new directions in channelized receivers and transmitters€¦ · version of edwin armstrong’s...

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 fC

Analog Digital

Multi-Channel FDM Input Signal

Single-Channel Base banded Output Signal

Receiver

Input Spectrum

ffC

Selec ted Narrow Band Signal

3

See!

Utah

4

First Generation DSP Receiver

f

f

f

f

Low Pass Filter

fs

-jte

1 2 3

3

4

4

2

1

AnalogSignal Processing

5

Signal Conditioning for DSP Receiver

f

f

ffs/2-fs/2

1

2

3

.... ....

Low Pass Filter

fs1 2 3 4

6

Replicate Analog Processing in DSP

Low Pass Filter

fs/M

M:1

-jne

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

....

....

....

....

....

....

....

....

Low Pass Filter

fs1 2 3 4

~ ~~ ~

IgnoringGood Advice!

7

ADC

M:1

s(t) s(n)

h(n)

r(n) r(nM)s(n)

e -j n

e -j n

LOCLK

LOWPASS FILTER

•Fundamental Operations•Select Frequency•Limit Bandwidth•Select Sample Rate

Digital Down Converter (DDC)

8

Spectral Description ofFundamental Operations

0

0

0

0

fs/2

fs/2

fs/2

fs/M

-fs/2

-fs/2

-fs/2

-fs/M

f

f

f

f

CHANNEL OF INTEREST

TRANSLATED SPECTRUM

FILTERED SPECTRUM

SPECTRAL REPLICATES AT DOWN-SAMPLED RATE

OUTPUT DIGITAL FILTER RESPONSE

INPUT ANALOG FILTER RESPONSE

.... ....

9

A Shell Game:Rearrange the Players!

Keep Your Eye on the Pea!

10

Signal and Filter are at Different FrequenciesWhich One to Move?

0

0

0

0

f

f

f

f

CHANNEL OF INTEREST

TRANSLATED SPECTRUM

TRANSLATED FILTER

FILTER RESPONSE

SecondOption

FirstOption

11

Down Sample Complex Digital IF

Band Pass Filter

fs/M

M:1

-jMnjn

ee

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

....

....

....

....

....

....

....

....

Low Pass Filter

fs1 2 3 4

~

~

~

~

No Spectral

Image

12

Fundamental Operation with Rearrange Operators

ADCM:1

s(t)

h(n)

s(n) r(n)r(n) r(nM)

e -j ne j n

e j n

LOCLK

BANDPASS FILTER

Up-Convert Filter, Filter Signal at IF, Down Convert Output of Filter

13

Equivalency Theorem

0

0

0 0

0 0

( )

( ) ( ) * ( )( ) ( )

( ) ( )

{ ( )* ( ) }

j n

j n k

k

j n j k

k

j n j n

r n s n e h ks 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 FilterTo Become Complex

Band-Pass Filter14

Signal Flow Description of Equivalency Theorem

Not Finished: Moving Down Converter from Input to Output Replaces 2-Multipliers (Complex Scalar) with 4-Multipliers (Complex Product)

15

Interchange Down Converter and Resampler

ADCM:1

s(t)

h(n)

s(n) r(nM)r(n)r(nM)

e -j Mne j n

e j n e j Mn

LOCLK

BANDPASS FILTER

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

SPECTRAL DESCRIPTION REORDERED FUNDAMENTAL OPERATION

0

0

0

0

fs/2

fs/2

fs/M

fs/M

-fs/2

-fs/2

-fs/M

-fs/M

f

f

f

f

CHANNEL OF INTEREST

TRANSLATED FILTERFILTERED SPECTRUM

ALIASED REPLICATES AT DOWN-SAMPLED RATE

TRANSLATED REPLICATES AT DOWN-SAMPLED RATE

INPUT ANALOG FILTER RESPONSE

....

....

....

....

17

Successive Transformations Turn Sampled Data Version of Edwin Armstrong’s Heterodyne Receiver

to Tuned Radio Frequency (TRF) Receiverand then to Aliased TRF Receiver.

DigitalBand-Pass

M-to-1

e -j kn

H(Ze )-j k

DigitalLow-Pass

M-to-1

H(Z)

e-j kn

DigitalBand-Pass

M-to-1

H(Ze )-jk

k

k

M θ = k 2π2πor θ = kM

Any Multiple of Output Sample Rate Aliases to Baseband

Armstrong Nyquist Equivalency Theorem

DigitalBand-Pass

M-to-1

-j Mkne

H(Ze )-jk

18

Let’s Keep Rearranging the Players!

