SLIDE 1
New Directions in Channelized Receivers and Transmitters
fred harris
2-December 2011
1
Motivation For Using Multirate Filters
2
New Directions in Channelized Receivers and Transmitters fred - - PowerPoint PPT Presentation
New Directions in Channelized Receivers and Transmitters fred - - PowerPoint PPT Presentation
New Directions in Channelized Receivers and Transmitters fred harris 2-December 2011 1 Motivation For Using Multirate Filters 2 Processing Task: Obtain Digital Samples of Complex Envelope Residing at Frequency f C Analog Digital R
SLIDE 2
SLIDE 3
Processing Task: Obtain Digital Samples
- f Complex Envelope Residing at
Frequency fC
Analog Digital Multi-Channel FDM Input S ignal S ingle-Channel Base banded Output S ignal
R eceiver
Input Spectrum
f fC
S
elected Narrow B and S ignal
3
SLIDE 4
See!
U tah
4
SLIDE 5
First Generation DSP Receiver
f f f f
L
- w P
ass Filter fs
- jt
e
1 2 3 3 4 4 2 1
Analog Signal Processing
5
SLIDE 6
Signal Conditioning for DSP Receiver
f f f fs/2
- fs/2
1 2 3
.... ....
L
- w P
ass Filter fs
1 2 3 4
6
SLIDE 7
Replicate Analog Processing in DSP
L
- w P
ass Filter fs/M M:1
- jn
e
5 6
f f f f fs/2 fs/2 fs/2 fs/M
- fs/2
- fs/2
- fs/2
- fs/M
- fs
fs
3 4 5 6
.... .... .... .... .... .... .... ....
L
- w P
ass Filter fs
1 2 3 4
~ ~ ~ ~
Ignoring Good Advice!
7
SLIDE 8
ADC M:1 s(t) s(n) h(n) r(n) r(nM) s(n) e -j n e -j n L O CLK L OWP AS S FIL T E R
- Fundamental Operations
- Select Frequency
- Limit Bandwidth
- Select Sample Rate
Digital Down Converter (DDC)
8
SLIDE 9
Spectral Description of Fundamental Operations
fs/2 fs/2 fs/2 fs/M
- fs/2
- fs/2
- fs/2
- fs/M
f f f f CHANNE L OF INT E R E S T T R ANS LAT E D S P E CT R U M F IL T E R E D S P E CT R U M S P E CT R AL R E P L ICAT E S AT DOWN-S AMP L E D R AT E OU T P U T DIGIT AL F IL T E R R E S P ONS E INP U T ANAL OG F IL T E R R E S P ONS E
.... ....
9
SLIDE 10
A Shell Game: Rearrange the Players! Keep Your Eye on the Pea!
10
SLIDE 11
Signal and Filter are at Different Frequencies Which One to Move?
f f f f CHANNE L OF INT E R E S T T R ANS LAT E D S P E CT R U M T R ANS LAT E D F IL T E R F IL T E R R E S P ONS E
Second Option First Option
11
SLIDE 12
Down Sample Complex Digital IF
B and P ass F ilter fs/M M:1
- jMn
jn e e
5 6
f f f f fs/2 fs/2 fs/M fs/M
- fs/2
- fs/2
- fs/M
- fs/M
- fs
fs
3 4 5 6
.... .... .... .... .... .... .... ....
L
- w P
ass Filter fs
1 2 3 4
~ ~ ~ ~
No Spectral Image
12
SLIDE 13
Fundamental Operation with Rearrange Operators
ADC M:1 s(t) h(n) s(n) r(n) r(n) r(nM) e -j n e j n e j n L O CLK BANDP AS S FIL T E R
Up-Convert Filter, Filter Signal at IF, Down Convert Output of Filter
13
SLIDE 14
Equivalency Theorem
( )
( ) ( ) * ( ) ( ) ( ) ( ) ( ) { ( )* ( ) }
j n j n k k j n j k k j n j n
r n s n e h k s n k e h k e s n k h k e e s n h n e
Down-Convert Signal at Input to Low-Pass Filter Down-Convert Signal at Output of Band-Pass Filter Up-Convert Low Pass Filter To Become Complex Band-Pass Filter
14
SLIDE 15
Signal Flow Description
- f Equivalency Theorem
Not Finished: Moving Down Converter from Input to Output Replaces 2-Multipliers (Complex Scalar) with 4-Multipliers (Complex Product)
15
SLIDE 16
Interchange Down Converter and Resampler
ADC M:1 s(t) h(n) s(n) r(nM) r(n) r(nM) e -j Mn e j n e j n e j Mn L O CL K BANDP AS S FIL T E R
Only Down Convert the Samples we Intend to Keep! Let the Resampler Alias the Center Frequency to Baseband 0 Aliases to M0 Mod(2) 0 Aliases to 0 if M0 = k2
16
SLIDE 17
SPECTRAL DESCRIPTION
REORDERED FUNDAMENTAL OPERATION
fs/2 fs/2 fs/M fs/M
- fs/2
- fs/2
- fs/M
- fs/M
f f f f CHANNE L OF INT E R E S T T R ANS LAT E D F IL T E R F IL T E R E D S P E CT R U M AL IAS E D R E P L ICA T E S A T DOWN-S AMP L E D R AT E T R ANS LAT E D R E P L ICAT E S AT DOWN-S AMP L E D R AT E INP U T ANAL OG F IL T E R R E S P ONS E
.... .... .... ....
17
SLIDE 18
Successive Transformations Turn Sampled Data Version of Edwin Armstrong’s Heterodyne Receiver to Tuned Radio Frequency (TRF) Receiver and then to Aliased TRF Receiver.
Digital
B and-P ass M-to-1
e -j kn
H(Z e )
- j
k Digital L
- w-P
ass M-to-1
H(Z
)
e-j kn
Digital
B and-P ass
M-to-1
H(Z e )
- j
k
k k
M θ = k 2π 2π
- r θ = k M
Any Multiple of Output Sample Rate Aliases to Baseband Armstrong Nyquist Equivalency Theorem Digital
B and-P ass
M-to-1
- j Mkn
e
H(Z e )
- j
k
18
SLIDE 19
Let’s Keep Rearranging the Players!
