Perfect Reconstruction Two- Channel FIR Filter Banks A perfect reconstruction two-channel FIR filter bank with linear-phase FIR filters can be designed if the power-complementary requirement e jω + e jω = between the two analysis filters and is not imposed Copyright, S. K. Mitra
Copyright, S. K. Mitra Perfect Reconstruction Two- Perfect Reconstruction Two- Channel FIR Filter Banks Channel FIR Filter Banks To develop the pertinent design equations we observe that the input-output relation of the -channel QMF bank can be expressed in matrix form as } { X Y + = } { X + + [ ] = X X Y
Copyright, S. K. Mitra 3 Perfect Reconstruction Two- Perfect Reconstruction Two- Channel FIR Filter Banks Channel FIR Filter Banks From the previous equation we obtain Combining the two matrix equations we get [ ] = X X Y = X X Y Y = ] [ X X T m m
Copyright, S. K. Mitra 4 Perfect Reconstruction Two- Perfect Reconstruction Two- Channel FIR Filter Banks Channel FIR Filter Banks where are called the modulation matrices = m = m
Perfect Reconstruction Two- Channel FIR Filter Banks 5 Now for perfect reconstruction we must have l Y = X and correspondingly l Y = X Substituting these relations in the equation Y Y m m = [ ] X X we observe that the PR condition is satisfied if l m m T [ ] = l T Copyright, S. K. Mitra
Perfect Reconstruction Two- Channel FIR Filter Banks Thus, knowing the analysis filters and, the synthesis filters and are determined from l m = m l [ ] After some algebra we arrive at T 6 Copyright, S. K. Mitra
Copyright, S. K. Mitra 7 Perfect Reconstruction Two- Perfect Reconstruction Two- Channel FIR Filter Banks Channel FIR Filter Banks where and is an odd positive integer ] det[ m = l ] det[ m = l ] det[ m = l
Perfect Reconstruction Two- Channel FIR Filter Banks 8 For FIR analysis filters and, the synthesis filters and will also be FIR filters if det[ ] = c where c is a real number and k is a positive integer In this case m lk k = c lk = c Copyright, S. K. Mitra
Orthogonal Filter Banks Let be an FIR filter of odd order N satisfying the power-symmetric condition + Choose Then = N = = N det[ m ] + = N 9 Copyright, S. K. Mitra
Copyright, S. K. Mitra Orthogonal Filter Banks Orthogonal Filter Banks Comparing the last equation with we observe that and k = N Using in with we get k m c ] = det[ = c = N k = c l k = c l = k = N l, = = N N
Orthogonal Filter Banks Note: If is a causal FIR filter, the other three filters are also causal FIR filters Moreover, e jω = e jω Thus, for a real coefficient transfer function if is a lowpass filter, then is a highpass filter In addition, jω = jω i e e, i =, i Copyright, S. K. Mitra
Orthogonal Filter Banks A perfect reconstruction power-symmetric filter bank is also called an orthogonal filter bank The filter design problem reduces to the design of a power-symmetric lowpass filter To this end, we can design a an even-order F = whose spectral factoriation yields Copyright, S. K. Mitra
Orthogonal Filter Banks 3 Now, the power-symmetric condition + = implies that F be a ero-phase half-band lowpass filter with a non-negative frequency response F e jω Such a half-band filter can be obtained by adding a constant term K to a ero-phase half-band filter Q such that F e jω = Q e jω + K for all ω Copyright, S. K. Mitra
Orthogonal Filter Banks Summariing, the steps for the design of a real-coefficient power-symmetric lowpass filter are: Design a ero-phase real-coefficient FIR half-band lowpass filter Q of order N with N an odd positive integer: 4 Q q[ n] = N n=n n Copyright, S. K. Mitra
Orthogonal Filter Banks 5 Let δ denote the peak stopband ripple of Q e jω Define F = Q + δ which guarantees that F e jω for all ω Note: If q[n] denotes the impulse response of Q, then the impulse response f [n] of F is given by q[ n] + δ, for n= f [n] = q[ n], for n 3 Determine the spectral factor of F Copyright, S. K. Mitra
