Ideal Filter Digital Signal Processing--FIR Digital Filter Design Dl@)- -8 |o≤0 .The frequency characteristics are ，o.≤o≤π periodic and even functions. dp(k)=sin(@k)/() .Phase are all considered zero. D:@)= ≤0 如 .Sharp edges at cutoff frequencies dnp(k)=6(k)-sin(@k)/(k) Infinite length sequences are all even (or symmetrical)and real-valued 1, 0。≤⊙≤0 Dgp(@)= o,≤⊙≤π functions. 0 0≤ld≤oa .High pass filter equals to all pass filter dpp(k)= sin(;k)-sin(@k) minus low pass filter 水 0 0。≤⊙≤0 Band pass equals to one low pass filter o,≤⊙≤π minus the other low pass filter. 0≤o≤oa Band stop filter equals to all pass filter dps(k)=5(k)- sin(k)-sin(@k 水 minus band pass filter 上游克通大
Digital Signal Processing—— FIR Digital Filter Design Ideal Filter •The frequency characteristics are periodic and even functions. •Phase are all considered zero. •Sharp edges at cutoff frequencies •Infinite length sequences are all even (or symmetrical) and real-valued functions. •High pass filter equals to all pass filter minus low pass filter •Band pass equals to one low pass filter minus the other low pass filter. •Band stop filter equals to all pass filter minus band pass filter
Ideal Filter Digital Signal Processing--FIR Digital Filter Design Differentiator .The frequency characteristics are D(@)=jo periodic and odd functions. dlk)=cos(a）_sin() .Only imaginary part occurs. k 2 .Infinite length sequences are both odd (or antisymmetrical)and real- Hilbert transformer valued functions. D(o）=-jsign(o) Both filters have d(0)=0. d(k)=1-cos(k) .All have linear phase property. 永 ↑D(O)j D(o)j differentiator ① 0 一π 0 Hilbert transformer 上浒充通大
Digital Signal Processing—— FIR Digital Filter Design Ideal Filter •The frequency characteristics are periodic and odd functions. •Only imaginary part occurs. •Infinite length sequences are both odd (or antisymmetrical) and realvalued functions. •Both filters have d(0)=0. •All have linear phase property. Differentiator Hilbert transformer
Design of linear-phase FIR filters using windows Digital Signal Processing--FIR Digital Filter Design H,(o)=∑h,(n)em h(n)=w(n)d (n -M) 1=0 h,(m)=2元H,(@)edo @)=nww-aewpw)g aamro-ale- h(n）=ha(n)w(n） (time domain） h(n),n=0,1,,M-1 0, otherwise 上游充通大
Digital Signal Processing—— FIR Digital Filter Design Design of linear-phase FIR filters using windows h(n)= w(n)d(n − M)
Rectangular windowing function Digital Signal Processing--FIR Digital Filter Design .Its spectrum W(w) w(n)= 0≤n≤N-1 >The period is w=2T. 0 otherwise >The height of the main lobe is N-1 m,(e)= 1-w N and its bottom width is 4T/N. 1=0 1-1 >The height of the first side lobe N is about 2N/3TTat w=3TT/N. W(o)= ∑ea sin(No/2 e-j(N-1)o/2 n=0 sin(@/2) >The attenuation of side lobe is approximately 13dB. H(o)= in(No/2) >The zero-cross points are sim(o/2） ωk=2kTN,k=士1，±2，， ↑H(o) ±(N-1). .Frequency leakage:because of sharp transition of window edges. .Frequency resolution:because of the 2T/N3dB-width of the main lobe. 上浒究通大粤
Digital Signal Processing—— FIR Digital Filter Design Rectangular windowing function •Its spectrum W(ω) The period is ω=2π. The height of the main lobe is N and its bottom width is 4π/N. The height of the first side lobe is about 2N/3πat ω=3π/N. The attenuation of side lobe is approximately 13dB. The zero-cross points are ωk=2kπ/N, k=±1, ±2,…, ±(N-1). •Frequency leakage: because of sharp transition of window edges. •Frequency resolution: because of the 2π/N3dB-width of the main lobe
Design of linear-phase FIR filters using windows Digital Signal Processing--FIR Digital Filter Design dk)=」Dw) -M≤k≤M 2π d=[d-M,...,d-2,d-1,do,di,d2,...,dm] h=d=[ho,...,hm-2,hM-1,hM,hM+1,hM+2,...,h2m] h(n)=d(n-M),n=0,1,.,N-1 rectangular ◆dk) ◆h(n) rectangular window window w(n) ●● ●最 …-5-4-3-2-1012345…k -2-10123456789…n 上府元通大￥
Design of linear-phase FIR filters using windows Digital Signal Processing—— FIR Digital Filter Design