Download Linear Phase Finite Impulse Response (FIR) Digital Filters and more Summaries Microelectronic Circuits in PDF only on Docsity! 1 Finite Impulse Response (FIR) Digital Filters (I) Types of linear phase FIR filters Yogananda Isukapalli 2 Key characteristic features of FIR filters 1. The basic FIR filter is characterized by the following two equations: å - = -= 1 0 )()()( N k knxkhny å - = -= 1 0 )()( N k kzkhzH where h(k), k=0,1,…,N-1, are the impulse response coefficients of the filter, H(z) is the transfer function and N the length of the filter. (1) (2) 2. FIR filters can have an exactly linear phase response. 3. FIR filters are simple to implement with all DSP processors available having architectures that are suited to FIR filtering. 5 H(ejw) = k e-jwa passband 0 otherwise Magnitude response = |H(ejw)| = k Phase response (q(w)) = < H(ejw) = -wa Follows: y[n] = kx[n-a] : Linear phase implies that the output is a replica of x[n] {LPF} with a time shift of a - p -wu -wl 0 wl wu p w - p -wu -wl 0 wl wu p w 6 Linear phase FIR filters • Symmetric impulse response will yield linear phase FIR filters. 1. Positive symmetry of impulse response: h(n) = h(N-n-1), n = 0,1,…, (N-1)/2 (N odd) n = 0,1,…, (N/2)-1 (N even) a = (N-1)/2 in eqn (5) 1. Negative symmetry of impulse response: h(n) = -h(N-n-1), a = (N-1)/2 b = p/2 in eqn (6) n = 0,1,…, (N-1)/2 (N odd) n = 0,1,…, (N/2)-1 (N even) 7 h[n] = h[N-1-n] n = 0,1,….(N-1)/2 h[0] = h[10] h[1] = h[9] h[2] = h[8] h[3] = h[7] h[4] = h[6] h[5] = h[5] Example) For positive symmetry and N = 11 odd length 10 Consider Frequency Response : +++= == --- = wjjwjwwjjw ez jw eeheheheH zHeH jw 22 ]]3[]1[[]2[)( ) 1T ( )()( wjwjwj eeheh 222 ]]4[]0[[ --+ [ ] |)(| ))2(cos(][2]2[[ 2cos]0[2cos]1[2]2[[ 2 1 0 2 2 wjwj n wj wj eHe nwnhhe whwhhe q- = - - = ú û ù ê ë é -+= ++= å Phase = -2w ( Linear Phase form) 11 Group Delay : )( phase dw dTg - = = 2 passband 2 w 0 wp p w H 0 wp p w Group delay is constant over the passband for linear phase filters. gT 12 Types of FIR linear phase systems 1. Type I FIR linear phase system The impulse response is positive symmetric and N an odd integer ],1[][ nNhnh --= 2/)1(0 -££ Nn The frequency response is å - = -= 2/)1( 0 ][)( N n jwnjw enheH å - = --= 2/)1( 0 2/)1( )cos(][)( N n Njwjw wnnaeeH ],)2/)1[((2][ ],2/)1[(]0[ nNhna Nha where --= -= .2/)1,...(2,1 -= Nn 15 4. Type IV FIR linear phase system The impulse response is negative-symmetric and N an even integer. ],1[][ nNhnh ---= 1)2/(0 -££ Nn The frequency response is å - = -= 1)2/( 0 ][)( N n jwnjw enheH ,)] 2 1(sin[][)( 2/ 1 2/)1( ïþ ï ý ü ïî ï í ì -= å = -- N n Njwjw nwnbjeeH ],2/[2][ nNhnb where -= .2/,...,2,1 Nn = 16 Fig: A summary of four types of linear phase FIR filters1 17 Fig: A comparison of the impulse of the four types of linear phase FIR filters1
