Narrow-Band Low-Pass Digital Differentiator Design Ivan Selesnick Polytechnic University Brooklyn, New York [email protected] http://taco.poly.edu/selesi 1 Ideal Lowpass Digital Differentiator The frequency response of the ideal lowpass digital differentiator is ( j! j ! j!j < !c HLP (e ) = (1) 0 !c < j!j < ¼ ² It is a narrow-band filter if !c is much smaller than ¼. ² A narrow-band filter should have a long impulse response. ² =) It is desirable to have simple design algorithms so that ill-conditioning and computational complexity is minimized. ² The window method for FIR filter design is a natural choice in this case. The design method described here gives an alternative approach. 2 EOG Example The next slide illustrates the result of filtering an EOG signal with: 1. a full-band differentiator and 2. a narrow-band lowpass differentiator Differentiation with the full-band differentiator yields an extremely noisy signal, while lowpass differentiation gives a more useful result. 3 EOG SIGNAL EOG SIGNAL AFTER FULLBAND DIFFERENTIATION EOG SIGNAL AFTER LOWPASS DIFFERENTIATION n 4 Design To avoid the undesirable amplification of noise in digital differentiation, lowpass differentiators can be used in place of full-band ones. Low-pass digital differentiator design: 1. Maxflat 2. Least-squares 3. Remez 4. Flat passband, Equiripple stopbands (Kaiser, Rabiner, Vaidyanathan) 5 Our Approach We describes a simple formulation for the non-iterative design of narrow-band FIR linear-phase lowpass digital differentiators. ² The filters are flat around dc and have equally spaced nulls in the stopband. ² The impulse response can be written as a sum of sines (Frequency sampling expression). ² The design problem is formulated so as to avoid the complexity or ill-conditioning of standard methods for the design of similar filters when those methods are used to design narrow-band filters with long impulse responses. 6 Analog Lowpass Differentiator The sinc function is given by sin(¼ f) sinc(f) := : ¼ f The function sinc(f) is symmetric (sinc(¡f) = sinc(f)) and equal to zero for f = §1; §2; §3;::: ; therefore if we define sk(f) := sinc(f ¡ k) ¡ sinc(f + k) then we have 1. sk(f) is antisymmetric, sinc(¡f) = ¡ sinc(f). 2. sk(f) = 0 for f 2 Z=f§kg. (sk(f) = 0 whenever f is an integer different from §k.) 7 Analog Lowpass Differentiator The digital filter design procedure we propose begins with an analog frequency response having the following form: XK A(f) = a(k; K) (sinc(f ¡ k) ¡ sinc(f + k)) k=1 Therefore, the frequency response A(f) has the following properties: 1. A(f) is antisymmetric, A(¡f) = ¡A(f). 2. A(f) = 0 for f = 0, and for f = §(K + 1); §(K + 2); §(K + 3);::: . ² The frequency response A(f) is zero at f = 0. ² The first null in the stopband depends on K. ² The exact behavior of A(f) depends on the coefficients a(k; K), however, the uniformly spaced nulls in the stopband ensures that the attenuation increases with frequency. 8 Problem Formulation The coefficients a(k; K) are to be determined so that the frequency response A(f) approximates f near f = 0 A(f) ¼ f: Given K, find a(k; K) for 1 · k · K such that the derivatives of A(f) at f = 0 match the derivatives of the ideal differentiator IdealDiff(f) := f at the point f = 0: A(1)(0) = 1 (2) A(i)(0) = 0; i = 3; 5;:::; 2 K ¡ 1: (3) ² The even derivatives are automatically zero because A(f) is an odd function, A(¡f) = ¡A(f). ² This is a linear system of equations with an equal number of equations and variables. ² The stopband of A(f) is neither equiripple nor maximally flat. 9 Example For example, when K = 1, we have 1 a(1; 1) = : 2 When K = 2, we have 1 1 a(1; 2) = ¡ + ¼2 6 9 4 2 a(2; 2) = ¡ + ¼2 3 9 When K = 3, we have 1 13 7 a(1; 3) = ¡ ¼2 + ¼4 48 288 480 16 16 14 a(2; 3) = ¡ ¼2 + ¼4 15 9 75 243 81 189 a(3; 3) = ¡ ¼2 + ¼4 80 32 800 10 Example When K = 4, we have 1 29 427 31 a(1; 4) = ¡ + ¼2 ¡ ¼4 + ¼6 720 4320 64800 18900 16 208 2366 496 a(2; 4) = ¡ + ¼2 ¡ ¼4 + ¼6 45 135 2025 4725 2187 2187 5103 2511 a(3; 4) = ¡ + ¼2 ¡ ¼4 + ¼6 560 160 800 4900 2048 2048 12544 15872 a(4; 4) = ¡ + ¼2 ¡ ¼4 + ¼6 315 135 2025 33075 Other values a(k; K) can be easily computed. 