This project addresses the challenges of designing and comparing various parameters of low pass Butterworth and Chebyshev1 Infinite Impulse Response (IIR) filters. The two digital filters were designed and simulated in MATLAB and the results obtained were analyzed and compared with each other.
The goals of the proposed project were achieved through the following objectives:
Design and simulation of digital low pass Butterworth and Chebychev1 IIR filters.
Comparison of these filters with respect to the following specifications:
Magnitude Frequency response.
Ripple Factor response.
Percentage of contribution
List of sections written or program/code
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Sai Guruva Reddy Avuthu
Program, Presentation and Report
Binu Baby Narakathu
Program, Presentation and Report
Harshita Kamala Nanjappa
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Program, Presentation and Report
Over the past decade, a lot of interest has been centered on the development of digital filters for applications in various areas such as digital control systems, pattern recognition, speech and image processing systems. Accuracy, stability, flexibility and reliability are some of the advantage of digital filter over traditional analog filters.
Depending on the frequency domain characteristics filters are also classified as low pass, high pass, band pass, band stop or band elimination and all pass filters as shown in Fig. 1.
Figure 1. Classification of filters.
Poles and zeros of any LTI system play an important role in the design of filters. For example for a low pass filter the poles should be placed near by the unit circle at a point corresponding to low frequencies (near Ï‰ = 0) and zeros should be placed near or onto the unit circle at points corresponding to high frequencies (near Ï‰ = Ï€). The opposite holds true for high pass filters.
Types of digital filters
All the digital filters belong to discrete time linear time invariantÂ (LTI) systems which are always linear, time invariant and usually causal and stable systems. The unit impulse response sequence of any causal LTI system is of either finite or infinite duration. Depending upon this property, the filters are classified as FIR or non recursive; and IIR or recursive filters. The general differential equation for any digital filter is given by Eq. 1.
Where y(n) is present output, x(n-i) is the corresponding input depending upon the value I, ai and bi are filter coefficients and N is the order of filter. IIR filters have one or more feedback coefficients; i.e. if the filter is excited with an impulse it continuously keeps giving an output. On the other hand a FIR filter has no non-zero feedback coefficient; i.e. if the filter is excited with an impulse then the output exists only for a finite number of cycles equal to the order of filter N. Fig. 2 shows the common IIR and FIR filter architectures.
Figure 2. IIR and FIR filter architectures.
For any given order, an ideal IIR filter has much sharper transition band when compared to a FIR filter, as IIR filter uses both poles and zeros as roots whereas the FIR filter uses only zeros. The zeros of FIR filters make its phase linear when compared to IIR filters whose phase is non-linear because of its poles.
A method of modifying, reshaping or manipulation of a frequency spectrum, according to a set of desired specifications, is known as "Filtering". The process of designing and implementing a filter network includes the enhancement of a wanted signal in relation to unwanted signals such as noise. An IIR filter, whose frequency response is approximated by a transfer function expressed as a ratio of polynomials, has shown to demonstrate better efficiency and robustness than a Finite Impulse Response (FIR) filter having the same number of coefficients [2,3]. The impulse response of an IIR system is defined as the response of a given dynamic system to some external stimuli as a function of time. IIR filters can be either analog or digital filters.
Pass band, stop band and transition band are the main specifications in filter design. Ideal filters have a rectangular pass band and no transition band. In any ideal filter, the desired frequencies are passed with no attenuation while the undesired frequencies are completely blocked. All the ideal filters are non causal and are usually not achievable. However, using practical techniques, such as Butterworth and Chebyshev the ideal filter characteristics can be achieved.
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Butterworth filters are designed to have a flat frequency response in the pass band so that it has a rectangular pass band. The transition band of Butterworth filter is high when compared to the Chebyshev filters. The general expression for the transfer function (H(s)) of an nth order Butterworth low pass filter is given by eq. 2.
The general shape of Butterworth low pass filter magnitude response with cutoff frequency (Ï‰) equal to 1 Hz is shown in Fig. 3. The numbers 1 through 5 represents the order of the low pass filter. As the order of filter increases the transition band decreases.
Chebyshev I filters
Chebyshev I filters are designed to have a relatively sharp transition from the pass band to the stop band. This sharpness is accomplished at the expense of ripples that are introduces into the response in the pass band. Specifically Chebyshev filters are obtained as an equiripple approximation to the pass band of an ideal low pass filter which is given by Eq. 3.
