It is known that the main goal of adaptive filtering is to match the filter's output with the desired signal. In the literature, the desired signal is assumed to be available at the receiver side while in real case it is only known at the transmitter side. Practically, how can we estimate the desired signal ?
Dear Oussama Gharbi,
In general, adaptive filters are systems applied to noisy data to obtain useful information at a certain instant "t", these systems being implemented in three configurations.
Filtering, that is, extracting this information at time "t" from noisy data measured up to and including time "t". The smoothing which will also use the data after time "t". The prediction that only uses data up to time ( t-τ ) to derive the information we are interested in at time "t".
Wiener filtering is inadequate for situations where the signal or noise is non-stationary. In such situations the optimal filter must be variable over time. The solution to this problem is provided by Kalman filtering.
In this case We can estimate if the signal or noise is non-stationary using Kalman filter.
For more details and information about this subject I suggest you to see links and attached file on topic.
https://www.sciencedirect.com/topics/engineering/adaptive-filter-design
https://arxiv.org/ftp/arxiv/papers/1706/1706.00897.pdf
https://link.springer.com/chapter/10.1007%2F1-84628-121-0_1
Chapter Applications of Adaptive Filtering
Best regards
I think this is a career, not a question. Less flippantly, there are volumes written on this question.
Dear Oussama Gharbi
The adaptive filter is the basis for signal optimization, which is done by designing the adaptive filter with the help of optimization technology to produce an optimal adaptive filter for signal processing while ensuring optimum stability and best affinity ratio as well as the lowest square error rate.
In my research paper, I talked about the adaptive filter and the ratio of convergence and stability. You can see it and do not hesitate to contact us if you have any questions about the digital filter in general.
https://iopscience.iop.org/article/10.1088/1742-6596/1963/1/012103
Best regards
Dear Oussama Gharbi
The answer to the question depends on various scenarios. I will try to explain by giving a few examples.
There are four signals/vectors in an adaptive filter: input signal x(n), adaptive filter coefficients w(n), output signal y(n) and desired signal d(n).
Output y(n) is the dot product (inner product) of x(n) and w(n).
The filter coefficients are adapted using some optimization algorithm (least mean square (LMS) ) based on error signal e(n), which is given as,
e(n)=d(n)-y(n)
Example 1: System modeling
Suppose you have a system with an unknown impulse response. Let this be h(n).
Now you can supply input x(n) to this system, observe the output and take this output as the desired signal d(n).
Now you can estimate the unknown system response by designing an adaptive filter with input x(n) and d(n). The system response can be approximated as h(n)=w(n).
Example 2: Signal separation
How to get a fetus' electrocardiogram (ECG) signal with the mother's ECG signal interfering? Both signals are in the same frequency band, so applying a normal filter and separating the signals is difficult.
The abdominal lead provides interfered ECG signal from the fetus and mother. The chest lead provides the mother's ECG signal.
Now set the adaptive filter parameters. Let d(n) be the abdominal ECG signal, and x(n) be the chest ECG signal. The fetus' ECG signal for this case is d(n)-y(n).
There are many examples of adaptive filters such as noise cancellation, prediction, echo canceling, channel separation, etc.
I hope this helped.
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