19
SLIDE 20
Linear Systems Commute and are Associative
x(n) x(n) (n) y(n) h(n) w(n) w(n) r(n) r(n) R (f)= H(f)G(f) h(n) H(f) h(n) g(n) G(f) g(n) f f f x(n) x(n) (n) v(n) g(n) w(n) w(n) r(n) r(n) R (f)= G(f)H(f) h(n) H(f) h(n) g(n) G(f) g(n) f f f
20
SLIDE 21
Linear systems Are Associative
X X Y W W L
1
L
2
L = L L
3 1 2
X X Y W W L
1
L
2
L = L L
3 1 2
Filter R esample R esampled F ilter
21
SLIDE 22
In Case you Couldn’t Wait to See the Proof
22
2
1 1 2 2 1 2 1 2 3 3 2 2 3 3 2 2 3 3 2 2 1 2 2 1 2 3 2 3
3
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) where ( ) ( ) ( )
k k k k k k k k k k
y n x n k h k dy n y n k g k x n k k h k g k x n k h k k g k x n k h k k g k x n k f k f n h n k g k
SLIDE 23
Filter and Output Resampler can Commute to Input Resampler and Resampled Filter
X(Z) X(Z) X(Z) Y (Z) Y (Z) y(n) y(n) Y (Z )
M
Y (Z )
M
Y (Z )
M
Z
- rX(Z )
M
y(nM) y(nM) y(nM) x(nM+ r) x(n) x(n) x(n) H(Z) M:1 M:1 M:1 Z H(Z )
- r
M r
Z H(Z)
- r
r
r r
23
SLIDE 24
Coefficient Assignment of Low-Pass Polyphase Partition
Extract Delays T
- First
Non-Zero Coefficient
For M-to-1 resample start at Index r and Increment by M For 3-to-1 resample start at index r and increment by 3
C0 C0 C0 C3 C3 C3 C4 C4 C4 C5 C5 C5 C6 C6 C6 C7 C7 C7 C8 C8 C8 C9 C9 C9 C10 C10 C10 C11 C11 C11 C1 C1 C1 C2 C2 C2
This mapping from 1-D to 2-D is used by Cooley-Tukey FFT. Polyphase Filters and CT
- FFT are kissing cousins!
24
SLIDE 25
Polyphase Partition of Low Pass Filter 1-Path to M-Path Transformation
H( Z) x(n) y(n) y(nM) M:1
1
( ) ( )
N n n
H Z h n Z
1 1 ( )
( ) ( )
M N r nM r n
H Z h r nM Z
1 1
( ) ( )
M N r nM r n
H Z Z h r nM Z
Z Z Z Z
- 1
- 2
- (M-2)
- (M-1)
H ( Z )
M
H ( Z )
1
M
H ( Z )
2
M
H ( Z )
M-2
M
H ( Z )
M-1
M
.... .... ....
x(n) y(n) y(nM) M:1
M-Path Partition Supports M-to-1 Down Sample Also Supports Rational Ratio M-to-Q and M-to-Q/P Down Sample!
25
SLIDE 26
Noble Identity: Commute M-units of Delay followed by M-to-1 Down Sample
M Delays 1 Delay M Delays 1 Delay
Input Clock, T Output Clock, MT
Z Z
- M
- 1
M:1 M:1 M-Units of Delay at Input Rate Same as 1-Unit of Delay at Output Rate H(Z ) H(Z )
- M
- 1
M:1 M:1
SLIDE 27
Interchange Filters and Resampler: Place Resampler at Input Rather Than at Output of Filter
Z Z Z Z
- 1
- 2
- (M-2)
- (M-1)
H ( Z ) H ( Z )
1
H ( Z )
2
H ( Z )
M-2
H ( Z )
M-1
.... .... ....
x(n) y(nM,k) M:1 M:1 M:1 M:1 M:1
SLIDE 28
Replace Delays with Commutator Perform Path Operations Sequentially
H ( Z ) H ( Z )
1
H ( Z )
2
H ( Z )
M-2
H ( Z )
M-1
.... ....
x(n) y(nM)
8-tap Coefficient B ank S elect
SLIDE 29
IF S T AGE RF S T AGE IF S T AGE DDS DDS S
- P
P
- S
Clock Clock CARR IER P L L T IMING P LL DAC ADC DIGIT AL L OW
- P
AS S DIGIT AL L OW
- P
AS S DIGIT AL L OW
- P
AS S DIGIT AL L OW
- P
AS S DIGIT AL L OW
- P
AS S DIGIT AL L OW
- P
AS S DIGIT AL L OW
- P
AS S DIGIT AL L OW
- P
AS S S hape & U psample Matched Filter Interpolate Decimate I-Q T able Detect
cos( n) cos( n) sin( n) sin( n)
Modulator Demodulator
f Analog Analog
Analog
Oscillator P A R F Carrier Gain Control Oscillator Carrier W aveform VGA L NA
R F Carrier
IF
Channel Channel Bits Bits
Transmitter Process, Up-Sample and Up-Convert Receiver Process, Down Convert and Down-Sample
Modulator Raises Sample Rate & Applies Heterodyne at High Output Sample Rate! De-Modulator Applies Heterodynes at High Input Rate & then Reduces
SLIDE 30
Conventional and Ubiquitous DDC
DDS DIGIT AL L OW
- P
AS S DIGIT AL L OW
- P
AS S
M:1 M:1
DDS CIC CIC M:1 2:1 2:1 2:1 2:1 M:1 S cale Factor CIC Correction CIC Correction Half Band Half Band Half Band Half Band
z z z z-1 z-1 z-1
- 1
- 1
- 1
- z
z-1
- 1 -
z z-1
- 1 -
M:1 Integrators Derivative Filters
SLIDE 31
Convert Two Parallel Paths into M Sequential Paths for each Path
DDS DIGIT AL L OW
- P
AS S DIGIT AL L OW
- P
AS S
M:1 M:1
… …
… … … …
0 0 1 1 M-1 M-1
DDS DDS
8-tap 8-tap Coefficient
SLIDE 32
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 Per Filter 2‐A 2‐Shifts 2‐A 2‐Shifts 2‐A 2‐Shifts 2‐A 2‐Shifts 4‐A 2‐Mult 4‐A 2‐Mult 4‐A 2‐Mult 4‐A 2‐Mult 6‐A 3‐Mult 10‐A 5‐Mult ___ Adds Ref to Input 2 2/2 2/4 2/8 4/16 4/32 4/64 4/128 6/256 10/512 4.26 Mult Ref to Input 2/16 2/32 2/64 2/128 3/256 5/512 0.27
