LPF Design Using Rectangular Window - Essay Example

Preliminary Discussion Preliminary Discussion The window function together with the filter length, M determines the width of the main lobe. Blackman function and the hamming have bigger width compared to the width of the rectangular window’s main lobe. The width of the main lobe influences transition width. An increase in M leads to a corresponding decrease in width, thus increasing steepness. For every window function, the side lobe attenuation function varies, but remains unchanged with changes in M. The anticipated main lobe widths, attenuation and MATLAB outcomes are indicated in Table 1. The width of the main lobe was ascertained through determination of the magnitude function’s (dB) x value and doubling it. Table 1 indicates that expected values are the same with program’s values. Experiment 1: LPF Design using Rectangular Window Rectangular window function was utilized to design low-pass filters in MATLAB with window length M=2, 45, 65 and ω=0.2 as cut-off frequency. The graphs for impulse response, pole-zero, and magnitude function (linear as well as dB) were drawn. Discussion In the first experiment, three LPFs of varying lengths, length M = 25, 45, 65 as well as a 0.2π cut-off frequency were designed. Many observations were made during filter design process. The rectangular function’s main lobe width is 4π/M. The main lobe’s width narrows as M increases, affecting transition width and increasing its gradient. Increasing M reduces a large transition width, which is an unwanted effect. The area below the side lobes remain unchanged with signals retaining the ripples. High side lobe as well as stopband attenuation of -13 db and -21 dB respectively makes the rectangular function undesirable to use. Using a different window function with lower side lobes as well as stop band attenuations is the only way to overcome. Poles-zeros plot significantly affect response frequency when ascertaining the poles/zeros position in the unit circle, thus remain very crucial for the experiment. Unit circle zeros make response to move towards zero, which are components of stopband range. Zeros that are components of passband range produce ripples during passband frequency, thus impact the signal, which consequently affects filter accuracy. Zeros ranging 0-π are crucial because those that are outside do not have impact on the filter response. The poles at zero only have impact on the height of the passband. Experiment 2: LPF Design using various window functions All window functions, including rectangular, hamming and Blackman, were used in MATLAB to design low-pass filters where length, M was 25 and cut-off frequency was ω=0.2. The graphs for impulse response, pole-zero, and magnitude function (linear and dB) were plotted for this experiment as indicated in the figure below: Discussion Experiment 2 entailed using all three window functions to make a LPF, applying similar resources as in experiment 1. Like in the first experiment, as M increases, the rectangular lobe width decreases. The major drawback of the function is that the area beneath remains unchanged and contains ripples despite the fact that the main lobe’s width decreases with increasing value of M. Window functions like Hamming and Blackman have low-level side lobe as well as side band attenuations, thus are very crucial alternatives. However, Hamming and Blackman window functions are characterized with large transition widths of 8π/M and 12π/M respectively. Using M=25, widths of the main lobe correspond to 1.0005 and 1.5000, which makes it large compared to the rectangular window, consequently increasing the filter’s transition area. This problem can be overcome by increasing size of M, which results into decrease in the main lobe length. An increase in M will lead to a decrease in the Hamming’s and Blackman’s window functions transition region. Experiment 3: LPF Design using Parks-McClellan Method In experiment 3, the concept of Parks-McClellan method was adopted in designing a LPF. In order to execute this method in MATLAB, the firpm command utilized in designing the filter where a length, M =25 was used and a pass-band as well as stop band edge frequencies, ωp = 0.18π and ωs = 0.22π were used. Discussion For the reason of lack of control over factors like pass band as well as stop band frequencies, windowing approach is not the best strategy for designing FIR filters. As such, Parks-McClellan method (PMC) was utilized for designing filters in this experiment. PMC utilizes Remez and Chebyshev approximation, thus emerges as the ideal approach to designing filters. The method keeps low maximum error between filter’s ideal and actual frequency response. Control over error approximation in PMC in various frequency bands makes it superior over windowing approach in designing FIR filters [1]. PMC method will include ripples of uniform weights over pass band as well as the stop band areas characterized with steeper transition width [1]. Applying PMC in MATLAB demanded the use of firpm command: b=firpm (n, f, a) f= [0, 0.18, 0.2, 1] a= [1, 1, 0, 0] The function works through value selection for n, f and a. The function responds with a coefficient b with n. For this experiment, n was represented by M-1 with M=25, n=24. Normalized frequency array ranging between 0-1 was represented by f. The parameters were assembled in an increasing manner such that f= [0, 0.18, 0.2, 1]. Array of expected amplitude was represented by a. Figure 15 indicates that uniformly weighed ripples make up the magnitude function. Therefore, filters designed under this method are characterized by equiripple behaviour [3] and thus also referred to as equiripple filters. Experiment 4: Band Pass filter Design using Parks-McClellan Method Same as experiment 3, the method used to device a band pass filter of length M = 25 was the Parks-McClellan. The pass band frequency of ωp1 = 0.2π, ωs1 = 0.3π, ωp2 = 0.6π and ωs2 = 0.7π was also used. The graphs of magnitude function (linear and dB), impulse response, and pole-zero were drawn as shown below: Discussion In the case of band pass filter, plotting pole zero resulted in a linear phase for the filter because the zeros yielded ripples in the pass band region. The zeros placed on the unit circle are usually stronger compared with those located to the outside of the circle. Reducing the spacing between zeros would result into steeper linear phase. Experiment 5: LPF Design using windowing technique on MATLAB and applying it on C6713 system Hamming window was utilized in this experiment to design a filter in MATLAB and a passband frequency ranging from 0-2000Hz as well as a 8000Hz sampling frequency and length M= 25 were used. The filter was applied on C6713 system and graphs of magnitude function (linear and dB), impulse response, and pole-zero were drawn as shown below: Discussion The output voltage decreased and after reaching 2000Hz mark remained constant as indicated in figure 17. The magnitude function showed that theory and practical indication of filter decay ant high frequencies are the same. The magnitude variance in function dB scale can be attributed to inaccurate measure of voltage by the oscilloscope. Read More