InterpolatorsAnd Interpolation

1

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

2

Spectrum of Interpolator andPeriodic Spectrum of

Zero-Packed Shaping Filter

-30 -20 -10 0 10 20 30-80

-60

-40

-20

0

Spectrum of Shaping Filter and 1-to-32 Interpolating Filter

Normalized Frequency

Gai

n (d

B)

-8 -6 -4 -2 0 2 4 6 8-80

-60

-40

-20

0

Zoom to Spectrum

Normalized Frequency

Gai

n (d

B)

3

Spectrum of 1-to-32 Interpolated Shaping Filter

-30 -20 -10 0 10 20 30-80

-60

-40

-20

0

Spectrum of Interpolated Shaping Filter

Normalized Frequency

Gai

n (d

B)

-8 -6 -4 -2 0 2 4 6 8-80

-60

-40

-20

0

Zoom to Spectrum

Normalized Frequency

Gai

n (d

B)

4

Polyphase Partition of M-PathResampling Filter

H (Z )

H (Z )

H (Z )

H (Z )

0

1

2

M-1

x(n) y(m)

N/M= 4

....

....

5

Efficient Hardware Implementation of 1-to-M Polyphase Interpolator

N/M= 4

H (Z ) r

......

x(n) y(m )

h(0+ nM)

h(1+ nM)

h(2+ nM)

h(M-1+ nM)

6

Interpolation Options

Initial Sample Grid,

Unit distance

Between Samples

Same Rate Samp le Grid,

Unit Distance

Between Sam ples

Higher Rate Sam ple Grid ,

Less Than Unit Distance

Between Samp les

Lower Rate Sam ple Grid,

More Than Unit Distance

Between Samp les

Interpola ted Samp le PositionsInitial Sample Positions

7

M-Path, 1-to-M/Q Interpolator

H (Z )

H (Z )

H (Z )

H (Z )

0

1

2

M-1

x(n)y(m )

N/M= 4

....

.... Q:1

Q:11:Mx(n) y(m)

H(Z)

Polyphase Filter

8

5/3, Rational Ratio Re-Sampling

3-to-1

phs(0)

phs(1)

phs(2)

phs(3)

phs(4)

x(n)

y (m)= x(n+ k/5)5

y(m)= x(n+ 3k/5)

n n+ 1 n+ 2 n+ 3

0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

x x x x xx x x xx x x x

In Out

n 0,3

n+ 1 1,4

n+ 2 2

K(m+1)=[k(m)+3] modulo(5)

9

Rational Ratio Interpolation. Example; up 8, down 3

n+ 1

m+ 1 m+ 3

m+ 6

n+ 2

m+ 2 m+ 4

m+ 7

m+ 5

m+ 8

n+ 3

n

m

Input Samples and availab le

1-to-8 Interpola ted Samp les

3-to-8 Interpolated Samples

(up 8, down 3)

10

Interpolation To Time Position Between Available Interpolation Points

(Arbitrary Ratio Interpolation)

n n+ 1

Desired Sample

Position n+ k/M+

Desired Sample Value

Available Sample Value

Nearest Available

Sample Position

n+ k/M

Input

Sample

Error

11

Zero Order Hold Model of

Nearest (Left) Neighbor Interpolation

n n+ 1

Desired Sample Position n+ k/M+

Error

Interpolated Sample Values

Zero-Order-Hold

Analog Levels

12

Spectrum of Up-Sampled Signal at Input and Output of Virtual DAC

BW= 1

BW= 1

0

0

N

N

f

f

2N

2N

Output Sample Rate

Output Sample Rate

DAC Response

13

Frequency Response of DAC at First Spectral

Null

NN-0.5 N+ 0.5

f

DAC ResponseH( f)= - f1N

1 1 1 1 | ( ) | : | ( ) | : 2

2 2 2

( 1) 7 2 , 8( ), 2 128

When signal is already 4-times oversampled

Need 32 stage up-sampler to suppress spectral artifact

s

to -

bH f f H

N N N

bN Say b bits N

48 dB

14

Shaping Filter: Time and Frequency Response, Four Times Over Sampled

-5 0 5 10 15 20 25 30 35 40 45 50-0.1

0

0.1

0.2

0.3time response of shaping filter

time

am

plit

ude

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-60

-40

-20

0

spectral response of shaping filter

frequency

log m

agnitude

15

Time and Frequency Response of 32/6.4 Left Neighbor Interpolator

0 50 100 150 200 250

-0.01

0

0.01

0.02

0.03

0.04

0.05

-10 -8 -6 -4 -2 0 2 4 6 8 10-80

-60

-40

-20

0

16

Time and Frequency Response of 32/6.37 Left Neighbor Interpolator

0 50 100 150 200 250

-0.01

0

0.01

0.02

0.03

0.04

0.05

-10 -8 -6 -4 -2 0 2 4 6 8 10-80

-60

-40

-20

0

17

Prototype Interpolator Length for 8-bit data, initially Over Sampled by 2.

