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日期:2020-12-18 10:59

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.

2. 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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ESE531 Spring 2018

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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ESE531 Spring 2018

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]|

2 ? 1)y[n]xn. Thus the corresponding

stochastic gradient algorithm becomes gn+1 = gn ? μ(|y[n]|

2 ? 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.

3. 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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ESE531 Spring 2018

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.

4. 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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