-Channel QMF Bans The basic structure of the -channel QMF ban is shown below The expressions for the -transforms of various intermediate signals in the above structure are given by Copyright, S. K. Mitra
Copyright, S. K. Mitra -Channel QMF Bans -Channel QMF Bans where Define the vector of down-sampled subband signals as X V W X W U l l l / / U V ^ [ ] T U U U... u
Copyright, S. K. Mitra 3 -Channel QMF Bans -Channel QMF Bans Define the modulation vector of the input signals as Define the analysis filter ban modulation matrix as T m W X W X X ]... [ x......... m W W W W W W......
Copyright, S. K. Mitra 4 -Channel QMF Bans -Channel QMF Bans Then we can write the set of equations as The output of the QMF ban is given by W X W U l l l / / ] [ / / m T m x u V G Y ^
-Channel QMF Bans In matrix form we can write Y g where T u [ G G... G ] T g 5 Copyright, S. K. Mitra
Alias-Free -Channel QMF Bans From the output equation Y gt u the modulated versions of the output signal are given by Y W T g W u W g T W u, 6 Copyright, S. K. Mitra
Copyright, S. K. Mitra 7 Alias-Free Alias-Free -Channel QMF -Channel QMF Bans Bans Define the modulation vector of the output signal as Define the synthesis filter ban modulation matrix as T m W Y W Y Y ]... [ y......... m W G W G W G W G W G W G G G G G......
Copyright, S. K. Mitra 8 Alias-Free Alias-Free -Channel QMF -Channel QMF Bans Bans Then the modulation vector of the output signal can be expressed as Combining the above and we arrive at m m u y G ] [ / / m T m x u ] [ m T m m m x G y
Alias-Free -Channel QMF Bans 9 Using the notation we can write m m T G [ ] m m y T x T is called the transfer matrix relating the input signal X and its modulated versions X W with the output signal Y and its modulated versions Y W T Copyright, S. K. Mitra
Alias-Free -Channel QMF Bans The filter ban is alias-free if the transfer matrix T is a diagonal matrix of the form... T diag[ T T W T W ] The first element T of the above diagonal matrix is called the distortion transfer function of the -channel filter ban Copyright, S. K. Mitra
Copyright, S. K. Mitra Alias-Free Alias-Free -Channel QMF -Channel QMF Bans Bans Substituting in X V W X W U l l l / / U V ^ V G Y ^
Alias-Free -Channel QMF Bans we arrive at l Y l al X W where a l l W G, l l On the unit circle the term X W becomes jω l j ωπl / X e W X e Copyright, S. K. Mitra
Alias-Free -Channel QMF Bans Thus, from l Y l al X W jω we observe that the output spectrum Y e jω is a weighted sum of X e and its j ωπl / uniformly shifted versions X e for l,,..., which are caused by the sampling rate alteration operations 3 Copyright, S. K. Mitra
Alias-Free -Channel QMF 4 l Bans The term X W is called the l-th aliasing term, with a l representing its gain at the output In general, the -channel QMF ban is a linear, time-varying system with a period It follows from l Y l al X W that the aliasing effect at the output can be completely eliminated if and only if a l l, Copyright, S. K. Mitra
Alias-Free -Channel QMF Bans Note: The aliasing cancellation condition given above must hold for all possible inputs If the aliasing cancellation condition holds then the -channel QMF ban becomes a linear, time-invariant system with an inputoutput relation given by Y TX 5 Copyright, S. K. Mitra
Alias-Free -Channel QMF Bans 6 The distortion transfer function T is given by T a G If T has a constant magnitude, then the - channel QMF ban is magnitude-preserving If T has a linear phase, then the -channel QMF ban is phase-preserving If T is a pure delay, then it is a perfect reconstruction filter ban Copyright, S. K. Mitra
Alias-Free -Channel QMF Bans Define A [ a a... a ] Then l al W G, l can be expressed as A m g 7 Copyright, S. K. Mitra
Alias-Free -Channel QMF Bans The aliasing cancellation condition can now be rewritten as m g t where T t [ a... ] 8 Copyright, S. K. Mitra
Alias-Free -Channel QMF Bans 9 ence, nowing the set of analysis filters { }, we can determine the desired set of synthesis filters { G } as m g [ ] t provided [det m ] Moreover, a perfect reconstruction QMF n ban results if we set T o in the expression for t Copyright, S. K. Mitra
Alias-Free -Channel QMF Bans In practice, the above approach is difficult to carry out for a number of reasons A more practical solution to the design of a perfect reconstruction QMF ban is based on a polyphase representation Copyright, S. K. Mitra
Polyphase Representation Consider the -band Type I polyphase representation of the -th analysis filter: l l El, A matrix representation of the above set of equations is given by e h E where h [... ] T Copyright, S. K. Mitra
Copyright, S. K. Mitra Polyphase Polyphase Representation Representation and E is called the Type I polyphase component matrix. T ]... [ e.........,,,,, E E E E E E E E E E......