11 Conversion to Digital Filter To convert the analog frequency response A(f) into a digital frequency response D(f), we can use the digital sinc function in place of the usual sinc function. The digital sinc function dsinc(f; N) can be written as sin(N¼ f) dsinc(f; N) := : (4) sin(¼ f) ² The digital sinc function defined in (4) is periodic in f with period 2: dsinc(f + 2) = dsinc(f): ² We have the following approximation: µ ¶ 1 f sinc(f) ¼ dsinc ;N for jfj < 0:5 N: N N for large values of N. 12 Sinc vs. Digital Sinc sinc(f) 1 0.5 0 −0.5 −1 −30 −20 −10 0 10 20 30 f dsinc(f/N)/N [N = 30] 1 0.5 0 −0.5 −1 −30 −20 −10 0 10 20 30 f 13 Sinc Minus Digital Sinc sinc(f)−dsinc(f/N)/N [N = 30] 1 0.5 0 −0.5 −1 −30 −20 −10 0 10 20 30 f sinc(f)−dsinc(f/N)/N [N = 30] 0.015 0.01 0.005 0 −0.005 −0.01 −0.015 −15 −10 −5 0 5 10 15 f 14 Sinc vs. Digital Sinc sinc(f) ¼ dsinc(f=N)=N; for jfj < 0:5N especially for large values of N. The design of digital differentiators described here is intended for long impulse responses. In this case, N is large and the approximation is valid. 15 Digital Lowpass Differentiators Consider the function D(f), based on the digital sinc function: · µ ¶ µ ¶¸ 1 XK f ¡ k f + k D(f) = a(k; K) dsinc ;N ¡ dsinc ;N N N N k=1 For N > K, we have the approximation D(f) ¼ A(f) for jfj < N=2: For example: with K = 3, N = 30: −3 x 10 A(f)−D(f) 2 1 0 −1 −2 −10 −5 0 5 10 f 16 Digital Lowpass Differentiators Consider the function D(f), based on the digital sinc function: · µ ¶ µ ¶¸ 1 XK f ¡ k f + k D(f) = a(k; K) dsinc ;N ¡ dsinc ;N N N N k=1 N ² The function D(2 ¼!) is then a 2 ¼ periodic function of ! and can therefore be used as the frequency response H(ej!) of a digital filter. ² Then H(ej!) will be approximately maximally flat at ! = 0. 17 Digital Lowpass Differentiators The impulse response h(n) is given by the inverse discrete-time Fourier transform j! of H(e ), ½ µ ¶¾ N¡1 N h(n) = IDTFT e¡j( 2 )! ¢ D ! 2 ¼ where the phase term is included to make h(n) causal. Then h(n) is a linear- phase FIR impulse response of length N: µ µ ¶¶ 1 XK 2¼k (N ¡ 1) h(n) = a(k; K) sin n ¡ N 2 N 2 k=1 for 0 · n · N ¡ 1. Once a table of values a(k; K) is computed, it can be used regardless of the length N of the impulse response h(n). The width of the passband is controlled by the parameter K, the number of flatness constraints at dc. Note that 2 (K ¡ 1) is the number of zeros of H(z) that lie away from the unit circle, as illustrated in the following examples. 18 Example K = 2 −3 x 10 IMPULSE RESPONSE 1 0.5 0 −0.5 −1 0 10 20 30 40 50 60 70 80 90 100 FREQUENCY RESPONSE 0.025 0.02 0.015 0.01 0.005 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 19 Example K = 3 −3 x 10 IMPULSE RESPONSE 1.5 1 0.5 0 −0.5 −1 −1.5 0 10 20 30 40 50 60 70 80 90 100 FREQUENCY RESPONSE 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 20 Example K = 4 −3 x 10 IMPULSE RESPONSE 3 2 1 0 −1 −2 −3 0 10 20 30 40 50 60 70 80 90 100 FREQUENCY RESPONSE 0.05 0.04 0.03 0.02 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 21 Example K = 5 −3 x 10 IMPULSE RESPONSE 4 2 0 −2 −4 0 10 20 30 40 50 60 70 80 90 100 FREQUENCY RESPONSE 0.06 0.05 0.04 0.03 0.02 0.01 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 22 Example K = 6 −3 x 10 IMPULSE RESPONSE 5 0 −5 0 10 20 30 40 50 60 70 80 90 100 FREQUENCY RESPONSE 0.08 0.06 0.04 0.02 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 23 Example K = 2 ZERO DIAGRAM FREQUENCY RESPONSE 0.025 1 0.02 0.5 0.015 100 0 0.01 −0.5 0.005 −1 0 −1 −0.5 0 0.5 1 0 0.05 0.1 0.15 0.2 Lowpass differentiator (K = 2, N = 101).