19

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

Linear systems Are Associative

X

X

Y W

W

L1 L2

L = L L3 1 2

X

X

Y W

W

L1 L2

L = L L3 1 2

Filter Resample

Resampled Filter

21

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

Filter and Output Resampler can Commuteto 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

Coefficient Assignment of Low-Pass Polyphase Partition

Extract Delays To First Non-Zero Coefficient

For M-to-1 resample start at Index r and Increment by MFor 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

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

00

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

This mapping from 1-D to 2-D is used by Cooley-Tukey FFT. Polyphase Filters and CT-FFT are kissing cousins!

24

Polyphase Partition of Low Pass Filter

1-Path to M-Path Transformation

H( Z) x(n) y(n) y(nM)M:1

1

0( ) ( )

Nn

nH Z h n Z

1 1

( )

0 0( ) ( )

M Nr nM

r nH Z h r nM Z

1 1

0 0( ) ( )

M Nr nM

r nH Z Z h r nM Z

Z

Z

Z

Z

-1

-2

-(M-2)

-(M-1)

H ( Z )0M

H ( Z )1M

H ( Z )2M

H ( Z )M-2M

H ( Z )M-1M

....

.... ....

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

Noble Identity: Commute M-units of Delay

followed by M-to-1 Down SampleM Delays

1 Delay

M Delays

1 Delay

Input Clock, T

Output Clock, MT

Z Z-M -1

M:1 M:1M-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

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 )0

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

Replace Delays with CommutatorPerform Path Operations Sequentially

H ( Z )0

H ( Z )1

H ( Z )2

H ( Z )M-2

H ( Z )M-1

.... ....

x(n) y(nM)

8-tap

Coeffic ient BankSelect

IFSTAGE

RFSTAGE

IFSTAGE

DDS

DDS

S-P

P-S

Clock

Clock

CARRIER PLL

TIMING PLL

DAC

ADC

DIGITALLOW-PASS

DIGITALLOW-PASS

DIGITALLOW-PASS

DIGITALLOW-PASS

DIGITALLOW-PASS

DIGITALLOW-PASS

DIGITALLOW-PASS

DIGITALLOW-PASS

Shape &Upsample

Matched Filter

Interpola te

Dec ima te

I-Q Table

Detect

cos( n)

cos( n)

sin( n)

sin( n)

Modulator

Demodulator

f

Analog Analog

Analog

Osc illator

PA

RF Carrier

GainControl

Osc illator

CarrierWaveform

VGALNA

RF Carrier

IFChannel Channel

Bits

Bits

Transmitter Process, Up-Sample and Up-ConvertReceiver 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

Conventional and Ubiquitous DDC

DDS

DIGITALLOW-PASS

DIGITALLOW-PASS

M:1

M:1

DDS

CIC

CIC

M:1 2:1

2:1

2:1

2:1M:1

Scale Factor

CIC Correc tion

CIC Correc tion

HalfBand

HalfBand

HalfBand

HalfBand

z z zz-1z-1z-1 -1 -1 -1- - -

zz-1 -1-

zz-1 -1-

M:1Integra tors Derivative Filters

Convert Two Parallel Paths into M Sequential Paths for each Path

DDS

DIGITALLOW-PASS

DIGITALLOW-PASS

M:1

M:1

……

……

……

0

0

1

1

M-1

M-1

DDS

DDS

8-tap

8-tap

Coeffic ient

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‐A2‐Shifts

2‐A2‐Shifts

2‐A2‐Shifts

2‐A2‐Shifts

4‐A2‐Mult

4‐A2‐Mult

4‐A2‐Mult

4‐A2‐Mult

6‐A3‐Mult

10‐A5‐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

0 0 0 0 2/16 2/32 2/64 2/128 3/256 5/512 0.27

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 1-80-60-40-20

0

5 10 15 20 25 30 35 40 45 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-80-60-40-20

