A New Class of Smooth Power Complementary Windows and Their Application to Audio Signal Processing

Audio Engineering Society Convention Paper Presented at the 119th Convention 2005 October 7–10 New York, New York USA This convention paper has been reproduced from the author's advance manuscript, without editing, corrections, or consideration by the Review Board. The AES takes no responsibility for the contents. Additional papers may be obtained by sending request and remittance to Audio Engineering Society, 60 East 42nd Street, New York, New York 10165-2520, USA; also see www.aes.org. All rights reserved. Reproduction of this paper, or any portion thereof, is not permitted without direct permission from the Journal of the Audio Engineering Society. A New Class of Smooth Power Complementary Windows and their Application to Audio Signal Processing Deepen Sinha1, Anibal J. S. Ferreira1,2 1 ATC Labs, New Jersey, USA 2 University of Porto, Portugal Correspondence should be addressed to D. Sinha ([email protected]) ABSTRACT In this paper we describe a new family of smooth power complementary windows which exhibit a very high level of localization in both time and frequency domain. This window family is parameterized by a "smoothness quotient". As the smoothness quotient increases the window becomes increasingly localized in time (most of the energy gets concentrated in the center half of the window) and frequency (far field rejection becomes increasing stronger to the order of 150 dB or higher). A closed form solution for such window function exists and the associated design procedure is described. The new class of windows is quite attractive for a number of applications as switching functions, equalization functions, or as windows for overlap-add and modulated filter banks. An extension to the family of smooth windows which exhibits improved near-field response in the frequency domain is also discussed. More information is available at http://www.atc-labs.com/technology/misc/windows. In many applications it is desirable to use a symmetric 1. INTRODUCTION window function nw )( (with a maximum value of 1.0) Windowing is a prevalent signal processing operation that satisfies an additional power complementary which has important implications to the performance of condition. To formalize these conditions, let’s define a a range of signal processing algorithms. Several window window function of size 2N. The symmetry condition functions have been proposed and utilized in signal implies processing. Some examples include Hanning Window, Hamming Window, Kaiser Window, Raised Cosine = − − = NnnNwnw −12,..,1,0),12()( (1) Window, etc. [1-4]. D. Sinha and A. Ferreira A New Class of Smooth Power Complementary The power complementary condition is indicated by the 2. THE CLASS OF SMOOTH WINDOWS constraint It is well known in the theory of signal approximation w( n )2 + w ( n + N )2 = 1.0for n= 0,1,.., N − 1 and splines [9] that the smoothness of a function increases as a function of the highest order continuous (2) differential it possesses. The increased smoothness of a function results in improved frequency response for the The above condition is at times also referred to as function. We construct the proposed family of smooth Princen-Bradley condition [6-7]. For the reasons that window based on such differentiability criterion. In will become obvious in the later sections we will refer particular the window function w() n is constructed as to this condition as the power complementary condition. samples of a continuous function w(t) with compact In choosing a particular window function the frequency support, in other words selectivity is an important consideration. The frequency selectivity is characterized by the stop-band energy of w( t )= 0∀ t ∉[ 0,1] the frequency response of the window function. Furthermore, in many applications the time selectivity and (3) of the window is also an important consideration. In the w( n )= w ( t )i+ 5.0 , i= 0,1,..,2 N − 1 t= framework of power complementary windows, time 2N localization is quantified by the fraction of window energy concentrated in the center half of the window. The window is said to be smooth to a degree p (also Obviously the most time localized window is the called the order of the smooth window) if it possesses a rectangular window albeit at the cost of extremely poor frequency selectivity. It may also be noted that the continuous differential of order p −1 . Here we describe popular Raised Cosine window [4, 6] is an example of a a procedure for constructing a power complementary power complementary window that has relatively