IIR Filters Results The responses of both filters, in terms of phase, are a little different, because the Chebyshev I filter, have phase distortion, and the Butterworth phase response have some linear parts. So, since these filters are IIR filters, they have phase distortion, that it s a disadvantage. Talking about the amplitude, the best filter of this two is the Chebyshev I filter, with this filter results in a fall of frequency response more pronounced at low frequencies and it allow a ripple in

14 the pass band as shown in the magnitude response above. In conclusion comparing the two types of filters, after seeing the magnitude and phase responses, we see that IIR filters are characterized by greater attenuation to frequencies that are not in the pass band, besides the most important feature of these filters is that the phase response is linear. One advantage of IIR filters is that we will need smaller number of coefficients for the specified design features, but today that s no problem because the microprocessor can use a large amount of coefficients. Then the best option is to use FIR filters. 6 BIBLIOGRAPHY Books: <> Digital and Kalman Filtering, S.M. Bozic England 1994, 2th Edition [Search: 01 August] <> Tratamiento Digital de Señales, Proakis Madrid 2007, 4th Edition [Search: 03 August] <> Tratamiento de Señales en Tiempo Discreto, Oppenheim and Shafer Madrid 2000, 2th Edition [Search: 03 August] Web Pages: <> Digital Filters. <> <> htttp://focus.ti.com/dsp <> Magazines: <> IEEE Transactions on Digital Signal Processing <> IEEE Transactions on Digital Signal Processing Magazine

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