Polyphase Representation 3 iewise, we can represent the synthesis filters in a -band Type II polyphase form: l G l Rl, In matrix form the above set of equations can be rewritten as T ~ g e R where g [ G G... G ] T Copyright, S. K. Mitra
Copyright, S. K. Mitra 4 Polyphase Polyphase Representation Representation and R is called the Type II polyphase component matrix.........,,,,, R R R R R R R R R R...... ]... [ T e e
Polyphase Representation The polyphase representations of the - channel analysis and the -channel synthesis filter bans are shown below 5 Analysis filter ban Synthesis filter ban Copyright, S. K. Mitra
Polyphase Representation Substituting the polyphase representations of the analysis and synthesis filter bans in the original structure of the -channel QMF ban, and maing use of the cascade equivalences we arrive at 6 Copyright, S. K. Mitra
Copyright, S. K. Mitra 7 Polyphase Polyphase Representation Representation From and it can be seen that......... m W W W W W W...... T ]... [ h ]... [ ] [ T m W W h h h
Polyphase Representation 8 e Maing use of h E in the previous equation we get m T [ ] E [ e e W... e W T Now, from... e [ ] we have W e W. W ] Copyright, S. K. Mitra
Polyphase Representation 9 where we have used the notation diag[... Maing use of the above notation in m T [ ] E [ e e W... e W we arrive at D E T where D is the conjugate transpose of the DFT matrix D ] ] Copyright, S. K. Mitra
Condition for Perfect Reconstruction Consider the -channel QMF structure repeated below for convenience 3 Copyright, S. K. Mitra
Condition for Perfect Reconstruction 3 Assume that the polyphase component matrices satisfy the relation R E ci where I is an identity matrix and c is a constant Then the QMF structure on the previous slide reduces to the one shown on the next slide Copyright, S. K. Mitra
Condition for Perfect Reconstruction 3 Note: The structure can be considered as a special case of the most general -channel QMF ban shown earlier if we set, G, Copyright, S. K. Mitra
Condition for Perfect Reconstruction Substituting, G, in a W l l G, l we get l l a l W, l 33 Copyright, S. K. Mitra
Condition for Perfect Reconstruction Now, W l l,, l l 34 ence, from the last equation on the previous slide it follows that a, a l for l As a result, T or in other words, the simplified QMF structure satisfies the perfect reconstruction property Copyright, S. K. Mitra
Condition for Perfect Reconstruction The analysis and synthesis filters of the perfect reconstruction -channel QMF ban can be easily determined from nown polyphase component matrices Example - The structure shown below is by construction a perfect reconstruction filter ban 35 Copyright, S. K. Mitra
Condition for Perfect Reconstruction 36 The output of the filter ban is simply y[ n] dx[ n ] Note: In this structure E 3 P and R 3 dp Consider P Copyright, S. K. Mitra
Copyright, S. K. Mitra 37 Condition for Perfect Condition for Perfect Reconstruction Reconstruction From we get ence, 3 e h E E E E E E E E E E,, + + +
Copyright, S. K. Mitra 38 Condition for Perfect Condition for Perfect Reconstruction Reconstruction With d 4 we have Then from we get which leads to dp 3 T R e g ~ [ ] G G G,, + + + G G + G
Polyphase Representation 39 For a given -channel analysis filter ban, the polyphase component matrix E is nown E E ci, ence, a perfect reconstruction -channel QMF ban can be designed by constructing a synthesis filter ban with a polyphase component matrix R [ ] E Copyright, S. K. Mitra
Polyphase Representation In general, it is not easy to compute the inverse of a rational matrix An alternative elegant approach is to design the analysis filter ban with an invertible polyphase matrix E For example, E can be chosen to be a paraunitary matrix satisfying the condition ~ EE ci, for all 4 Copyright, S. K. Mitra
Polyphase Representation ~ Note: E is the paraconjugate of E given by the transpose of E, with each coefficient replaced by its conjugate A perfect reconstruction -channel QMF ban is then obtained by choosing ~ R E 4 Copyright, S. K. Mitra
Polyphase Representation 4 For the design of a perfect reconstruction - channel QMF ban, the matrix E can be expressed in a product form E ER ER... E E where E is a constant unitary matrix, and E [ *] T [ * T l I vl vl + vl vl] In the above v l is a column vector of order with unit norm, i.e., [ v * ] T l v l Copyright, S. K. Mitra
Polyphase Representation 43 With E expressed in the product form, one can set up an appropriate objective function that can be minimied to arrive at a set of analysis filters meeting the desired specifications To this end, a suitable objective function is given by φ ω e j th stopband dω Copyright, S. K. Mitra
Polyphase Representation 44 The optimiation parameters are the elements of vl and E Example- Consider the design of a 3- channel FIR perfect reconstruction QMF ban with a passband width π/3 The passband width of the lowpass filter is from to π/3, that of the bandpass filter is from π/3 to π/3, and that of the highpass filter is from π/3 to π Copyright, S. K. Mitra
Polyphase Representation The objective function to be minimied here is thus of the form φ π +ε π π +ε π 3 jω ε 3 jω π ε e jω dω + 3 e jω e dω + e + 3 dω The gain responses of the 3 analysis filters of length 5 are shown on the next slide π dω 45 Copyright, S. K. Mitra
Polyphase Representation - Gain, db - -3-4 -5 46-6..4.6.8 ω/π The coefficients of the corresponding synthesis filters are given by g [ n] h [4 n],,,3 Copyright, S. K. Mitra