Convert Two Parallel Paths into M Sequential Paths for each Path 0 1 M:1 … … … DIGIT AL L OW -P AS S M-1 M:1 0 DIGIT AL L OW -P AS S 1 … … … DDS DDS M-1 8-tap 8-tap DDS Coefficient
Replace CIC with Cascade 2-to-1 Half Band FIR Filters Filter Number 1 2 3 4 5 6 7 8 9 10 Total Number Taps 3 3 3 3 7 7 7 7 11 19 70 Operations 2 ‐ A 2 ‐ A 2 ‐ A 2 ‐ A 4 ‐ A 4 ‐ A 4 ‐ A 4 ‐ A 6 ‐ A 10 ‐ A ___ Per Filter 2 ‐ Shifts 2 ‐ Shifts 2 ‐ Shifts 2 ‐ Shifts 2 ‐ Mult 2 ‐ Mult 2 ‐ Mult 2 ‐ Mult 3 ‐ Mult 5 ‐ Mult Adds Ref to 2 2/2 2/4 2/8 4/16 4/32 4/64 4/128 6/256 10/512 4.26 Input Mult Ref to 0 0 0 0 2/16 2/32 2/64 2/128 3/256 5/512 0.27 Input
Impulse and Frequency Response of Last Stage Referred to Earlier Stages 0 -20 -40 -60 -80 2 4 6 8 10 12 14 16 18 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 -20 -40 -60 -80 5 10 15 20 25 30 35 40 45 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 -20 -40 -60 -80 10 20 30 40 50 60 70 80 90 -4 -3 -2 -1 0 1 2 3 4 0 -20 -40 -60 -80 20 40 60 80 100 120 140 160 180 200 -8 -6 -4 -2 0 2 4 6 8 0 -20 -40 -60 -80 50 100 150 200 250 300 350 400 15 10 5 0 5 10 15
Replace CIC with Cascade 2-to-1 Half Band Linear Phase IIR Filters Filter Number 1 2 3 4 5 6 7 8 9 10 Total Number Taps 1 1 1 1 3 3 3 3 3 4 23 Operations 3 ‐ A 3 ‐ A 3 ‐ A 3 ‐ A 7 ‐ A 7 ‐ A 7 ‐ A 7 ‐ A 7 ‐ A 9 ‐ A ___ Per Filter 1 ‐ Mult 1 ‐ Mult 1 ‐ Mult 1 ‐ Mult 3 ‐ Mult 3 ‐ Mult 3 ‐ Mult 3 ‐ Mult 3 ‐ Mult 4 ‐ Mult Adds Ref to 3/2 3/4 3/8 3/16 7/32 7/64 7/128 7/256 7/512 9/1024 3.25 Input Mult Ref to 1/2 1/4 1/8 1/16 3/32 3/64 3/128 3/256 3/512 4/1024 1.12 Input
Impulse and Frequency Response of Last Stage Referred to Earlier Stages 0 -20 -40 -60 -80 5 10 15 20 25 30 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 -20 -40 -60 -80 10 20 30 40 50 60 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 -20 -40 -60 -80 20 40 60 80 100 120 -4 -3 -2 -1 0 1 2 3 4 0 -20 -40 -60 -80 50 100 150 200 -8 -6 -4 -2 0 2 4 6 8 0 -20 -40 -60 -80 100 150 200 250 300 350 400 450 15 10 5 0 5 10 15
Impulse, Frequency, & Group Delay Response of 2-Path Linear Phase, Recursive Half-Band Filter Impulse Response 0 2 4 6 8 10 12 14 16 18 20 Frequency Response -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Group Delay 0 8 0 6 0 4 0 2 0 0 2 0 4 0 6 0 8 1