0

10 20 30 40 50 60 70 80 90 -4 -3 -2 -1 0 1 2 3 4-80-60-40-20

0

20 40 60 80 100 120 140 160 180 200 -8 -6 -4 -2 0 2 4 6 8-80-60-40-20

0

50 100 150 200 250 300 350 400 15 10 5 0 5 10 15-80-60-40-20

0

Impulse and Frequency Response of Last Stage Referred to Earlier Stages

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‐A1‐Mult

3‐A1‐Mult

3‐A1‐Mult

3‐A1‐Mult

7‐A3‐Mult

7‐A3‐Mult

7‐A3‐Mult

7‐A3‐Mult

7‐A3‐Mult

9‐A4‐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

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 0.2 0.4 0.6 0.8 1-80-60-40-20

0

10 20 30 40 50 60 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-80-60-40-20

0

20 40 60 80 100 120 -4 -3 -2 -1 0 1 2 3 4-80-60-40-20

0

50 100 150 200 -8 -6 -4 -2 0 2 4 6 8-80-60-40-20

0

100 150 200 250 300 350 400 450 15 10 5 0 5 10 15-80-60-40-20

0

0 2 4 6 8 10 12 14 16 18 20

Impulse Response

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Frequency Response

0 8 0 6 0 4 0 2 0 0 2 0 4 0 6 0 8 1

Group Delay

Impulse, Frequency, & Group Delay Response of 2-PathLinear Phase, Recursive Half-Band Filter

0 0

h1 h1

0 0

h5 h5

0 0 0

h7h7

0

h9 h9

01

h3h3

-

-0

0

1

1

Low-Pass

Low-Pass

High-Pass

High-Pass

Half Band Filter: 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 0.50 0.25-0.25

LP

HP

2-Point DFT

o-1Resampling 4-Path Down-Sample Polyphase Filter and 4-Point IFFT Extracts Signal Component

From One-of-Four Selected Nyquist Zones

Half Band FiltersCentered on

Cardinal DirectionsEach Reduces

BW 2-to-1 and Reduces

Sample Rate 2-to-1

0

2

1

3

Low-PassFilter

Pos FreqHilbert Filter

High-PassFilter

Neg FreqHilbert Filter

)

1

1

1

1H (Z )0

2

H (Z )12

Z H (Z )-1

22

Z H (Z )-1

32

4-Point IFFT

4-Path, 2-to-1 Down-Samplewith 4 Possible Trivial Phase Shifters

4-Path Polyphase FilterPath-0 Not Used

2.5-Multiplies per Input

4-Phase Rotatorsfsk/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-1Down-Sample

0

0

h1

h1

0

0

h5

h5

0

0 0

h13

h13

0

h9

h9

0

1

h17

h17

-

-

-

-

-

-

-

-

- -

-

--

-

-

-

-

-

-

-

c0

c 1

c 2

c 3

Four Bands Centered on the Cardinal Directions

Bands Centeredon 0 and 180 (DC and fS/2)

Alias To DC When Down-Sampled

2-to-1

Bands Centeredon +90 and -90 (+fS/4 and –fS/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

Alias FreePass Band

Alias FreePass Band

TransitionBandwidth

TransitionBandwidth

Folded BandwidthDue to 2-to-1 Down Sample

Low-PassBand-0

Positive Freq-Pass Band-1

Negative Freq-PassBand-3

High-PassBand-2

Any NarrowbandSignal Must Reside

in One of the 4 Alias Free Band Intervals.

The Alias FreeBand Intervals Overlap!

Pole-Zero Diagrams of Four Nyquist Zone Filters

-2 -1 0 1 2

-2

-1

0

1

2

-2 -1 0 1 2

-2

-1

0

1

2

-2 -1 0 1 2

-2

-1

0

1

2

-2 -1 0 1 2

-2

-1

0

1

2

Frequency Responses of Four Nyquist Zone Filters

-0.5 0 0.5

-80

-60

-40

-20

0

Frequency Response

Log

Mag

(dB

)

-0.5 0 0.5

-80

-60

-40

-20

0

Frequency Response

-80

-60

-40

-20

0

Log

Mag

(dB

)

-80

-60

-40

-20

0

ctra of Signal Aliased to Different Sampled Data Frequencies in Successive 2-to-1 Sample Rate Reductions.

Most Efficient MultistageHalf-Band Digital Down-Converter

Channelizer 4-Path 2-to-1 Down Sample




......

......