f

f

0

0

2

2

-2

-2

-0.5

-0.5

0.5

0.5

1.5

1.5

-1.5

-1.5

-4

-4

4

4

f= 1

f= 1

DCDCDC DC

To Obtain 128 Over Sample, M=64, N=(128/1)(66/22)=384N/M=6: Need 64 6-tap filters in Polyphase Interpolator

18

Prototype Interpolator Length for 8-bit data, initially Over Sampled by 4.

f

f

0 8-8

-8

-0.5 0.5 3.5

3.5

-1.5

-3.5

-4

-4

4

4 8

f= 3

f= 3

DCDC

DCDC

0-0.5 0.5

To Obtain 128 Over Sample, M=32, N=(128/3)(66/22)=128N/M=4: Need 32 4-tap filters in Polyphase Interpolator

19

Address Control: Modulo Accumulator

...

Mod(M) Int(--)Z-1

d-acc

acc(m)

-

k(m)

(m)

Filterx(n) y(m)

Polyphase

Weights

n n+ 1 n+ 2

Input

Samples

Output

Samples

0 1 2 3 4 5 6 9 8 9 0 1 2 3 4 5 6 9 0 1 2........ ....

TINTOUT

Input Time index ”n”Polyphase index ”k”

Fractional Offset: d-acc

- Out In

In Out

T fd acc M M

T f

On Overflow, Insert New Input

Fractional Part(For later use)

20

Two Neighbor, Linear Slope Interpolator

n

n+ 1

Desired Sample

Position k+

Desired

Sample

Value

Interpolated

Sample Value

Left Available

Interpolation

Sample Position

Left Available

Interpolated

Sample Value

Right Available

Interpolation

Sample Position

Right Available

Interpolated

Sample Value

Input

Sample

value

Input

Sample

value

Linear

Interpolator

n+ k/Mn+ (k+ 1)/M

21

Equivalent Interpolating Kernel

k-1 k+ 2

TRI(k)TRI(k+ 1)

k k+ 1

x(k)

x(k+ )

x(k)

x(k+ 1)

x(k+ 1)

M M M M

22

Spectrum of Up-Sampled Signal at Input and Output of Virtual Linear

Interpolator

BW= 1

BW= 1

0

0

N

N

f

f

2N

2N

Output Sample Rate

Output Sample Rate

Triangle Spectral

Response

Repeated

Spectral Zeros

23

Frequency Response at First Spectral Null of Linear Interpolator

NN-0.5 N+ 0.5

f

Triangle

Response

H( f)= f1N

[ ]2

2 2 211 1 1 / 2

| ( ) | : | ( ) | : 2 : 22 2 2

( / 2 1) 7 2 , 16( ), 2 128

When signal is already 4-times ov

1

2

b bH f f H

N N N

bN Say b bits N

N

ersampled

Need 32 stage up-sampler to suppress spectral artifacts by -96 dB

24

Estimate y(n+k/M) & y(n+k/M) With 3 Arms of Polyphase Filter

PHS-(k-2)

PHS-(k)

PHS-(k-1)

PHS-(k+ 1)

PHS-(k+ 2)

y(n+ k/M)

y(n)

- y(n+ k/M)

.

.

25

Estimate y(n+k/M) & y(n+k/M) With 2 Polyphase Filters

Polyphase

Derivative

Matched

Filter

Polyphase

Matched

Filter

y(n+ k/M)

y(n)

y(n+ k/M)

....

....

.

k

k

.

26

y(n+k/M) & y(n+k/M) With 2 Efficient Polyphase Filters

1-Stage Filter

1-Stage Filter

y(n+ k/M)

y(n+ k/M)y(n)

.

Polyphase

Matched Filter

Coeffic ients

Polyphase

Derivative

Matched Filter

Coeffic ients

Coeffic ient

Selection

.

27

Interpolation with Polyphase Low-pass Filter and Polyphase Derivative Filter

for Local Slope Correction

Mod(M) Int(--)Z-1

d-acc

acc(m)

-

k(m)

k(m)

(m)

Filter

Filter

x(n)x(n)

x(n)

y(n+ k/M)y(n+ k/M+ /M) =

y(n+ k/M)+ y(n+ k/M)

y(n+ k/M)

h (n)k

dh (n)k

.

