Adaptive Filters – Processing Structures Gerhard Schmidt Christian-Albrechts-Universität zu Kiel Faculty of Engineering Electrical Engineering and Information Technology Digital Signal Processing and System Theory Slide 1
Contents of the Lecture Today: Introduction and Motivation Adaptive Filters Operating in Subbands Filter Design for Prototype Lowpass Filters Examples Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 2 Slide 2
Application Example – Echo Cancellation Problem and Objective Objective: Application example: Reduction of the computational complexity x ( n ) x ( n ) with nearly the same convergence properties by at least 75 %. If an additional delay is necessary, it should be lower than 30 ms. Ansatz: b h ( n ) d ( n ) x ( n ) Echo s ( n ) cancellation filter b d ( n ) b ( n ) h ( n ) b y ( n ) e ( n ) h ¹ ( n ) + s ( n ) d ( n ) b d ¹ ( n ) + + e ¹ ( n ) y ¹ ( n ) e ( n ) + b ( n ) y ( n ) Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 3 Slide 3
Application Example – Echo Cancellation Boundary Conditions Boundary conditions: The loudspeaker-enclosure-microphone system is modeled by an adaptive Filter with 4000 coefficients at a sampling rate of 8000 Hz. The filter should be adapted using the NLMS algorithm. Computational complexity: A convolution and an adaptation with 4000 elements has to be performed 8000 times per second. Assuming that a multiplication and an addition can be performed in one cycle on the target hardware, about million instructions per second (MIPS) are required. Pros and cons: + Rather simple and efficient algorithmic realization, only very little program memory is required. A very good temporal resolution (for control purposes) is achieved. + _ Very high computational complexity. _ Frequency selective control is not possible. Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 4 Slide 4
Application Example – Echo Cancellation Ansatz – Part 1 n n + 1 n + 2 y ( n ¡ 1) Fullband y ( n ) x ( n ) processing x ( n +1) y ( n + 1) x ( n + 2) Synthesis y ( n ¡ 1) Subband y ( n ¡ 2) processing Synthesis Analysis y ( n ) x ( n ) x ( n + 1) y ( n + 1) Analysis x ( n + 2) x ( n + 3) Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 5 Slide 5
Application Example – Echo Cancellation Ansatz – Part 2 Anti-aliasing Sub- Subband Up- Anti-imaging filters sampling processing sampling filters r 1 Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 6 Slide 6
Application Example – Echo Cancellation Computational Complexity Boundary conditions: Design of a subband system with the following parameters: subbands (with equal bandwidth) (same subsampling rate for all subbands) (average complexity ratio of complex and real operations) Computational complexity: MIPS (fullband) MIPS (subband) Reduction of about 83 % Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 7 Slide 7
Literature Hints Books Basic text: E. Hänsler / G. Schmidt: Acoustic Echo and Noise Control – Chapter 9 (Echo Cancellation), Wiley, 2004 E. Hänsler / G. Schmidt: Acoustic Echo and Noise Control – Appendix B (Filterbank Design), Wiley, 2004 Further details: P. P. Vaidyanathan: Multirate Systems and Filter Banks, Prentice-Hall, 1993 R. E. Crochiere, L. R. Rabiner: Multirate Digital Signal Processing, Prentice-Hall, 1983 H. G. Göckler, A. Groth: Multiratensysteme, Schlembach-Verlag, 2003/2004 (in German) Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 8 Slide 8
Basic Elements of Filterbanks and Multirate Systems Repetition … Known elements: Addition (signal + signal) Multiplication (signal * constant) Multiplication (signal * exponential series) Delay Further necessary elements: Subsampling Upsampling Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 9 Slide 9
Basics Up- and Downsampling – Part 1 (Derivation on the blackboard) Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 10 Slide 10
Basics Up- and Downsampling – Part 2 Downsampling (r = 2) Upsampling (r = 2) Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 11 Slide 11
Basics Up- and Downsampling – Part 3 Downsampling (r = 2) Upsampling (r = 2) Signal components above the max. allowed (Nyquist) frequency Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 12 Slide 12
Subband Systems How We Will Proceed … Problem (Reducing the complexity of adaptive filters) Efficient Subband Structures (by inserting a few restrictions) Design of so-called „prototype lowpass filters“ Verification (Complexity reduction and adaptation performance) Solution (Complexity reduction but also an additional delay) Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 13 Slide 13
Subband Systems Basic Structure Analysis filterbank Control of the subband adaptive filters Echo cancellation Loudspeaker- enclosure- microphone system + Residual echo Analysis Synthesis and noise suppression filterbank filterbank Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 14 Slide 14
Subband Systems Basic Structure of the Analysis Filterbank Lowpass, bandpass and Sub- highpass filters sampling Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 15 Slide 15
Subband Systems Restrictions That Lead to Efficient Implementations The same subsampling rate and the same prototype filter in all channels: Using the same subsampling rate for all channels/subbands: Realizing the bandpass and the highpass filters as frequency shifted version of a lowpass filter: Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 16 Slide 16
Subband Systems Structure of the Analysis Filterbank (with Restrictions) Lowpass, bandpass and Sub- highpass filters sampling Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 17 Slide 17
Subband Systems Analysis of the Filterbank Structure – Part 1 Signal after subsampling: Assuming a causal prototype lowpass filter: Signal after anti-aliasing filtering: Inserting results in: Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 18 Slide 18
Subband Systems Analysis of the Filterbank Structure – Part 2 Previous result: Splitting the summation index: Inserting results in: Exchanging the order of the sums Resolving the exponential term Simplification Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 19 Slide 19
Subband Systems Analysis of the Filterbank Structure – Part 3 Previous result: Specialties: … does not depend on ! … is a weighted inverse DFT and can be realized efficiently as an inverse FFT! Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 20 Slide 20
Subband Systems DFT-Modulated Polyphase Analysis Filterbank IDFT Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 21 Slide 21
Subband Systems Filterbank Versus DFT – Part 1 Input signal Windowing Window function Windowed frame Filterbanks allow for lengths of window functions larger than the DFT size. This can lead to improved frequency resolution. Transform Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 22 Slide 22
Subband Systems Filterbank Versus DFT – Part 2 Impulse responses Magnitude frequency responses Hann window Hann window Prototype lowpass filter Prototype lowpass filter Both filters were designed for a DFT order Coefficient index of 256. Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 23 Slide 23
Subband Systems Comparison of the Basic and the Efficient Structure Direct implementation: Filterlength of the (FIR-) lowpass filter We have to compute 2 r M N operations (real-complex) convolutions with coefficients = 49152 operations per input frame. Efficient implementation: Example for M=16, r=12, N=128 We have to compute 1 (real-real) convolution with coefficients and N + M log 2 M operations 1 IFFT of order = 192 operations per input frame. Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 24 Slide 24
Subband Systems Synthesis Filterbanks – Part 1 A comparable structure can be derived for the synthesis filterbank (details e.g. in E. Hänsler, G. Schmidt: Acoustic Echo and Noise Control, Wiley, 2004) Up- Lowpass, bandpass and sampling highpass filters Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 25 Slide 25
Subband Systems Synthesis Filterbanks – Part 2 IDFT Digital Signal Processing and System Theory| Adaptive Filters | Processing Structures Slide 26 Slide 26
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