Filter

Channel Select DDS

0 5 10 15 20

0

0.5

1

Impulse Response: Half Band Filter 2 2 22.5 [1 ]2 4 8

1 1 12.5 2.5 [1 ]2 4 8

2.5 3 7.5

N

RL I-Q I-Q I-Q

Spectrum of Input Signal and Zoom to Spectral Segment

0 5 10 15 20 25 30 35 40 45100

-50

0

Frequency / MHz

Received Signal Spectrum

17.9 17.92 17.94 17.96 17.98 18 18.02 18.04 18.06 18.08 18.1100

-50

0

F / MH

Zoom-in

Spectra: Last Four Stages Processing Chain. Dotted Line Indicates Center Frequency of

Desired Spectral Component

0.2 -0.1 0 0.1 0.2 0.3Frequency/MHz

Stage 6

-0.15 -0.1 -0.05 0 0.05 0.1 0.15-100

-80

-60

-40

-20

0

Frequency/MHz

Stage 7

-0.05 0 0.05-100

-80

-60

-40

-20

0

Frequency/MHz

Stage 8

-0.04 -0.02 0 0.02 0.04-100

-80

-60

-40

-20

0

Frequency/MHz

Stage 9

+0.56= -0.440.283

-0.86= 0.14

ampled Data Frequency Locations on Successive Aliases

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

pectrum at Input and Output of Final Heterodyne and Filter Stage

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-100

-50

0

Frequency / MHz

Mag

nitu

de /

dB

Post Processing--Input Signal Spectrum (ch # 600)

-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04-100

-50

0

Frequency / MHz

Mag

nitu

de /

dB

Post Processing--Down Convert (ch # 600)

-100

-50

0

Mag

nitu

de /

dB

Post Processing--Filtering (ch # 600)

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 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

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

8

0

2Group Delay: Two Path, 4-Coefficient, Linear Phase 2-Path Filter

0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.59

8

1Group Delay: Detail

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0

0

Frequency response

Most Efficient MultistageHalf-Band Digital Down-Converter





......

......

Filter

Channel Select DDS

2 2 22.0 [1 ]2 4 8

1 1 12.0 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

Polyphase Partition of Band Pass Filter1-Path to M-Path Transformation

21 1

0 0

( ) ( )M Nj k r nMM

r n

G Z e Z h r nM Z

1 1

0 0( ) ( ) ( ) ( ) ( )k k k

N Nj n j jn n

n nG Z h n e Z h n e Z H e Z

1 1

( ) ( )

0 0( ) ( ) k

M Nj r nM r nM

r nG Z h r nM e Z

1 1

0 0( ) ( )k k

M Nj j nMr nM

r nG Z e Z h r nM e Z

k

k

θ = k 2π2πθ = kM

Modulation Theorem of Z-Transform

Polyphase Band Pass Filter and M-to-1 Resampler

Z

Z

Z

Z

-1

-2

-(M-2)

-(M-1)

H ( Z )0M

H ( Z )1M

H ( Z )2M

H ( Z )M-2M

H ( Z )M-1M

....

.... ....

x(n) y(n) y(nM)M:1e

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

Apply Noble Identity to Polyphase Partition

Z

Z

Z

Z

-1

-2

-(M-2)

-(M-1)

H ( Z )0

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-EquationsWe can the separate Aliases!

Move Phase Spinners to

Output of Polyphase Filter Paths

Z

Z

Z

Z

-1

-2

-(M-2)

-(M-1)

H ( Z )0

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

Polyphase Partition with Commutator Replacing the “r” Delays in the “r-th” Path

H ( Z )0

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 assignPhase Spinners to Select Desired Center FrequencyTill after Down SamplingAnd Path Processing

This Means thatThe Processing for every Channelis the same till the Phase Spinner

No longer LTI, Filter now hasM-Different Impulse Responses!Now LTV or PTV Filter.

Armstrong to Tuned RF with Alias Down Conversion to Polyphase Receiver

DigitalBand-Pass

M-to-1

H(Ze )-jk

DigitalLow-Pass

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 operations at low output rate rather than at high input rate!

M-Path DigitalPolyphase

M-to-1

H(Z)r

e-j 2

M rk

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

— — — — — —

Polyphase Partition 1-D filter becomes2-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

——————

——————

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)

CLK

BANDPASS FILTERPOLYPHASE PARTITION

LOWPASS FILTERPOLYPHASE PARTITION

PHASE ROTATORSALIASED HETERODYNE

ej r kM

... ...

.....

.....

.. this is verystuff....

