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LECTURE 06: APPLICATIONS OF ADAPTIVE FILTERS. Objectives: Adaptive Noise Cancellation ANC W/O External Reference Adaptive Line Enhancement Least Squares Solution Convergence - PowerPoint PPT PresentationTRANSCRIPT
ECE 8443 – Pattern RecognitionECE 8423 – Adaptive Signal Processing
• Objectives:Adaptive Noise CancellationANC W/O External ReferenceAdaptive Line EnhancementLeast Squares SolutionConvergence
• Resources:RK: ANC TutorialMATLAB: ANC ToolboxCNX: Interference CancellationWIKI: Weiner FilteringRICE: Weiner Filtering
• URL: .../publications/courses/ece_8423/lectures/current/lecture_06.ppt• MP3: .../publications/courses/ece_8423/lectures/current/lecture_06.mp3
LECTURE 06: APPLICATIONS OF ADAPTIVE FILTERS
ECE 8423: Lecture 06, Slide 2
• ANC is concerned with the enhancement of noise corrupted signals.
• Requires no a priori knowledge of the signal or noise.
• Variation of optimal filtering which uses a secondary, or reference, signal.
• This reference measurement should contain no signal and noise that is correlated with the original noise.
• Relation to adaptive filtering paradigm fromprevious lecture: the desired signal is replaced by the primary signal (signal + noise), and the system input is replaced by the reference.
• The output is the error signal.
Adaptive Noise Cancellation
• One of the most important applications of adaptive filtering is adaptive noise cancellation (ANC).
• Originally proposed byB. Widrow in 1975.
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ECE 8423: Lecture 06, Slide 3
Reference Signal is Critical
• The signal is measured in a noisy ambient environment (e.g., a hands-free microphone in a car).
• The noise is measured from a different location, typically close to the noise source (e.g., under the driver’s seat in a car).
• The noise estimate need not be the exact noise signal added to the speech signal, but should be related through a linear filtering process. In practice, this is often not the case.
• The error signal can be written as:
• Squaring and takingthe expectation: )()()( 0 nvnsnd
)(ny
)(ne
+–
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}))()({(
)}({)}({
0
20
22
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)()()()( 0 nynvnsne
ECE 8423: Lecture 06, Slide 4
Filter Analysis
• It is assumed that s(n) is uncorrelated with v0(n) and v1(n).
• Hence, for a fixed filter, s(n) is also uncorrelated with .
• The expectation of the error reduces to:
• Recall we can also solve for thefilter using the z-transform:
• For our ANC system:
• The noises v0(n) and v1(n) are correlated in some way. We assume they can be modeled using a LINEAR filter (and this is a big assumption!):
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ECE 8423: Lecture 06, Slide 5
Estimating The Filter
• We note that:
• Further:
• Therefore, the adaptive filter modelsthe unknown transmission path, h(n):
• To verify this:
• In practice, there can be signal leakage betweend(n) and x(n). Assuming we can model this as a linear filtering:
• Effectively a signal to noise (SNR) ratio.
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ECE 8423: Lecture 06, Slide 6
Additional Comments on ANC
• We can characterize the performance of the system in terms of the signal to noiseratios of the system output and the system reference:
• That is, the SNR of the output (the error) is inversely proportional to the SNR of the reference (x(n) contaminated by noise).
• Hence, the more leakage, the worse the ANC result.
• This analysis assumes an idealized approximation of the adaptive filter as a fixed, infinite, two-sided Wiener filter. It neglects issues of stability, convergence, and steady-state error.
• Nevertheless, one-sided, or causal filters, do extremely well in this process. Many simplifications and optimizations of this approach have been implemented over the years to produce very effective technology (e.g., noise-cancelling headphones).
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1)(
jref
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ECE 8423: Lecture 06, Slide 7
Adaptive Line Enhancement
• In some applications, thesignal of interest is anextremely narrowbandsignal that can be modeledas a sum of sinewaves.
• To cancel noise in such cases,we invoke a special form of an adaptive filter known as adaptive line enhancement (ALE).
• We can write the output, y(n):
• The goal of the system is to maximize the SNR:
• The input signals may be written as:
• Recall the normal equation:
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i
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• Assume a solution of the form:
)cos()(* 0 iTBif
ECE 8423: Lecture 06, Slide 8
Solution of the Normal Equation
• For our application:
• Hence, our normal equation becomes:
• Combining the normal equation with our assumed solution:
• Noting that and
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ECE 8423: Lecture 06, Slide 9
Solution of the Normal Equation (Cont.)
• The solution to this equation:
is provided by:
• We can write this in terms of the SNR:
• We have essentially verified that the matched filter solution is optimal in a least squares sense (see the textbook).
• As we would expect from a least squares solution, the results are independent of the phase of the signal.
• All that is needed for the least squares solution is the auto and cross-correlation coefficients, which can be estimated from the data.
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ECE 8423: Lecture 06, Slide 10
SNR Gain
• The output of the filter can be computed via convolution:
• The output consists of two terms:
• The output noise power is:
• The output SNR is:
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ECE 8423: Lecture 06, Slide 11
Convergence of the ALE Filter
• The convergence properties of this filter are very similar to what we derived in the general LMS case:
• For the ALE system:
• However, estimating the noise power may be problematic, especially when the noise is nonstationary, so conservative values of the adaptation constant are often chosen in practice.
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ECE 8423: Lecture 06, Slide 12
• We introduced two historically significant adaptive filters: adaptive noise cancellation (ANC) and adaptive line enhancement (ALE).
• We derived the estimation equations and the SNR properties of these filters.
• We discussed what happens when less than ideal conditions are encountered.
Summary