2-to-1 Resampling 2-Path Polyphase Filter and Digital Down-Converter 2-to-1 Half Band F ilter: h(n) 0 h(2n) L ow-P as s 1 High-P ass h(2n+ 1) - 0 0 0 0 0 1 0 0 0 0 0 L ow-P as s h 3 h 5 h 5 h 3 h 1 h 7 h 9 h 9 h 7 h 1 1 High-P ass - L P 2-Point DFT f -0.25 0 0.25 0.5 -0.5 HP
o-1Resampling 4-Path Down-Sample Polyphase Filter and 4-Point IFFT Extracts Signal Component From One-of-Four Selected Nyquist Zones 0 1 2 H (Z ) L ow-P ass 0 ) Filter Half Band Filters Centered on 1 1 2 H (Z ) P os Freq Cardinal Directions 1 4-P oint Hilbert Filter Each Reduces IFFT 2 BW 2-to-1 1 -1 2 High-P ass Z H (Z ) and Reduces 2 Filter Sample Rate 2-to-1 1 3 -1 2 Neg Freq Z H (Z ) 3 Hilbert Filter
4-Path, 2-to-1 Down-Sample with 4 Possible Trivial Phase Shifters c 0 - - - - - 0 0 0 0 0 - c 1 - - - - h 9 h 1 h 5 h 13 h 17 c 2 - - - - - 1 0 0 0 0 c 3 - - - - - h 17 h 13 h 9 h 1 h 5 4-Phase Rotators fs k/4: {c 0 c 1 c 2 c 3 } 4-Path Polyphase Filter fs 0/4: {1 1 1 1} 2-to-1 Path-0 Not Used fs 1/4: {1 Down-Sample j -1 -j} 2.5-Multiplies per Input fs 2/4: {1 -1 1 -1} fs 3/4: {1 -j -1 j}
Four Bands Centered on the Cardinal Directions Bands Centered on 0 and 180 (DC and f S /2) Alias To DC When Down-Sampled 2-to-1 Bands Centered on +90 and -90 (+f S /4 and –f S /4) Alias To fs/2 When Down-Sampled 2-to-1
Spectra: Four Half Band Filters on Unit Circle Showing Alias Free Pass, Transition, and Aliased Bands T ransition Alias Free Folded B andwidth Bandwidth P ass Band Due to 2-to-1 Down S ample Alias Free Any Narrowband P ass Band Signal Must Reside in One of the 4 T ransition Bandwidth Alias Free Band Intervals. P ositive Freq-P ass L ow-P ass The Alias Free B and-1 Band-0 Band Intervals Overlap! High-P ass Negative F req-P ass Band-2 Band-3
Pole-Zero Diagrams of Four Nyquist Zone Filters 2 2 1 1 0 0 -1 -1 -2 -2 -2 -1 0 1 2 -2 -1 0 1 2 2 2 1 1 0 0 -1 -1 -2 -2 -2 -1 0 1 2 -2 -1 0 1 2
Frequency Responses of Four Nyquist Zone Filters Frequency Response Frequency Response 0 0 -20 -20 Log Mag (dB) -40 -40 -60 -60 -80 -80 -0.5 0 0.5 -0.5 0 0.5 0 0 -20 -20 Log Mag (dB) -40 -40 -60 -60 -80 -80
ctra of Signal Aliased to Different Sampled Data Frequencies in Successive 2-to-1 Sample Rate Reductions.