SLIDE 33
2 4 6 8 10 12 14 16 18
- 1
- 0.8
- 0.6
- 0.4
- 0.2
0.2 0.4 0.6 0.8 1
- 80
- 60
- 40
- 20
5 10 15 20 25 30 35 40 45
- 2
- 1.5
- 1
- 0.5
0.5 1 1.5 2
- 80
- 60
- 40
- 20
10 20 30 40 50 60 70 80 90
- 4
- 3
- 2
- 1
1 2 3 4
- 80
- 60
- 40
- 20
20 40 60 80 100 120 140 160 180 200
- 8
- 6
- 4
- 2
2 4 6 8
- 80
- 60
- 40
- 20
50 100 150 200 250 300 350 400 15 10 5 5 10 15
- 80
- 60
- 40
- 20
Impulse and Frequency Response of Last Stage Referred to Earlier Stages
SLIDE 34
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 Per Filter 3‐A 1‐Mult 3‐A 1‐Mult 3‐A 1‐Mult 3‐A 1‐Mult 7‐A 3‐Mult 7‐A 3‐Mult 7‐A 3‐Mult 7‐A 3‐Mult 7‐A 3‐Mult 9‐A 4‐Mult ___ Adds Ref to Input 3/2 3/4 3/8 3/16 7/32 7/64 7/128 7/256 7/512 9/1024 3.25 Mult Ref to Input 1/2 1/4 1/8 1/16 3/32 3/64 3/128 3/256 3/512 4/1024 1.12
SLIDE 35
Impulse and Frequency Response of Last Stage Referred to Earlier Stages
5 10 15 20 25 30
- 1
- 0.8
- 0.6
- 0.4
- 0.2
0.2 0.4 0.6 0.8 1
- 80
- 60
- 40
- 20
10 20 30 40 50 60
- 2
- 1.5
- 1
- 0.5
0.5 1 1.5 2
- 80
- 60
- 40
- 20
20 40 60 80 100 120
- 4
- 3
- 2
- 1
1 2 3 4
- 80
- 60
- 40
- 20
50 100 150 200
- 8
- 6
- 4
- 2
2 4 6 8
- 80
- 60
- 40
- 20
100 150 200 250 300 350 400 450 15 10 5 5 10 15
- 80
- 60
- 40
- 20
SLIDE 36
2 4 6 8 10 12 14 16 18 20
Impulse Response
- 0.8
- 0.6
- 0.4
- 0.2
0.2 0.4 0.6 0.8 1
Frequency Response
0 8 0 6 0 4 0 2 0 2 0 4 0 6 0 8 1
Group Delay
Impulse, Frequency, & Group Delay Response of 2-Path Linear Phase, Recursive Half-Band Filter
SLIDE 37
h1 h1 h5 h5 h7 h7 h9 h9 1 h3 h3
- 1
1 L
- w-P
as s L
- w-P
as s High-P ass High-P ass Half Band F ilter: h(n) h(2n) h(2n+ 1) 2-to-1
2-to-1 Resampling 2-Path Polyphase Filter and Digital Down-Converter
f
- 0.5
0.5 0.25
- 0.25
L P HP
2-Point DFT
SLIDE 38
- -1Resampling 4-Path Down-Sample Polyphase Filter and
4-Point IFFT Extracts Signal Component From One-of-Four Selected Nyquist Zones
Half Band Filters Centered on Cardinal Directions Each Reduces BW 2-to-1 and Reduces Sample Rate 2-to-1
2 1 3 L
- w-P
ass Filter P
- s Freq
Hilbert Filter High-P ass Filter Neg Freq Hilbert Filter
)
1 1 1 1 H (Z )
2
H (Z )
1
2
Z H (Z )
- 1
2
2
Z H (Z )
- 1
3
2
4-P
- int
IFFT
SLIDE 39
4-Path, 2-to-1 Down-Sample with 4 Possible Trivial Phase Shifters
4-Path Polyphase Filter Path-0 Not Used 2.5-Multiplies per Input 4-Phase Rotators fsk/4: {c0 c1 c2 c3} fs0/4: {1 1 1 1} fs1/4: {1 j -1 -j} fs2/4: {1 -1 1 -1} fs3/4: {1 -j -1 j} 2-to-1 Down-Sample
h1 h1 h5 h5 h13 h13 h9 h9 1 h17 h17
- c 0
c 1 c 2 c 3
SLIDE 40
Four Bands Centered on the Cardinal Directions
Bands Centered
- n 0 and 180
(DC and fS/2) Alias To DC When Down-Sampled 2-to-1 Bands Centered
- n +90 and -90
(+fS/4 and –fS/4) Alias To fs/2 When Down-Sampled 2-to-1
SLIDE 41
Spectra: Four Half Band Filters on Unit Circle
Showing Alias Free Pass, Transition, and Aliased Bands
Alias Free P ass Band Alias Free P ass Band T ransition Bandwidth T ransition Bandwidth Folded B andwidth Due to 2-to-1 Down S ample L
- w-P
ass Band-0 P
- sitive Freq-P
ass B and-1 Negative F req-P ass Band-3 High-P ass Band-2
Any Narrowband Signal Must Reside in One of the 4 Alias Free Band Intervals. The Alias Free Band Intervals Overlap!
SLIDE 42
Pole-Zero Diagrams of Four Nyquist Zone Filters
- 2
- 1
1 2
- 2
- 1
1 2
- 2
- 1
1 2
- 2
- 1
1 2
- 2
- 1
1 2
- 2
- 1
1 2
- 2
- 1
1 2
- 2
- 1
1 2
SLIDE 43
Frequency Responses of Four Nyquist Zone Filters
- 0.5
0.5
- 80
- 60
- 40
- 20
Frequency Response Log Mag (dB)
- 0.5
0.5
- 80
- 60
- 40
- 20
Frequency Response
- 80
- 60
- 40
- 20
Log Mag (dB)
- 80
- 60
- 40
- 20
SLIDE 44
ctra of Signal Aliased to Different Sampled Data Frequencies in Successive 2-to-1 Sample Rate Reductions.
SLIDE 45
Most Efficient Multistage Half-Band Digital Down-Converter
Channelizer 4-P ath 2-to-1 Down S ample Channelizer 4-P ath 2-to-1 Down S ample Channelizer 4-P ath 2-to-1 Down S ample Channelizer 4-P ath 2-to-1 Down S ample
...... ......
