ESE531信号处理

ESE531 Spring 2018
University of Pennsylvania
Department of Electrical and System Engineering
Digital Signal Processing
ESE531, Spring 2018 Final Project: Adaptive Filtering Friday, Mar. 30
Due: Tuesday, April 24th, 11:59pm
Project Teams: You can work in groups of no more than 2 for this project. You may work
alone. Each group must turn in one report with the clear contributions of each member
clearly delineated. Everyone is responsible for understanding the design and report in its
entirety, and the instructor reserves the right to interview the students to verify this.
You are to complete all three parts of the handout: Part A, Part B, and Part C. Part D may
be turned in for extra credit.

  1. Part A: ADAPTIVE NOTCH FILTER.
    DO NOT use high level Matlab commands that may be available in the Signal Processing
    and other Matlab toolboxes for adaptive filtering in this part. It is easy and much more
    instructive to write your own Matlab code to implement these.
    A simple real IIR notch filter is a second order filter with two conjugate zeros on the
    unit circle and two conjugate poles inside the unit circle,
    The notch is at real frequency ω0 and the closeness of the poles to the unit circle
    determines the notch sharpness. This filter can be used to reject a strong interfering
    sinusoid that is contaminating a desired signal. For unknown interfering frequency we
    want to build an adaptive notch filter, based on the principle that the filter output is
    minimized when the notch is at the correct location. Note that the adaptation is with
    respect to the parameter value a, with r generally set at a fixed value. Since this is
    not an FIR filter, we cannot directly apply the LMS algorithm derived for adaptive
    FIR filters. However it is possible to develop a simple algorithm for the adaptive notch
    filter.
    We can decompose H(z) as a cascade of the FIR part followed by the all-pole filter,
    and we can write the output sequence y[n] when input sequence is x[n] in terms of an
    intermediate sequence e[n]:
    ESE531 Spring 2018
    The gradient of y[n]
  2. with respect to a is 2y[n]
    dy[n]
    da where the derivative is not simple;
    thus, as an approximation we replace dy[n]
    da with de[n]
    da = x[n−1]. To minimize E(y[n]
    2
    ),
    we approximate its gradient as 2y[n]x[n − 1]. The adaptive notch filter update of the
    parameter a (for fixed choice of r) is therefore: a[n + 1] = a[n] − µy[n]x[n − 1]. Since
    a = −2cos(ω0) we have −2 ≤ a < 2 . We should impose this constraint for the updates
    and reset a[n] = 0 if it is out of bounds.
    (a) Create a simple simulation of an adaptive notch filter for a desired signal with
    unwanted additive sinusoidal interference. The desired signal may be some real
    sequence perhaps with a small amount of additive noise; you can use a singlefrequency
    desired signal as one possibility, but experiment with other types of
    desired signal also. The interference will be a single strong frequency (low signalto-interference
    power ratio). Start with a = 0 (ω0 =
    π
    2
    , mid-point of frequency
    band). Consider different small values of µ , different fixed values of r (of the
    order of 0.85 to 0.98), different interfering frequencies and powers. Note that
    µ will have to be quite small. Give plots and results on frequency response,
    convergence, spectra, etc. to show how well your adaptive filter works.
    (b) Consider one case where the single interfering sinusoid has a frequency that is
    changing slowly. For example, you might define the interference as a sinusoid
    with a slowly linearly increasing frequency or some other slowly changing profile.
    Examine the tracking ability of the adaptive notch filter and give your results and
    comments. (Note that the instantaneous frequency of a sinusoid cos(2π · φ(t)) is

    dt )
    (c) Can you extend your scheme of part (a) to a filter scheme creating two notches
    to reject two sinusoidal components? In particular, you might consider a cascade
    of two single-notch second-order notch filters, and adapt the a parameter of each
    (i.e. a = a1 and a = a2). Explain your approach and give your results for a test
    case of two interfering sinusoids on a desired signal.
  3. Part B: ADAPTIVE EQUALIZATION
    DO NOT use high level Matlab commands that may be available in the Signal Processing
    and other Matlab toolboxes for adaptive filtering in this part. It is easy and much more
    instructive to write your own Matlab code to implement these.
