Y. C. Jenq 1

Non-uniform Sampling Signals and Systems

(A/D & D/A Converters)

Department of Electrical & Computer Engineering

Portland State University

P. O. Box 751

Portland, OR 97207

jenq@ece.pdx.edu

Y. C. Jenq 2

Outlines

Non-uniform Sampling Signals Digital Spectrum of Non-uniformly

Sampled Signal Timing Error Estimation Reconstruction of Digital Spectrum

Y. C. Jenq 3

Non-uniform Sampling

time, t

Waveform amplitude, x(t) with FT = Xc()

t0 t1 t5t4t3t2 t7t6 t8

T = nominal sampling periodn = tn- nT, rn = n / T

T

1 2

M=4

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Non-uniform Sampling Clock

t0 t1 t5t4t3t2 t7t6 t8

T

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Non-uniform Sampling Examples

Random Equivalent–time Sampling Interleaved ADC Array Direct Digital Synthesizer

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Random Equivalent-Time Sampling

Triggering LevelTriggering Time Instances

Sampling Time Instances

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Random Equivalent-Time Sampling

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Interleaved ADC Arrays

ADC

ADC

ADC

ADC

Signal in

Delay elements

Memory

Memory

Memory

MemoryOR with a 4-phase clock

Sampling Clock

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Interleaved ADC ArraysADC

ADC

ADC

ADC

Signal in

Memory

Memory

Memory

Memory

4-phase clock

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Direct Digital Synthesizer (DDS)

WaveformMemory

WaveformMemory

Phase AccumulatorPhase Accumulator

D/A Converter

D/A Converter

Low-Pass Filter

Low-Pass Filter

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Direct Digital Synthesizer (DDS)

WaveformMemory

WaveformMemory

D/A Converter

D/A Converter

Low-Pass Filter

Low-Pass Filter

Integer Part Fraction

Integer Part Fraction

+

Address Accumulator

Address Increment Register

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Direct Digital Synthesizer (DDS)Waveform Memory

Fs: Master Clock Frequencyf: Sine Wave FrequencyTL: Table Length

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Direct Digital Synthesizer (DDS)Frequency Resolution

Integer Part Fraction

W + L/M

B bits

Frequency Resolution = Fs/2B-1

Sine wave Frequency f = (W+L/M)Fs/TL

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Non-uniform Sampling Model

T = nominal sampling period tn = nT + n , and n is periodic with period M.

Let n = k M + m where k ranges from –∞ to +∞ and m ranges from 0 to (M-1), Then

tn = ( k M + m )T + (kM+m)

= k M T + m T + m

= k M T + m T + rm T

where rm = m/T

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Yih-Chyun Jenq, “Digital Spectra of Non-uniformly Sampled Signals - Fundamentals and High-Speed Waveform Digitizers,” IEEE Transactions on Instrumentation and Measurement, vol. 37, no. 2, June 1988.

Yih-Chyun Jenq, “Digital Spectra of Non-uniformly Sampled Signals: A Robust Time Offset Estimation Algorithm for Ultra High-Speed Waveform Digitizers Using Interleaving,” IEEE Transactions on Instrumentation and Measurement, vol. 39, no. 1, February 1990

Digital Spectrum of Non-uniformly Sampled Signals

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Digital Spectrum of Non-uniformly Sampled SignalsIf we use x(tn) to compute the digital spectrum, Xd(), as if the data points were sampled uniformly, i.e.,

Xd() = n x(tn) e-jn

Then, it can be shown that

Xd(T) = (1/T)k A(k,) Xc[-k(2/MT)]Where

A(k,) = (1/M)m=0,(M-1) e-j[-k(2/MT]rmTe-jkm(2/M)

Notice that A(k,) is the m-point DFT of e-j[-k(2/MT]rmT

Y. C. Jenq 17

Digital Spectrum of Non-uniformly Sampled Sinusoid

Input Signal x(t) = exp(jot), And Xc()=2()

