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Torben Larsen et al
Aalborg University, Denmark
TechOnline
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Torben
Larsen, Christian Rye
Iversen, and Troels Emil
Kolding of Aalborg University's Center for PersonKommunikation,
RF Integrated Systems and Circuits Group (Iversen is also with
Siemens Mobile Phones) all have backgrounds in RF signal analysis,
circuits design, and measurement techniques. The authors have
written numerous papers on noise in electronic circuits.
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This article presents a time-domain-based
technique to determine noise factor (or noise temperature) of a
device with a low desired output frequency. The input frequency (or
frequencies) can be either low or high. We base our technique on
the Y-factor method, where cold and hot noise source temperatures
are applied to determine the device noise level. Combining this
with Fourier stochastic process techniques provides a means to
determine spot noise quantities, such as noise factor or noise
temperature, of the device. The technique is very flexible, since
the frequency resolution can be made arbitrarily small and it is
directly possible to investigate convergence of the results. The
technique has been tested for a direct-conversion receiver for GSM
where measurements agree very well with expected results based on
known receiver performance.
Electronic noise remains one of the most critical design
parameters in analog systems. However, evaluation of noise
performance of circuits that perform a frequency translation from
RF to baseband is not easy to do. Many current systems exhibit such
behavior, for example, direct-conversion and low-IF receivers. A
common approach to noise evaluation is to employ a low-frequency
spectrum analyzer and conduct manual calculation of the noise
temperature/figure. However, solutions in the frequency domain are
associated with limited flexibility, particularly in terms of
post-processing.
This article investigates the applicability of device noise
measurements based on a time-domain waveform sampler and a noise
diode. High-performance, low-cost data acquisition boards have
become available for the PC platform and readily support sampling
rates in the megahertz range. This paper describes a technique that
maps time-domain sampling of the noise waveforms into popular noise
quantities such as the noise temperature and noise figure. The technique is based on the Y-factor method and
sampling of the noise output waveforms for a hot and a cold noise
source at the system input. The two sets of time-domain waveform measurements
are divided into a number of time slots. A characteristic feature
of the method is that the frequency resolution can be made
arbitrarily small by appropriate choice of time-slot length. The
noise quantities are then calculated versus frequency for each time
slot by means of power spectral density or Fourier coefficients. We then apply statistical techniques to validate
the convergence of the results, which can vary significantly with
frequency. An improvement over frequency-domain-based solutions is
that the proposed method yields arbitrarily good frequency
resolution on noise quantities, validation of results with
convergence analysis, and the possibility for digital
post-filtering of the data.
Noise Measurement Method
Measurement Setup
We need to determine the noise temperature (or figure) of a linear
stage based on time-domain sampling of the output waveform. The
measurement setup is shown in Figure 1. You can switch the
noise diode to generate noise at two different noise temperatures:
a cold temperature (Tc) and a hot temperature
(Th). The signal from the noise source goes directly to
the input to the device under test (DUT). The continuous
time-domain signal from the output of the DUT is so(t)
and the continuous power spectral density of this signal is
No(f). Next, the signal is transferred to the data
acquisition board (DAQ) where the signal is amplified, converted to
digital form (sampled and quantized), and applied to a buffer for
subsequent storage on a PC. The output signal from the DAQ in
discrete time is s'o(tn) and the
corresponding discrete power spectral density is
No(xp) at
frequency xp. It is assumed
that the ADC is excited such that we can ignore quantization
errors. This assumption can be fulfilled by proper low-noise
pre-amplification of the signal.
Figure 1: In this diagram of the measurement
set-up, the functional block DAQ represents the data acquisition
board
Y-Factor Method
First, the noise diode is set to generate noise at the cold
temperature (ambient temperature in most cases). This yields an
available noise power density at the output of the DUT that is:
(1)
Similarly, the available noise power density at the output from
the DUT with the hot noise diode applied is:
(2)
where k = 1.3806 · 10-23 [J/K],
Te(xp) is the
effective (broadband) input noise temperature of the DUT referred
to the output frequency xp,
and Ga(xp) is the
broadband gain referred to the output frequency xp when a broadband noise source is
applied at the input. This defines the Y-factor as:
(3)
The main issue is to determine the Y-factor from time-domain
samples.
Determining the Y-Factor
As shown in the previous subsection, the key to characterizing the
noise performance of the DUT is to determine the Y-factor at the
DUT output. The Y-factor is the ratio of two available noise power
densities. The noise power densities are not readily available
through the setup in Figure 1.
