Analog Devices AN501 Application Notes
AN-501
d
dt
v (t ) = A 2 πf = t
JITTER
a
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APPLICATION NOTE
Aperture Uncertainty and ADC System Performance
by Brad Brannon
Aperture Uncertainty
One of the key concerns when performing IF sampling is
that of aperture jitter or aperture uncertainty. The terms
aperture jitter and aperture uncertainty are frequently
interchanged in text. In this application, they have the
same meaning. Aperture uncertainty is the sample-tosample variation in the encode process. Aperture uncertainty has three residual effects: the first is an increase in
system noise, the second is an uncertainty in the actual
phase of the sampled signal itself and third is
intersymbol interference. To achieve required noise performance, aperture uncertainty of less than 1 ps is
required when IF sampling. In terms of phase accuracy
and intersymbol interference, the effects of aperture
uncertainty are small. In a worst case scenario of 1 ps
rms at an IF of 250 MHz, the phase uncertainty of error is
0.09 degrees rms. This is quite acceptable even for a
demanding specification such as GSM. The focus of this
analysis will therefore be on overall noise contribution
due to aperture uncertainty.
dV
ENCODE
dt
Figure 1. RMS Jitter vs. RMS Noise
In a sine wave, the maximum slew rate is at the zero
crossing. At this point, the slew rate is defined by the
first derivative of the sine function evaluated at t = 0.
v (t ) = A sin(2 πft )
t
evaluated at
the equation simplifies to:
The units of slew rate are volts per second and yields
how fast the signal is slewing through the zero crossing
of the input signal. In a sampling system, a reference
clock is used to sample the input signal. If the sample
clock has aperture uncertainty, an error voltage is generated. This error voltage can be determined by multiplying the input slew rate by the jitter.
V
ERROR
By analyzing the units, it can be seen that this yields unit
of volts. Usually, aperture uncertainty is expressed in
seconds rms and, therefore, the error voltage would be
in volts rms. Additional analysis of Equation 3 shows
that as analog input frequency increases, the rms error
voltage also increases in direct proportion to the aperture uncertainty.
Contribution to Overall System Performance
In IF sampling converters, clock purity is of extreme
importance. As with the mixing process, the input signal
is multiplied by a local oscillator or in this case, a sampling clock. Since multiplication in time is convolution in
the frequency domain, the spectrum of the sample clock
is convolved with the spectrum of the input signal. Since
aperture uncertainty is wideband noise on the clock, it
shows up as wideband noise in the sampled spectrum
as well. And since an ADC is a sampling system, the
spectrum is periodic and repeated around the sample
rate. This wideband noise therefore degrades the noise
= 0, the cosine function evaluates to 1 and
= Slew Rate × t
d
v (t ) = A 2 πf cos (2 πft )
dt
JITTER
(1)
(2)
(3)
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AN-501
floor performance of the ADC. The theoretical SNR for
an ADC, as limited by aperture uncertainty, is determined by the following equation.
SNR ( f t rms)
=−
20 log 2 π
analog JITTER
[]
(4)
If Equation 4 is evaluated for an analog input of 201 MHz
and 0.7 ps rms “jitter,” the theoretical SNR is limited
to 61 dB. Therefore, systems that require very high
dynamic range and very high analog input frequencies
also require a very low jitter encode source. When using
standard TTL/CMOS clock oscillators modules, 0.7 ps rms
has been verified for both the ADC and oscillator. Better
numbers can be achieved with low noise modules.
When considering overall system performance, a more
generalized equation may be used. This equation builds
on the previous equation but includes the effects of
thermal noise and differential nonlinearity.
SNR 20 log (2 f t rms)
=− +
f
analog
t
JITTER
π
analog JITTER
= analog IF Frequency
rms
= aperture uncertainty
2
+
N
2
1e2V rms
+
noise
2
1/2
2
N
(5)
ε = average DNL of converter (~ 0.4 LSB)
V
rms
noise
N
= thermal noise in LSBs
= number of bits
Although this is a simple equation, it provides much
insight into the noise performance that can be expected
from a data converter.
Measurement of Sub-Picosecond Aperture Uncertainty
Aperture uncertainty is easily measured by looking at
degraded SNR performance as a function of analog
input frequency. Since SNR degrades as analog input
frequency increases due to jitter, two FFTs are required
for the calculation. The first FFT should be done at a sufficiently low analog frequency where the effects of aperture uncertainty are negligible. Record the SNR
excluding all harmonics and higher order spurs. Then
solve Equation 5, above, for general converter performance by assuming that thermal noise is rolled up into
the quantization noise and jitter is neglected. This gives
the equation below.
–SNR
ε=2N×10
SNR
is the low frequency SNR
N
is the number of converter bits
20
– 1
(6)
ε = average DNL (+ thermal noise)
Then an FFT is done at very high frequency. The high frequency should be chosen to be near the 3 dB bandwidth
of the converter. Again, the SNR without harmonics
should be measured.
At this high frequency, we can assume that jitter is a
contributor to noise. From the previous data measurement we know the average quantization and thermal
noise; we can solve the general form equation for jitter
as shown.
2
SNR
–
20
10
t rms
JITTER
SNR
is the high frequency SNR
N
is the number of converter bits
=
f
π
2
IF
2
ε
+
1
–
N
2
(7)
ε = average DNL from above and thermal noise
f
is the IF analog input frequency
IF
Putting the Calculations to the Test
The following data was collected using the AD9042ST/
PCB evaluation board. No modifications were made.
The clock oscillator (M1280, manufactured by MF Electronics) supplied with the evaluation board was used to
generate the encode signal which was delivered to the
AD9042 differentially via a transformer (Mini-Circuits
T1-1). The analog input was generated by a Rohde &
Schwarz synthesizer. For more information about the
evaluation board, please see the AD9042 data sheet.
0.00
–10.00
–20.00
–30.00
–40.00
–50.00
–60.00
–70.00
–80.00
–90.00
–100.00
–110.00
Figure 2. 2.3 MHz FFT
Figure 2 is a 16K FFT of the AD9042 sampling a 2.3 MHz
sine wave at 40.96 MSPS. Since we must exclude higher
order harmonics from the SNR calculation, × represents
the unintegrated noise floor, or the mean noise floor.
Instead of integrating all of the noise spikes, this number is summed across the entire spectrum, thus eliminating the higher (and lower) order harmonics. Using
Equation 8:
SNR
= –(–108 + 10 log (8192)) (8)
SNR
is found to be 69 dB. When this is used to
solve Equation 6 for ε the average DNL (and thermal
noise) for this converter is 0.4533 LSBs.
–2–
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