good window with an arbitrary degree of smoothness, p . As frequency selectivity but poor time localizing. In will be seen below, higher degree of smoothness for a general frequency selectivity and temporal localization window function (satisfying the power complementary are two conflicting requirements and it is difficult to condition) results in improved temporal localization (in design a window function that has good performance addition to improved far-field frequency response). with respect to both of these performance criteria. To describe the design procedure we first define a Here we describe a new family of smooth window function P ()ξ which may be considered as a complex functions satisfying the above constraints. The new 0 class of windows is designed using the realization that periodic extension of w()t with conjugate symmetry; the power complementary condition (2) is a time i.e., domain dual of an identical frequency domain condition used in the design of wavelets with high degree of PP()()−ξ = * ξ regularity [2, 8, 16]. This leads to a systematic design 0 0 procedure for a family of smooth windows P0 (ξ+ 2 π ) = P0 ( ξ ) parameterized by a smoothness quotient. The window (4) family is unique in the sense that as the smoothness and quotient is increased the window becomes better w()() t= P ξ ∀t ∈ ]1,0[ localized in both time and frequency domain. It may be 0 ξ= π (2t − 1) noted that although we have used this family of windows in our work for a few years, such windows The power symmetry condition in (2) puts the following appear to be otherwise not well known in the signal constraint on P ()ξ ; i.e., processing literature. 0 2 2 PP0 (ξ )+0 ( ξ + π ) = 1 (5) AES 119th Convention, New York, New York, 2005 October 7–10 Page 2 of 10 D. Sinha and A. Ferreira A New Class of Smooth Power Complementary And smoothness quotient, p , translates into the p−1 −1 k 2 ⎛ N−1 + k ⎞ [z−2 + z ] Q() z = ⎜ ⎟ (9) condition that P ()ξ has a zero order p at ξ = π . ∑⎜ ⎟ k 0 k −0 ⎝ k ⎠ 4 Readers familiar with the theory of wavelets and filter banks will readily recognize that the above formulation The procedure for constructing the smooth power makes this family of windows the time domain dual of complementary window of an order p is as follows: the well known orthogonal wavelet filter bank used in 2 band QMF analysis that satisfy a regularity or flatness Step 1: For the chosen p form the polynomial constraint [2, 8]. A solution for P ()ξ that satisfies (4) 0 2 can be found using a famous result due to Daubechies Q() z as in (7) above. [8]. In particular we reproduce Proposition 4.5 from this reference [8] as follows: 2 Step 2: Find all the roots of Q() z . It may be noted Proposition [Daubechies, 1988]: Any Trigonometric that all the roots will occur in complex polynomial solution of equation (4) above is of the form conjugate pairs which are also mirror imaged across the unit circle, in other words if p z= ρ e jφ is a root then so will be ⎡1 jξ ⎤ P ()ξ =1 + e Q() e jξ (6) 0 ⎢ ()⎥ 1 1 ⎣2 ⎦ z= ρ e− jφ,, e j φ and e − jφ (this results ρ ρ where p 1 is the order of zeros it possesses at 2 ≥ from the fact that Q() z is a zero phase, real ξ = π and Q is a polynomial such that polynomial). p−1 jξ 2 ⎛ N−1 + k ⎞ 2k 1 Step 3: For each set of roots choose one pair which is Q() e = ∑⎜ ⎟sin ξ + on the same side of the unit circle. Form a k−0 ⎝ k ⎠ 2 (7) polynomial Q() z by combining all the chosen ⎡ 2N 1 ⎤ 1 ⎢sin ξR( cos ξ ) ⎣ 2 ⎦⎥ 2 roots. We have Q()() z= Q z (this is spectral factorization). ⎛n⎞ where R is an odd polynomial. In the above ⎜ ⎟ ⎝k ⎠ Step 4: Combine Q() z with the p th order zero in (6) implies the Choice function [4]. The polynomial R is to form one possible polynomial, P() z . The not arbitrary, and for our purpose (as will be explained 0 below) may be set to 0. taps of this P0 () z form a FIR filter p( n ). A solution for P0 ()ξ may be found by using spectral Step 5: P0 () z can be found by performing a Fourier jξ factorization technique on the polynomial, Q() e . For analysis of the sequence p( n ). The window this we substitute coefficients are then computed by sampling the Fourier transform of p(n). as per equations 1 1 − (3) and (4). 1 z2 − z 2 sin ξ = (8) 2 2 where z is a free complex variable. We then have Windows designed using the procedure for p = 3,2,1 are shown in Figure 1. The magnitude response of these windows is included in Figure 2(a), AES 119th Convention, New York, New York, 2005 October 7–10 Page 3 of 10 D. Sinha and A. Ferreira A New Class of Smooth Power Complementary (b), (c) respectively.

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