Orthogonal Filter Banks 6 Example- Consider the FIR filter N N + F = + R where R is a polynomial in of degree N with N odd F can be made a half-band filter by choosing R appropriately This class of half-band filters has been called the binomial or maxflat filter Copyright, S. K. Mitra
Orthogonal Filter Banks 7 The filter F has a frequency response that is maximally flat at ω = and at ω = π For N = 3, R = + 4 6 resulting in 3 3 F = + 9 + 6 + 9 6 which is seen to be a symmetric polynomial with 4 eros located at =, a ero at = 3, and a ero at = + 3 Copyright, S. K. Mitra
Orthogonal Filter Banks 8 The minimum-phase spectral factor is therefore the lowpass analysis filter =. 345 + [ 3 The corresponding highpass filter is given by 3 = =. 345. 679 +. 464. 73 + =. 345 +. 73 +. 464. 68 ] 3 3 Copyright, S. K. Mitra
Orthogonal Filter Banks The two synthesis filters are given by 3 = =. 683. 679 +. 464 +. 73 + 3 = Magnitude responses of the two analysis filters are shown on the next slide =. 683. 73 +. 464 +. 679 3 3 9 Copyright, S. K. Mitra
Orthogonal Filter Banks Magnitude.8.6.4...4.6.8 ω/π Comments: The order of F is of the form 4K+, where K is a positive integer Order of is N = K+, which is odd as required Copyright, S. K. Mitra
Orthogonal Filter Banks Zeros of F appear with mirror-image symmetry in the -plane with the eros on the unit circle being of even multiplicity Any appropriate half of these eros can be grouped to form the spectral factor For example, a minimum-phase can be formed by grouping all the eros inside the unit circle along with half of the eros on the unit circle Copyright, S. K. Mitra
Orthogonal Filter Banks Likewise, a maximum-phase can be formed by grouping all the eros outside the unit circle along with half of the eros on the unit circle owever, it is not possible to form a spectral factor with linear phase 3 The stopband edge frequency is the same for F and Copyright, S. K. Mitra
Orthogonal Filter Banks 4 If the desired minimum stopband attenuation of is α s db, then the minimum stopband attenuation of F is + 6. db α s Example - Design a lowpass real-coefficient power-symmetric filter with the following specifications: ωs =. 63π, and = db α s 3 Copyright, S. K. Mitra
Orthogonal Filter Banks 4 The specifications of the corresponding erophase half-band filter F are therefore: ωs =. 63π and α s = 4 db The desired stopband ripple is thus δ s =. which is also the passband ripple The passband edge is atω p = π. 63π =. 37π Using the function remeord we first estimate the order of F and then using the function reme design Q Copyright, S. K. Mitra
Orthogonal Filter Banks 5 The order of F is found to be 4 implying that the order of is 7 which is odd as desired To determine the coefficients of F we add err the maximum stopband ripple to the central coefficient q[7] Next, using the function roots we determine the roots of F which should theretically exhibit mirror-image symmetry with double roots on the unit circle Copyright, S. K. Mitra
Orthogonal Filter Banks owever, the algorithm s numerically quite sensitive and it is found that a slightly larger value than err should be added to ensure double eros of F on the unit circle Choosing the roots inside the unit circle along with one set of unit circle roots we get the minimum-phase spectral factor 6 Copyright, S. K. Mitra
Orthogonal Filter Banks The ero locations of F and are shown below Zero locations of F Zero locations of Imaginary Part 7.5 -.5-4 - Real Part Imaginary Part.5 -.5-7 - Real Part Copyright, S. K. Mitra
Orthogonal Filter Banks The gain responses of the two analysis filters are shown below ain, db - - -3-4 -5..4.6.8 ω/π 8 Copyright, S. K. Mitra
Orthogonal Filter Banks 9 Separate realiations of the two filters and would require N+ multipliers and N two-input adders owever, a computationally efficient realiation requiring N+ multipliers and N two-input adders can be developed by exploiting the relation N = Copyright, S. K. Mitra
