A Time-Domain-Based Technique for Noise Measurements

	ABOUT THE AUTHORS 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.

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 (T_c) and a hot temperature (T_h). 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 s_o(t) and the continuous power spectral density of this signal is N_o(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(t_n) and the corresponding discrete power spectral density is N_o(x_p) at frequency x_p. 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], T_e(x_p) is the effective (broadband) input noise temperature of the DUT referred to the output frequency x_p, and G_a(x_p) is the broadband gain referred to the output frequency x_p 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 u_o(t) at frequency x_p 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 x_p 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/f_s, with f_s 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 s_o,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 T_M(x_p) 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 T_amb = 22°C. Thus the cold temperature is T_c = 290 + T_amb = 312K. The hot temperature is determined from ENR as T_h = 290 · (10^ENR/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.