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 PolyphasePartition Prior to 6-to-1 Resampling

Frequency Response of 6-Path PolyphasePartition Prior to 6-to-1 Resampling

Phase Response of 6-Path PolyphasePartition Prior to 6-to-1 Resampling

Overlay Phase Response of 6-Path Polyphase Partition Prior to 6-to-1 Resampling

0 Nyquist

Zone +1 Nyquist

Zone +2 Nyquist

Zone

-1 Nyquist

Zone

-2 Nyquist

Zone

Phase Aligned 2/6

PhaseShifts

-2/6PhaseShifts

-2/3PhaseShifts

2/3PhaseShifts

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

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

rr

Standard DDC

Polyphase DDC

2-Polyphase Filters

1-Polyphase 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

……

……

……

0

0

1

1

M-1

M-1

DDS

H (Z )

H (Z )

H (Z )

H (Z )

0

1

2

M-1

......

x(n) y(nM,k)

e Mj 1k2

e Mj 0k2

e Mj 2k2

e Mj (M-1)k2

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 TapFIR Filter

20 MHz InputSample Ra te

20 MHz Output Sample Rate

Polyphase Partition of Low‐Pass Filter

… … …0

1

2

49

48

50-to-1

400 Taps

8 Taps

20 kHz

20 MHz

400 kHz

400 kHz

PolyphaseLow Pass Filter

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 Taps 400 Taps20 MHz 20 MHz

20 MHz 20 MHz

400 kHz

400 kHz



8-taps 8-taps

8-tap 8-tap

20 MHz 20 MHz400 kHz

Coeffic ient Bank

Coeffic ient BankSelec t Selec t

Efficient Polyphase Filter Implementation

400 TapFIR Filter

20 MHz InputSample Ra te

20 MHz Output Sample Rate

16‐Ops/Input

60‐Ops/Input

Two Processing in Boxes:ow can you tell which is which from outside box?

400-TapLowpass Filter

20MHz

20MHz

20MHz

400 kHz20MHz8-Tap Filter

8-Tap Filter

Coeffic ient Bank

Coeffic ient Bank

State MachineSelec tSelec t

White Box

White Box

Polyphase Partition of Low‐Pass Filter

H (Z ) H (Z )

H (Z ) H (Z )

H (Z ) H (Z )

H (Z ) H (Z )

0 0

1 1

2 2

M-1 M-1

......

......

x(n)y(nM)

y(n)

f f f

Polyphase Partition of Band Pass Filter

H (Z ) H (Z )

H (Z ) H (Z )

H (Z ) H (Z )

H (Z ) H (Z )

0 0

1 1

2 2

M-1 M-1

......

......

x(n)y(nM,k)

y(n,k)

e Mj 1k2

e Mj 1k2

e Mj 0k2

e Mj 0k2

e Mj 2k2

e Mj 2k2

e Mj (M-1)k2

e Mj (M-1)k2

f f f

olyphase Partition of Two Band Pass Filters

H (Z ) H (Z )

H (Z ) H (Z )

H (Z ) H (Z )

H (Z ) H (Z )

0 0

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 Mj rk12 e Mj rk2

2

e Mj rk12 e Mj rk2

2

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 [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!

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)0

h (n)2

h (n)= h(r+ nM)r

Polyphase Pa rtition

h (n)M-2

h (n)1

h (n)3

h (n)M-1

FDM TDM

M-PNT IFFT

..... ..........

.....

.. this is verystuff....

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

rr

Standard DDC

Polyphase DDC

1

1

1

1

1

2

2

2

2

2

2-Polyphase Filters

2-Polyphase Filters

1-Polyphase Filter

Up-sampling by Zero Packing and Filtering

Spectra Of Input, of Zero-Packed, and of Low-Pass Filtered Zero-Packed Signal

y(m)

x(m)

x(n)

0

0

0

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

....

....

....

....

....

....

FilterRejec tedSpec trum

Rejec tedSpec trum

Spectra Of Input, of Zero-Packed, and of Band Pass Filtered Zero-Pack Signal

m)

m)

n)

0

0

63

3

91 2

1

74

4

10 20

2

85

5

2515

m

n

m

fs

fs

fs

f

f

f

5

5

.... .... ....


