1. Finite-duration impulse response (FIR) filter (Order=M-1, Length=M) The impulse response The system function The difference equation (Linear convolution) The frequency response (DTFT) Linear-phase FIR filters

2. Linear-phase FIR filter (Order=M-1, Length=M)  = 0,  /2.  is a constant Four types Linear-phase FIR filters

3. Type-1 linear-phase FIR filter (Symmetrical, M odd)  = 0.  = (M-1)/2 (integer) Symmetric about  (the index of symmetry) Linear-phase FIR filters

4. Type-2 linear-phase FIR filter (Symmetrical, M even)  = 0.  = (M-1)/2 (non-integer) Linear-phase FIR filters

5. Type-3 linear-phase FIR filter (Antisymmetrical, M odd)  =  /2.  = (M-1)/2 (integer) Linear-phase FIR filters

6. Type-4 linear-phase FIR filter (Antisymmetrical, M even)  =  /2.  = (M-1)/2 (non-integer) Linear-phase FIR filters

7. Design Specs (LPF) FIR LPF filter specifications: (a) Absolute (b) Relative Band [0,  p ]: pass band Band [  s,  ]: stop band Band [  p,  s ]: transition band  1 : Absolute ripple in pass band  2 : Absolute ripple in stop band R p : Relative ripple in pass band (in dB) A s : Relative ripple in stop band (in dB)

8. Design Specs (LPF)

9. Window Design Techniques Shifting

10. Window Design Techniques General Design Procedures: : Ideal frequency response (given) Step 1 Step 2 Step 3 Window function symmetric about  over 0 otherwise

11. Window Design Techniques Rectangular Window Exact transition width =  s -  p = 1.8  /M Min. stopband attenuation = 21dB MATLAB function: w=boxcar (M)

12. Window Design Techniques Bartlett Window Exact transition width =  s -  p = 6.1  /M Min. stopband attenuation = 25dB MATLAB function: w=bartlett (M)

13. Window Design Techniques Hann Window Exact transition width =  s -  p = 6.2  /M Min. stopband attenuation = 44dB MATLAB function: w=hann (M)

14. Window Design Techniques Hamming Window Exact transition width =  s -  p = 6.6  /M Min. stopband attenuation = 53dB MATLAB function: w=hamming (M)

15. Window Design Techniques Blackman Window Exact transition width =  s -  p = 11  /M Min. stopband attenuation = 74dB MATLAB function: w=blackman (M)

16. Window Design Techniques LPF Design function hd=ideal_lp(wc,M) %hd: ideal LPF impulse response between 0 and M-1 %wc: cut-off frequencies in radians %M: length of the filter alpha=(M-1)/2; n=[0:M-1]; m=n-alpha; fc=wc/pi; hd=fc*sinc(fc*m);

17. Window Design Techniques LPF Design %Example 1 in FIR filter design wp=0.2*pi; ws=0.3*pi; tr_width=ws-wp; M=ceil(6.6*pi/tr_width)+1; n=[0:M-1]; wc=(ws+wp)/2; %ideal cutoff frequency hd=ideal_lp(wc,M); w_hamming=(hamming(M))'; h=hd.*w_hamming; figure(1);stem(n,h); title('h(n)') figure(2);freqz(h,[1]) Example:  p =0.2   s =0.3  A s =50 dB

18. Window Design Techniques

20. BPS example:  1s =0.2 ,  1p =0.35 ,  2p =0.65 ,  2s =0.8  A s =60 dB Two transition band widths must be the same!

21. Window Design Techniques %Example 2 in FIR filter design % BPS design wp1=0.35*pi; ws1=0.2*pi; wp2=0.65*pi; ws2=0.8*pi; %only one transition bandwidth allowed in window design tr_width=min(wp1-ws1,ws2-wp2); M=ceil(11*pi/tr_width)+1; n=[0:M-1]; wc1=(ws1+wp1)/2; %ideal cutoff frequency 1 wc2=(ws2+wp2)/2; %ideal cutoff frequency 2 hd=ideal_lp(wc2,M)-ideal_lp(wc1,M); w_blackman=(blackman(M))'; h=hd.*w_blackman; figure(1);stem(n,h); title('h(n)') figure(2);freqz(h,[1])

22. Window Design Techniques

24. Example: Digital differentiator %Example 2 in FIR filter design % Digital differentiator design M=21;alpha=(M-1)/2;n=0:M-1; hd=(cos(pi*(n-alpha)))./(n-alpha);hd(alpha+1)=0; w_ham=(hamming(M))'; h=hd.*w_ham; [H,W]=freqz(h,[1]); plot(W/pi,abs(H)); title('Digital differentiator: |H(\omega)|')

25. Window Design Techniques

27. >> t=linspace(-2,2,1000); >> xt=sin(2*pi*t);yt=conv(xt,h); >> subplot(2,1,1);plot(xt);title('x(t)=sin(2\pit)') >> subplot(2,1,2);plot(yt(22:end));title('y(t)=Dx(t)')