Splitting the Unit Delay - IEEE Signal Processing ... - IEEE Xplore

constant with high precision when the fractional delay ischanged, unlike with previous designs. However, this comesat the cost of a worse phase delay response, which is particularlysevere with the 4-tap example (see Fig. A3b).Other transition functions are not considered here. Theimpulse response of a raised-cosine transition function iseasily obtained by modifying the result of [12].4) General Least Squares FIR Approximationof a Complex Frequency ResponseThe smooth-transition-band-method reduces the error of thefractional delay filter in an advantageous way, but it has someshortcomings. First, the impulse response is still infinitelylong although it decays fast, and the truncation always causessome error. Second, the desired response must be definedexplicitly in the whole Nyquist band. While the method hasthe clear design advantage that the inverse Fourier transformcan be used to determine the coefficients explicitly, it neverthelessforces approximation resources to be wasted outsidethe approximation band where they are actually not needed.In principle, the delay filter with the smallest LS error inthe defined approximation band is accomplished by definingthe response only in that part of the frequency band and byleaving the rest out of the error measure as a "don't care"band. This scheme also enables frequency-domain weightingof the LS error; it results in the following error formulation(alternative formulations employing eigenfilter techniqueshave been introduced in [76] and [SS]).Fourier transform as H(e'w) and the error measure Eq. 24becomeswhere the superscript '*' denotes complex conjugation. Thiscan further be put into the following formwhereE4 = h'Ph- 2hTpl + po (28)1po = - jW(o)IHjd(ejw)12 doZ Owhere the error is defined in the lowpass frequency band [0,an] only and W(o) is the nonnegative frequency-domainweighting function. Note that W(o) has nothing to do withthe time-domain window w(n) (Eq. 22).We now derive the solution for a general H,d(e'"). Toformulate the solution in compact form, let us introducevector notation asThe error measure (Eq. 28) is quadratic with a uniqueminimum-enor solution that is found by setting its derivativewith respect to h to zero. This results in the following normalequationwhich is solved formally by matrix inversion, i.e.,h= [WO) WNIT (254for the filter coefficients and for the discrete-time Fouriertransform, respectively, and the matrixcos 0 ... cos(No) 1r 11 ... cos[N-1103where the superscript 'H' stands for the Hermitian operation,i.e., transposition with conjugation. Now we can express theHence, the optimal solution is obtained by determining theintegrals involved in Eqs. 29 (usually numerically) and solvingthe set of (N+ 1) linear equations (Eq. 32). The arithmeticcomplexity (i.e., the required number of multiplications andadditions) of a matrix inversion is in general proportional to(N + 1)3 so that the computational costs are considerablyincreased over the previous methods, where the coefficientsare obtained in explicit form. Furthermore, numerical problemsmay arise, particularly in narrowband approximation[60, 131. This is the price to be paid for the increased flexibilityof the design.The numerical problems are indeed regrettable as thegeneral least squares approximation of a complex response38 IEEE SIGNAL PROCESSING MAGAZINE JANUARY 1996