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Complex Sampling Considerations

Walter Brisken National Radio Astronomy Observatory (Socorro, NM) 8th US VLBI Meeting Arecibo Observatory 2010 November 9

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Conventional analog complex sampler

Many pitfalls for analog equipment

(vR , vI ) (I , Q) form one complex sample per clock cycle Bandwidth must be relatively small complered to LO frequency Different samplers may have different characteristics Each branch can incur different band-pass and delays

Digital electronics avoid these problems; it's time to reconsider

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Sampling and Quantization
These are two separate processes often performed by the same device. Sampling Descretization of a continuous-in-time signal into a finite number of samples Nyquist-Shannon Theorem: lossless for real sampling if t Lossless for complex sampling if t
1 1 2

Quantization Approximation of a range by a finite set of values Each value can be represented by a code for internal representation A reconstructed signal will differ from the original due to introduction of quantization noise This process is a form of lossy compression The rest of this talk is really about quantization

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Metrics to evaluate quantization schemes
Quantization efficiency Q = corr (v , v )2 = ^ Effective Numb er of Bits (entropy)
nstate 2

vv ^ v
2

v2 ^

ENOB = - Packing efficiency
pack

Pi log2 P
i=1

i

=

nbit bits of storage 2 Q 2
Q pack nbit
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Sensitivity to input p ower (smaller is better) s=-

VLBI wants maximal


Real 2-state (1-bit) quantization
Divide real axis into 2 regions

0

1

Code 0 1
v

No free parameters Symmetry & reasoning invoked

-

0

Range - to 0 0 to

Value - 2/ 2/

Frac. 50% 50%

Values determined so as to minimize quantization noise Effective number of bits, ENOB = 1
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Quantization efficiency Q = 64%


Quantization noise

2-state quantization noise is quite non-Gaussian

P (v )

-

0



v

This noise is anti-correlated with unquantized values corr(v , v ) = -0.60 (for 2-state quantization)
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Real 4-state (2-bit) quantization
Divide real axis into 4 regions

00

10

01

11

Only 2 free parameters Optimal values: v0 = 0.96 ; R = 3.3359 - Q = 88% ENOB = 1.92 = 0.4780 determined so as to minimize quantization noise Note: Different conventions for the codes exist (e.g., VLBA, Mark5B)
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-v0

0

v0

v

Code 00 10 01 11

Range - to -v -v0 to 0 0 to v0 v0 to

0

Value -R - R

Frac. 17% 33% 33% 17%


Elementary complex 4-state quantization

Divide complex plane into 4 regions
vI

01

11

00 10

vR

Code 00 01 10 11

Value 2/ (-1 - i) 2/ (-1 + i) 2/ (1 - i) 2/ (1 + i)

Frac. 25% 25% 25% 25%

Q = 64%, same as for 2-state real sampling; ENOB = 2
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Elementary complex 16-state quantization
Divide complex plane into 16 regions
vI D C B A H G F E L K J I P O N M vR

Code A, D, M , P B , C, N , O E , H, I , L F, G, J, K

Value ±R ± Ri ±R ± i ± ± Ri ±R ± Ri

Frac. 2.9% 5.6% 5.6% 10.9%

Q = 88%, same as for 4-state real sampling; ENOB = 3.85 Analogous to 4-state real (v0 = 0.96 , R = 3.3359, = 0.4780 ) There are additional free parameters not being exploited here!
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Digression: Voronoi Tesselations



Formed given a set of sites (black dots) Region around each site consists of all points closest to that site Defines optimal sampling for uniform probability Proves to be a useful starting point for Gaussian distributions
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Optimized complex 8-state quantization
vI

H F D C A E B G vR

Co D, B , C, A,

de E F, G H

Site ±R1 ±R2 ± R3 i ±R4 i

Frac. 18% 9.5% 13%

R = (0.47650, 1.37156, 1.39495, 0.84048) Values here are also the Voronoi Sites Four free parameters, Rn ; optimal Q = 80%, ENOB = 2.94 This parameterization is based on unconstrained 16 D.o.F. search 2% better signal-to-noise per bit than 2- or 4-state real sampling Suffers from low (94%) VDIF pack due to 3-bit word size
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Optimized complex 16-state quantization
vI O M G D B F K H E C A P N L I J vR

Code H F, I , K E , G, L B , J, O A, C, D, M, N, P

Site 0 R1 e2i -R2 e2 R3 e2i

n 3 in 3 n 3

Frac. 8% 9% 9% 4%
i
n 3

(R4 ± R5 i)e2 for n = 0, 1, 2

4%

R = (0.77958, 0.80659, 2.05868, 1.56032, 1.27047) This is optimized with a 5 parameter search Parameters based on uncontrained 32 parameter optimization Q = 89%, ENOB = 3.84 1% better signal-to-noise than 4-state real sampling
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Sensitivity to input levels

1

'eta1bit' 'eta2bit' 'eta3bitcomplex' 'eta4bitcomplex'

u u u u

1:2 1:2 1:2 1:2

0.8

0.6

0.4

0.2

0



Q

0.5

1

1.5

2

2.5

3

3.5

4

/0
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Summary of performance metrics

Quantization type 2-state real 4-state real 4-state complex 16-state complex opt. 8-state complex opt. 16-state complex

Q 0.64 0.88 0.64 0.88 0.80 0.89



pack

1 1 1 1 0.94 1

ENOB 1 1.92 1 3.85 2.94 3.84

s 0 0.35 0 0.35 0.20 0.20

Q pack nbit

0.64 0.62 0.64 0.62 0.61 0.63

This value would be 0.65 if not for severe packing penalties.

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Conventions need establishment

Sign of the pre-quantized imaginary part Defines sense of spectrum analogous to USB / LSB Needs to be similarly specified in control files Enco ding scheme needs to b e defined Elementary quantization codes explicitly covered in VDIF How should nbit b e generalized? I suggest n
bit

= log2 n

state

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Final remarks
Caution: coarse complex quantization not well understoo d Equivalent of van Vleck correction needs thought Effect of non-flat bandpass not understood

Implemention can likely b e sloppy with reasonable results Extention of ideas All concepts about complex samples can be applied to consecutive pairs of real samples Improvements for oversampled data very likely since consecutive values are highly correlated The 2-D concept used here can be extended to higher dimensions with increased performance expected
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