Paraunitary System 3 A p-input, q-output LTI discrete-time system with a transfer matrix T pq is called a paraunitary system if T pq is a paraunitary matrix, i.e., ~ T T = ci pq pq Note: T ~ pq is the paraconjugate of T pq given by the transpose of T pq with each coefficient replaced by its conjugate Ip is an p p identity matrix, c is a real constant p Copyright, S. K. Mitra
Paraunitary Filter Banks A causal, stable paraunitary system is also a lossless system It can be shown that the modulation matrix m = of a power-symmetric filter bank is a paraunitary matrix 3 Copyright, S. K. Mitra
Paraunitary Filter Banks 3 ence, a power-symmetric filter bank has also been referred to as a paraunitary filter bank The cascade of two paraunitary systems with transfer matrices pq T and T qr is also paraunitary The above property can be utilied in designing a paraunitary filter bank without resorting to spectral factoriation Copyright, S. K. Mitra
Power-Symmetric FIR Cascaded Lattice Structure Consider a real-coefficient FIR transfer function N of order N satisfying the power-symmetric condition N N + N N = K We shall show now that N can be realied in the form of a cascaded lattice structure as shown on the next slide N 33 Copyright, S. K. Mitra
Copyright, S. K. Mitra 34 Power-Symmetric FIR Power-Symmetric FIR Cascaded Lattice Structure Cascaded Lattice Structure Define, X X i i = X Y i i = i N
Power-Symmetric FIR Cascaded Lattice Structure 35 From the figure we observe that = X + k X Therefore, X Y = kx + X = + k, = k + It can be easily shown that = Copyright, S. K. Mitra
Power-Symmetric FIR Cascaded Lattice Structure Next from the figure it follows that i = i + ki i i = kii + i It can easily be shown that i i = i provided i = i i 36 Copyright, S. K. Mitra
Power-Symmetric FIR Cascaded Lattice Structure 37 We have shown that i = i holds for i = ence the above relation holds for all odd integer values of i N must be an odd integer It is a simple exercise to show that both i and i satisfy the power-symmetry condition + = K i i i i i Copyright, S. K. Mitra i
Power-Symmetric FIR Cascaded Lattice Structure 38 In addition, i and i are powercomplementary, i.e., + ki i = kii + i To develop the synthesis equation we express i and i in terms of i and i : + i i i i + ki i = kii + k = k i i Copyright, S. K. Mitra
Power-Symmetric FIR Cascaded Lattice Structure 39 Note: At the i-th step, the coefficient is chosen to eliminate the coefficient of i, the highest power of in k k i For this choice of the coefficient of also vanishes making i a polynomial of degree i The synthesis process begins with i = N and compute N using N = N i i k i N i Copyright, S. K. Mitra
Power-Symmetric FIR Cascaded Lattice Structure Next, the transfer functions N and N are determined using the synthesis equations k = k + i i i i + ki i = kii + This process is repeated until all coefficients of the lattice have been determined i i 4 Copyright, S. K. Mitra
Power-Symmetric FIR Cascaded Lattice Structure 4 Example - Consider 5 = +. 3 +.. 376 4 5. 6 +. It can be easily verified that 5 satisfies the power-symmetric condition Next we form 5 5 = 5 =.. 6 3 4 +. 376 +.. 3 + 3 5 Copyright, S. K. Mitra
Power-Symmetric FIR Cascaded Lattice Structure 4 To determine we first form 5 5 k55 = +. k5 +. 3 +. 6k5 5 +. 376. +.. 376k k 4 5 +. +. 6 +. 3k k 5 To cancel the coefficient of in the above we choose k = 5. 5 5 5 3 Copyright, S. K. Mitra
Power-Symmetric FIR Cascaded Lattice Structure Then. 4 3 = k 5 5 [ k ] =. 4 +. 3 +. 48. 46 We next form 3 3 = 3 =. 4 +.. 3 + Continuing the above process we get k =. 4, k = 3 5 3. 5 3 3 43 Copyright, S. K. Mitra
Power-Symmetric FIR Banks Using the method outlined for the realiation of a power-symmetric transfer function, we can develop a cascaded lattice realiation of the -channel paraunitary QMF bank Three important properties of the QMF lattice structure are structurally induced 44 Copyright, S. K. Mitra