    Here you will use adaptive filtering to equalize or invert an unknown channel, with (1)
    the help of a training sequence and also (2) blindly, without training.
    Training Mode Equalization:
    Consider a source sending continuous-time pulses of amplitude A or −A to represent
    bits 1 and 0. The sequence of such pulses passing through a channel can undergo
    distortion causing them to spread out and overlap with each other, creating what
    is called inter-symbol interference or ISI. In addition, any front-end filtering at the
    receiver may cause further pulse spreading.
    The channel may be modeled in discrete-time as follows: an input sequence s[n] (random
    sequence) of ±A amplitudes passes through a discrete-time LTI system (channel)
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    with unit-sample response sequence h[n] to produce the observed sequence of channel
    output samples x[n] in the presence of noise. Thus we have x[n] = h[n] ∗ s[n] + w[n]
    where w[n] is a sequence of zero-mean, independent, Gaussian variates representing
    additive noise. The channel may be assumed to be an FIR channel of order L (length
    L + 1), and its unit-sample response h[n] is unknown. At the channel output we use
    an FIR adaptive equalizer filter of larger length M + 1 operating on x[n] to try to
    invert the channel, to ideally get at its output a delayed version of the original input
    sequence s[n], after the adaptive filter has converged. In order to implement the LMS
    training algorithm for the equalizer, we need a copy of the actual input sequence for
    use as the desired (delayed) output signal during an initial training phase. After the
    training phase, the equalizer should have converged to a good approximation of an
    inverse filter.
    (a) Produce a random ±1 amplitude training sequence of length 1000. Assume
    some channel unit-sample response h[n]; you may use for example something like
    h = [0.3, 1, 0.7, 0.3, 0.2] for your initial trials, but you should also test your implementation
    for different unknown channels of such short lengths ( ≤ 7). At
    the output of the channel (after convolution of input sequence with h[n]) add
    Gaussian noise to simulate a more realistic noisy output condition. You may use
    an SNR of around 20-35 dB. Now implement an adaptive filter operating on the
    noisy channel output x[n], using a training sequence which is an appropriately
    delayed version of the input sequence. The delay will take care of any unknown
    channel delay (for example, for the channel given above the channel delay may
    be 2 units,) and equalizer filter delay. Examine the performance you get with different
    combinations of equalizer filter order M (choose this between 10 and 30),
    filter step-size µ , adaptive filter initialization, SNR, channel impulse response
    h[n], etc.
    Examine the impulse and frequency response and pole-zero plot of the channel,
    and the impulse and frequency responses of the equalizer upon convergence, and
    provide plots and results that show how well your adaptive equalizer works under
    different channel and noise conditions. Plot also the impulse response and frequency
    response of the equivalent LTI system between input and output, i.e. the
    result of the channel and equalizer in cascade, after the equalizer has reached reasonable
    convergence. Determine if output decisions about the transmitted bits are
    better after equalization, compared to the un-equalized case. (You can check the
    output of the system using the equalizer obtained after convergence, by sending
    several thousand further inputs into the channel.) Comment on your findings.
    Blind Equalization:
    It is not always possible to have an initial training phase, during which an equalizer
    at the receiver knows the input, to form an error to drive its LMS training algorithm.
    The transmitter may be continuously sending data through a channel, and it may be
    left to the receiver to figure out for itself how to equalize the channel, without the
    benefit of a known training input sequence.
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    In the context of the simple scenario of part 3(a), the situation now is that the receiver
    knows only that the transmitted pulse amplitudes are two-level ±1 values (more
    generally ±A amplitudes). It has to use only this general knowledge about the nature
    of the input to learn how to equalize the channel. It has no knowledge of an explicit
    transmitted sequence that it can use for training, and we say it is operating in the
    blind mode.
    A simple approach to blind equalization in this setting is based on the use of the
    constant- modulus property of the input; the modulus or absolute value of the input
    amplitude sequence is a constant (= 1 or more generally A). The constant-modulus
    (CM) blind equalization algorithm attempts to equalize the channel by iterative adjustments
    to the equalizer with the objective of minimizing a measure of deviation of
    the equalizer output modulus values from a constant modulus value.
