FIR Filter Design

The objective of this project is to use terzetto divers(prenominal) rule methods to goal a low-pass interpenetrate that brings specifications given, and then compargon these three different methods through different parameters. In this project, seven dawns should be designed employ Matlab. And we comp atomic number 18 them on chastise case don, largest hit weight coefficient, supreme passband and stopband misunderstanding, magnitude frequency response, impulse response, root word remain and nobodys/poles location. Fin completelyy, use these filters to do filtering, and then compare their responses to the predicted champion.Discussion of Results Part 1 Window mode(a) Use fir1 procedure to synthesize an FIR that meets specifications using a boxcar window. lash gain = 1.8372 Largest dab weight coefficient = 0.3694 supreme passband misapprehension = 0.1678 level best stopband error = 0.0795(b) Use Hann window to synthesize an FIR that meets specifications.Worst ga in = 1.4154 Largest tip off weight coefficient = 0.3496Maximal passband error = 0.0052 Maximal stopband error = 0.2385** interpenetrate 1 is the unwindowed design, and separate out 2 is the windowed design.From the comparison above, we merchant ship touch that the unwindowed design has a more tiny passband and stopband edge, but the windowed one has a small maximum passband error as we expected. Also, the windowed one has a larger fading on stopband than the unwindowed one. The host delay responses of devil designs are the same.(c) Use Kaiser window to synthesize an FIR that meets specificationsWorst gain = 1.6900 Largest tap weight coefficient = 0.3500 N = 21 (which is in 20 in matlab)Maximal passband error = 0.0706 Maximal stopband error = 0.0852** get through 1 is the unwindowed design, and Filter 2 is the kaiser design.From the comparison above, we can see that both designs have hypercritical passband and stopband edges, but the kaiser one has a littler maximal passba nd error as we expected. Also, the kaiser one has a smaller fading on stopband compare with the unwindowed one. The group delay responses of two designs are different, the Kaiser one only has 20th order, so the group delay is 10, not 11 as the unwindowed one.(d)The zeros of the three windowed designs** Filter 1 is the boxcar design, and Filter 2 is the Hann design, Filter 3 is the Kaiser design.From course above, we can see that Hann design has a zero far from whole circle, which is like to the slower attenuation compared to the other two designs. The zeros of boxcar design are similar to the Kaiser design. Part 2 LMS Method(a) victimisation Matlabs firls function to meet the veritable design specification.Worst gain = 1.5990 Largest tap weight coefficient = 0.3477Maximal passband error = 0.0403 Maximal stopband error = 0.1137** Filter 1 is the 2(a) design, and Filter 2 is the boxcar design.From the comparison above, we can see that the boxcar design has a more critical passba nd and stopband edge, but the LMS one has a smaller maximal passband error as we expected. Also, the LMS one has a larger attenuation on stopband than the boxcar one. The group delay responses of two designs are the same.(b) Using Matlabs fircls1 function to meet the original design specification.Worst gain = 1.6771 Largest tap weight coefficient = 0.3464Maximal passband error = 0.0516 Maximal stopband error = 0.0782** Filter 1 is the 2(a) design, and Filter 2 is the 2(b) design.From the comparison above, we can see that the 2(b) design has a more critical passband and stopband edge, but the 2(a) one has a smaller maximal passband error. Also, the 2(a) one has a larger attenuation on stopband than the 2(b) one. The group delay responses of two designs are the same.(c)The zeros of the two LMS designs** Filter 1 is the 2(a) design, and Filter 2 is the 2(b) design.From figure above, we can see that 2(b) design has a zero far from unit circle, which is corresponding to the slower attenu ation compared to the other design. The zeros around the unit circle are similar to each other. Part 3 Equiripple Method(a) Using Matlabs firgr function to meet the original design specification (uniform error weight)Worst gain = 1.6646 Largest tap weight coefficient = 0.3500Maximal passband error = 0.0538 Maximal stopband error = 0.0538** Filter 1 is the 3(a) design, and Filter 2 is the boxcar design.From the comparison above, we can see that the boxcar design has a more critical passband and stopband edge, but the 3(a) one has a smaller maximal passband error. Also, the boxcar one has a larger attenuation on stopband than the 3(a) one. The group delay responses of two designs are the same.(b) Using Matlabs firpm function to meet the original design specificationWorst gain = 1.6639 Largest tap weight coefficient = 0.3476Maximal passband error = 0.0638 Maximal stopband error = 0.0594** Filter 1 is the 3(a) design, and Filter 2 is the 3(b) design.From the comparison above, we can see that the 3(b) design has a more critical passband and stopband edge. And the stopband error is 0.0488 (which is consistent with 0.0538*(1-20%)=0.04304), the passband error is 0.0639 (which is consistent with 0.0538/(1-20%)=0.06725). The group delay responses of two designs are the same.(c) The zeros of the two equiripple designs** Filter 1 is the 3(a) design, and Filter 2 is the 3(b) design.From figure above, we can see that 3(a) design has a zero far from unit circle, which is corresponding to the slower attenuation compared to the other design (almost no attenuation on the figure shown ). There is only one zero stay outside the unit circle for 3(b) design, which is the minimum phase design. Part 4 Testing(a)Table the features for the 7 designed FIRsFeaturesFilter 1Filter 2Filter 3Filter 4Filter 5Filter 6Filter 7Maximum gain1.83721.41541.69001.59901.67711.66461.6639Maximum passband one-dimensional0.16780.00520.07060.04030.05160.05380.0638Maximum passband error(dB)-15.5052-45.756 8-23.0266-27.8855-25.7472-25.3838-23.9007Maximum stopband linear0.07950.23850.08520.11370.07820.05380.0594Maximum stopband error(dB)-21.9886-12.4495-21.3913-18.8858-22.1339-25.3838-24.5274Group delay11111011111111Largest tap weight coefficient0.36940.34960.35000.34770.34640.35000.3476(b) From the figure followed, we can figure out that the group delay is 22-11=11 samples regardless of the input frequency.(c) Compare the original, mirror, and complement FIRs impulse, magnitude frequency, and group delay response**Filter 1 is the original filter, Filter 2 is the mirror filter, and Filter 3 is the complement filter.(d) Maximal product is 1.8372, which equals to the worst gain prediction of this filter. Part 5 Run-time Architecture(a) N = 8, M=1 N = 12, M=1 N = 16, M=1Round off errorN=8 N=12N=16From the comparison above, we can see clearly that as the value of N increases, the round-off error decreases.Bits of precision is N-1-1=N-2(b) Choose two 12-bit deal out space which has memory cycle time of 12 ns, so the maximum run-time filter speed is 1/ (12ns/cycle*16 bits) =1/ (192 ns/filter cycle) =5.21*106 filter cycles/sec Part 6 Experimentation(a) The maximal of the output time-series is 1.1341. It is reasonable, because it is smaller than the worst case gain which is 1.8372. So this agrees with the predicted filter response.(b) The tweedle function makes a short, high-pitched respectable, and it sounds four times, which is corresponding to the 4*fs. When all the .wav files are played, we can hear obviously that the frequency of output sound is much lower than the frequency of input sound, which means that the filter did filter high-frequency components out.From the figure above, we can see the high-frequency components are gone, which agrees with the predicted filter response, a low-pass filter. thickThrough this project, the detailed processes of designing a filter by three different methods have been understood. And we know more about all the parameters whic h would pretend properties of the filters, and how to use different methods to design them and make best trade-off in the midst of each other.