Power-Symmetric FIR Banks 45 The QMF lattice guarantees perfect reconstruction independent of the lattice parameters It exhibits very small coefficient sensitivity to lattice parameters as each stage remains lossless under coefficient quantiation 3 Computational complexity is about onehalf that of any other realiation as it requires N+/ total number of multipliers for an order-n filter Copyright, S. K. Mitra
Power-Symmetric FIR Banks 46 Example- Consider the analysis filter of the previous example: 7 =. 33+. 5935 +. 334 3 4. 78. 3767 +. 3 6 +. 79. 49 7 We place a multiplier h[] =.33 at the input and synthesie a cascade lattice structure for the normalied transfer function / [ ] 7 h 5 Copyright, S. K. Mitra
Power-Symmetric FIR Banks 47 The lattice coefficients obtained for the normalied analysis transfer function are: k7 =. 565, k5 =. 354, k =. 48393, k = 6 3. Note: Because of the numerical problem, the coefficients of the spectral factor obtained in the previous example are not very accurate Copyright, S. K. Mitra
Power-Symmetric FIR Banks i As a result, the coefficients of of the transfer function i generated from the transfer function i using the relation i = [ ] i k + k i i i is not exactly ero, and has been set to ero at each iteration 48 Copyright, S. K. Mitra
Power-Symmetric FIR Banks Two interesting properties of the cascaded lattice QMF bank can be seen by examining its multiplier coefficient values Signs of coefficients alternate between stages The values of the coefficients decrease with increasing i { k i } 49 Copyright, S. K. Mitra
Power-Symmetric FIR Banks The QMF lattice structure can be used directly to design the power-symmetric analysis filter using an iterative computer-aided optimiation technique oal: Determine the lattice parameters by minimiing the energy in the stopband of k i 5 Copyright, S. K. Mitra
Power-Symmetric FIR Banks The pertinent objective function is given by φ = π ω s e jω dω Note: The power-symmetric property ensures good passband response 5 Copyright, S. K. Mitra
Biorthogonal FIR Banks 5 In the design of an orthogonal -channel filter bank, the analysis filter is chosen as a spectral factor of the ero-phase even-order half-band filter F = Note: The two spectral factors and of F have the same magnitude response Copyright, S. K. Mitra
Biorthogonal FIR Banks As a result, it is not possible to design perfect reconstruction filter banks with linear-phase analysis and synthesis filters owever, it is possible to maintain the perfect reconstruction condition with linearphase filters by choosing a different factoriation scheme 53 Copyright, S. K. Mitra
Biorthogonal FIR Banks 54 To this end, the causal half-band filter N F of order N is factoried in the form N F = where and are linear-phase filters The determinant of the modulation matrix m is now given by det m ] = = N [ Copyright, S. K. Mitra
Biorthogonal FIR Banks 55 Note: The determinant of the modulation matrix satisfies the perfect reconstruction condition The filter bank designed using the factoriation scheme N F = is called a biorthogonal filter bank The two synthesis filters are given by =, = Copyright, S. K. Mitra
Biorthogonal FIR Banks 56 Example - The half-band filter F = 3 + 4 + 4 6 can be factored several different ways to yield linear-phase analysis filters and For example, one choice yields = + + 6 + 3 4 8 = + Copyright, S. K. Mitra
Biorthogonal FIR Banks Since the length of is 5 and the length of is 3, the above set of analysis filters is known as the 5/3 filterpair of Daubechies A plot of the gain responses of the 5/3 filterpair is shown below 57 Magnitude.8.6.4...4.6.8 ω/π Copyright, S. K. Mitra
Biorthogonal FIR Banks Another choice yields the 4/4 filter-pair of Daubechies = + 3 + 3 + 3 8 + 3 = 3 + 3 A plot of the gain responses of the 4/4 filterpair is shown below.5 Magnitude.5 58..4.6.8 ω/π Copyright, S. K. Mitra