Y. C. Jenq1 Non-uniform Sampling Signals and Systems (A/D & D/A Converters) Y. C. Jenq Department of...
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Transcript of Y. C. Jenq1 Non-uniform Sampling Signals and Systems (A/D & D/A Converters) Y. C. Jenq Department of...
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
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
Y. C. Jenq 4
Non-uniform Sampling Clock
t0 t1 t5t4t3t2 t7t6 t8
T
Y. C. Jenq 5
Non-uniform Sampling Examples
Random Equivalent–time Sampling Interleaved ADC Array Direct Digital Synthesizer
Y. C. Jenq 6
Random Equivalent-Time Sampling
Triggering LevelTriggering Time Instances
Sampling Time Instances
Y. C. Jenq 7
Random Equivalent-Time Sampling
Y. C. Jenq 8
Interleaved ADC Arrays
ADC
ADC
ADC
ADC
Signal in
Delay elements
Memory
Memory
Memory
MemoryOR with a 4-phase clock
Sampling Clock
Y. C. Jenq 9
Interleaved ADC ArraysADC
ADC
ADC
ADC
Signal in
Memory
Memory
Memory
Memory
4-phase clock
Y. C. Jenq 10
Direct Digital Synthesizer (DDS)
WaveformMemory
WaveformMemory
Phase AccumulatorPhase Accumulator
D/A Converter
D/A Converter
Low-Pass Filter
Low-Pass Filter
Y. C. Jenq 11
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
Y. C. Jenq 12
Direct Digital Synthesizer (DDS)Waveform Memory
Fs: Master Clock Frequencyf: Sine Wave FrequencyTL: Table Length
Y. C. Jenq 13
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
Y. C. Jenq 14
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
Y. C. Jenq 15
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
Y. C. Jenq 16
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
Y. C. Jenq 18
Digital Spectrum of Non-uniformly Sampled Sinusoid
A(0)
A(1)A(2)
A(3)
M=4
Y. C. Jenq 19
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
Y. C. Jenq 20
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)
Y. C. Jenq 21
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?
Y. C. Jenq 22
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
Y. C. Jenq 23
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!
Y. C. Jenq 24
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
Y. C. Jenq 25
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
Y. C. Jenq 26
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.
Y. C. Jenq 27
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:
Y. C. Jenq 28
Reconstruction of Digital Spectrumwith Residual Timing Error
Yih-Chyun Jenq, “Improveing Timing Offset Estimation by Aliasing Sampling,” IMTC’05, May 2005, Ottawa, Canada.