I need 200 words on affects the performance of the adaptive filter.

Instructions

Do Adaptive notches filter using MATLAB 

It will be MATLAB based right, without using filter functions 

Please check attached file  

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Attachment 1;

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ECE 529, Spring 2019

Mini-Project #1: Adaptive Filtering

Due Thu. Nov. 14 at 5:00 p.m. MST

Do not use any existing adaptive filtering functions for this assignment. Instead, write your own

source code.

Part 1: Adaptive Notch Filter

Implement an adaptive notch filter to reduce additive, sinusoidal noise that is interfering with a

desired signal. Assume the frequency of the sinusoidal noise is unknown and slowly varying. A

second-order IIR notch filter (a.k.a. bandstop filter) has transfer function,

𝐻𝐻(𝑧𝑧) = �1 − 𝑒𝑒𝑗𝑗𝜔𝜔0𝑧𝑧−1��1 − 𝑒𝑒−𝑗𝑗𝜔𝜔0𝑧𝑧−1�

(1 − 𝑟𝑟𝑟𝑟𝑗𝑗𝜔𝜔0𝑧𝑧−1)(1 − 𝑟𝑟𝑟𝑟−𝑗𝑗𝜔𝜔0𝑧𝑧−1) = 1 + 𝑎𝑎𝑧𝑧−1 + 𝑧𝑧−2

1 + 𝑟𝑟𝑟𝑟𝑧𝑧−1 + 𝑟𝑟2𝑧𝑧−2

where 𝑎𝑎 = −2 cos 𝜔𝜔0 and 0 ≤ 𝑟𝑟 ≤ 1. The corresponding difference equation is

𝑦𝑦(𝑛𝑛) = 𝑥𝑥(𝑛𝑛) + 𝑎𝑎 𝑥𝑥(𝑛𝑛 − 1) + 𝑥𝑥(𝑛𝑛 − 2) − 𝑟𝑟 𝑎𝑎 𝑦𝑦(𝑛𝑛 − 1) − 𝑟𝑟2 𝑦𝑦(𝑛𝑛 − 2) (Eq. 1)

The notch is centered at digital frequency 𝜔𝜔0. The notch becomes sharper as 𝑟𝑟 approaches 1.

We can adapt the notch frequency automatically by solving for the value of 𝜔𝜔0 that minimizes

the power in the output signal. (If the frequency is selected correctly, then the sinusoidal noise

will be canceled, thereby minimizing the output power.) Set 𝑟𝑟 to a fixed value. If the noise

frequency varies over a narrow range, then 𝑟𝑟 should be set closer to 1 in order to generate a

sharper notch. If the frequency varies over a larger range, then 𝑟𝑟 should be set to a smaller

value.

Define the error signal to be 𝑒𝑒(𝑛𝑛) = 𝑦𝑦(𝑛𝑛) − 𝑠𝑠(𝑛𝑛). We want to adapt the filter in order to

minimize the power in the error signal. Assuming that the signal and noise are uncorrelated, we

can equivalently adapt the filter to minimize the total output power, 𝑦𝑦2(𝑛𝑛). The traditional

gradient descent method requires knowing the derivative of the output power, which we

calculate as 2𝑦𝑦(𝑛𝑛)

𝑑𝑑𝑑𝑑(𝑛𝑛)

𝑑𝑑𝑑𝑑 . And 𝑑𝑑𝑑𝑑(𝑛𝑛)

𝑑𝑑𝑑𝑑 can be obtained by differentiating Eq. 1, which gives us

𝑑𝑑𝑑𝑑(𝑛𝑛)

𝑑𝑑𝑑𝑑 ≈ 𝑥𝑥(𝑛𝑛 − 1) − 𝑟𝑟 𝑦𝑦(𝑛𝑛 − 1)

If we use this in a gradient descent iteration to update the value of 𝑎𝑎, then we have

𝑎𝑎(𝑛𝑛 + 1) = 𝑎𝑎(𝑛𝑛) − 𝜇𝜇 𝑦𝑦(𝑛𝑛)

𝑑𝑑𝑑𝑑(𝑛𝑛)

𝑑𝑑𝑑𝑑

𝑎𝑎(𝑛𝑛 + 1) = 𝑎𝑎(𝑛𝑛) − 𝜇𝜇 𝑦𝑦(𝑛𝑛) [𝑥𝑥(𝑛𝑛 − 1) − 𝑟𝑟 𝑦𝑦(𝑛𝑛 − 1)] (Eq. 4)

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where 𝜇𝜇 is a very small constant that serves as a convergence factor. Since 𝑎𝑎 = −2 cos 𝜔𝜔0, we

should impose a constraint that −2 ≤ 𝑎𝑎 ≤ 2. Initialize the value to be 𝑎𝑎 = −2 cos(𝜋𝜋⁄2) = 0. If

the update equation yields |𝑎𝑎| > 2, then we should reset 𝑎𝑎(𝑛𝑛) = 0.