Most Efficient Multistage Half-Band Digital Down-Converter I-Q RL I-Q I-Q Channelizer Channelizer Channelizer Channelizer 4-P ath 4-P ath 4-P ath 4-P ath ...... 2-to-1 2-to-1 2-to-1 2-to-1 Filter Down Down Down Down S ample S ample S ample S ample ...... Channel DDS S elect 2 2 2 Impulse Response: Half Band Filter N 2.5 [1 ] 2 4 8 1 1 1 1 2.5 2.5 [1 ] 2 4 8 0.5 2.5 3 7.5 0 0 5 10 15 20
Spectrum of Input Signal and Zoom to Spectral Segment Received Signal Spectrum 0 -50 100 0 5 10 15 20 25 30 35 40 45 Frequency / MHz Zoom-in 0 -50 100 17.9 17.92 17.94 17.96 17.98 18 18.02 18.04 18.06 18.08 18.1 F / MH
Spectra: Last Four Stages Processing Chain. Dotted Line Indicates Center Frequency of Desired Spectral Component -0.86 +0.56 0.28 = 0.14 = -0.44 3 Stage 6 Stage 7 Stage 8 Stage 9 0 0 0 -20 -20 -20 -40 -40 -40 -60 -60 -60 -80 -80 -80 -100 -100 -100 -0.05 0 0.05 -0.04 -0.02 0 0.02 0.04 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 -0.1 0 0.1 0.2 0.3 Frequency/MHz Frequency/MHz Frequency/MHz Frequency/MHz
ampled Data Frequency Locations on Successive Aliases 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1
pectrum at Input and Output of Final Heterodyne and Filter Stage Post Processing--Input Signal Spectrum (ch # 600) B / d 0 e d itu -50 n g a M -100 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 Frequency / MHz Post Processing--Down Convert (ch # 600) B / d 0 e d itu -50 n g a M -100 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 Frequency / MHz Post Processing--Filtering (ch # 600) B / d 0 e d itu -50 n g a M -100
A 375-to-1 down-sample: 90 MHz to 240 kHz with a 30 kHz output BW 80 dB dynamic range. Require 6 CIC stages. The gain of each stage is 375: Gain of 6 stages becomes (375) 6 or 2.8 . 10 15 or 52 bits growth in the CIC integrators. With 16-bit input data integrator bit width is 16+52 or 68. Six integrators in both I & Q paths would be circulating 816 bits per input sample which if converted to the 16-bit width required of the arithmetic in the half-band filters proves to be same number of bits to manipulate 48 arithmetic operations per input sample. Number of operations for the I-Q half band filter chain is on the order of 8-multiply and 16 adds per input sample which represents a workload 1/6 of the CIC chain. The efficient cascade CIC filter chain can be replaced with an even more efficient cascade four-path half band filter chain.
Linear Phase IIR Filter Group Delay: Two Path, 4-Coefficient, Linear Phase 2-Path Filter 2 0 8 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Group Delay: Detail 1 8 9 0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Frequency response 0 0 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Most Efficient Multistage Half-Band Digital Down-Converter I-Q RL I-Q I-Q Channelizer Channelizer Channelizer Channelizer 4-P ath 4-P ath 4-P ath 4-P ath ...... 2-to-1 2-to-1 2-to-1 2-to-1 Filter Down Down Down Down S ample S ample S ample S ample ...... Channel DDS S elect Impulse Response, Two-Path, 4-Coefficient, Linear Phase IIR 2 2 2 N 2.0 [1 ] 2 4 8 1 1 1 2.0 2.0 [1 ] 2 4 8 2.0 3 6.0 5 10 15 20 25 30
Polyphase Partition of Band Pass Filter 1-Path to M-Path Transformation Modulation Theorem of Z-Transform 1 1 N N ( ) ( ) ( )( ) ( ) j n j j n n G Z h n e Z h n e Z H e Z k k k 0 0 n n 1 1 M N ( ) ( ) ( ) ( ) j r nM r nM G Z h r nM e Z k 0 0 r n 1 1 M N ( ) ( ) j j nM r nM G Z e Z h r nM e Z k k 0 0 r n θ = k 2 π k 2 1 1 2 π M N j k θ = k M ( ) ( ) r nM M G Z e Z h r nM Z k 0 0 r n
Polyphase Band Pass Filter and M-to-1 Resampler 2 j k0 M:1 M e y(nM) y(n) x(n) M H ( Z ) 0 2 j k1 M e M -1 H ( Z ) Z 1 2 j k2 M e M -2 H ( Z ) Z 2 .... .... .... 2 j k(M-2) M e -(M-2) M H ( Z ) Z M-2 2 j k(M-1) M e M -(M-1) H ( Z ) Z M-1
Apply Noble Identity to Polyphase Partition 2 j k0 M:1 M e y(nM,k) x(n) H ( Z ) 0 2 j k1 M:1 M e -1 H ( Z ) We Reduce Sample Rate Z 1 2 M-to-1 Prior to Reducing Bandwidth j k2 M:1 M e -2 H ( Z ) Z 2 (Nyquist is Raising His Eyebrows!) .... .... We Intentionally Alias the Spectrum. .... 2 (Were you Paying Attention j k(M-2) M:1 M e in school when they discussed the -(M-2) H ( Z ) Z importance of anti-aliasing filters?) M-2 2 j k(M-1) M:1 M e M-fold Aliasing! -(M-1) H ( Z ) Z M-1 M-Unknowns! M-Paths supply M-Equations We can the separate Aliases!
