EE123 Digital Signal Processing

Transcription

1 Announcements EE23 Digital Signal Processing ecture 5 ast Time D.T processing of C.T signals C.T processing of D.T signals (ha????) D.T are represented as bandlimited C.T signals Fractional delay Resampling Today: Resampling Interchanging operations multi-rate processing 2 DownSampling Much like C/D conversion Expect similar effects: Aliasing mitigate by antialiasing filter Changing Sampling-rate via D.T Processing d (e j! )= M i=0 j(w/m e 2i/M) Finely sampled signal almost continuous Downsample in that case is like sampling! d M=2 3 4

2 Changing Sampling-rate via D.T Processing j(w/m e d (e ) = M i=0 j! Anti-Aliasing 2i/M ) /M x d [n] = x [nm ] M=3 d x [n] PF M=3 d 5 6 U U ;.::') /J 4/ IJ crij} ;.::') /J 4/ IJ crij} oml UpSampling Up-sampling -s - s oml Much like D/C converter Upsample by A OT almost continuous '"\r ( l/l---, ( l/l---, '"\r..)( [-) '"'"""" ' / (/..)( ii[-) =.=.'"'"""" C C' (. '- Tl I./ Tl infej& infej &. '- I./ Intuition: Recall our D/C model: xs(t) xc(t) Approximate xs(t) by placing zeros between samples Convolve with a sinc to obtain xc(t) It 7 It 8

3 Up-Sampling Up-Sampling It xi [n] = xe [n] sinc(n/) xe [n] = x[k] [n k] k= xi [n] = x[k]sinc( n k= 9 0 Frequency Domain Interpretation Frequency Domain Interpretation xe [n] PF xi [n] gain= / e (ej! ) = sinc(n/) DTFT = m= gain= /M k= PF xe [n] k ) xe [n] e {z } xi [n] j!n 0 only for n=m (integer m) xe [m] e {z } j!m = (ej! ) =x[m] Compress DTFT by a factor of! 2

4 c (j) c (j) N (e j! ) i (e j! ) =T/ N (e j! ) i (e j! ) =T/ /T e (e j! ) expanding expanding e (e j! ) 3 4 c (j) c (j) N (e j! ) i (e j! ) /T e (e j! ) expanding =T/ /T 5 N (e j! ) i (e j! ) /T e (e j! ) expanding /T =T/ /T 6

5 c (j) N (e j! ) i (e j! ) /T e (e j! ) expanding /T =T/ /T 7 Practical Upsampling Can interpolate with simple, practical filters. See ab! =3, linear interpolation /T ideal 3 sinc 2 8 Resampling by non-integer T = TM/ (upsample, downsample M) = 2, M=3, T =3/2T (fig 4.30) c (j) Or, PF gain= / PF /M (e j! ) N Subsampling M=3 PF min{ /, /M } expanding =2 P filtering What would happen if change order? e (e j! ) i = H d e 9 20

6 = 2, M=3, T =3/2T (fig 4.30) (e j! ) expanding =2 e (e j! ) /T /T c (j) N Subsampling M=3 P filtering i = H d e 2/(3T) 2/T Multi-Rate Signal Processing What if we want to resample by.0t? Expand by =00 Filter π/0 ($$$$$) Downsample by M=0 Fortunately there are ways around it! Called multi-rate Uses compressors, expanders and filtering 2 22 Interchanging Operations# Interchanging Operations# Note: expander H(e j! )(e j! ) H(e j! )(e j! ) compressor not TI! (e j! ) H(ej! )(e j! ) Note: expander H(e j! )(e j! ) H(e j! )(e j! ) compressor not TI! (e j! ) H(ej! )(e j! ) H(z ) (e j! ) H(ej! )(e j! ) 23 24

7 Interchanging Filter Expander Q: Can we move expander from eft to Right (with xform)?? H(z ) A: Yes, if is rational No, otherwise Compressor Compressor Proof: Claim: H(z M ) y [n] v[n] Proof: 27 v(e}") m/(-'j,x (e Jv) Jv after compressor v(e}") m/(-'j,x (e ) q: Jvfs-. (Ol"-!Pto q: (Ol"-!Pon.b Jvfs-. 4. UtI -Y-") IJ li\lj. 28! to

8 Compressor Multi-Rate Filtering Claim: -J.ly-,) H(z M ) >,,) I,V) - btta 6J 'Jg M::-})- \(,] y [n] i F-EJ} ") xun v[n] Proof: )lchl-fj'llohjjt t,)(t.] f\\ vv(e}") m/(-'jm/(-'j,x (e Jv) m/(-'j,x v(e}"),x (e Jv) (e}") (e Jv) q: Jvfs-. (Ol"-!Pto Jvfs-. (Ol"-!Pto Jvfs-. on.b 4. q: UtI -Y-") IJ(Ol"-!Pli\lJ. on.b 4. UtI -Y-") IJ li\lj.! q:! on.b 4.! after compressor UtI -Y-") to IJ li\lj