Filter Channel S elect DDS
5 10 15 20 0.5 1 Impulse Response: Half Band Filter
2 2 2 2.5 [1 ] 2 4 8 1 1 1 2.5 2.5 [1 ] 2 4 8 2.5 3 7.5 N
RL I-Q I-Q I-Q
SLIDE 46
Spectrum of Input Signal and Zoom to Spectral Segment
5 10 15 20 25 30 35 40 45 100
- 50
Frequency / MHz Received Signal Spectrum 17.9 17.92 17.94 17.96 17.98 18 18.02 18.04 18.06 18.08 18.1 100
- 50
F / MH Zoom-in
SLIDE 47
Spectra: Last Four Stages Processing Chain. Dotted Line Indicates Center Frequency of Desired Spectral Component
0.2
- 0.1
0.1 0.2 0.3 Frequency/MHz Stage 6
- 0.15 -0.1 -0.05
0.05 0.1 0.15
- 100
- 80
- 60
- 40
- 20
Frequency/MHz Stage 7
- 0.05
0.05
- 100
- 80
- 60
- 40
- 20
Frequency/MHz Stage 8
- 0.04
- 0.02
0.02 0.04
- 100
- 80
- 60
- 40
- 20
Frequency/MHz Stage 9
+0.56 = -0.44 0.28 3
- 0.86
= 0.14
SLIDE 48
ampled Data Frequency Locations on Successive Aliases
- 1
- 0.5
0.5 1
- 1
- 0.5
0.5 1
- 1
- 0.5
0.5 1
- 1
- 0.5
0.5 1
- 1
- 0.5
0.5 1
- 1
- 0.5
0.5 1
SLIDE 49
pectrum at Input and Output of Final Heterodyne and Filter Stage
- 0.04
- 0.03
- 0.02
- 0.01
0.01 0.02 0.03 0.04
- 100
- 50
Frequency / MHz M a g n itu d e / d B Post Processing--Input Signal Spectrum (ch # 600)
- 0.04
- 0.03
- 0.02
- 0.01
0.01 0.02 0.03 0.04
- 100
- 50
Frequency / MHz M a g n itu d e / d B Post Processing--Down Convert (ch # 600)
- 100
- 50
M a g n itu d e / d B Post Processing--Filtering (ch # 600)
SLIDE 50
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 .1015 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
- f bits to manipulate 48 arithmetic operations per input sample.
Number of operations for the I-Q half band filter chain is on the
- rder 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.
SLIDE 51
Linear Phase IIR Filter
- 0.5
- 0.4
- 0.3
- 0.2
- 0.1
0.1 0.2 0.3 0.4 0.5 8 2
Group Delay: Two Path, 4-Coefficient, Linear Phase 2-Path Filter
0.5
- 0.4
- 0.3
- 0.2
- 0.1
0.1 0.2 0.3 0.4 0.5 9 8 1
Group Delay: Detail
- 0.5
- 0.4
- 0.3
- 0.2
- 0.1
0.1 0.2 0.3 0.4 0.5
Frequency response
SLIDE 52
Most Efficient Multistage Half-Band Digital Down-Converter
Channelizer 4-P ath 2-to-1 Down S ample Channelizer 4-P ath 2-to-1 Down S ample Channelizer 4-P ath 2-to-1 Down S ample Channelizer 4-P ath 2-to-1 Down S ample
...... ......
Filter Channel S elect DDS
2 2 2 2.0 [1 ] 2 4 8 1 1 1 2.0 2.0 [1 ] 2 4 8 2.0 3 6.0 N
RL I-Q I-Q I-Q
5 10 15 20 25 30
Impulse Response, Two-Path, 4-Coefficient, Linear Phase IIR
SLIDE 53
Polyphase Partition of Band Pass Filter 1-Path to M-Path Transformation
2 1 1
( ) ( )
M N j k r nM M r n
G Z e Z h r nM Z
1 1
( ) ( ) ( )( ) ( )
k k k
N N j n j j n n n n
G Z h n e Z h n e Z H e Z
1 1 ( ) ( )
( ) ( )
k
M N j r nM r nM r n
G Z h r nM e Z
1 1
( ) ( )
k k
M N j j nM r nM r n
G Z e Z h r nM e Z
k k
θ = k 2π 2π θ = k M
Modulation Theorem of Z-Transform
SLIDE 54
Polyphase Band Pass Filter and M-to-1 Resampler
Z Z Z Z
- 1
- 2
- (M-2)
- (M-1)
H ( Z )
M
H ( Z )
1
M
H ( Z )
2
M
H ( Z )
M-2
M
H ( Z )
M-1
M
.... .... ....
x(n) y(n) y(nM) M:1 e e e e e j k0 j k1 j k2 j k(M-2) j k(M-1) 2 2 2 2 2
M M M M M
SLIDE 55
Apply Noble Identity to Polyphase Partition
Z Z Z Z
- 1
- 2
- (M-2)
- (M-1)
H ( Z ) H ( Z )
1
H ( Z )
2
H ( Z )
M-2
H ( Z )
M-1
.... .... ....
x(n) y(nM,k) M:1 M:1 M:1 M:1 M:1 e e e e e j k0 j k1 j k2 j k(M-2) j k(M-1) 2 2 2 2 2
M M M M M
We Reduce Sample Rate M-to-1 Prior to Reducing Bandwidth (Nyquist is Raising His Eyebrows!) We Intentionally Alias the Spectrum. (Were you Paying Attention in school when they discussed the importance of anti-aliasing filters?) M-fold Aliasing! M-Unknowns! M-Paths supply M-Equations We can the separate Aliases!
SLIDE 56
Move Phase Spinners to Output of Polyphase Filter Paths
Z Z Z Z
- 1
- 2
- (M-2)
- (M-1)
H ( Z ) H ( Z )
1
H ( Z )
2
H ( Z )
M-2
H ( Z )
M-1
.... .... ....
x(n) y(nM,k) M:1 M:1 M:1 M:1 M:1 e e e e e j k0 j k1 j k2 j k(M-2) j k(M-1) 2 2 2 2 2
M M M M M
W t Ph S i f f l ibl
SLIDE 57
Polyphase Partition with Commutator Replacing the “r” Delays in the “r-th” Path
H ( Z ) H ( Z )
1
H ( Z )
2
H ( Z )
M-2
H ( Z )
M-1
.... ....
x(n) y(nM,k) e e e e e j k0 j k1 j k2 j k(M-2) j k(M-1) 2 2 2 2 2
M M M M M
Note: We don’t assign Phase Spinners to Select Desired Center Frequency Till after Down Sampling And Path Processing This Means that The Processing for every Channel is the same till the Phase Spinner No longer LTI, Filter now has M-Different Impulse Responses! Now LTV or PTV Filter.
