Diniz, da Silva and Netto FIR Filter Approximations Paulo S. R. Diniz Eduardo A. B. da Silva Sergio L. Netto diniz,eduardo,[email protected] September 2010 1 Diniz, da Silva and Netto Contents Ideal characteristics of standard filters • – Lowpass, highpass, bandpass, and bandstop filters – Differentiators – Hilbert transformers FIR filter approximation by frequency sampling • FIR filter approximation by window functions • – Rectangular window – Tr iangular windows – Hamming and Hann windows – Blackman window – Kaiser window – Dolph-Chebyshev window 2 Diniz, da Silva and Netto Contents (cont.) Maximally-flat FIR filter approximation • FIR filter approximation by optimization • – Weighted-least-squares method – Chebyshev method – WLS-Chebyshev method Do-it-yourself: FIR filter approximations • 3 Diniz, da Silva and Netto Ideal characteristics of standard filters As seen in Section 2.8, the relationship between H(e jω) and h(n) is given by the • following pair of equations: jω − jωn H(e )= ∞ h(n)e (2) n"=− 1 ∞π h(n)= H(e jω)e jωndω (3) 2π #−π In what follows, we determine H(e jω) and h(n) related to ideal standard filters. • 8 Diniz, da Silva and Netto Lowpass, highpass, bandpass, and bandstop filters The ideal magnitude responses of some standard digital filters are depicted below. • j! j! |H(e )| |H(e )| 1 1 2 0 (a) !c " " ! (b) !c " 2" ! j! j! | H(e )| | H(e ) | 1 1 0 !c !c " 2" ! 0 ! !c " 2" ! (c) 1 2 (d) c1 2 Figure 1: Ideal magnitude responses: (a) lowpass; (b) highpass; (c) bandpass; (d) band- stop filters. 9 Diniz, da Silva and Netto Lowpass, highpass, bandpass, and bandstop filters For instance, the lowpass filter, as seen in Figure 1a, is described by • 1, for |ω| ωc |H(e jω)| = ≤ (4) 0, for ωc < |ω| π ≤ Using (3), the impulse response for the ideal lowpass filter is • ωc , for n = 0 ωc 1 jωn π h(n)= e dω = (5) 2π #−ω sin(ωcn) c , for n = 0 πn " One should note that in the above inverse transform calculations we have supposed • that the phase of the filter is zero. 10 Diniz, da Silva and Netto Lowpass, highpass, bandpass, and bandstop filters From Section 4.2.3, we have that the phase of an FIR filter must be of the form • − jω M e 2 ,whereM is an integer. M Therefore, for M even, it suffices to shift the above impulse response by 2 • M samples. However, for M odd, 2 is not an integer, and the impulse response must be computed as ωc 1 − jω M jωn h(n)= e 2 e dω 2π #−ωc ωc 1 jω n− M = e ( 2 )dω 2π #−ωc M sin ωc n − = 2 (6) π n − M ! " 2 #$ " # 11 Diniz, da Silva and Netto Lowpass, highpass, bandpass, and bandstop filters Likewise, for bandstop filters, the ideal magnitude response, depicted in Figure 1d, is • given by 1, for 0 |ω| ωc1 ≤ ≤ |H(e jω)| = | | (7) 0, for ωc1 < ω < ωc2 1, for ωc2 |ω| π ≤ ≤ Then, using (3), the impulse response for such an ideal filter is • ω π −ω 1 c1 c2 h(n)= e jωndω + e jωndω + e jωndω 2π # # # % −ωc1 ωc2 −π & ωc − ωc 1 + 1 2 , for n = 0 = π (8) 1 [sin(ωc n) − sin(ωc n)] , for n = 0 πn 1 2 " 12 Diniz, da Silva and Netto Differentiators An ideal discrete-time differentiator is a linear system that, when samples of a • band-limited continuous signal are used as input, the outputsamplesrepresentthe derivative of the continuous signal. π π More precisely, given a continuous-time signal xa(t) band-limited to − , , • T T when its corresponding sampled version x(n)=x (nT) is input to an ideal a ! # differentiator, it produces the output signal, y(n),suchthat dxa(t) y(n)= (9) dt 't=nT ' ' If the Fourier transform of the continuous-time signal' is denoted by Xa(jΩ),we • have that the Fourier transform of its derivative is jΩXa(jΩ). 14 Diniz, da Silva and Netto Differentiators Therefore, an ideal discrete-time differentiator is characterized by a frequency • response, up to a multiplicative constant, of the form H(e jω)=jω, for −π ω < π (10) ≤ The magnitude and phase responses of a differentiator are depicted in Figure 2. • 15 Diniz, da Silva and Netto Differentiators | H (e j ! ) | ... ... (a) – 2" – " " 2" ! # (! ) _" 2 ... ... – 2 " – " " 2 " ! – _" 2 (b) Figure 2: Characteristics of an ideal discrete-time differentiator: (a) magnitude response; (b) phase response. 