Move Phase Spinners to Output of Polyphase Filter Paths 2 j k0 M M:1 e y(nM,k) x(n) H ( Z ) 0 2 j k1 M M:1 e -1 H ( Z ) Z 1 2 j k2 M M:1 e -2 H ( Z ) Z 2 .... .... .... 2 j k(M-2) M:1 M e -(M-2) H ( Z ) Z M-2 2 j k(M-1) M M:1 e -(M-1) H ( Z ) Z M-1 W t Ph S i f f l ibl
Polyphase Partition with Commutator Replacing the “r” Delays in the “r-th” Path 2 j k0 M e y(nM,k) x(n) H ( Z ) 0 Note: We don’t assign 2 j k1 M Phase Spinners to Select e Desired Center Frequency H ( Z ) 1 Till after Down Sampling 2 j k2 M And Path Processing e H ( Z ) 2 This Means that The Processing for every Channel .... .... is the same till the Phase Spinner 2 j k(M-2) M e No longer LTI, Filter now has H ( Z ) M-2 M-Different Impulse Responses! 2 j k(M-1) M Now LTV or PTV Filter. e H ( Z ) M-1
Armstrong to Tuned RF with Alias Down Conversion to Polyphase Receiver Digital e-j kn Digital B and-P ass L ow-P ass -j H(Z ) k H(Z e ) M-to-1 M-to-1 -j 2 M-P ath Digital M rk e P olyphase H(Z ) r M-to-1 Rather than selecting center frequency at input and reduce sample rate at output, we reverse the order, reduce sample rate at input and select center frequency at output. We perform arithmetic operations at low output rate rather than at high input rate!
Down Sample 6-to-1 n n-1 n-2 n-3 n-4 n-5 n-6 n-7 n-8 n-9 n-10 n-11 n-12 n-13 n-14 — — — — — — n n-1 n-2 n-3 n-4 n-5 n-6 n-7 n-8 n n-1 n-2 n-3 n-4 n-5 n-6 n-7 n-8 n+ 6 n+ 5 n+ 4 n+ 3 n+ 2 n+ 1
Polyphase Partition 1-D filter becomes 2-D M-Path Filter — — n n-6 n-12 n n-6 n-12 n-6 n+ 6 n n-12 — — n-1 n-7 n-13 n-7 n+ 5 n-1 n-13 n-1 n-7 n-13 — — n-8 n-2 n-14 n-2 n-8 n+ 4 n-14 n-2 n-8 n-14 — — n-3 n-9 n-15 n-9 n+ 3 n-3 n-15 n-3 n-9 n-15 — — n-10 n-4 n-16 n-4 n-10 n+ 2 n-16 n-4 n-10 n-16 — — n-5 n-11 n-17 n-5 n-11 n+ 1 n-17 n-5 n-11 n-17
Reorder Filter and Resample L OW P AS S FIL T E R .. this is very P OL Y P HAS E P AR T IT ION s(t) s(n) stuff.... r(nM,k) ADC h(0+ nM) h(1+ nM) ... ... CL K h(r+ nM) ..... ..... r(nM) h(M-1+ nM) j r k M e P HAS E R OT AT OR S AL IAS E D HE T E R ODY NE B ANDP AS S FIL T E R P OL Y P HAS E P AR T IT ION
Phase and Gain Response (3-Versions of Filter) Prototype Filter, Polyphase Filter Prior to Resampling, Polyphase Filter after Resampling
Impulse Response and Frequency Response of Prototype Low Pass FIR Filter
Impulse Response of 6-Path Polyphase Partition Prior to 6-to-1 Resampling
Frequency Response of 6-Path Polyphase Partition Prior to 6-to-1 Resampling
Phase Response of 6-Path Polyphase Partition Prior to 6-to-1 Resampling
