11. Adaptive Filters Flashcards
In which respect do adaptive filters differ from classical filters?
The classical filters discussed so far in the course all have fixed, i.e. constant coefficients.
Adaptive filters on the other hand, autonomously adjust their coefficients to the surrounding environment offering (nearly) optimal performance at each moment in time.
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Adaptive filters are said to be self-designing, self-learning and robust to changing environments. What does this mean?
Typical for an adaptive filter is that it is self-designing and self-learning. This means that after initialization (usually accomplished by setting all the filter coefficients to zero) and starting from complete ignorance about the environment, the filter is capable of autonomously finding the coefficients that make it perform optimally for the environment it is in and for the signals it sees at the input.
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Most adaptive filters are of the FIR type. Why?
Unlike with standard digital filters, the coefficients of an adaptive filter vary over time. They are usually updated every sample instance k, i.e. every time a new sample arrives at the filter input. As a result, the poles/zeros of the filter change position over time. This may lead to undesired behaviour once poles move outside the unit circle and the filter becomes unstable. This is an issue with adaptive IIR filters only as adaptive FIR filters are stable by construction. This explains why most practical adaptive filters are of the FIR type.
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What are the in-/output(s) of an adaptive filter? Visualize.
An adaptive filter has two inputs, the filter input x[k] and the desired input signal d[k], and one main output, the error signal e[k]. Have a look at the scheme on slide 3 of LectureAdaptive.pdf.
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What happens at every sample instance?
Every sample instance k the adaptive filter performs two operations: a filtering operation to calculate the error signal e[k] and a filter coefficient update.
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What is the aim of the adaptive filter and how does it achieve its goal?
The adaptive filter tries to make the error signal |e[k]| as small as possible by minimizing the expected energy of e[k]. By taking advantage of the correlation properties of (and also between) the input signals x[k] and d[k] the optimal filter weights wopt[n] can be found in an iterative way.
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Make sure you understand the filter coefficient update equation that can be found on slide 6 of LectureAdaptive.pdf and in the formulary. Interpret. Recall that the formula defines the LMS adaptive filter.
What does the acronym LMS stand for?
Least Mean Square
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What is the stepsize parameter µ used for?
How do you select an appropriate value for µ?
What happens is µ is (too) large?
What happens if µ is close to zero?
The formulas on slide 7 of LectureAdaptive.pdf can be found in the formulary.
Recall that the stepsize in the LMS weight update formula controls the speed of adaptation : the larger µ, the faster wk[n] goes to wopt[n], and hence, |e[k]| goes towards 0. In the most general case, stepsize µ is time dependent.
Parameter µ needs to be selected with care, though, and has to be a positive real number. If µ ≥ 0 is small the filter converges slowly and may not be able to track changes in the environment. If
µ = 0, the filter coefficients are said to be frozen, i.e. wk+1[n] = wk[n] for all k, n. A large stepsize on the other hand, forces the adaptive filter to change its coefficients rapidly. This may cause overcorrection and make the filter diverge.
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How does the normalized LMS adaptive filter relate to the LMS adaptive filter?
The update equations of NLMS can be found in the formulary. Why is µ0 divided by Sum m x^2[k − m]?
The normalized LMS algorithm (NLMS) is an extension to the LMS adaptive filter. It has a stepsize that is inversely proportional to the energy in the input signal buffer by construction. Hence,
µ does not need to be continuously updated according to the energy of the current input signal buffer.
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Have a look at the plots on slide 9 of LectureAdaptive.pdf and explain what happens. Also play around with DemoNLMS.exe.
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How many arithmetic operations are required (per second) by the (N)LMS adaptive filter?
How much memory is needed for a real-time implementation?
How large is the processing delay?
- N+1 multiplications and N+1 additions to compute e[k]
- 1 multiplication to compute µ·e[k]
- (N+1) times 1 multiplication and 1 addition for the weight updates
Hence, if the sampling rate is fs, the total number of arithmetic operations per second amounts to (4N+5)·fs.
For a real-time implementation in total N+1 storage registers are needed for the filter weights, N cells for past inputs x[k–n], and about 4 cells for e[k], d[k], x[k], µ, … .
The block processing delay of the (N)LMS adaptive filter is typically one sample.
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Make sure you understand how the scheme on slide 11 of LectureAdaptive.pdf works. Give an example of a practical application where this scheme can be used. Show how d, x and e are connected.
What kind of input signal x[k] would you apply to the scheme?
Keep in mind that the unknown system can be exactly modelled only if the unknown system is FIR and if the order of the adaptive filter is higher than or equal to the order of the unknown system.
Practical applications are :
• transmission channel identification in mobile telephony with the aid of a training sequence
• adaptive measurement of the acoustic impulse response of a room
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How does interference cancellation or noise suppression work?
Have a look at slide 13 of LectureAdaptive.pdf and at demo beam.html. Give an example of a practical application where the scheme of slide 13 can be used. Show how d, x and e are connected. You are not supposed to reproduce the information you find on demo beam.html at the exam.
Typical applications are :
• line and acoustic echo cancellation in (hands-free) communications
• multi-microphone noise reduction (teleconferencing, hearing aids, …)
• the removal of 50 Hz (hum)
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What is acoustic echo cancellation? Explain how the adaptive filter can be used to enhance the recorded signal. Have a look at slide 14 of LectureAdaptive.pdf.
Disadvantage of a standard hands-free setup is that the radio signal x is recorded by the microphone and is sent over the telephone line. To avoid this, an adaptive filter is applied that models the acoustics in the car cabin. In this way, the radio signal is removed from the microphone recording whilst the desired speech signal s is preserved.
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How does the equalization of a data transmission channel with a training sequence work? Have a look at slide 16 of LectureAdaptive.pdf.
Solution : every now and then, during training, a known data sequence strain[k] = s[k] = d[k] is transmitted so that the adaptive filtering weights can be updated to compensate for the spectral distortion caused by hk[n]. At other times, useful (e.g. GSM) data s[k] is transmitted. In that case, the filter coefficients are frozen (stepsize µ = 0).
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