Rejec tedSpec trum

1

0( ) ( )

Nn

nH Z h n Z

C0 C3 C4 C5 C6 C7 C8 C9 C12C10 C13C11 C14C1 C2

1-to-5

11( )

0 0( ) ( )

NM M

r nM

r nH Z h r nM Z

C0

C3

C4

C5

C6

C7

C8

C12

C9

C13

C10

C14

C11C1

C2

0

0

0 0

0

0

0

0

0

0 0

0

0

0

00

0

00

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1-to-5

Polyphase Partition of Resampling Filter

11

0 0

( ) ( )

NM M

r nM

r n

H Z Z h r nM Z

C0

C3

C4

C5

C6

C7

C8

C12

C9

C13

C10

C14

C11C1

C2

0 0 0

0 0

0

0

0 0

0

0

0

00

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0 0

0

0

0

0

0

0

00

0

0

0

0

0

0

0

0

0

0

0

1-to-5

C0

C3

C4

C5

C6

C7

C8

C12

C9

C13

C10

C14

C11C1

C2

0 0 0

0 0

0

0

0 0

0

0

0

00

0

0

0

00

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0 0

0

0

0

0

0

0

0 0

0

0

0

0

0

0

0

1-to-5

Factor Delays and Rearrange

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:1x(n) y(m) y(m)x(n)

Noble Identity: Interchange M-Delays with M-to-1 Resample

C0

C3

C4

C5

C6

C7

C8

C12

C9

C13

C10

C14

C11C1

C2

0

0

0

0

0

0

0

0

0

0

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

C11C1

C2

Interchange Filter and Resampler

Replace Up Samplers, Delays, and Summerwith M-Port Output

Commutator

21

0

( ) ( )N j k n nM

n

G Z h n e Z

1 21 ( ) ( )

0 0

12 21

0 0

121

0 0

( ) ( )

( ) ( )

( ) ( )

NM M j r nM k r nMM

r n

NM Mj r k j nMr nMM M

r nN

M Mj r kr nMM

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

Low-Pass to Band-Pass1-to-M Up-Sampling Filter

M-Path, M-Channel Channelizer: Spinners are in IFFT

......

fs

fBW

fsM

f

F0 F1 F2 FM-2 FM-1

M-Point IFFT Supplies Phase Spinners to FormUp Converters to all Multiples of Input Sample Rate

All Output Channels Centered on Multiples of Input Sample Rate

Example: Multiples of 6-MHz

h (5n)0 h (5n)0

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 25 5kr 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

y(m)

x(m)

x(n)

x(n)

0 31 42 5

m

n

n

m

fs

fs

fs

fs

f

f

f

f

5

5


Rejec tedSpec trum

Input Spectrum

Shifted Spectrum

Up-Sampled Spectrum

Distorted Spectrum

y(m)

x(m)

x(n)

x(n)

0 31 42 5

m

n

n

m

fs

fs

fs

fs

f

f

f

f

5

5

2 FiltersRejec tedSpec trum

Rejec tedSpec trum

Input Spectrum

Shifted Spectrum

Up-Sampled Spectrum

Two Filter Spectra

M-Channel Polyphase Channelizer:

M-path Filter and M-point FFT

fs

h (n)0

h (n)2

h (n)= h(r+ nM)r

Polyphase Partition

h (n)M-2

h (n)1

h (n)3

h (n)M-1

FDM TDM

M-PNT FFT

..... ..........

Critically SampledfS=fC

NyquistSampling

fS=2fC

QRT NyquistFilterfS=2fC

QRT NyquistFilterfS=4fC

NyquistSampling

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 = fsBW

f = fsBW

f = fsBW

f = 2fsBW

f = fsBW

f = fsBW

f = fsBW

f = fsBW

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

f f

BW BW

Sample Rate= 2 Channel SpacingSample Rate = Channel SpacingChannel Spac ing Channel Spac ing

AliasedTransition Bandwidth

Filter Sampled at Rate to Avoid Band Edge Aliasing

Prototype Low‐Pass Filter for 120 Channel Channelizer

(M 1)

H ( Z )02

H ( Z )12

H ( Z )22

H ( Z )2

H ( Z )2

H ( Z )2

2

.... ....

y(n )y(n) M2

Z

Z

Z

Z

Z

-1

-2

....

....

....

.... ....

....

M

MM

M

MM

2

22

2

22

M2

:1

-( - 1)

-( 1)+ ( 1)+

( - 1)

-( )( )

M‐Path Polyphase Filter andM/2‐to‐1 Down Sampling

(M 1)

H ( Z )02

H ( Z )12

H ( Z )22

H ( Z )2

H ( Z )2

H ( Z )2

.... ....

y(n )M2

Z

Z

Z

Z

Z

-1

-2

....

