Interpolators And Interpolation

Transcription

1 Interpolators And Interpolation 1

2 Applications Fixed Up-Sampler Interpolators Fixed Down-Sample Filters Reduced Cost Filtering When Large Ratio of Sample Rate to Bandwidth Timing Recovery Re-Sampling of Time Series Timing Recovery Re-Sampling of Matched Filter Clock Domain Alignment

3 Spectrum of Interpolator and Periodic Spectrum of Zero-Packed Shaping Filter Spectrum of Shaping Filter and 1-to-3 Interpolating Filter Gain (db) Normalized Frequency Zoom to Spectrum Gain (db) Normalized Frequency 3

4 Spectrum of 1-to-3 Interpolated Shaping Filter Spectrum of Interpolated Shaping Filter Gain (db) Normalized Frequency Zoom to Spectrum Gain (db) Normalized Frequency 4

5 Polyphase Partition of M-Path Resampling Filter N/M= 4 H (Z ) H (Z ) 1 x(n) H (Z ) y(m) H (Z ) M-1 5

6 Efficient Hardware Implementation of 1-to-M Polyphase Interpolator N/M= 4 x(n) y(m ) H r (Z ) h(+ nm) h(1+ nm) h(+ nm)... h(m-1+ nm) 6

7 Interpolation Options Initial Sa m ple Positions Interpola ted Sam p le Positions Initial Sa mple Grid, Unit dista nce Between Sa mples Same Rate Samp le Grid, Unit Dista nce Between Sam ples Higher Rate Sam ple Grid, Less Tha n Unit Distance Between Samp les Lower Rate Sam ple Grid, More Tha n Unit Distance Between Samp les 7

8 M-Path, 1-to-M/Q Interpolator Polypha se Filter x(n) 1:M H(Z) Q:1 y(m) N/M= 4 H (Z ) H (Z ) 1 x(n) H (Z ) Q:1 y(m ) H (Z ) M-1 8

9 5/3, Rational Ratio Re-Sampling K(m+1)=[k(m)+3] modulo(5) x(n) phs() phs(1) phs() phs(3) y (m)= x(n+ k/5) 5 3-to-1 y(m)= x(n+ 3k/5) In Out n,3 n+ 1 1,4 n+ phs(4) n n+ 1 n+ n x x x x x x x x x x x x x 9

10 Rational Ratio Interpolation. Example; up 8, down 3 Input Sa mples and availab le 1-to-8 Interpola ted Samp les n+ 3 n n+ 1 n+ 3-to-8 Interpolated Sa m ples (up 8, down 3) m+ 6 m+ 7 m+ 8 m m+ 1 m+ m+ 3 m+ 4 m+ 5 1

11 Interpolation To Time Position Between Available Interpolation Points (Arbitrary Ratio Interpolation) Desired Sample Value Error Available Sam ple Value n n+ 1 Input Sample Nearest Available Sample Position n+ k/m Desired Sample Position n+ k/m+ 11

12 Zero Order Hold Model of Nearest (Left) Neighbor Interpolation Interpolated Sam ple Values Zero-Order-Hold Analog Levels Error n n+ 1 Desired Sample Position n+ k/m+ 1

13 Spectrum of Up-Sampled Signal at Input and Output of Virtual DAC BW= 1 f DAC Response N Output Sam ple Rate N BW= 1 N Output Sam ple Rate N f 13

14 Frequency Response of DAC at First Spectral Null H( f)= - 1 N f DAC Response f N-.5 N N b H ( f ) f : H ( ) : N N N ( b 1) 7 N, Say b 8( bits), N 18 When signal is already 4-times oversampled Need 3 stage up-sampler to suppress spectral artifacts to -48 db 14

15 Shaping Filter: Time and Frequency Response, Four Times Over Sampled.3 time response of shaping filter. amplitude time spectral response of shaping filter log magnitude frequency 15

16 Time and Frequency Response of 3/6.4 Left Neighbor Interpolator

17 Time and Frequency Response of 3/6.37 Left Neighbor Interpolator

18 Prototype Interpolator Length for 8- bit data, initially Over Sampled by. f= 1 f f= 1 DC DC DC DC f To Obtain 18 Over Sample, M=64, N=(18/1)(66/)=384 N/M=6: Need 64 6-tap filters in Polyphase Interpolator 18

19 Prototype Interpolator Length for 8- bit data, initially Over Sampled by 4. f= 3 f f= 3 DC DC DC DC f To Obtain 18 Over Sample, M=3, N=(18/3)(66/)=18 N/M=4: Need 3 4-tap filters in Polyphase Interpolator 19

20 ... Address Control: Modulo Accumulator T OUT T IN Input Sam ples Outp ut Sam ples n n+ 1 n Input Time index n Polyp hase index k On Overflow, Insert New Input x(n) Filter y(m) Fractional Offset: d-acc Polyp hase Weig hts k(m) TOut fin d -acc M M T f In Out d-acc Mod(M) Z -1 acc(m) Int(--) - (m) Fractional Part (For later use)

21 Two Neighbor, Linear Slope Interpolator Left Available Interpolated Sample Value Desired Sample Value Right Available Interpolated Sample Value Input Sample value Linear Interpolator n Interpolated Sample Value Input Sample value n+ k/m Left Available Interpolation Sample Position n+ (k+ 1)/M Desired Sample Position k+ n+ 1 Right Available Interpolation Sample Position 1

