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
jenq@ece.pdx.edu
Y. C. Jenq
1
Outlines
Non-uniform Sampling Signals
 Digital Spectrum of Non-uniformly
Sampled Signal
 Timing Error Estimation
 Reconstruction of Digital Spectrum

Y. C. Jenq
2
Non-uniform Sampling
Waveform amplitude, x(t) with FT = Xc(W)
M=4
T = nominal sampling period
Dn = tn- nT, rn = Dn / T
T
D1
t0
t1
D2
t2
t3
t4
t5
Y. C. Jenq
t6
t7
t8
time, t
3
Non-uniform Sampling Clock
T
t0
t1
t2
t3
t4
Y. C. Jenq
t5
t6
t7
t8
4
Non-uniform Sampling Examples
Random Equivalent–time Sampling
 Interleaved ADC Array
 Direct Digital Synthesizer

Y. C. Jenq
5
Random Equivalent-Time Sampling
Triggering Level
Triggering Time Instances
Sampling Time Instances
Y. C. Jenq
6
Random Equivalent-Time Sampling
Y. C. Jenq
7
Interleaved ADC Arrays
Sampling Clock
Signal in
ADC
Memory
ADC
Memory
ADC
Memory
ADC
OR with a 4-phase clock
Memory
Delay elements
Y. C. Jenq
8
Interleaved ADC Arrays
ADC
Memory
ADC
Memory
ADC
Memory
ADC
Memory
Signal in
4-phase clock
Y. C. Jenq
9
Direct Digital Synthesizer (DDS)
Waveform
Memory
D/A
Converter
Low-Pass
Filter
Phase Accumulator
Y. C. Jenq
10
Direct Digital Synthesizer (DDS)
Waveform
Memory
D/A
Converter
Integer Part
Low-Pass
Filter
Fraction
Address
Accumulator
Fraction
Address
Increment
Register
+
Integer Part
Y. C. Jenq
11
Direct Digital Synthesizer (DDS)
Waveform Memory
Fs:
f:
TL:
Master Clock Frequency
Sine Wave Frequency
Table Length
Y. C. Jenq
12
Direct Digital Synthesizer (DDS)
Frequency Resolution
W + L/M
Integer Part
Fraction
B bits
Frequency Resolution = Fs/2B-1
Sine wave Frequency f = (W+L/M)Fs/TL
Y. C. Jenq
13
Non-uniform Sampling Model
T = nominal sampling period
tn = nT + Dn , and Dn 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 + D(kM+m)
= k M T + m T + Dm
= k M T + m T + rm T
where rm = Dm/T
Y. C. Jenq
14
Digital Spectrum of
Non-uniformly Sampled Signals

