Project 1: Filter Design and Filter Banks
Project 1: Filter Design and Filter Banks
Thursday, March 16th
Project: You should work individually and are required to complete Part A and Part B for full credit. Part C and D are extra credit. All work should be your own including and text, code, figures and simulation tests.
1. Part A: FIR FILTER DESIGN 2-WAYS In this problem you will design FIR filters using two different methods discussed in lecture.
(a) Design an FIR filter using Window
Ideally, a low-pass filter with cutoff frequency ωc should have a frequency response of ( 1, |ω| ≤ ωc Hideal (ejω ) = 0, ωc < ω ≤ π and a corresponding impulse response of ω nccfor − ∞ < n < ∞ hideal [n] = sincππ
However, no real filter can have this frequency response because hideal [n] is infinite in duration. One method for creating a realizable approximation to an ideal filter is to truncate this impulse response. A truncated impulse response is of finite duration, yet the filter is still noncausal. In order to make the FIR filter causal, it must be shifted to the right by M units.
To examine the effect of filter size on the frequency characteristics of the filter, write a Matlab function LPFtrunc(N) that computes the truncated and shifted impulse response of size N for a low pass filter with a cutoff frequency of ωc = 2.0. For each of the following filters, plot the magnitude of the filters DTFT in decibels.
Hints: The magnitude of the response in decibels is given by |HdB (ejω )| = 20log10 |H(ejω )|. Note that the log command in Matlab computes the natural logarithm. Therefore, use the log10 command to compute decibels. To get an accurate representation of the DTFT make sure that you compute at least 512 sample points using the command [X,w]=DTFT(filterResponse,512).
i. Design two filters with size N=21 and N=101. • Submit the plots of the magnitude response for the two filters (not in decibels). On each of the plots, mark the passband, the transition band and the stopband. • Submit the plots of the magnitude response in decibels for the two filters. • Explain how the filter size effects the passband and stopband ripple. Why does it have this effect? ii. Design two filters with size N=21 with an added window. Use the functions blackman and hamming to window your impulse response after truncation. • Submit the plots of the magnitude response for the two windowed filters (not in decibels). On each of the plots, mark the passband, the transition band and the stopband. • Submit the plots of the magnitude response in decibels for the two filters. • Compare your plots to the truncated filter from above. Explain how the window type effects the passband and stopband ripple and transition band. iii. Now download the noisy speech signal nspeech2.mat, and load it into the Matlab workspace. Apply the two filters with no additional windows with the N=21 and N=101 sizes to this signal. Since these are FIR filters, you can simply convolve them with the audio signal. Listen carefully to the unfiltered and filtered signals, and note the result. Can you hear a difference between the two filtered signals? In order to hear the filtered signals better, you may want to multiply each of them by 2 or 3 before using sound. (b) Design a symmetric low-pass FIR filter using firpmord and firpm in Matlab to meet the design specifications given below:
Compute the DTFT of the filters response for at least 512 points, and use this result to compute the passband and stopband ripple of the filter that was designed. Adjust the filter length until the minimum order which meets the design constraints is found. Plot the magnitude of the DTFT in dB of the final filter design.
Submit the final measured values of filter length, passband ripple, and stopband ripple. How accurate was the filter order computation using Matlabs firpmord? Submit the plot of the filters DTFT.
2. Part B: MULTI-CHANNEL FIR FILTER-BANK IN ONE DIMENSION You are to implement a perfect reconstruction filter-bank (PR-FB) using Matlab, and apply it to one-dimensional signals (audio). You can use Matlab routines such as firpr2chfb and mfilt.firdecim, etc., for this purpose.
(a) Design a 4-channel octave band scheme, in which the full-band is decomposed into 2 low frequency bands (1/8 of full-band each), an intermediate 1/4 of full-band, and the upper 1/2 high-frequency band. Use reasonable parameter values (filter length and band edges for the filters). Filter length should not be unreasonably long. (b) To test the filter-bank, use a simple test signal such as a number of sinusoids at different frequencies in each of the bands, or any other test signal of your choice. Submit plots to show your test signals to demonstrate the filter-bank operation.