Figure 2: Output of the DUT and input to the
DAQ
Figure 2 shows the output of the DUT and the input of the
DAQ. The Y-factor is described by available noise power density at
the DUT output. This available noise power density is given by:
(4)
where  is the two-sided Fourier coefficient of
uo(t) at frequency xp that is evaluated in a time
interval 2T = 1/x, and  denotes the ensemble average. Taking the
output/input impedances of the DUT and DAQ into account, you can
relate the available noise power density to the s-signal as:
(5)
where a is a real constant depending
on DUT output and DAQ input impedances. Since the Y-factor
describes the ratio of hot/cold noise powers, this constant is
irrelevant—it appears in both the numerator and denominator of
the Y-factor. So far all noise power densities are evaluated at the
DAQ input. The DAQ board amplifies, samples and quantizes the input
signal. Ignoring the quantization error, the available noise power
density referred to the DUT output is:
(6)
where a' takes into account a as well as signal amplification in the DAQ
board, and 
is the two-sided Fourier coefficient at frequency xp at the DAQ output. You can write
the Y-factor as:
(7)
where the s'o Fourier coefficients are for the
hot/cold case, respectively.
Determining the Fourier Coefficients
Observing the DAQ output versus time may result in the output
waveforms in Figure 3.
Figure 3: Illustration of output signal from DAQ in
continuous time
In Figure 3, the time-domain signal is divided into M
slots (or ensembles), each with a time duration of (2N+1)DT. This indicates that each time slot is
represented by 2N+1 samples, with two subsequent samples separated
in time by DT = 1/fs, with
fs being the sample frequency. We assume ergodicity and
stationarity for the underlying noise processes, which lets us
estimate the ensembles by considering each time slot as one
ensemble in the stochastic process. One time slot is then processed
(Figure 4).
Figure 4: The DAQ output signal in time-slot m
Regarding signal representation, it is essential to note that
any finite energy signal can be written as a Fourier series when
observed in a limited time interval. The representation in this interval is exact, but
may be incorrect outside the interval. Since the following analysis
only considers the representation within the observation interval,
we use a Fourier series approach. The continuous time domain signal
so,m(f) gives the Fourier series coefficient:
(8)
where:
(9)
The frequency resolution can be made arbitrarily small by
choosing many samples for each ensemble. To utilize this desirable
feature it is, of course, necessary to have sufficient storage
capacity. In most cases, this is no problem since DAQ boards can
typically be acquired with large amounts of on-board memory, or the
boards are capable of streaming data directly to a disk at the full
sampling rate. The DAQ signals are discrete in time with:
(10)
according to Figure 4. This translates the integration to
summation in the Fourier series transformation as:
(11)
Result Convergence
We now evaluate the Y-factor as more and more time slots
(ensembles) are included in the evaluation. The Y-factor obtained
considering a total of M time slots is then:
(12)
We now can represent the noise figure as:
(13)
where TM(xp) is
the effective input noise temperature after including M ensembles
in the calculation. We can determine the effective input noise
temperature from the corresponding Y-factor as:
(14)
We see that the noise figure converges with increasing M. This
factor is very useful for validating the measurements.
Measurement Example
The method this paper outlines was used to measure the noise
factor of a 900MHz GSM direct-conversion receiver. The receiver
contains a 200kHz low-pass filter at the output. A noise diode with
ENR=15.15dB at 1GHz was employed, and the ambient temperature was
Tamb = 22°C. Thus the cold temperature is
Tc = 290 + Tamb = 312K. The hot temperature
is determined from ENR as Th = 290 ·
(10ENR/10+1) = 9461K where ENR is in dB. The sampling
frequency of the data acquisition board was 2.166MHz. The value of
N was 512, and thus the frequency resolution is 2.12kHz. A National
Instruments NI6110E data-acquisition board was used for acquisition
of time domain samples. Direct streaming of data to disk is a
possibility with this card, which provides excellent possibilities
to test convergence since this requires large amounts of data.
Figure 5 shows the (spot) noise figure versus frequency.
Note that there is some variation of the curve versus frequency.
One reason for this variation is that the receiver is driven in
burst mode to ensure that AGC and dc-offset compensation work as
intended. This means that discontinuities in the data appear. In
the passband of the low-pass filter, the average noise factor is
5.1dB. From receiver performance, we know that the noise factor is
5.0dB, thus indicating very good agreement with our results.
Figure 5: Spot noise factor versus frequency
Figure 6 shows the noise factor convergence versus number
of ensembles for three different frequencies. Reasonable
convergence is obtained after 600-700 ensembles. This corresponds
to a total of approximately 660,000 noise samples, or approximately
10.3MB of storage for both cold and hot data.
Figure 6: Noise factor convergence versus number of
ensembles at three different frequencies
Conclusions
This article describes a technique for determining noise
performance quantities from time-domain sampling of noise waveforms
using a hot/cold noise diode. The technique is very flexible, since
it relies on sampling using a general-purpose data-acquisition
board and it provides possibilities for arbitrary frequency
resolution and convergence check. We have shown a measurement
example that indicates good agreement between the presented
technique and alternative estimation of the noise figure of a test
device.
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