FIR Filter Design

The objective of this project is to use lead different earn methods to tendency a low-pass try that meets specifications given, and then analyse these three different methods through different parameters. In this project, seven interpenetrates should be designed using Matlab. And we compare them on worst case march on, largest tip off freight coefficient, maximum passband and stopband misplay, magnitude frequency response, impulse response, assembly fit and aughts/poles location. Finally, use these filters to do filtering, and then compare their responses to the predicted unrivaled.Discussion of Results> Part 1 Window order(a) aim fir1 solve to synthesize an FIR that meets specifications using a boxcar window. bruise gain = 1.8372 Largest faucet weight coefficient = 0.3694 maximal passband fallacy = 0.1678 supreme stopband wrongdoing = 0.0795(b) Use Hann window to synthesize an FIR that meets specifications. smite gain = 1.4154 Largest angle weight coefficie nt = 0.3496Maximal passband wrongdoing = 0.0052 Maximal stopband error = 0.2385** riddle 1 is the unwindowed design, and percolate 2 is the windowed design.From the analogy above, we sens cop that the unwindowed design has a more critical passband and stopband edge, but the windowed unity has a smaller maximal passband error as we expected. Also, the windowed maven has a large attenuation on stopband than the unwindowed one. The group discipline responses of both 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** tense up 1 is the unwindowed design, and Filter 2 is the kaiser design.From the comparison above, we can go steady that both designs have critical passband and stopband edges, but the kaiser one has a smaller maximal passband error as we expected. Also, the kaiser one has a smaller attenuation on stopband compare with the unwindowed one. The group contain 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 figure above, we can see that Hann design has a zero farthermost from whole circle, which is identical 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) utilize Matlabs firls function to meet the authoritative 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 passband and stopband edge, but the LMS one has a smaller maximal passband error as we expected. Also, the LMS one has a big 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 attenuation 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 stays turn upside the unit circle for 3(b) design, which is the minimum figure 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 linear0.16780.00520.07060.04030.05160.05380.0638Maximum passband error(dB)-15.5052-45.7568-23.0266-27.8855-25.7472 -25.3838-23.9007Maximum stopband linear0.07950.23850.08520.11370.07820.05380.0594Maximum stopband error(dB)-21.9886-12.4495-21.3913-18.8858-22.1339-25.3838-24.5274 conclave 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 balance 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 output is 1.8372, which equals to the worst gain prediction of this filter.> Part 5 Run-time computer 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) look at two 12-bit address space which has memory cycle time o f 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 experiment(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 chirp function makes a short, high-pitched sound, 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 core 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.SummaryThrough this project, the detailed processes of designing a filter by three different methods have been understood. And we know more close to all the parameters which would affect prope rties of the filters, and how to use different methods to design them and make outperform trade-off between each other.