Then Xd() = (2/T) k A(k) [-o-k(2/MT)]

where A(k) =m=0,(M-1)(1/M)ejrmoTe-jkm(2/M)

Notice that A(k) is no longer a function of

and A(k) is a M-point DFT of ejrmoT, m=0, 1,…,M-1

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Digital Spectrum of Non-uniformly Sampled Sinusoid

A(0)

A(1)A(2)

A(3)

M=4

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Digital Spectrum of Non-uniformly Sampled Sinusoid

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-20

-10

0

10

20

30

40

50

60Spectrum of Non-uniformly Sampled Data

Digital Frequency

Mag

nitu

de S

pect

rum

in d

BM=8

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Estimation of Timing Errors - rm

A(k) =m=0,(M-1)[(1/M)exp(jrmoT)]e-jkm(2/M)

A(0)

A(1)A(2)

A(3)

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Reconstruction of Digital Spectrum

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-60

-40

-20

0

20

40

60Spectrum Reconstruction from Non-uniformly Sampled Data

Digital Frequency

Mag

nitu

de S

pect

rum

in d

B

* : Reconstructed Spectrum, 10-bit quantization

Once the timing errors are known, can we reconstruct the correct digital spectrum?

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Selecting Test Frequencies

A(0)

A(1)A(2)

A(3)

Higher frequency more sensitive to timing errorUsing FFT spurious harmonics should be on the binsWindowing function selection

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Estimation of rm- Synchronous Case Residual Timing Error

timing offset error

RMS value beforeAdjust-ment

30%

20%

10%

5%

4x10-11

3x10-11

2x10-11

0.9x10-11

RMS value after

(4 bits)

RMS value after

(6 bits)

RMS value after

(8 bits)

RMS value after

(10 bits)

RMS value after

(∞ bits)

2.4x10-12

3.1x10-12

2.3x10-12

2.6x10-12

4.4x10-13

5.6x10-13

6.1x10-13

5.4x10-13

1.1x10-13

1.6x10-13

1.3x10-13

1.4x10-13

2.9x10-14

3.0x10-14

2.7x10-14

3.6x10-14

2.6x10-24

2.2x10-24

1.8x10-24

2.0x10-24

Residual timing errors are independent of initial timing errors!

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Estimation of rm- Synchronous Case

4 6 8 10 12 14 1610

-6

10-5

10-4

10-3

10-2

10-1

Sensitivity of Timing Error Estimation Algorithm

Number of Effective Bits in A/D Converter

Res

idua

l Tim

ing

Err

or:

rm

* : Starting rm ~ 0.5

+ : Starting rm ~ 0.05

o : Starting rm ~ 0.005

Sensitivity to Quantization Noise in A/D Converter

Residual TimingError is relatively independent of initial timing error, but it is quite sensitive to the effective-bit of ADC

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1

10-1

10-2

10-3

10-4

10-5

4 6 8 10 bits

Residual Timing Error: RMS rm

Residual Timing Error

One order of magnitude

improvement per 3 effective bits increase

Residual RMS rm ~ 10-3 at 7 Bits

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Perfect Reconstruction of Digital Spectrum

Yih-Chyun Jenq, “Perfect Reconstruction of Digital Spectrum from Non-uniformly Sampled Signals,” IEEE Transactions on Instrumentation and Measurement, vol. 46, no. 3, 1997.

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Reconstruction of Digital Spectrumwith Residual Timing Error

S/N ~ 20*log(1/) -16 dB

SNR = 6.02* (number of bits) + 1.76 dB

(Residual ~ (Initial /1000 at 7 Bits and improve one order of magnitude

per 4 bits increase

= standard deviation of rm

Reconstruction noise due to quantization error:

Reconstruction noise due to residual timing error:

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Reconstruction of Digital Spectrumwith Residual Timing Error

Yih-Chyun Jenq, “Improveing Timing Offset Estimation by Aliasing Sampling,” IMTC’05, May 2005, Ottawa, Canada.