.

Derivative Polyphase Filter

dh=conv(h,[1 0 -1]*M/2

dh=dh(2:length(dh)-1);

28

Input Shaping Filter at4-Samples per Symbol

29

Spectra of 64/10.49 Interpolated Signal

0 200 400 600 800 1000 1200

-0.2

0

0.2

0.4

0.6

0.8

1

Interpolated Shaping Filter

-20 -15 -10 -5 0 5 10 15 20-120

-100

-80

-60

-40

-20

0

Frequency Response

Normalized Frequency (f/fsym

)

Lo

g m

ag

nitu

de

(d

B)

30

Signal Conditioning and Processing

Spectral Centers 1.7 MHz SeparationChannel BW: 1.7 MHzChannels Span 30 MHz ( 17 Channels)24-Channel Channelizer: 24*1.7=40.8MHz12-to-1 Down Sample in ChannelizerOutput Sample Rate; 3.4 MHz/Channel

160 MHz

ADC Half

Band

Filter

Half

Band

Filter

Interpolate

Filter

DDS

DDS

Phase

Accumulator

160

MHz81.6

MHz

40.8

MHz

40.8

MHz

163.2

MHz

3.4

MHz

3.4

MHz 3.4

MHz

22 22

2

2

24-Path

Polyphase

Filter

24-PNT

FFT

Interp

Bank

...

16

Ch

an

ne

ls

Circ

ula

r Bu

ffe

r

12-to-1

31

Wide Dynamic Range Resampler

32

Spectra from 24-channel Channelizer at 3.4 MHz Sample Rate

-1.5 -1 -0.5 0 0.5 1 1.5

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1.5 -1 -0.5 0 0.5 1 1.5

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1.5 -1 -0.5 0 0.5 1 1.5

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1.5 -1 -0.5 0 0.5 1 1.5

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1.5 -1 -0.5 0 0.5 1 1.5

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1.5 -1 -0.5 0 0.5 1 1.5

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

-1 0 1

-150

-100

-50

0

33

Time Series from 24-channel Channelizer at 3.4 MHz Sample Rate

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

0 50 100 150 200-2

0

2

34

Equipment Bay: 192-Stereo FM Modulators

35

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 192 MP3 Stereo Channels.

• 1 Virtex V-4 for 192 Digital Stereo FM Modulators and 256 Channel Channelizer @ 293 kHz Bandwidth per channel. (60% of Chip)

36

Prototype Analog Stereo FM Modulator

dbx

Encode

dbx

Encode

50- sec

Pre-emph

75- sec

Pre-emph

50- sec

Pre-emph

LPF

14 kHz

LPF

14 kHz

LPF

7.5 kHz

BPF

15-50 kHz

BPF

60-90 kHz

VCO

32 kHz

VCO

80 kHz

100. .

0. .

3.2 MHz

3.2 MHz

Left

Right

SCA

L+ R

L - R IF

Output

37

DSP Based Stereo FM Modulator

dbx

Encoder

dbx

Encoder

LPF

14-kHz

LPF

14-kHz

LPF

14-kHz

BPF

35-kHz

BPF

30-kHz

DDS FM-MOD

&

Up-Converter

DDS FM-MOD

&

Up-Converter

50-usec

Pre-emph

50-usec

Pre-emph

75-usec

Pre-emph

48-to-293

Arb itrary

Re-Sample

48-to-293

Arb itrary

Re-Sample

48-to-293

Arb itrary

Re-Sample

Gain

Gain

Gain

Gain

IIR

IIR

KACC

KACC

IIR IIR

IIR

IIRIIR

IIRSCA

Left

Right

(L+ R)

(L-R)

32 kHz

32 kHz

CORDIC

CORDIC

Satellite Clock Domain Transceiver Clock Domain

38

256 Channel Channelizer for 50-MHz

Digital IF Sampled at 225.024 MHz

Radix-2 Butterfly of two 128-Point FFT’s

25

6 C

ha

nn

els

1:2

Up

_Sa

mp

ler

25

6 C

ha

nn

els

Ad

de

r

OddSamples

EvenSamples

12

8 P

oin

t FFT

12

8 P

oin

t FFT

12

8 P

ath

Po

lyp

ha

se F

ilte

r

1

1-T

ap

s Pe

r Pa

th1

28 P

ath

Po

lyp

ha

se F

ilte

r

1

1-T

ap

s Pe

r Pa

th

Ha

lf Ba

nd

Ph

ase

Shift

1-to-3Up-Sample

DDS

Quantize DAC

50 MHz

225.024 Mhz

225.024 Mhz

75.008 Mhz

39

40

41

42