Overlay Phase Response of 6-Path Polyphase Partition Prior to 6-to-1 Resampling -2 Nyquist Zone -1 Nyquist Zone -2 /3 0 Phase Nyquist Shifts -2 /6 Zone +1 Phase Nyquist Shifts Zone +2 Phase Nyquist Aligned 2 /6 Zone Phase Shifts 2 /3 Phase Shifts
De-Trended Overlay Phase Response: 6-Path Partition Prior to 6-to-1 Resampling
3-D Paddle-Wheel Phase Profiles, 6-Path Partition Prior to 6-to-1 Resampling
Overlay 3-D Paddle-Wheel Phase Profiles, 6-Path Partition Prior to 6-to-1 Resampling
Overlay 3-D Paddle-Wheel Phase Profiles, Showing Phase Shifts in +1 Nyquist Zone
Overlay 3-D Paddle-Wheel Phase Profiles, Phase Shifted to Align Phases in +1 Nyquist Zone
PolyChanDemo
Single Channel Armstrong and Multirate Aliased Polyphase Receiver 2 j nk S tandard DDC e M y(n,k) y(nM,k) x(n) H(Z ) 2 1 2 2 M-to-1 2-P olyphase Filters P olyphase DDC 2 j nk e M y (nM) x(n) x(n) r y(nM,k) H (Z ) r M M 2 1 M-to-1 1-P olyphase Filter
ide Input Down Conversion to Output of Filter Where it anishes Due to Down Sampling. Rotators in Filter Factor Out and are Applied to Path Outputs Rather than to Coefficients. Advantage: Real sequence is made complex at output of Filter Rather than at Input to Filter j 0k2 M e 0 H (Z ) 0 1 j 1k2 M e … … … H (Z ) 1 M-1 j 2k2 M e x(n) y(nM,k) 0 H (Z ) 2 1 . . . . . . j (M-1)k2 … … … M e DDS M-1 H (Z ) M-1
Bad Mismatch: Sample Rate Large Compared to Transition Bandwidth 200 kHz 0.1 dB 80 dB -6 dB /Octave f 200 kHz 20 MHz 20 MHz Input 400 T ap 20 MHz Output S ample R ate FIR Filter S ample R ate Nyquist Rate for Filter is 200 kHz + 200 kHz = 400 kHz or fs/50
Polyphase Partition of Low ‐ Pass Filter 400 T aps 20 kHz 400 kHz P olyphase L ow P ass Filter 50-to-1 8 T aps 0 1 2 … … … 20 MHz 400 kHz 48 49
Cascade Polyphase Filter Down ‐ Sampling and Up ‐ Sampling 400 T aps 400 T aps 400 kHz 20 MHz 20 MHz P olyphase P olyphase L ow P ass Filter L ow P ass Filter 50-to-1 1-to-50 8-taps 8-taps 0 0 1 1 2 2 … … … … … … 20 MHz 20 MHz 400 kHz 48 48 49 49
Efficient Polyphase Filter Implementation 20 MHz Input 400 T ap 20 MHz Output S ample R ate FIR Filter S ample R ate 20 MHz 20 MHz 400 kHz 8-tap 8-tap Coeffic ient Coeffic ient S elec t S elec t B ank B ank
Two Processing in Boxes: ow can you tell which is which from outside box? White Box 20MHz 20MHz 400-T ap L owpass 60 ‐ Ops/Input Filter White Box 400 kHz 20MHz 20MHz 8-T ap 8-T ap F ilter F ilter 16 ‐ Ops/Input Coefficient Coefficient Bank Bank S elect S elect S tate Machine
Polyphase Partition of Low ‐ Pass Filter H (Z ) H (Z ) 0 0 H (Z ) H (Z ) 1 1 x(n) y(nM) H (Z ) H (Z ) 2 2 y(n) ...... ...... H (Z ) H (Z ) M-1 M-1 f f f
