Non-uniform Sampling Signals and Systems (A/D & D/A Converters)
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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 [email protected]
description
Non-uniform Sampling Signals and Systems (A/D & D/A Converters). Y. C. Jenq Department of Electrical & Computer Engineering Portland State University P. O. Box 751 Portland, OR 97207 [email protected]. Outlines. Non-uniform Sampling Signals Digital Spectrum of Non-uniformly Sampled Signal - PowerPoint PPT Presentation
Transcript of Non-uniform Sampling Signals and Systems (A/D & D/A Converters)
Y. C. Jenq 1
Non-uniform Sampling Signals and Systems
(A/D & D/A Converters)
Department of Electrical & Computer EngineeringPortland State University
P. O. Box 751Portland, OR [email protected]
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
Phase Accumulator
D/A Converter
Low-Pass Filter
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Direct Digital Synthesizer (DDS)
WaveformMemory
D/A Converter
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
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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
Erro
r: 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.