22 Equivalent Interpolating Kernel x(k) TRI(k) x(k) x(k+ ) x(k+ 1) x(k+ 1) TRI(k+ 1) k-1 k k+ 1 k+ M M M M

23 Spectrum of Up-Sampled Signal at Input and Output of Virtual Linear Interpolator BW= 1 f N Output Sample Rate N Triangle Spectral Response Repeated Spectral Zeros BW= 1 N Output Sample Rate N f 3

24 Frequency Response at First Spectral Null of Linear Interpolator Tria ngle Response H( f)= [ 1 f] N N-.5 N N+.5 f b 1 b/ H ( f ) f : H ( ) : : N N N N ( b / 1) 7 N, Say b 16( bits), N 18 When signal is already 4-times oversampled Need 3 stage up-sampler to suppress spectral artifacts by -96 db 4

25 . Estimate y(n+k/m) & y(n+k/m) With 3 Arms of Polyphase Filter. y(n) PHS-(k-) PHS-(k-1) PHS-(k) PHS-(k+ 1) - y(n+ k/m) y(n+ k/m) PHS-(k+ ) 5

26 Estimate y(n+k/m) & y(n+k/m) With Polyphase Filters. y(n+ k/m) y(n) Polyphase Matched Filter k. y(n+ k/m) Polyphase Derivative Matched Filter k 6

27 .. y(n+k/m) & y(n+k/m) With Efficient Polyphase Filters y(n) 1-Stage Filter y(n+ k/m) Coeffic ient Selection Polyphase Matched Filter Coeffic ients 1-Stage Filter y(n+ k/m) Polyphase Derivative Matched Filter Coeffic ients 7

28 Interpolation with Polyphase Low-pass Filter and Polyphase Derivative Filter for Local Slope Correction x(n) x(n) Filter y(n+ k/m) y(n+ k/m+ /M) = y(n+ k/m)+ y(n+ k/m). Derivative Polyphase Filter dh=conv(h,[1-1]*m/ dh=dh(:length(dh)-1); h (n) k x(n) k(m) Filter. y(n+ k/m) dh (n) k k(m) d-acc Mod(M) Z -1 acc(m) Int(--) - (m) 8

29 Input Shaping Filter at 4-Samples per Symbol 9

30 Spectra of 64/1.49 Interpolated Signal Interpolated Shaping Filter Frequency Response Log magnitude (db) Normalized Frequency (f/f sym ) 3

31 Signal Conditioning and Processing ADC 16 MHz 16 MHz Interpolate Filter Interp Bank 163. MHz Half Band Filter 81.6 MHz Half Band Filter DDS 4.8 MHz DDS 4.8 MHz 4-Path Polyphase Filter 1-to MHz Circular Buffer 3.4 MHz 4-PNT FFT 3.4 MHz Channels Phase Acc umulator Spectral Centers 1.7 MHz Separation Channel BW: 1.7 MHz Channels Span 3 MHz ( 17 Channels) 4-Channel Channelizer: 4*1.7=4.8MHz 1-to-1 Down Sample in Channelizer Output Sample Rate; 3.4 MHz/Channel 31

32 Wide Dynamic Range Resampler 3

33 Spectra from 4-channel Channelizer at 3.4 MHz Sample Rate

34 Time Series from 4-channel Channelizer at 3.4 MHz Sample Rate

35 Equipment Bay: 19-Stereo FM Modulators 35

36 Conversation with Client! How big a room will we need to house the DSP version of this Transceiver? Answer: I think it will fit on one chip. Response: Don t be Absurd, You Can t Pack a Room into a Single Chip! Results: 48-Analog Devices Blackfin Processors to Demodulate 19 MP3 Stereo Channels. 1 Virtex V-4 for 19 Digital Stereo FM Modulators and 56 Channel 93 khz Bandwidth per channel. (6% of Chip) 36

37 .... Prototype Analog Stereo FM Modulator Left L+ R 75- sec Pre-emph LPF 14 khz 3. MHz 1 Right L - R dbx Encode 5- sec Pre-emph LPF 14 khz 3. MHz VCO 3 khz BPF 15-5 khz IF Output SCA dbx Encode 5- sec Pre-emph LPF 7.5 khz VCO 8 khz BPF 6-9 khz 37

38 DSP Based Stereo FM Modulator Left Right SCA (L+ R) (L-R) dbx Enc oder dbx Enc oder IIR 75-usec Pre-em ph IIR 5-usec Pre-em ph IIR 5-usec Pre-emph IIR LPF 14-kHz IIR LPF 14-kHz IIR LPF 14-kHz Satellite Cloc k Dom ain 48-to-93 Arb itrary Re-Sam ple 48-to-93 Arb itrary Re-Sam ple 48-to-93 Arb itrary Re-Sam ple K ACC K ACC CORDIC DDS FM-MOD & Up-Converter 3 khz CORDIC DDS FM-MOD & Up-Converter 3 khz IIR BPF 35-kHz IIR BPF 3-kHz Transceiver Cloc k Dom ain Gain Gain Gain Gain 38

39 56 Channels 1: Up_Sampler 18 Point FFT 18 Point FFT Half Band Phase Shift 18 Path Polyphase Filter 11-Taps Per Path 56 Channels Adder 18 Path Polyphase Filter 11-Taps Per Path 56 Channel Channelizer for 5-MHz Digital IF Sampled at 5.4 MHz 75.8 Mhz Odd Samp les 5.4 Mhz 1-to-3 Up-Sample Qua ntize DAC Even Samp les 5 MHz DDS 5.4 Mhz Radix- Butterfly of two 18-Point FFT s 39

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