SLIDE 58
Armstrong to Tuned RF with Alias Down Conversion to Polyphase Receiver
Digital
B and-P ass
M-to-1
H(Z e )
- j
k
Digital L
- w-P
ass M-to-1
H(Z
)
e-j kn
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
- perations at low output rate rather than at high input rate!
M-P ath Digital P
- lyphase
M-to-1
H(Z )
r
e
- j 2
M rk
SLIDE 59
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 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 n-10 n-11 n-12 n-13 n-14
— — — — — —
SLIDE 60
Polyphase Partition 1-D filter becomes 2-D M-Path Filter
n n-1 n-2 n-3 n-4 n-5 n n-1 n-2 n-3 n-4 n-5 n n-1 n-2 n-3 n-4 n-5 n-12 n-13 n-14 n-15 n-16 n-17 n-12 n-13 n-14 n-15 n-16 n-17 n-12 n-13 n-14 n-15 n-16 n-17 n+ 6 n+ 5 n+ 4 n+ 3 n+ 2 n+ 1 n-6 n-7 n-8 n-9 n-10 n-11 n-6 n-7 n-8 n-9 n-10 n-11 n-6 n-7 n-8 n-9 n-10 n-11
— — — — — — — — — — — —
SLIDE 61
Reorder Filter and Resample
ADC s(t) h(r+ nM) h(M-1+ nM) h(0+ nM) h(1+ nM) s(n) r(nM) r(nM,k) CL K B ANDP AS S FIL T E R P OL Y P HAS E P AR T IT ION L OW P AS S FIL T E R P OL Y P HAS E P AR T IT ION P HAS E R OT AT OR S AL IAS E D HE T E R ODY NE e j r k
M
... ... ..... .....
.. this is very stuff....
SLIDE 62
Phase and Gain Response (3-Versions of Filter) Prototype Filter, Polyphase Filter Prior to Resampling, Polyphase Filter after Resampling
SLIDE 63
Impulse Response and Frequency Response of Prototype Low Pass FIR Filter
SLIDE 64
Impulse Response of 6-Path Polyphase Partition Prior to 6-to-1 Resampling
SLIDE 65
Frequency Response of 6-Path Polyphase Partition Prior to 6-to-1 Resampling
SLIDE 66
Phase Response of 6-Path Polyphase Partition Prior to 6-to-1 Resampling
SLIDE 67
Overlay Phase Response of 6-Path Polyphase Partition Prior to 6-to-1 Resampling
Nyquist Zone +1 Nyquist Zone +2 Nyquist Zone
- 1
Nyquist Zone
- 2
Nyquist Zone Phase Aligned 2/6 Phase Shifts
- 2/6
Phase Shifts
- 2/3
Phase Shifts 2/3 Phase Shifts
SLIDE 68
De-Trended Overlay Phase Response: 6-Path Partition Prior to 6-to-1 Resampling
SLIDE 69
3-D Paddle-Wheel Phase Profiles, 6-Path Partition Prior to 6-to-1 Resampling
SLIDE 70
Overlay 3-D Paddle-Wheel Phase Profiles, 6-Path Partition Prior to 6-to-1 Resampling
SLIDE 71
Overlay 3-D Paddle-Wheel Phase Profiles, Showing Phase Shifts in +1 Nyquist Zone
SLIDE 72
Overlay 3-D Paddle-Wheel Phase Profiles, Phase Shifted to Align Phases in +1 Nyquist Zone
SLIDE 73
PolyChanDemo
SLIDE 74
Single Channel Armstrong and Multirate Aliased Polyphase Receiver
H(Z ) H (Z )
y(nM,k) y(nM,k) y (nM) y(n,k)
e
M-to-1 M-to-1
1 1 2 2 2 2 M M
x(n) x(n) x(n)
e
M M j nk j nk 2 2
r r S tandard DDC P
- lyphase DDC
2-P
- lyphase
Filters 1-P
- lyphase
Filter
SLIDE 75
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
… …
… … … …
0 0 1 1 M-1 M-1
DDS
H (Z ) H (Z ) H (Z ) H (Z )
1 2 M-1
. . . . . .
x(n) y(nM,k)
e
M j 1k2
e
M j 0k2
e
M j 2k2
e
M j (M-1)k2
SLIDE 76
Bad Mismatch: Sample Rate Large Compared to Transition Bandwidth
Nyquist Rate for Filter is 200 kHz + 200 kHz = 400 kHz or fs/50
200 kHz 80 dB 0.1 dB 200 kHz 20 MHz
f
- 6 dB
/Octave 400 T ap FIR Filter 20 MHz Input S ample R ate 20 MHz Output S ample R ate
SLIDE 77
Polyphase Partition of Low‐Pass Filter
…
… …
0 1 2 49 48
50-to-1 400 T aps 8 T aps 20 kHz 20 MHz 400 kHz 400 kHz P
- lyphase
L
- w P
ass Filter
SLIDE 78
Cascade Polyphase Filter Down‐Sampling and Up‐Sampling
… …
… … … …
0 0 1 1 2 2 49 49 48 48
50-to-1 1-to-50 400 T aps 400 T aps 20 MHz 20 MHz 20 MHz 20 MHz 400 kHz 400 kHz P
- lyphase
L
- w P
ass Filter P
- lyphase
L
- w P
ass Filter 8-taps 8-taps
SLIDE 79
8-tap 8-tap 20 MHz 20 MHz 400 kHz Coeffic ient B ank Coeffic ient B ank S elec t S elec t
Efficient Polyphase Filter Implementation
400 T ap FIR Filter 20 MHz Input S ample R ate 20 MHz Output S ample R ate
SLIDE 80
16‐Ops/Input 60‐Ops/Input
Two Processing in Boxes:
- w can you tell which is which from outside box?
