08_CM0268_Filters

# 08_CM0268_Filters - Filtering Filtering in a broad sense is...

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CM0268 MATLAB DSP GRAPHICS 1 232 JJ II J I Back Close Filtering Filtering in a broad sense is selecting portion(s) of data for some processing. In many multimedia contexts this involves the removal of data from a signal — This is essential in almost all aspects of lossy multimedia data representations. We will look at ﬁltering in the frequency space very soon, but ﬁrst we consider ﬁltering via impulse responses. We will look at: IIR Systems : Inﬁnite impulse response systems FIR Systems : Finite impulse response systems

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CM0268 MATLAB DSP GRAPHICS 1 233 JJ II J I Back Close Inﬁnite Impulse Response (IIR) Systems If h ( n ) is an inﬁnite impulse response function then the digital system is called and IIR system. Example: The algorithm is represented by the difference equation: y ( n ) = x ( n ) - a 1 y ( n - 1) - a 2 y ( n - 2) This produces the opposite signal ﬂow graph + y ( n ) T T × × y ( n - 1) = x H 1 ( n ) y ( n - 2) = x H 2 ( n ) - a 1 - a 2 x ( n )
CM0268 MATLAB DSP GRAPHICS 1 234 JJ II J I Back Close Inﬁnite Impulse Response (IIR)Systems Explained The following happens: The output signal y ( n ) is fed back through a series of delays Each delay is weighted Fed back weighted delay summed and passed to new output. Such a feedback system is called a recursive system + y ( n ) T T × × y ( n - 1) = x H 1 ( n ) y ( n - 2) = x H 2 ( n ) - a 1 - a 2 x ( n )

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CM0268 MATLAB DSP GRAPHICS 1 235 JJ II J I Back Close Z-transform of IIR If we apply the Z-transform we get: Y ( z ) = X ( z ) - a 1 z - 1 Y ( z ) - a 2 z - 2 Y ( z ) X ( z ) = Y ( z )(1 + a 1 z - 1 + a 2 z - 2 ) Solving for Y ( z ) /X ( z ) gives H ( z ) our transfer function: H ( z ) = Y ( z ) X ( z ) = 1 1 + a 1 z - 1 + a 2 z - 2
CM0268 MATLAB DSP GRAPHICS 1 236 JJ II J I Back Close A Complete IIR System x ( n ) TT T x ( n - 1) x ( n - 2) x ( n - N + 1) ××××× b 0 b 1 b 2 b N - 2 b N - 1 +++++ y ( n ) ×××× - a M - a M - 1 - a M - 2 - a 1 T y ( n - M ) y ( n - 1) Here we extend: The input delay line up to N - 1 elements and The output delay line by M elements.

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CM0268 MATLAB DSP GRAPHICS 1 237 JJ II J I Back Close Complete IIR System Algorithm x ( n ) TT T x ( n - 1) x ( n - 2) x ( n - N + 1) ××××× b 0 b 1 b 2 b N - 2 b N - 1 +++++ y ( n ) ×××× - a M - a M - 1 - a M - 2 - a 1 T y ( n - M ) y ( n - 1) We can represent the IIR system algorithm by the difference equation: y ( n ) = - M X k =1 a k y ( n - k ) + N - 1 X k =0 b k x ( n - k )
CM0268 MATLAB DSP GRAPHICS 1 238 JJ II J I Back Close Complete IIR system Transfer Function The Z-transform of the difference equation is: Y ( z ) = - M X k =1 a k z - k Y ( z ) + N - 1 X k =0 b k z - k X ( z ) and the resulting transfer function is: H ( z ) = N - 1 k =0 b k z - k 1 + M k =1 a k z - k

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CM0268 MATLAB DSP GRAPHICS 1 239 JJ II J I Back Close Filtering with IIR We have two ﬁlter banks deﬁned by vectors: A = { a k } , B = { b k } . These can be applied in a sample-by-sample algorithm: MATLAB provides a generic filter(B,A,X) function which ﬁlters the data in vector X with the ﬁlter described by vectors A and B to create the ﬁltered data Y .
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## This note was uploaded on 01/24/2012 for the course CM 0268 taught by Professor Davidmarshall during the Winter '11 term at Cardiff University.

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08_CM0268_Filters - Filtering Filtering in a broad sense is...

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