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
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
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