400-T ap L
- wpass
Filter 20MHz 20MHz 20MHz 400 kHz 20MHz 8-T ap F ilter 8-T ap F ilter
Coefficient Bank Coefficient Bank S tate Machine S elect S elect White Box White Box
SLIDE 81
Polyphase Partition of Low‐Pass Filter
H (Z ) H (Z ) H (Z ) H (Z ) H (Z ) H (Z ) H (Z ) H (Z )
1 1 2 2 M-1 M-1
...... ......
x(n) y(nM) y(n)
f f f
SLIDE 82
Polyphase Partition of Band Pass Filter
H (Z ) H (Z ) H (Z ) H (Z ) H (Z ) H (Z ) H (Z ) H (Z )
1 1 2 2 M-1 M-1
...... ......
x(n) y(nM,k) y(n,k)
e
M j 1k2
e
M j 1k2
e
M j 0k2
e
M j 0k2
e
M j 2k2
e
M j 2k2
e
M j (M-1)k2
e
M j (M-1)k2
f f f
SLIDE 83
- lyphase Partition of T
wo Band Pass Filters
H (Z ) H (Z ) H (Z ) H (Z ) H (Z ) H (Z ) H (Z ) H (Z )
1 1 2 2 M-1 M-1
...... ......
y(nM,k )
1
y(nM,k )
2
y(n,k )+ y(n+ k )
1 2
e
M j rk12
e
M j rk22
e
M j rk12 e M j rk22
f
SLIDE 84
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 [log2(M)/2]M Complex Phase Rotators
hen k > Log2(M)/2 Build all channels and discard the channels you don’t need! M=16, Log2(16)/2 = 2: thus if you want 2 or more, Build them all! M=128, Log2(128)/2 = 3.5: thus if you want 4 or more, Build them all! M=1024, Log2(1024)/2 = 5: thus if you want 5 or more, Build them all!
SLIDE 85
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
fs h (n) h (n)
2
h (n)= h(r+ nM)
r
P
- lyphase
P artition h (n)
M-2
h (n)
1
h (n)
3
h (n)
M-1
FDM T DM
M-PNT IFF T
..... ..... ..... .....
.. this is very stuff....
SLIDE 86
Dual Channel Armstrong and Multirate Aliased Polyphase Receiver
H(Z ) H(Z ) H (Z )
y(nM,k ) y(nM,k ) y(nM,k ) y(nM k ) y (nM) y(n,k ) y(n,k )
e e
M-to-1 M-to-1 M-to-1
1 1 1 2 2 2 2 2 2 2 M M
x(n) x(n) x(n)
e e
M M M M j nk j nk j nk j nk 2 2 2 2
r r S tandard DDC P
- lyphase DDC
1 1 1 1 1 2 2 2 2 2
2-P
- lyphase
Filters 2-P
- lyphase
Filters 1-P
- lyphase
Filter
SLIDE 87
Up-sampling by Zero Packing and Filtering
SLIDE 88
Spectra Of Input, of Zero-Packed, and of Low-Pass Filtered Zero-Packed Signal
y(m) x(m) x(n)
6 6 3 3 3 9 9 1 1 2 2 1 7 7 4 4 4 10 10 20 20 2 8 8 5 5 5 25 25 15 15 m n m
fs fs fs f f f 5 5
.... .... .... .... .... ....
F ilter R ejected S pectrum R ejected S pectrum
SLIDE 89
Spectra Of Input, of Zero-Packed, and of Band Pass Filtered Zero-Pack Signal
m) m) n)
6 3 3 9 1 2 1 7 4 4 10 20 2 8 5 5 25 15 m n m
fs fs fs f f f 5 5
.... .... ....
F ilter R ejected S pectrum R ejected S pectrum
SLIDE 90
1
( ) ( )
N n n
H Z h n Z
C0 C3 C4 C5 C6 C7 C8 C9 C12 C10 C13 C11 C14 C1 C2
1-to-5
1 1 ( )
( ) ( )
N M M r nM r n
H Z h r nM Z
C0 C3 C4 C5 C6 C7 C8 C12 C9 C13 C10 C14 C11 C1 C2
1-to-5
Polyphase Partition of Resampling Filter
SLIDE 91
1 1
( ) ( )
N M M r nM r n
H Z Z h r nM Z
C0 C3 C4 C5 C6 C7 C8 C12 C9 C13 C10 C14 C11 C1 C2
1-to-5
C0 C3 C4 C5 C6 C7 C8 C12 C9 C13 C10 C14 C11 C1 C2
1-to-5
Factor Delays and Rearrange
SLIDE 92
M Delays 1 Delay M Delays 1 Delay
Input Clock, T Output Clock, MT Z H(Z ) Z H(Z )
- M
- M
- 1
- 1
M:1 M:1 M:1 M:1 x(n) y(m) y(m) x(n)
Noble Identity: Interchange M-Delays with M-to-1 Resample
SLIDE 93
C0 C3 C4 C5 C6 C7 C8 C12 C9 C13 C10 C14 C11 C1 C2
1-to-5 1-to-5 1-to-5 1-to-5 1-to-5
C0 C3 C4 C5 C6 C7 C8 C12 C9 C13 C10 C14 C11 C1 C2
Interchange Filter and Resampler
Replace Up Samplers, Delays, and Summer with M-Port Output Commutator
SLIDE 94
2 1
( ) ( )
N j k n n M n
G Z h n e Z
1 2 1 ( ) ( ) 1 2 2 1 1 2 1
( ) ( ) ( ) ( ) ( ) ( )
N M M j r nM k r nM M r n N M M j rk j nM r nM M M r n N M M j rk r nM M r n
G Z h r nM e Z G Z Z e h r nM e Z G Z Z e h r nM Z
Low-Pass Replaced by Band-Pass
Spin The Delays, Don’t Touch the M-Path Partitioned Weights
SLIDE 95
Low-Pass to Band-Pass 1-to-M Up-Sampling Filter
SLIDE 96
M-Path, M-Channel Channelizer: Spinners are in IFFT
SLIDE 97
......
fs fB
W
fs M f
F0 F1 F2 FM-2 FM-1
M-Point IFFT Supplies Phase Spinners to Form Up Converters to all Multiples of Input Sample Rate
All Output Channels Centered on Multiples of Input Sample Rate Example: Multiples of 6-MHz
SLIDE 98
h (5n) h (5n) h (5n)
2
h (5n)
2
h (5n)
4
h (5n)
4
h ( 5n)
1
h ( 5n)
1
h ( 5n)
3
h ( 5n)
3
x(n) y(m,k) y(m,k) e e e
j j j n 2 2 5 5 kr kr
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 Frequency
Input Offset Observed at Output
SLIDE 99
y(m) x(m) x(n) x(n)
3 1 4 2 5 m n n m
fs fs fs fs f f f f 5 5 F ilter R ejected S pectrum R ejected S pectrum
Input Spectrum Shifted Spectrum Up-Sampled Spectrum Distorted Spectrum
SLIDE 100
y(m) x(m) x(n) x(n)
3 1 4 2 5 m n n m
fs fs fs fs f f f f 5 5 2 F ilters R ejected S pectrum R ejected S pectrum
Input Spectrum Shifted Spectrum Up-Sampled Spectrum T wo Filter Spectra
SLIDE 101
M-Channel Polyphase Channelizer: M-path Filter and M-point FFT
fs h (n) h (n)
2
h (n)= h(r+ nM)
r
P
- lyphase
P artition h (n)
M-2
h (n)
1
h (n)
3
h (n)
M-1
FDM T DM
M-P NT FF T
..... ..... .....