Project Submission: Your report submission for this project will consist of two parts. • Project Report: You should submit by the due date a single typed file project report in .pdf format explaining what you did and the results you obtained, including figures, test cases, interpretations and comments, as well as responses to any specific questions asked above. Anything handwritten (either on paper or a tablet) will not be looked at.You do not need to include your code in your report unless you refer to it in your write-up. If you must include it do so in an Appendix. The report must be uploaded to Gradescope by midnight on the due date.
• MATLAB Code and Other Soft Files: Also submit (upload) by the due date in Gradescope all supporting material (all your Matlab code files, test input/output files, and any other results files). Ideally all placed in a compressed .zip file. Please make sure you follow the proper procedure for submitting files through Gradescope.
NOTE: You must submit the report and code in two different assignment submissions on Gradescope. The late penalty will be determined by the report submission time.
3. Part C: FILTER-BANK EQUALIZER Use the filter-bank from Part B on an audible waveform such as a segment of music or speech. (a) Use the filter-bank to implement a simple equalizer for the speech or music; by this we mean impart different gains on the 4 subband signals before reconstruction, to enhance certain frequency bands relative to others (e.g. bass boost, highfrequency boost, etc). Submit plots to show your test signals to demonstrate the equalizer performance. (b) Find out if introduction of small relative delays in the four subbands prior to reconstruction affects your perception of the audio (i.e. determine to what extent differential phase shifts or delays are important in perception of speech or music). Show any tests conducted, describe the audible results and any conclusions you find.
4. Part D: MULTI-CHANNEL FIR FILTER-BANK IN TWO DIMENSIONS Extending what you did in Part B, you are to implement a perfect reconstruction filterbank (PR-FB) using Matlab, and apply it to images. You can use Matlab routines such as firpr2chfb and mfilt.firdecim, etc., for this purpose. Image Processing and related toolbox functions in Matlab may be useful (e.g. imread, imshow, mat2gray, hist, imhist, etc.) (a) Design a one-dimensional 2-channel PR-FB. This can be applied for separable (rows and columns) sub-band analysis/synthesis of two-dimensional images. For image processing, strict spatial frequency separation may not be as critical as in other applications. Experiment with a small filter order (between 6 and 12) and a longer one. Use a reasonable choice for filter band-edge. Your design will yield the analysis and synthesis filters H0, H1, G0, and G1. (You might want to first test the filter bank on some simple one-dimensional test signal.) (b) You will now use a grayscale 8-bit, 512x512 image to test image sub-band decomposition. A couple of possible test images (man.gif, owl.gif) are posted on the course calendar; any other reasonable test image can be used. Apply your 2-channel analysis filter bank of part (a) to your test image(s), to get decomposition into (256x256) sub-band components (Low-Low or LL, Low-High, HighLow, and High-High). The image filtering can be implemented as separable rowcolumn filtering operations. Test for exact reconstruction when synthesizing the image back from these four sub-bands. (c) Perform reconstruction or synthesis as above, but this time with all pixels of the LH, HL, and HH sub-band images set to 0. Comment on the nature of the reconstruction you see. (d) Now instead of using single zero levels as in (c) for all pixels of the LH, HL, and HH sub-band images, experiment with less severe compression by using one bit (two levels) per pixel quantization for these three sub-band images, and also with three level (and, if you want, four level or 2-bit) per pixel quantization. To design reasonable quantization schemes (levels and thresholds) you may consider histograms of pixel values for the sub-bands. (e) Do a further decomposition of the Low-Low (LL) image obtained in part (b) by applying the same 2-channel analysis filter bank to it. (This corresponds for 1D signals to octave- band filtering into 3 sub-bands). Now apply considerations similar to those of parts (c) and (d) to see if reasonable reconstructions can be obtained with simple quantization. Note: There can be different delays or shifts for different sub-band octave components.