Where Îµ2= 10r/10 - 1, TN(Ï‰) is the Chebyshev I polynomial of order n and R is the pass band ripple (in dB). The general shape of Chebyshev I low pass filter magnitude response with cutoff frequency (Ï‰) equal to 1 Hz is shown in Fig. 4.
Figure 3. Frequency response of Butterworth filters .
Figure 4. Frequency response of Chebyshev I filter .
Figure 5 shows a flow chart of the process performed to obtain the objectives of this project. The algorithm starts by designing a Butterworth and Chebyshev 1 low pass digital filter. Then the user is asked to enter: the order, cut-off frequency (in Hz) and sampling frequency (in Hz) for the Butterworth and Chebyshev 1 low pass digital filters, and ripple factor (in dB) for the Chebyshev 1 low pass digital filter. The algorithm then plots the graphs for the magnitude and phase response for the Butterworth and Chebyshev 1 low pass digital filters and the responses are compared.
Figure 5. Flow diagram.
Results and Discussion
Figure 6 shows the user input screen in the command window of the MATLAB software platform.
Figure 6. User input screen.
The Butterworth filter response for magnitude and phase are shown in Fig. 7. As observed in the figure (Fig. 7(a)), as the order of the filter increases (N, 2N and 3N) the transition time from the pass band to the stop band decreases. The phase response was more non-linear as the order was increased from N to 3N, as shown in Fig. 7(b).
Figure 7. (a) Magnitude and (b) Phase response of Butterworth filter, for varying filter orders (N, 2N and 3N).
The Chebyshev I filter response for magnitude and phase are shown in Fig. 8. As observed in the figure (Fig. 8(a)), as the order of the filter increases (N, 2N and 3N) the transition time from the pass band to the stop band decreases. The phase response was more non-linear as the order was increased from N to 3N, as shown in Fig. 8(b).
Figure 8. (a) Magnitude and (b) Phase response of Chebyshev I filter, for varying filter orders (N, 2N and 3N).
The Chebyshev I filter response for magnitude and phase are shown in Fig. 9. As observed in the figure (Fig. 9(a)), as the ripple factor of the filter increases (R, 2R and 3R) the magnitude of the ripples in pass band increases. The phase response was more non-linear as the ripple factor was increased from R to 3R, as shown in Fig. 9(b).
Figure 9. (a) Magnitude and (b) Phase response of Chebyshev I filter, for varying ripple factors (R, 2R and 3R).
The Butterworth and Chebyshev I filter response for magnitude and phase are shown in Fig. 10. As observed in the figure (Fig. 10(a)), the pass band of the Butterworth filter is more maximally flat when compared to the Chebyshev I filter which has ripples in the pass band. However, the transition time of the Chebyshev I filter is lower when compared to the Butterworth filter. The phase response of the Butterworth filter is more linear when compared to the Chebyshev I filter, as shown in Fig. 10(b).
Figure 10. (a) Magnitude and (b) Phase response comparison of Butterworth and Chebyshev I filters.
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A Butterworth and Chebyshev I filter was designed and simulated in MATLAB and various parameters of the filters were analyzed and studied in this project work. It was observed that as the order of the filters was increased (N, 2N and 3N), the transition band decreased for the magnitude response of both the Butterworth and Chebyshev I filters. The phase response was found to be more non-linear as the order of the filters was increased (N, 2N and 3N). As the ripple factor of the Chebyshev I filter was increased (R, 2R and 3R), the magnitude of the ripples in the pass band increased and the phase response was more non-linear. The pass band of the Butterworth filter was more maximally flat and the transition time was higher when compared to the Chebyshev I filter which had ripples in the pass band. The phase response of the Butterworth filter was more linear when compared to the Chebyshev I filter.
N. Karaboga, "Digital IIR Filter Design Using Differential Evolution Algorithm", EURASIP Journal on Applied Signal Processing, vol. 8, pp. 1269-1276, 2005.
C.K.S. Pun, S.C. Chan, K.S. Yeung and K.L. Ho, "On the design and implementation of FIR and IIR digital filters with variable frequency characteristics", IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 49(11), pp. 689- 703, 2002.