LRA DSP. Multi-Rate DSP. Applications: Oversampling, Undersampling, Quadrature Mirror Filters. Professor L R Arnaut 1

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1 ulti-rate Application: Overampling, Underampling, Quadrature irror Filter Profeor L R Arnaut

2 ulti-rate Overampling Profeor L R Arnaut

3 Optimal Sampling v. Overampling Sampling at Nyquit rate F =F B Allow perfect recontruction in principle, but Pre-ampling anti-aliaing filter mut have very teep roll-off: High-order analogue filter: expenive, difficult, imprecie, large phae ditortion, Sampling at F >>F B & decimation to F B Larger eparation between image eaier filtering of aliae (lower-order filter Cheaper analogue component; eaier digital than analogue VLSI filter; greater digital complexity Profeor L R Arnaut 3

4 Overampling Noie Reduction Quantiation tep (b-bit ADC, range R: Noie power denity (per unit ampling bandwidth: N Q / Q Total in-band noie power: Low P in for high F : F ' >> F p N = σ = = F / F / F Q FB ( R / B Pin = pn ( f df = = 6F B P in b ' << σ N 6F FB F / P in p N p N σ N [W/H] P in R Q = b F B F F B F Profeor L R Arnaut 4

5 Overampling: Effective Reolution Equivalent β-bit ADC operating at F =F B giving ame noie power a b-bit ADC operating at F >>F B over ame range R: i.e., ( R / β = b b FB F / = ( R / ( F /(F = overampling ratio, β-b = reolution increae β F F log B log( + = b + B B / Profeor L R Arnaut 5

Overampling Ratio Example: = 6 : β=b+, i.e., tandard 6-bit ADC @ F =F B ha equivalent reolution w.r.t. noie power a an overampling 6-bit ADC @ F = 6 F B or a overampling -bit ADC @ F =F B Thu,.

6 Overampling Ratio Example: = 6 : β=b+, i.e., tandard 6-bit F =F B ha equivalent reolution w.r.t. noie power a an overampling 6-bit F = 6 F B or a overampling -bit F =F B Thu,.5 bit length reduction of ADC per doubling of Profeor L R Arnaut 6

7 Σ (Sigma-Delta Converter (b-bit multiplier (analogue (F B Analogue -bit (F B th band (F B integrator ADC Digital LPF b F B (digital -bit DAC Σ quantier downampler Analogue integrator x(n-y(n- + w(n -High-rate overampling allow for differential encoding: only bit needed to quantify change between input and delayed output of -bit ADC, for cloely paced conecutive ample in output of Σ quantier (bitmap of ample increment -th order alia digital LPF eliminate out-of-band quantiation noie (harp cut-off; heavy comp.; remedy: perform multiplication at later tage (down-rate F B intead -Long word length of digital o/p determine overall overampling rate - Profeor L R Arnaut 7

8 Σ Converter Accumulator (integrator & quantier output: w( n = x( n y( n + w( n y( n = w( n + e( n y( n = w( n + e( n = x( n + [ e( n e( n ] noie tranfer function: H ( = Output pdf (one-ided pectrum: p y ( f = H + w(n x(n y(n - ωt - - e DAC (n ( jωt e p ( f = 4in p ( f N N e ADC (n Σ quantier Profeor L R Arnaut 8

9 Σ Overampling Precie pdf of output noie depend on pdf & pectral characteritic of {x(n} Aume: {e(n} i random, uncorrelated, white: N p N = σ = F / / / pdf of output noie: p y ( f = in Q F ( π f 3 T Q F x(n w(n - - e DAC (n + e ADC (n Σ quantier y(n Profeor L R Arnaut 9

10 Overampling: Performance Comparion For efficient overampling >>, f << F : Output noie power for Σ modulator: p y ( f ( π f / F 3F ( Improvement over tandard overampling: Q = F π Q B Py = py( f df = 9F π Q 3 3F P 3 log in = log = [ log ( P y π F 3 B 3 f ] db Profeor L R Arnaut

11 Overampling: Performance Comparion A P in = log = [ log( ] db P =: A=4.8 db y =: A=54.8 db = 6 : A=4.8 db.5 bit length reduction of ADC per doubling of (relative to tandard overampling: i due to noie haping by H(f : noie reduction if f<f B Profeor L R Arnaut

12 Overampling: Application Example: meaurement of indoor E wave propagation: GS Profeor L R Arnaut

13 Overampling: Application GS band (95 H (λ/ = 6. cm cm cm cell ie 5 cm 5 cm cell ie Profeor L R Arnaut 3

14 Overampling: Application IS band (45 H (λ/ = 6 cm cm cm cell ie 5 cm 5 cm cell ie Profeor L R Arnaut 4

15 ulti-rate Underampling Profeor L R Arnaut 5

16 Nyquit Condition Alia-free (ubampling of (dicrete function x Spectrum (Fourier tranformation: convolution + + X ( f = X ( f ϕ δ ( ϕ kf dϕ Thu, n + + k = k = ( t = x( t δ ( t = x( t δ ( t kt = x( kt δ ( t kt T n T k = + + = X ( f ϕ δ ( φ kf T k = + X n( f = X ( f kf T k = No pectral overlap (aliaing, trobocopy iff dϕ F F B Profeor L R Arnaut 6