Polyphase Partition of Band Pass Filter j 0k2 j 0k2 M M e e H (Z ) H (Z ) 0 0 j 1k2 j 1k2 M M e e H (Z ) H (Z ) 1 1 j 2k2 j 2k2 M M e e x(n) y(nM,k) H (Z ) H (Z ) 2 2 y(n,k) ...... ...... j (M-1)k2 j (M-1)k2 M e M e H (Z ) H (Z ) M-1 M-1 f f f
olyphase Partition of T wo Band Pass Filters j rk 2 2 j rk 2 2 j rk 1 2 j rk 1 2 e M M e M e M e y(nM,k ) 2 H (Z ) H (Z ) 0 0 H (Z ) H (Z ) 1 1 H (Z ) H (Z ) 2 2 y(n,k )+ y(n+ k ) 1 2 ...... ...... y(nM,k ) 1 H (Z ) H (Z ) M-1 M-1 f
Workload for Multiple M-Path Filters • 1-Channel M-to-1 Down Sample • 1-Filter and M Complex Phase Rotators • 2-Channels M-to-1 Down Sample • 1-Filter and 2M Complex Phase Rotators • K-Channels M-to-1 Down Sample 1-Filter and kM Complex Phase Rotators • • M-channels M-to-1 Down Sample (use FFT) • 1-Filter and [log 2 (M)/2]M Complex Phase Rotators hen k > Log 2 (M)/2 Build all channels and discard the channels you don’t need! M=16, Log 2 (16)/2 = 2: thus if you want 2 or more, Build them all! M=128, Log 2 (128)/2 = 3.5: thus if you want 4 or more, Build them all! M=1024, Log 2 (1024)/2 = 5: thus if you want 5 or more, Build them all!
M-Channel Channelizer: Resampled M-Path Narrowband Filter Channels Alias to Baseband: Phase Aligned Sums Separate Aliases: rk Performed at Low Output Rate Rather Than at High Input Rate. One Input Filter Services M-Output Channels P olyphase .. this is very P artition stuff.... h (n) 0 h (n) 1 h (n) fs 2 M-PNT h (n) 3 IFF T ..... ..... ..... ..... FDM T DM h (n) M-2 h (n) M-1 h (n)= h(r+ nM) r
Dual Channel Armstrong and Multirate Aliased Polyphase Receiver 2 S tandard DDC j nk e 1 M y(n,k ) y(nM,k ) x(n) 1 1 H(Z ) 2 1 2 2 2-P olyphase M-to-1 2 j nk 2 Filters e M y(n,k ) y(nM,k ) 2 2 H(Z ) 2 2 1 2 2-P olyphase M-to-1 Filters P olyphase DDC 2 j nk 1 e M y (nM) x(n) x(n) r y(nM,k ) H (Z ) r 1 M M 2 1 M-to-1 1-P olyphase 2 j nk Filter 2 e M y(nM k ) 2
Up-sampling by Zero Packing and Filtering
Spectra Of Input, of Zero-Packed, and of Low-Pass Filtered Zero-Packed Signal x(n) f n 0 2 5 1 4 f s 3 x(m) m f .... .... .... 20 25 0 1 2 3 5 6 7 8 9 10 15 4 f s 5 R ejected R ejected F ilter S pectrum S pectrum y(m) m f .... .... .... 20 25 0 8 9 15 1 2 3 4 5 6 7 10 f s 5
Spectra Of Input, of Zero-Packed, and of Band Pass Filtered Zero-Pack Signal n) f n 2 0 5 f s 1 4 3 m) f m .... .... .... 20 25 0 1 2 3 5 6 7 8 9 15 4 10 f s 5 R ejected R ejected F ilter S pectrum S pectrum m) m f f s 5