SLIDE 102
Critically Sampled fS=fC Nyquist Sampling fS=2fC QRT Nyquist Filter fS=2fC
QRT Nyquist Filter fS=4fC Nyquist Sampling fS=4fC
Various Filter-Channelizer Configurations
f f f f f f f f f f fs fs fs fs fs f = fs
f = fs
f = fs
f = fs
f = fs
f = fs
B W
f = fs
B W
f = fs
B W
f = 2fs
B W
f = fs
B W
f = fs
B W
f = fs
B W
f = fs
B W
ch(k-1) ch(k-1) ch(k-1) ch(k-1) ch(k-1) ch(k+ 1) ch(k+ 1) ch(k+ 1) ch(k+ 1) ch(k+ 1) ch(k) ch(k) ch(k) ch(k) ch(k) 0.8 2.0 2.0 4.0 1.0 1.0 1.0 1.0 2.0 0.1 dB 0.1 dB 3.0 or 6.0 dB 6.0 dB 0.1 dB
SLIDE 103
f f
B W B W S ample R ate= 2 Channel S pacing S ample R ate = Channel S pacing Channel S pac ing Channel S pac ing Aliased T ransition B andwidth
Filter Sampled at Rate to Avoid Band Edge Aliasing
SLIDE 104
Prototype Low‐Pass Filter for 120 Channel Channelizer
SLIDE 105
(M 1)
H ( Z )
2
H ( Z )
1 2
H ( Z )
2 2
H ( Z )
2
H ( Z )
2
H ( Z )
2 2
.... ....
y(n ) y(n)
M 2
Z Z Z Z Z
- 1
- 2
.... .... .... .... ....
....
M M M M M M 2 2 2 2 2 2M 2 :1
- ( - 1)
- (
+
( 1)+
( - 1)- ( )
M‐Path Polyphase Filter and M/2‐to‐1 Down Sampling
SLIDE 106
(M 1)
H ( Z )
2
H ( Z )
1
2
H ( Z )
2
2
H ( Z )
2
H ( Z )
2
H ( Z )
2
.... ....
y(n )
M 2
Z Z Z Z Z
- 1
- 2
.... .... .... .... ....
....
M
M M M M M M
2
2 2 2 2 2 2
:1
M 2 :1 M 2 :1 M 2 :1 M 2 :1 M 2 :1 M 2 :1
- ( -1)
- (
1)
+
( 1)
+
( -1)
- ( )
( )
Use Noble Identity to Pull M/2‐to‐1 Resampler Through Path Filter
Path Filters: Polynomials in ZM Converted to Polynomials in Z2
SLIDE 107
H ( Z )
2
H ( Z )
1
2
H ( Z )
2
2
H ( Z )
2
H ( Z )
2
H ( Z )
2
H ( Z )
2
.... ....
y(n )
M 2
Z Z Z Z Z Z
- 1
- 1
- 1
- 1
- 1
- 2
.... .... .... ....
....
M
M M M M M
2
2 2 2 2
:1
M 2 :1 M 2 :1 M 2 :1 M 2 :1 M 2 :1 M 2 :1
- ( -1)
- (
- 1)
( 1)
+
( -1) ( )
Use Noble Identity to Pull M/2‐to‐1 Resampler Through Delays in Lower Half of Paths
SLIDE 108
Z Z Z
- 1
- 1
- 1
H ( Z ) H ( Z )
1
H ( Z )
2
H ( Z )
( -1)
H ( Z )
( )
H ( Z )
M-2
H ( Z )
M 1
.... .... .... .... ....
x(n) y(n )
M 2
M 2 M 2 2 2 2 2 2 2 2
Replace Delays and M/2‐to‐1 Resamplers with Dual Input M/2 Path Commutator
SLIDE 109
H (Z )
k 2
Z H (Z )
- 1
2 (k+ M/2)
h(k) h(k+M/2) h(k+M) h(k+3 ) h(k+2M) h(k+5 ) h(k+3M) h(k+7 ) h(k+4M) h(k+9 ) M 2 M 2 M 2 M 2
Fold Unit Delays in lower half Filter Paths Into Filter Polynomials in Z2
SLIDE 110
fs f fs fs fs fs fs fs fs fs M M M M M M M M M 2 2
- 2
- 2
- M-to-1 R
esample M/2-to-1 R esample
..... ..... ..... .....
f f
M‐to‐1 Down Sample Aliases Multiples of Output Sample Rate to DC M/2‐to‐1 Down Sample Aliases Odd Multiples of Output Sample Rate to Half sample Rate
SLIDE 111
0 5T T 1.5T T T 0.5T 0.5T mT (m+1)T
fs 2 fs/M M Path Poly Phase Filter in Z M-PNT IFFT
.....
flg=0
flg=0
flg=1 f l g = 1
Circular Buffer
2
Circular Buffer Between Polyphase Filter and IFFT Aligns Shifting Input Origin with IFFT’s Origin
SLIDE 112
1 1 2 2 3 M/2-1 M-1 M/2 M/2+ 1 M-2 M-1
M M M
.... ........ …. M-P
ath P
- lyphase Filter
M-P
- int IFFT
F DM
T DM
M-P ath Input Data B uffer
Circular Output B
uffer
S tate E ngine
M/2‐to‐1 Analysis Channelizer
SLIDE 113
1 1 2 2 3 M/2-1 M-1 M/2 M/2+ 1 M-2 M-1
M M M
.... ........ …. M-P
ath P
- lyphase Filter
M-P
- int IFFT
FDM T DM
M-P ath Input Data B uffer
Circular Output B
uffer
S tate E ngine
1‐to‐M/2 Synthesis Channelizer
SLIDE 114
.. ..