17 Underampling: : Aliaing Profeor L R Arnaut 7

18 Underampling: : Baeband F F B Applie to baeband ignal (DC-coupled: F B i larget frequency component in ignal Shannon ampling theorem: otivation: avoid pectral overlap of baeband frequency repone that are periodically continued due to ampling operation For bandpa (narrowband modulated ignal (e.g., radio- and optical communication, IF filter, etc.: condition i too conervative: large pectral gap occur becaue F c >> F B -F c Profeor L R Arnaut 8

19 Underampling: Bandpa Aliaing of bandpa ignal i avoided if baeband can be folded periodically around carrier frequency without cauing overlap Range of permiible ample frequencie: Fu Fl F F i.e., u n n n Fu Fl Yield additional permiible lower ampling rate for narrowband ignal without aliaing Practically ueful (lower computation Profeor L R Arnaut 9

20 Underampling: : Application Example : digitiation of analogue F audio F l = 88 H, F u = 8 H (n=4 (n=3 n 5 (n= (nonero gap (n= 54 7 F l F u F l F u 58.7 =88 =8 =76 =6 n=: claical ( Nyquit rate f (H n=: between modulated (I u = ignal and doubled (I l = ignal n=3: D u =/3, I l = n=5: 43.H F 44H 86.4H...5F H Profeor L R Arnaut

21 ulti-rate Quadrature irror Filter for Subband Coding Profeor L R Arnaut

22 Subband Coding Problem tatement: Efficient tranmiion of realitic peech or video ignal Contain mot energy at relative low frequencie (time/pace Coding cheme to be tailored to aign more bit to LF band Solution: Subband coding: divide total frequency band in unequal ubband; narrowet ubband for interval with highet energy (equaliation of power acro band each ubband i encoded eparately i alternative to companding (pre-ditortion Profeor L R Arnaut

23 Subband Coding Example: S(f 5 3 bit/ample 9f m /96 (I/D=/3 9f m /3 (I/D=9/6 f m / (D= (approximate equaliation - ulti-rate converion by factor I/D after each frequency ubdiviion (LPF/HPF - Reduced bitrate of digitied ignal (bandwidth compreion due to nonuniform coding (variable number of bit per ample f m Profeor L R Arnaut 3

24 Subband Coding Implementation: Brickwall Filter: Quadrature irror Filter (QF: Phyically aliaing for decimated ubband can be removed unrealiable by judiciou choice of H (ω and H (ω Profeor L R Arnaut 4

25 Two-Channel QF Implementation (analyer / yntheier: (for I/D=/ Profeor L R Arnaut 5

26 LRA QF Analyer: Two Two-Channel QF: Analyi Channel QF: Analyi D = (, exp exp = = D k D D k D j X k D j H D X π π ( + = π ω π ω ω ω ω X H X H X ( + = π ω π ω ω ω ω X H X H X Profeor L R Arnaut 6

27 Two-Channel QF: Synthei QF Syntheier: I V ( = Y (, I = ( ω = G ( ω Y ( ω G ( ω ( ω Y + Y Cacaded QF analyer-yntheier: ( = X ( ω Y ( ω ( ω Y, = X ω Aliaing (k= ( ω = [ H ( ω G ( ω + H ( ω G ( ω ] X ( ω + [ H ( ω π G ( ω + H ( ω π G ( ω ] X ( ω π Y Profeor L R Arnaut 7

28 QF Anti-Aliaing Aliaing Elimination of aliaing for any input ignal: H e.g. ( ω π G ( ω + H ( ω π G ( ω = ( ω H ( ω π G ( ω = ( ω π G, H = reult in time-invariant filter example: alia-free ymmetric ubband coding ( ω H ( ω G ( ω = ( ω π G, H = Profeor L R Arnaut 8

29 QF Perfect Recontruction Ditortion-free & alia-free recontruction: H H ( ω G ( ω + H ( ω G ( ω = Dexp( jkω, D = ( ω H ( ω π H ( ω H ( ω π = Dexp( jk ω Example: ymmetric ubband H ( ω H ( ω π = Dexp( jkω i.e., H ( ω H ( ω π independent of ω (all-pa filter, but may exhibit phae ditortion! It can be hown: linear-phae FIR QF caue amplitude ditortion Profeor L R Arnaut 9

30 LRA branche; in analyer, in yntheier Output k th analyer branch (BPF+D: Output yntheier (I+BPF: Profeor L R Arnaut 3 -Channel QF Bank Channel QF Bank ( ( ( = = k k k Y G Y ( (, exp exp / / D m j X m j H X m k k = = = π π ( ( = = = exp exp m k k k m j X m j H G Y π π (, exp = = m m m j X L π ( = = ( exp k k k m G m j H L π

31 -Channel QF Alia-free QF: = L ( ω X ( ω iff Lm ( X exp j =, X ( Y ω ( m= π m =, m i.e. ( L m Ditortion-free & alia-free QF: L ( ω independent of ω (all-pa filter Profeor L R Arnaut 3

Chapter 2 Sampling and Quantization. In order to investigate sampling and quantization, the difference between analog

Chapter 2 Sampling and Quantization. In order to investigate sampling and quantization, the difference between analog Chapter Sampling and Quantization.1 Analog and Digital Signal In order to invetigate ampling and quantization, the difference between analog and digital ignal mut be undertood. Analog ignal conit of continuou