Polyphase Partition of Resampling Filter 1 N ( ) ( ) n H Z h n Z 0 n C 0 C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 10 C 11 C 12 C 13 C 14 1-to-5 N 1 1 M M ( ) ( ) ( ) r nM H Z h r nM Z 0 0 r n C 0 0 0 0 0 C 5 0 0 0 0 C 10 0 0 0 0 0 C 1 0 0 0 0 C 6 0 0 0 0 C 11 0 0 0 0 0 C 2 0 0 0 0 C 7 0 0 0 0 C 12 0 0 1-to-5 0 0 0 C 3 0 0 0 0 C 8 0 0 0 0 C 13 0 0 0 0 0 C 4 0 0 0 0 C 9 0 0 0 0 C 14
Factor Delays and Rearrange N 1 1 M M ( ) ( ) r nM H Z Z h r nM Z 0 0 r n C 0 C 5 0 C 10 0 0 0 0 0 0 0 0 0 0 0 C 1 C 6 0 0 0 0 0 0 0 0 0 C 11 0 0 0 0 0 C 7 C 2 0 0 0 0 0 C 12 0 0 0 0 0 1-to-5 C 8 0 C 3 0 C 13 0 0 0 0 0 0 0 0 0 0 0 C 4 C 9 0 0 0 0 0 0 0 C 14 0 0 0 0 C 0 C 5 0 C 10 0 0 0 0 0 0 0 0 C 1 0 C 6 0 0 C 11 0 0 0 0 0 0 C 7 0 C 2 0 0 0 C 12 0 0 0 0 0 1-to-5 C 8 0 0 C 3 0 0 0 0 C 13 0 0 0 0 0 0 0 C 4 0 C 9 0 C 14 0 0 0 0 0 0 0 0
Noble Identity: Interchange M-Delays with M-to-1 Resample M Delays M Delays Input Clock, T 1 Delay 1 Delay Output Clock, MT M:1 M:1 -M -1 Z Z M:1 M:1 y(m) x(n) y(m) x(n) -M -1 H(Z ) H(Z )
Interchange Filter and Resampler C 10 C 0 C 5 1-to-5 0 Replace Up Samplers, C 1 C 6 C 11 1-to-5 Delays, and Summer 0 C 12 C 2 C 7 0 1-to-5 with M-Port Output 0 0 C 13 C 3 0 C 8 1-to-5 Commutator 0 0 0 C 14 0 C 4 C 9 1-to-5 C 10 C 0 C 5 C 1 C 6 C 11 C 12 C 2 C 7 C 13 C 3 C 8 C 4 C 14 C 9
Low-Pass Replaced by Band-Pass 2 1 N j k n ( ) ( ) n M G Z h n e Z 0 n N 1 2 1 M M ( ) j r nM k ( ) ( ) ( ) r nM G Z h r nM e M Z 0 0 r n N 1 2 2 1 M M j rk j nM ( ) ( ) r nM M M G Z Z e h r nM e Z 0 0 r n N 1 2 1 M M j rk ( ) ( ) r nM G Z Z e M h r nM Z 0 0 r n Spin The Delays, Don’t Touch the M-Path Partitioned Weights
Low-Pass to Band-Pass 1-to-M Up-Sampling Filter
M-Path, M-Channel Channelizer: Spinners are in IFFT
M-Point IFFT Supplies Phase Spinners to Form Up Converters to all Multiples of Input Sample Rate f B W F 2 F 1 F 0 F M-1 F M-2 ...... f fs M fs All Output Channels Centered on Multiples of Input Sample Rate Example: Multiples of 6-MHz
Heterodyne Input Signal a Small Frequency Offset from DC: Channelizer Aliases DC to Channel Center and offset signal from DC is Offset from Channel Center Input Offset Observed 2 2 Input at Output kr j j kr e 5 e 5 Offset Frequency h (5n) h (5n) 0 0 j n h ( 5n) e h ( 5n) 1 1 y(m,k) x(n) y(m,k) h (5n) h (5n) 2 2 h ( 5n) h ( 5n) 3 3 h (5n) h (5n) 4 4
Input Spectrum x(n) n f 0 2 5 f s 1 4 3 Shifted Spectrum x(n) n f f s Up-Sampled Spectrum x(m) m f f s 5 Distorted Spectrum R ejected R ejected F ilter S pectrum S pectrum y(m) m f f s 5
Input Spectrum x(n) n f 0 2 5 f s 1 4 3 Shifted Spectrum x(n) f n f s Up-Sampled Spectrum x(m) f m f s 5 T wo Filter Spectra R ejected R ejected 2 F ilters S pectrum S pectrum y(m) m f f s 5
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