Yih-Chyun Jenq, “Digital Spectra of Nonuniformly 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 Nonuniformly 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
Y. C. Jenq
15
Digital Spectrum of
Non-uniformly Sampled Signals
If we use x(tn) to compute the digital spectrum, Xd(w),
as if the data points were sampled uniformly, i.e.,
Xd(w) = Sn x(tn) e-jwn
Then, it can be shown that
Xd(WT) = (1/T)Sk A(k,W) Xc[W-k(2p/MT)]
Where
A(k,W) = (1/M)Sm=0,(M-1) e-j[W-k(2p/MT]rmTe-jkm(2p/M)
Notice that A(k,W) is the m-point DFT of e-j[W-k(2p/MT]rmT
Y. C. Jenq
16
Digital Spectrum of Non-uniformly
Sampled Sinusoid
Input Signal
And
x(t) = exp(jWot),
Xc(W)=2pd(W-Wo )
Then Xd(WT) = (2p/T) Sk A(k) d[W-Wo-k(2p/MT)]
where
A(k) =Sm=0,(M-1)(1/M)ejrmWoTe-jkm(2p/M)
Notice that A(k) is no longer a function of W,
and A(k) is a M-point DFT of ejrmWoT, m=0, 1,…,M-1
Y. C. Jenq
17
Digital Spectrum of Non-uniformly
Sampled Sinusoid
M=4
A(0)
A(2)
A(1)
Y. C. Jenq
A(3)
18
Digital Spectrum of Non-uniformly
Sampled Sinusoid
M=8
Spectrum of Non-uniformly Sampled Data
60
50
Magnitude Spectrum in dB
40
30
20
10
0
-10
-20
0
0.1
0.2
0.3
0.4
0.5
0.6
Digital Frequency
Y. C. Jenq
0.7
0.8
0.9
1
19
Estimation of Timing Errors - rm
A(k) =Sm=0,(M-1)[(1/M)exp(jrmWoT)]e-jkm(2p/M)
A(0)
A(2)
A(1)
Y. C. Jenq
A(3)
20
Reconstruction of Digital Spectrum
Spectrum Reconstruction from Non-uniformly Sampled Data
60
* : Reconstructed Spectrum, 10-bit quantization
40
Magnitude Spectrum in dB
Once the timing
errors are
known,
can we
reconstruct the
correct digital
spectrum?
20
0
-20
-40
-60
-0.5
-0.4
-0.3
Y. C. Jenq
-0.2
-0.1
0
0.1
Digital Frequency
0.2
0.3
0.4
0.5
21
Selecting Test Frequencies
Higher frequency  more sensitive to timing error
Using FFT spurious harmonics should be on the bins
Windowing function selection
A(0)
A(2)
A(1)
Y. C. Jenq
A(3)
22
Estimation of rm- Synchronous Case
Residual Timing Error
timing
offset
error
RMS
value
before
Adjustment
RMS
value
after
(4 bits)
RMS
value
after
(6 bits)
RMS
value
after
(8 bits)
RMS
value
after
(10 bits)
RMS
value
after
(∞ bits)
30%
4x10-11
2.4x10-12 4.4x10-13 1.1x10-13 2.9x10-14 2.6x10-24
20%
3x10-11
3.1x10-12 5.6x10-13 1.6x10-13 3.0x10-14 2.2x10-24
10%
2x10-11
2.3x10-12 6.1x10-13 1.3x10-13 2.7x10-14 1.8x10-24
5%
0.9x10-11
2.6x10-12 5.4x10-13 1.4x10-13 3.6x10-14 2.0x10-24
Residual timing errors are independent of initial timing errors!
Y. C. Jenq
23
Estimation of rm- Synchronous Case
Sensitivity to Quantization Noise in A/D Converter
Sensitivity of Timing Error Estimation Algorithm
-1
10
Residual Timing
+ : Starting rm ~ 0.05
-2
10
Residual Timing Error: rm
Error is relatively
independent of
initial timing
error, but it is
quite sensitive to
the effective-bit of
ADC
* : Starting rm ~ 0.5
o : Starting rm ~ 0.005
-3
10
-4
10
-5
10
-6
10
4
6
8
10
12
Number of Effective Bits in A/D Converter
Y. C. Jenq
14
16
24
Residual Timing Error
1
Residual Timing Error: RMS rm
Residual RMS rm
10-1
-3
~ 10 at 7 Bits
10-2
One order of
magnitude
improvement
per 3 effective
bits increase
10-3
10-4
10-5
4
Y. C. Jenq
6
8
10
bits
25
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
26
Reconstruction of Digital Spectrum
with Residual Timing Error
Reconstruction noise due to residual timing error:
S/N ~ 20*log(1/s) -16 dB
s = standard deviation of rm
(Residual s) ~ (Initial s)/1000 at 7 Bits
and improve one order of magnitude
per 4 bits increase
Reconstruction noise due to quantization error:
SNR = 6.02* (number of bits) + 1.76 dB
Y. C. Jenq
27
Reconstruction of Digital Spectrum
with Residual Timing Error

Yih-Chyun Jenq, “Improveing Timing Offset
Estimation by Aliasing Sampling,”
IMTC’05, May 2005, Ottawa, Canada.
Y. C. Jenq
28