1. 2003-10-14Dan Ellis 1 ELEN E4810: Digital Signal Processing Topic 6: Filters - Introduction 1.Simple Filters 2.Ideal Filters 3.Linear Phase and FIR filter types

2. 2003-10-14Dan Ellis 2 1. Simple Filters Filter = system for altering signal in some ‘useful’ way LSI systems: are characterized by H(z) (or h[n] ) have different gains (& phase shifts) at different frequencies can be designed systematically for specific filtering tasks

3. 2003-10-14Dan Ellis 3 FIR & IIR FIR = finite impulse response  no feedback in block diagram  no poles (only zeros) IIR = infinite impulse response  poles (and perhaps zeros)

4. 2003-10-14Dan Ellis 4 Simple FIR Lowpass h L [n] = { 1 / 2 1 / 2 } (2 pt moving avg.)   ZP zero at z = -1 n 1234 -2 1/21/2 hL[n]hL[n]   HL(ej)HL(ej) 1 / 2 sample delay e j  /2 +e -j  /2

5. 2003-10-14Dan Ellis 5 Filters are often characterized by their cutoff frequency  c : Cutoff frequency is most often defined as the half-power point i.e. If then Simple FIR Lowpass cc H(ej)H(ej)

6. 2003-10-14Dan Ellis 6 deciBels Filter magnitude responses are very often described in deciBels (dB) dB is simply a scaled log value: Half-power also known as 3dB point: power = level 2

7. 2003-10-14Dan Ellis 7 We often plot magnitudes in dB: A gain of 0 corresponds to -  dB deciBels

8. 2003-10-14Dan Ellis 8 Simple FIR Highpass h H [n] = { 1 / 2 - 1 / 2 } 3dB point  c =  / 2 ( again)  ZP zero at z = 1   HH(ej)HH(ej) n 234 -2 1/21/2 hH[n]hH[n] -1/2-1/2 cc

9. 2003-10-14Dan Ellis 9 FIR Lowpass and Highpass Note: h L [n] = { 1 / 2 1 / 2 } h H [n] = { 1 / 2 - 1 / 2 } i.e. i.e. 180° rotation of the z-plane,   shift of frequency response  z  j -j  -z  -j j  HL(ej)HL(ej)   HH(ej)HH(ej)

10. 2003-10-14Dan Ellis 10 Simple IIR Lowpass IIR  feedback, zeros and poles, conditional stability, h[n] less useful scale to make gain = 1 at  = 0  K = (1 -  )/2 pole-zero diagram frequency response FR on log-log axes

11. 2003-10-14Dan Ellis 11 Simple IIR Lowpass Cutoff freq.  c from max = 1 using K=(1-  )/2 Design Equation

12. 2003-10-14Dan Ellis 12 Simple IIR Highpass Pass  =   H HP (-1) = 1  K = (1+  )/2 Design Equation: (again)

13. 2003-10-14Dan Ellis 13 Highpass and Lowpass Consider lowpass filter: Then: However, (unless H(e j  ) is pure real - not for IIR) Highpass  1  H LP z   1  H z  c/w (-1) n h[n] just another z poly

14. 2003-10-14Dan Ellis 14 Simple IIR Bandpass where Center freq 3dB bandwidth Design

15. 2003-10-14Dan Ellis 15 Simple Filter Example Design a second-order IIR bandpass filter with  c = 0.4 , 3dB b/w of 0.1   M sensitive..

16. 2003-10-14Dan Ellis 16 Simple IIR Bandstop Design eqns: zeros at  c same poles as H BP

17. 2003-10-14Dan Ellis 17 Cascading Filters Repeating a filter (cascade connection) makes its characteristics more abrupt: Repeated roots in z-plane: H(ej)H(ej)H(ej)H(ej)H(ej)H(ej)H(ej)H(ej)  H(ej)H(ej)  H(ej)H(ej)  ZP

18. 2003-10-14Dan Ellis 18 1/81/8 1/41/4 Cascading Filters Cascade systems are higher order e.g. longer (finite) impulse response: In general, cascade filters will not be optimal (...) for a given order n 234 -2 1/21/2 h[n]h[n] -1/2-1/2 n 234 -2 -1/2-1/2 n 24 -2 -3/8-3/8 h[n]h[n]h[n]h[n]h[n]h[n]h[n]h[n]h[n]h[n]

19. 2003-10-14Dan Ellis 19 Cascading Filters Cascading filters improves rolloff slope: But: 3dB cutoff frequency will change (gain at  c  3N dB) cc