....

....

.... ....

....

M

M

MM

M

MM

2

2

22

2

22

:1

M2

:1

M2

:1

M2

:1

M2

:1

M2

:1

M2

:1

-( -1)

-( 1)+ ( 1)+

( -1)

-( )( )

Use Noble Identity to PullM/2‐to‐1 Resampler Through Path Filter

Path Filters:Polynomials in

ZMConverted toPolynomials in

Z2

H ( Z )02

H ( Z )12

H ( Z )22

H ( Z )2

H ( Z )2

H ( Z )2

H ( Z )2

.... ....

y(n )M2

Z

Z

Z

Z

Z

Z

-1

-1

-1

-1

-1

-2

....

....

....

....

....

M

M

M

M

M

M

2

2

2

2

2

:1

M2

:1

M2

:1

M2

:1

M2

:1

M2

:1

M2

:1

-( -1)

-( -1)

( 1)+

( -1)

( )

Use Noble Identity to Pull M/2‐to‐1 ResamplerThrough Delays in Lower Half of Paths

Z

Z

Z

-1

-1

-1

H ( Z )0

H ( Z )1

H ( Z )2

H ( Z )( -1)

H ( Z )( )

H ( Z )M-2

H ( Z )M 1

....

....

........

....

x(n)

y(n )M2

M2

M2

2

2

2

2

2

2

2

Replace Delays and M/2‐to‐1 Resamplers withDual Input M/2 Path Commutator

H (Z )k2

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 )M2

M2

M2

M2

Fold Unit Delays in lower half Filter PathsInto Filter Polynomials in Z2

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 Resample

M/2-to-1 Resample

.....

.....

.....

.....

f

f

M‐to‐1 Down Sample Aliases Multiples ofOutput Sample Rate to DC

M/2‐to‐1 Down Sample Aliases Odd Multiples ofOutput Sample Rate to Half sample Rate

0 5T T

1.5T

T

T

0.5T

0.5T

0mT

(m+1)T

fs

2 fs/M M Path PolyPhaseFilter in Z

M-PNT IFFT

.....

flg=0

flg=0

flg=1

flg=1

Circular Buffer

2

Circular Buffer Between Polyphase Filter and IFFTAligns Shifting Input Origin with IFFT’s Origin

0 01 12 2

3

M/2-1

M-1

M/2M/2+ 1

M-2M-1

M MM

....

......

..

….

M-P

ath

Polyp

hase

Filte

r

M-P

oint

IFFT

FDM

TDM

M-P

ath

Inpu

t Dat

a Bu

ffer

C

ircul

ar O

utpu

t Buf

fer

State Engine

M/2‐to‐1 Analysis Channelizer

001122

3

M/2-1

M-1

M/2

M/2+ 1

M-2M-1

MM M

....

......

..

….

M-P

ath

Polyp

hase

Filte

r

M-P

oint

IFFT

FDM

TDM

M-P

ath

Inpu

t Dat

a Bu

ffer

C

ircul

ar O

utpu

t Buf

fer

State Engine

1‐to‐M/2 Synthesis Channelizer

....

.... .... .... ....

....

....

MPoint IFFT

MPoint IFFT

MPathFilter

MPathFilter

nM/2 Point Shift M-PointCircular Buffer


xx

xx

Frequency Domain Filtering With Cascade M/2‐to‐1 Analysis and 1‐to‐M/2 Synthesis Channelizers

Impulse response and Frequency Response25‐Enabled Ports: 2.4 MHz Bandwidth

Impulse Response and Frequency Response40‐Enabled Ports: 3.9 MHz Bandwidth

Mixed, Arbitrary BandwidthChannelizers

Mixed Bandwidth Signals presented to Channel Synthesizer

ompose Broadband Signals Using Short Analysis rs and Present Components to Synthesizer

....

....

....

....

....

....

....

....

....

....

....

....

Sele

ctor

Input Channel Configuration

MM MMNN

Anal. 0

Anal. N-1

Bw0P0

PN-1BWN-1

2-to-M, M-Channel Synthesis Channelizer2fs Mfs

01

2

M/2-1

M/2

M/2+ 1

M-1

....

….