.. .. .. ..
.. ..
.. .. .. .. .. ..
M P
- int
IF FT M P
- int
IF FT M P ath F ilter M P ath F ilter
nM/2 P
- int
S hift M-P
- int
Circular Buffer nM/2 P
- int
S hift M-P
- int
Circular Buffer
x x x x
Frequency Domain Filtering With Cascade M/2‐to‐1 Analysis and 1‐to‐M/2 Synthesis Channelizers
SLIDE 115
Impulse response and Frequency Response 25‐Enabled Ports: 2.4 MHz Bandwidth
SLIDE 116
Impulse Response and Frequency Response 40‐Enabled Ports: 3.9 MHz Bandwidth
SLIDE 117
Mixed, Arbitrary Bandwidth Channelizers
SLIDE 118
SLIDE 119
Mixed Bandwidth Signals presented to Channel Synthesizer
SLIDE 120
- mpose Broadband Signals Using Short Analysis
rs and Present Components to Synthesizer
.... .... .... .... .... .... .... .... .... .... .... ....
S elector
Input Channel Configuration M M M M N N
Anal. Anal. N-1 B w0 P P
N-1
B WN-1 2-to-M, M-Channel S ynthesis Channelizer 2fs Mfs
1 2 M/2-1 M/2 M/2+ 1 M-1
.... …. M-P
ath P
- lyphase Filter
M-P
- int IFFT
M-P ath Input Data B uffer
Circular Output B
uffer
S tate E ngine
Analysis Channleizers
SLIDE 121
0-MHz Input Signal Partitioned into ive 10-MHz Sub Channels: fS=20 MHz
SLIDE 122
Multiple Partitioned Input Bands Presented to Synthesizer
SLIDE 123
Assembled Multiple BW Channels in Single Synthesis Channelizer
SLIDE 124
Reassemble Decomposed Broadband Signals Using Short Synthesis Filters formed by Multiple Channel Analysis Channelizer
1 2 M/2-1 M/2 M/2+ 1 M-1
....
.... .... .... .... .... .... .... .... .... .... .... ....
…. M-P
ath P
- lyphase Filter
M-P
- int IFFT
S elector M-P ath Input Data B uffer
Circular Output B
uffer
S tate E ngine M M M M N N
S ynth. S ynth. N-1 B w0 P P
N-1
B WN-1 M/2-to-1, M-Channel Channelizer fs
SLIDE 125
Partitioned spectral Components from ingle Multi-Channel Analyzer
- 10
10
- 40
40 FLTR 0
- 10
10
- 40
40 FLTR 1
- 10
10
- 40
40 FLTR 2
- 10
10
- 40
40 FLTR 3
- 10
10
- 40
40 FLTR 4
- 10
10
- 40
40 FLTR 5
- 10
10
- 40
40 FLTR 6
- 10
10
- 40
40 FLTR 7
- 10
10
- 40
40 FLTR 8
- 10
10
- 40
40 FLTR 9
- 10
10
- 40
40 FLTR 10
- 10
10
- 40
40 FLTR 11
- 10
10
- 40
40 FLTR 12
- 10
10
- 40
40 FLTR 13
- 10
10
- 40
40 FLTR 14
- 10
10
- 40
40 FLTR 15
- 10
10
- 40
40 FLTR 16
- 10
10
- 40
40 FLTR 17
- 10
10
- 40
40 FLTR 18
- 10
10
- 40
40 FLTR 19
- 10
10
- 40
40 FLTR 20
- 10
10
- 40
40 FLTR 21
- 10
10
- 40
40 FLTR 22
- 10
10
- 40
40 FLTR 23
- 10
10
- 40
40 FLTR 24
- 10
10
- 40
40 FLTR 25
- 10
10
- 40
40 FLTR 26
- 10
10
- 40
40 FLTR 27
- 10
10
- 40
40 FLTR 28
- 10
10
- 40
40 FLTR 29
- 10
10
- 40
40 FLTR 30
- 10
10
- 40
40 FLTR 31
- 10
10
- 40
40 FLTR 32
- 10
10
- 40
40 FLTR 33
- 10
10
- 40
40 FLTR 34
- 10
10
- 40
40 FLTR 35
- 10
10
- 40
40 FLTR 36
- 10
10
- 40
40 FLTR 37
- 10
10
- 40
40 FLTR 38
- 10
10
- 40
40 FLTR 39
- 40
40 FLTR 40
- 40
40 FLTR 41
- 40
40 FLTR 42
- 40
40 FLTR 43
- 40
40 FLTR 44
- 40
40 FLTR 45
- 40
40 FLTR 46
- 40
40 FLTR 47
SLIDE 126
Reassembled Wide band Channels rom Short Synthesis Channelizers
- 20
20
- 40
40 Sig 3 Magnitude (dB)
- 20
20
- 40
40 Sig 6
- 20
20
- 40
40 Sig 8
- 20
20
- 40
40 Sig 4 Frequency (MHz) Magnitude (dB)
- 20
20
- 40
40 Sig 5 Frequency (MHz)
- 20
20
- 40
40 Sig 7 Frequency (MHz)
- 30
- 20
- 10
10 20 30
- 40
40 Sig 1 F (MH ) Magnitude (dB)
- 30
- 20
- 10
10 20 30
- 40
40 Sig 2 F (MH )
SLIDE 127
Signal Fidelity Preserved under Multiple Sub- Channel Disassembly and Reassembly
SLIDE 128
SLIDE 129
.. ..
.. ..
.. ..
.. .. .. .. .. ..
M P
- int
IF FT M P
- int
IF FT M P ath F ilter M P ath F ilter
nM/2 P
- int
S hift M-P
- int
Circular Buffer nM/2 P
- int
S hift M-P
- int
Circular Buffer
Cascade M/2‐to‐1 Analysis and 1‐to‐M/2 Synthesis Channelizers Frequency Domain Filtering and Spectral Shuffle
SLIDE 130
SLIDE 131
HERE POINTED HAIR BOSS. THIS REPORT EXPLAINS HOW A SMALL FREQUENCY OFFSET AT THE INPUT SAMPLE RATE IS CONVERTED TO THE SAME FREQUENCY OFFSET FROM THE CHANNEL CENTER FREQUENCY AT THE HIGH OUTPUT SAMPLE RATE.
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Suspicions Confirmed!
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