(a) Implement a computer simulation of an adaptive notch filter, using Eq. 1 to perform the

filtering and using Eq. 4 to update the value of 𝑎𝑎 at each iteration. Add some sinusoidal

interference 𝑣𝑣(𝑛𝑛) to a desired signal 𝑠𝑠(𝑛𝑛) to obtain an input signal 𝑥𝑥(𝑛𝑛) to experiment

with. Here are some helpful MATLAB functions:

rng(1); % set the seed for the random number generator

v = randn(N,1); % generate N samples of pseudorandom white Gaussian noise

To convert white Gaussian noise to “colored” noise having a limited bandwidth, one

method is to send the white Gaussian noise through a lowpass filter, such as

𝐻𝐻(𝑧𝑧) = 1 + 𝑧𝑧−1

1 − 𝑎𝑎1𝑧𝑧−1

where −1 < 𝑎𝑎1 < 1. For example, 𝑎𝑎1 = 0.9 does lots of smoothing and 𝑎𝑎1 = −0.9 does

very little. Note: filtering the signal will affect the signal power.

Scale the signal components to achieve a reasonable signal-to-noise ratio (e.g., 0 dB to

30 dB):

SNR = 20 log10( var(𝑠𝑠(𝑛𝑛)) / var(𝑣𝑣(𝑛𝑛) )

However, for debugging purposes, you may want to try omitting 𝑠𝑠(𝑛𝑛) to see if the filter

can cancel the sinusoidal interference for the noise-only case. Try various fixed values of

𝜇𝜇 (e.g., 0.0005 to 0.05) and 𝑟𝑟 (e. g. , 0.85 to 0.98). Investigate whether the bandwidth of

𝑥𝑥(𝑛𝑛) (i.e., the amount of high-frequency components) affects the effectiveness of the

filtering. Plot “𝑎𝑎 versus 𝑛𝑛” and “𝑒𝑒2(𝑛𝑛) versus 𝑛𝑛”, where 𝑒𝑒(𝑛𝑛) = 𝑦𝑦(𝑛𝑛) − 𝑠𝑠(𝑛𝑛). Show how

the plots are affected if you change the value of 𝜇𝜇. Discuss what happens when you vary

𝑟𝑟, 𝜔𝜔0, and the bandwidth of 𝑥𝑥(𝑛𝑛). Plot the final 256 samples of 𝑠𝑠(𝑛𝑛), 𝑥𝑥(𝑛𝑛), and 𝑦𝑦(𝑛𝑛).

Also compare plots of the magnitude Fourier transform of the last 256 samples of 𝑥𝑥(𝑛𝑛)

and 𝑦𝑦(𝑛𝑛). For example, in MATLAB you can use Xmag = abs(fft(x)).

(b) Modify the sinusoidal noise so that its frequency is slowly varying rather than fixed.

Show how that affects the effectiveness of the adaptive filter. Note that the

instantaneous frequency of cos(2𝜋𝜋𝜋𝜋(𝑡𝑡)) is 𝑑𝑑𝑑𝑑⁄𝑑𝑑𝑑𝑑.

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Part 2: LMS Adaptive Filter 

Learn about the adaptive LMS filter by consulting various external references. For example, you

could start with Wikipedia: Least mean squares filter and consult its references for further

information, or just google to find numerous useful tutorials. Also learn about “adaptive noise

cancellation,” which is a well-known application of the LMS filter.

Implement a computer simulation of the LMS adaptive filter, and apply it to an adaptive noise

cancellation problem. Show experimental results and discuss how the adaptive filter performs.

Include plots showing how the performance varies as you vary the convergence factor, the filter

length, and any other parameters. Be sure to vary the bandwidth of the noise to see how that

affects the performance of the adaptive filter.