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Unformatted text preview: MODULE 15 NONLINEAR DIGITAL FILTERS • Noise Smoothing • Median Filter • Order Statistic Filters • The Teager Operator Page 15.1 INDEX Nonlinear Digital Filters Limitation of Linear Filtering Impulse Noise The Median Filter Order Statistics Endpoint Filtering Global Properties of Median Filter Local Properties of Median Filter Order Statistic Filters Example OS Filters Noise Smoothing OS Filters OS Filter Design Optimal (MMSE) OS Filter Derivation of Optimal OS Filter Examples of Optimal OS Filters Comparison with Linear Filtering Robust OS Filters Robust Inner Mean Filters Teager Operator The ContinuousTime Teager Operator ContinuousTime Energy Separation Algorithm The DiscreteTime Teager Operator DiscreteTime Energy Separation Algorithm MAIN INDEX Page 15.2 15. NONLINEAR DIGITAL FILTERS index Noise Smoothing • One of the principal reasons for filtering a signal is to remove noise due to: transmission errors bit errors quantization electrical (thermal) noise • The many linear filtering techniques we have studied are based upon frequency selection the goal is to either: filter away the noise spectrum Φ ε ( e j ϖ ) select the signal spectrum X ( e j ϖ ) • Of course, these are dual tasks. The Wiener Filter approach attempts a balance between these tasks. • Often the balance is very poor  when the signal and noise processes have significant spectral overlap . Page 15.3 Signal vs. Noise index • A simple signal x ( n ) and it's spectrum  X ( e j ϖ )  : n x ( n ) ϖ X ( e j ϖ ) • Generated white noise ε ( n ) and it's spectrum  Ε ( e j ϖ )  : ε ( n ) n ϖ Ε ( e j ϖ ) • The sum y ( n ) = x ( n ) + ε ( n ) and it's spectrum  Y ( e j ϖ )  : y ( n ) Y ( e j ϖ ) n ϖ • Question : How to separate signal from noise ? Page 15.4 Limitation of Linear Filtering index • A lowpass filter suppresses the high frequencies , hence much of the noise. • Here z ( n ) = y ( n ) * w ( n ), and w ( n ) is a 16point Hamming window: z ( n ) n • However, the high signal frequencies are also suppressed . This may severely distort the signal. • Whether the blurring is a problem depends on the application. • If sharp, sustained transitions are important, such as: edges in digital signals sudden utterances in speech signals then the linear solution may be inadequate . • Fact : It is generally impossible to remove highfrequency noise from a signal with edges, without blurring the edges  using linear filtering. Some nonlinear filters work, however. Page 15.5 Impulse Noise index • Another problem with linear filtering is impulse noise . • Often, bit errors occur in digitally transmitted signals. Bit errors in significant bits produce large, isolated errors....
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 Spring '04
 AlanC.Bovik
 Digital Signal Processing, Signal Processing, OS, Finite impulse response, Median filter

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