16 Diniz, da Silva and Netto Differentiators Using equation (3), the corresponding impulse response is given by • 1 π h(n)= jωe jωndω 2π#−π 0, for n = 0 = π 1 ω 1 (−1)n e jωn − = , for n = 0 2π n jn2 n " ( ) *+'−π ' ' (11) ' One should note that if a differentiator is to be approximatedbyalinear-phaseFIR • filter, one should necessarily use either a Type-III or a Type-IV form. 17 Diniz, da Silva and Netto Hilbert transformers These equations provide a relation between the Fourier transforms of the real and • imaginary parts of a signal whose Fourier transform is null for −π ω <0.Itthus ≤ implies that the ideal Hilbert transformer has the followingtransferfunction: −j, for 0 ω < π H(e jω)= ≤ (17) j, for − π ω <0 ≤ The magnitude and phase components of such a frequency response are depicted in • Figure 3. 22 Diniz, da Silva and Netto Hilbert transformers | H (e j ! ) | ... ... (a) –2 " – " " 2 " ! #(!) _" ... 2 −2" −" " 2" ! −_" ... 2 (b) Figure 3: Characteristics of an ideal Hilbert transformer: (a) magnitude response; (b) phase response. 23 Diniz, da Silva and Netto Hilbert transformers Using equation (3), the corresponding impulse response for the ideal Hilbert • transformer is given by π 0 0, for n = 0 1 jωn jωn h(n)= −je dω+ je dω = 2π #0 #−π 1 n % & [1−(−1) ] , for n = 0 πn " (18) By examining equation (17) we conclude, as in the case of the differentiator, that a • Hilbert transformer must be approximated, when using a linear-phase FIR filter, by either a Type-III or Type-IV structure. 24 Diniz, da Silva and Netto Summary Table 1: Ideal frequency characteristics and correspondingimpulseresponsesforlow- pass, highpass, bandpass, and bandstop filters, as well as fordifferentiatorsand Hilbert transformers. Filter type Magnitude response Impulse response |H(e jω )| h(n) ωc , for n = 0 1, for 0 ≤ |ω| ≤ ωc π Lowpass 0, for ω < |ω| ≤ π 1 c sin(ω n), for n "= 0 πn c ωc 1 − , for n = 0 0, for 0 ≤ |ω| < ωc π Highpass 1 1, for ωc ≤ |ω| ≤ π − sin(ω n), for n "= 0 πn c 27 Diniz, da Silva and Netto Summary Filter type Magnitude response Impulse response |H(e jω )| h(n) for | | 0, 0 ≤ ω < ωc1 (ωc2 − ωc1 ) , for n = 0 Bandpass for | | π 1, ωc1 ≤ ω ≤ ωc2 1 [sin(ωc2 n) − sin(ωc1 n)] , for n "= 0 0, for ωc2 < |ω| ≤ π πn for | | 1, 0 ≤ ω ≤ ωc1 (ωc2 − ωc1 ) 1 − , for n = 0 Bandstop for | | π 0, ωc1 < ω < ωc2 1 [sin(ωc1 n) − sin(ωc2 n)] , for n "= 0 1, for ωc2 ≤ |ω| ≤ π πn 28 Diniz, da Silva and Netto Summary Filter type Frequency response Impulse response H(e jω ) h(n) 0, for n = 0 Differentiator jω, for −π ≤ ω < π (−1)n , for n "= 0 n Hilbert −j, for 0 ≤ ω < π 0, for n = 0 1 n transformer j, for − π ≤ ω <0 [1 − (−1) ] , for n "= 0 πn 29 Diniz, da Silva and Netto FIR filter approximation by frequency sampling In general, the problem of FIR filter design is to find a finite-length impulse response • h(n),whoseFouriertransformH(e jω) approximates a given frequency response well enough. As seen in Section 3.2, one way of achieving such a goal is by noting that the DFT of • alength-N sequence h(n) corresponds to samples of its Fourier transform at the 2πk frequencies ω = N ,thatis N−1 − H(e jω)= h(n)e jωn (19) n"=0 and then N−1 2πk − 2πkn H(e j N )= h(n)e j N , for k = 0, 1, . ., (N − 1) (20) n"=0 30 Diniz, da Silva and Netto FIR filter approximation by frequency sampling It is then natural to consider designing a length-N FIR filter by finding an h(n) • whose DFT corresponds exactly to samples of the desired frequency response. In other words, h(n) can be determined by sampling the desired frequency • j 2π k response at the N points e N and finding its inverse DFT. This method is generally referred to as the frequency sampling approach. More precisely, if the desired frequency response is given by D(ω),onemustfirst • find ( ) ωsk A(k)e jθ k = D , for k = 0, 1, . ., (N − 1) (21) N ) * where A(k) and θ(k) are samples of the desired amplitude and phase responses, respectively. 31 Diniz, da Silva and Netto FIR filter approximation by frequency sampling Type I: Even order M and symmetrical impulse response. In this case, the phase • and amplitude responses must satisfy πkM θ(k)=− , for 0 k M (22) M + 1 ≤ ≤ M A(k)=A(M − k + 1), for 1 k (23) ≤ ≤ 2 and then, the impulse response is given by M 1 2 πk(1 + 2n) h(n)= A(0)+2 (−1)kA(k) cos (24) M + 1 M + 1 k"=1 for n = 0, 1, . ., M. 33 Diniz, da Silva and Netto Table 2: Impulse responses for linear-phase FIR filters with frequency sampling approach. Filter type Impulse response Condition h(n),for n = 0, 1, .