M-P

ath

Polyp

hase

Filte

r

M-P

oint

IFFT

M-P

ath

Inpu

t Dat

a Bu

ffer

C

ircul

ar O

utpu

t Buf

fer

State Engine

AnalysisChannleizers

0-MHz Input Signal Partitioned into ive 10-MHz Sub Channels: fS=20 MHz

Multiple Partitioned Input Bands Presented to Synthesizer

Assembled Multiple BW Channels in Single Synthesis Channelizer

Reassemble Decomposed Broadband Signals Using Short Synthesis Filters formed by Multiple Channel Analysis Channelizer

01

2

M/2-1M/2

M/2+ 1

M-1

....

....

....

....

....

....

....

....

....

....

....

....

....

….

M-P

ath

Polyp

hase

Filte

r

M-P

oint

IFFT

Sele

ctor

M-P

ath

Inpu

t Dat

a Bu

ffer

C

ircul

ar O

utpu

t Buf

fer

State Engine

M M M M NN

Synth. 0

Synth. N-1

Bw0P0

PN-1BWN-1

M/2-to-1, M-Channel Channelizer

fs

Partitioned spectral Components from ingle Multi-Channel Analyzer

-10 0 10-40

040

FLTR 0

-10 0 10-40

040

FLTR 1

-10 0 10-40

040

FLTR 2

-10 0 10-40

040

FLTR 3

-10 0 10-40

040

FLTR 4

-10 0 10-40

040

FLTR 5

-10 0 10-40

040

FLTR 6

-10 0 10-40

040

FLTR 7

-10 0 10-40

040

FLTR 8

-10 0 10-40

040

FLTR 9

-10 0 10-40

040

FLTR 10

-10 0 10-40

040

FLTR 11

-10 0 10-40

040

FLTR 12

-10 0 10-40

040

FLTR 13

-10 0 10-40

040

FLTR 14

-10 0 10-40

040

FLTR 15

-10 0 10-40

040

FLTR 16

-10 0 10-40

040

FLTR 17

-10 0 10-40

040

FLTR 18

-10 0 10-40

040

FLTR 19

-10 0 10-40

040

FLTR 20

-10 0 10-40

040

FLTR 21

-10 0 10-40

040

FLTR 22

-10 0 10-40

040

FLTR 23

-10 0 10-40

040

FLTR 24

-10 0 10-40

040

FLTR 25

-10 0 10-40

040

FLTR 26

-10 0 10-40

040

FLTR 27

-10 0 10-40

040

FLTR 28

-10 0 10-40

040

FLTR 29

-10 0 10-40

040

FLTR 30

-10 0 10-40

040

FLTR 31

-10 0 10-40

040

FLTR 32

-10 0 10-40

040

FLTR 33

-10 0 10-40

040

FLTR 34

-10 0 10-40

040

FLTR 35

-10 0 10-40

040

FLTR 36

-10 0 10-40

040

FLTR 37

-10 0 10-40

040

FLTR 38

-10 0 10-40

040

FLTR 39

-400

40

FLTR 40

-400

40

FLTR 41

-400

40

FLTR 42

-400

40

FLTR 43

-400

40

FLTR 44

-400

40

FLTR 45

-400

40

FLTR 46

-400

40

FLTR 47

Reassembled Wide band Channels rom Short Synthesis Channelizers

-20 0 20-40

0

40

Sig 3

Mag

nitu

de (d

B)

-20 0 20-40

0

40

Sig 6

-20 0 20-40

0

40

Sig 8

-20 0 20-40

0

40

Sig 4

Frequency (MHz)

Mag

nitu

de (d

B)

-20 0 20-40

0

40

Sig 5

Frequency (MHz)-20 0 20

-40

0

40

Sig 7

Frequency (MHz)

-30 -20 -10 0 10 20 30-40

0

40

Sig 1

F (MH )

Mag

nitu

de (d

B)

-30 -20 -10 0 10 20 30-40

0

40

Sig 2

F (MH )

Signal Fidelity Preserved under Multiple Sub-Channel Disassembly and Reassembly

....

.... .... ....

....

....

MPoint IFFT

MPoint IFFT

MPathFilter

MPathFilter



Cascade M/2‐to‐1 Analysis and 1‐to‐M/2 Synthesis Channelizers

Frequency Domain Filtering and Spectral Shuffle

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.

Suspicions Confirmed!

Is There any Doubt???

new directions in channelized receivers and transmitters€¦ · version of edwin armstrong’s...

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