4. Digital Filter
A system that performs mathematical operations on
a sampled, discrete-time signal to reduce or enhance
certain aspects of that signal
5. Noise Cancellation
Method for reducing unwanted sound by
addition of a second sound specifically
designed to cancel the first
6. Adaptive Filter
A system with a linear filter that has a transfer function
controlled by variable parameters and a means to
adjust those parameters according to an optimization
algorithm
7. Adaptive Noise Cancellation
Input x(n), a noise source N1(n), is compared with a desired
signal d(n), which consists of a signal s(n) corrupted by another
noise N0(n). The adaptive filter coefficients adapt to cause the
error signal to be a noiseless version of the signal s(n)
8. LMS Algorithm
An adaptive filtering algorithm used to mimic a desired filter
by finding the filter coefficients that relate to producing the
least mean square of the error signal
Consists of 2 basic processes,
1. Filtering Process
2. Adaptive Process
9. LMS: Gradient Approximation Method
2 Input Signals,
yk = sk + nk
Where sk is desired signal
& nk is noise
xk, is a measure of contaminating signal which
is someway correlated with nk
xk gives estimate of nk, consider it 𝑛 𝑘
Estimating sk, by subtracting the noise estimate,
𝑠 𝑘 = 𝑦 𝑘 − 𝑛 𝑘
10. LMS: Algorithm
𝑥 𝑘 =
𝑥 𝑘
𝑥 𝑘−1
⋮
⋮
𝑥 𝑘−(𝑁−1)
, vector correlated with
noise
𝑤 =
𝑤0
𝑤1
⋮
⋮
𝑤 𝑁−1
, set of adjustable weights with
N coefficients
• Approximations to
minimum mean
square error gives,
𝑠 𝑘 = 𝑒 𝑘 = 𝑦 𝑘 − 𝑛 𝑘
= 𝑦 𝑘 − 𝑤 𝑇 𝑥 𝑘
• Using methods of
steepest descent
method we get,
𝑤 𝑛+1 = 𝑤 𝑛 + 𝜇𝑒 𝑘 𝑥 𝑘
22. Limitations of this Project
Works only on real valued signals, not on complex
valued signals
Noise is taken as reference
23. Reference
“Adaptive Filtering: Fundamentals of Least Mean Squares with
MATLAB” by Alexander D. Poularikas
“Adaptive Filter Theory” by Symon Haykin
DSP System Toolbox
(https://www.mathworks.com/help/dsp/adaptive-filters.html)