    Referring to the LMS adaptive algorithm description, we now have input sequence
    x[n] to an equalizer that produces output y[n].
    coefficient vector at time n. Differentiating this with respect to gn we easily find
    that the gradient de2
    n
    dgn
    is proportional to (|y[n]|
  4. − 1)y[n]xn. Thus the corresponding
    stochastic gradient algorithm becomes gn+1 = gn − µ(|y[n]|
  5. − 1)y[n]xn.
    (b) Implement this blind adaptive equalizer. Does the blind equalizer converge to a
    reasonably equalized condition? You will have to try different settings for µ and
    equalizer order M. (You will need possibly many tens of thousands of iterations,
    and will need to experiment with rather small value of µ , perhaps of the order
    of 10−4 depending on the specifics of your other parameters.) Use short channel
    impulse response lengths (around 5) and dont use equalizer lengths that are too
    long (around 20 maximum). Try different SNRs, but expect poor results if the
    SNR is not high.
    Provide plots and results, and give explanations/comments as in the case of part
    (a) above. Also provide a one-dimensional scatter plot of the output amplitudes
    after equalizer convergence, to get a visual sense of how well the equalizer is able
    to bring the output amplitudes close to the desired two amplitudes.
  6. Part C: FILTER DESIGN USING LMS ALGORITHM
    DO NOT use high level Matlab commands that may be available in the Signal Processing
    and other Matlab toolboxes for adaptive filtering in this part. It is easy and much more
    instructive to write your own Matlab code to implement these.
    Suppose a desired frequency response Hd(e
    jω) is specified (with even magnitude and
    odd phase function). You want to design a real, causal, FIR impulse response {h[0], h[1], ..., h[M]}
    for an M-th order FIR filter which gives a good approximation to this Hd(e
    jω). The
    desired frequency response may not be of a standard type such as a low-pass or bandpass
    specification. For this we may form a long, real, test input signal sequence
    {xin[n], n = 0, 1, ..., K} consisting of a large number L of individual frequencies spanning
    the range (0, 0.5) . The corresponding ideal desired output {xd[n], n = 0, 1, ..., K}
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    is obtained using the magnitude scaling and phase shift specified by Hd(e
    jω) for each
    such input frequency. The impulse response coefficients of an FIR filter of order M
    may then be adaptively learned using the LMS algorithm to get approximately this
    desired output sequence xd[n] when driven by the test input sequence xin[n] .
    (a) Let the desired magnitude response be:
    and the FIR filter of order M is to have linear phase.
    Form reasonable test input and corresponding desired output sequences, containing
    an appropriate number L of individual frequencies spread over the (0,0.5)
    interval. Use the LMS adaptive FIR filter scheme to find a good design for a
    causal FIR filter of order M that will produce a good approximation of the desired
    output. Note that you will have to experiment with different combinations of
    step-size (this has to be generally small), filter length M, number of test frequencies
    L, and length of the test sequences N (this will generally be large). Give plots
    of your designed filter(s) characteristics (impulse response, frequency response).
    Discuss briefly/comment on any specific aspects of your method or results that
    you want to highlight.
    (b) Now suppose the desired response is
    Hd(e
    j2πf ) = (
    j2πf, for 0 ≤ |f| ≤ 0.3
    0, for 0.3 ≤ |f| ≤ 0.5
    any additional linear phase term is allowed. The real FIR filter of order M approximating
    this is to have generalized linear phase. Repeat your procedure of
    part (a) for this case. After you have obtained your FIR filter, test it on some
    interesting inputs.
  7. Part D: MULTI-CHANNEL FIR FILTER-BANK.
    You are to implement a perfect reconstruction filter-bank (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.)
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    ESE531 Spring 2018
    (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 1-
    D 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.
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    ESE531 Spring 2018
    Project Submission: Your report submission for this project will consist of two parts.
    • Project Report:
    You should submit by the due date a single filed report (preferably pdf) 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. Please explain
    briefly your Matlab code; include a copy of all your Matlab code in your report in an
    Appendix. The report must be uploaded to Canvas by midnight on the due date.
    • MATLAB Code and Other Soft Files: Also submit (upload) by the due date through
    the Assignments Area on the ESE 531 Canvas Site 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 Canvas.

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