20. 2003-10-14Dan Ellis 20 2. Ideal filters Typical filter requirements: gain = 1 for wanted parts (pass band) gain = 0 for unwanted parts (stop band) “Ideal” characteristics would be like: no phase distortion etc. What is this filter? can calculate IR h[n] as IDFT of ideal response...  |H(e j  )| “brickwall LP filter”

21. 2003-10-14Dan Ellis 21 Ideal Lowpass Filter Given ideal H(e j  ) : (assume  )   cc  c Ideal lowpass filter

22. 2003-10-14Dan Ellis 22 Ideal Lowpass Filter Problems! doubly infinite ( n = - ..  ) no rational polynomial  very long FIR excellent frequency-domain characteristics  poor time-domain characteristics (blurring, ringing – a general problem)

23. 2003-10-14Dan Ellis 23 lower-order realization (less computation) better time-domain properties (less ringing) easier to design... Practical filter specifications |H(e j  )| / dB  -40 1 Pass bandStop band Transition allow PB ripples allow SB ripples allow transition band

24. 2003-10-14Dan Ellis 24 | H(e j  ) | alone can hide phase distortion differing delays for adjacent frequencies can mangle the signal Prefer filters with a flat phase response e.g.  “zero phase filter” A filter with constant delay  p = D at all freq’s has  D  “linear phase” Linear phase can ‘shift’ to zero phase 3. Linear-phase Filters pure-real (zero-phase) portion

25. 2003-10-14Dan Ellis 25 Time reversal filtering v[n] = x[n]  h[n]  V(e j  ) = H(e j  )X(e j  ) u[n] = v[-n]  U(e j  ) = V(e -j  ) = V * (e j  ) w[n] = u[n]  h[n]  W(e j  ) = H(e j  )U(e j  ) y[n] = w[-n]  Y(e j  ) = W * (e j  ) = (H(e j  )(H(e j  )X(e j  )) * ) *  Y(e j  ) = X(e j  )  H(e j  )  2 Achieves zero-phase result Not causal! Need whole signal first H(z)H(z) x[n]x[n] Time reversal H(z)H(z) y[n]y[n] v[n]v[n]u[n]u[n]w[n]w[n] if v real

26. 2003-10-14Dan Ellis 26 Linear Phase FIR filters (Anti)Symmetric FIR filters are almost the only way to get zero/linear phase 4 types: Odd lengthEven length Symmetric Anti- symmetric Type 1Type 2 Type 3 Type 4 n n n n

27. 2003-10-14Dan Ellis 27 Linear Phase FIR: Type 1 Length L odd  order N = L - 1 even Symmetric  h[n] = h[N - n] ( h[N/2] unique) linear phase D = -  (  )/  = N/2 pure-real H(  ) from cosine basis ~

28. 2003-10-14Dan Ellis 28 Linear Phase FIR: Type 1 Where are the N zeros? thus for a zero  Reciprocal zeros (as well as cpx conj) Reciprocal pair Conjugate reciprocal constellation No reciprocal on u.circle

29. 2003-10-14Dan Ellis 29 Linear Phase FIR: Type 2 Length L even  order N = L - 1 odd Symmetric  h[n] = h[N - n] (no unique point) Non-integer delay of N/2 samples H(  ) from double-length cosine basis ~ always zero at 

30. 2003-10-14Dan Ellis 30 Linear Phase FIR: Type 2 Zeros: at z = -1, odd LPF-like

31. 2003-10-14Dan Ellis 31 Linear Phase FIR: Type 3 Length L odd  order N = L - 1 even Antisymmetric  h[n] = -h[N - n]  h[N/2]= -h[N/2] = 0  (  ) =  /2 -  ·N/2 Antisymmetric   /2 phase shift in addition to linear phase

32. 2003-10-14Dan Ellis 32 Linear Phase FIR: Type 3 Zeros:

33. 2003-10-14Dan Ellis 33 Linear Phase FIR: Type 4 Length L even  order N = L - 1 odd Antisymmetric  h[n] = -h[N - n] (no center point) fractional-sample delay  offset offset sine basis

34. 2003-10-14Dan Ellis 34 Linear Phase FIR: Type 4 Zeros: ( H(-1) OK because N is odd)

35. 2003-10-14Dan Ellis 35 4 Linear Phase FIR Types Odd lengthEven length Antisymmetric Symmetric 12 34 h[n]h[n] n  H()H() ~ ZP h[n]h[n] n  H()H() ~ DD  h[n]h[n] n  H()H() ~ D  h[n]h[n] n  H()H() ~ D 