Aerospace and Electronic Systems Magazine December 2017 - 47

To propagate the signals through the radio frequency (RF)
components in front of the analogue to digital conversion (ADC)
input, we follow the theory in Earl [3]. For each component we
express its output voltage as a Maclaurin series expansion of the
input voltage, vin = (vr + ve + vi + vs ) thus,
vo ( t ) = a1vin ( t ) + a2vin2 ( t ) + a3vin3 (t )

(1)

We assume that the series can be truncated at this point with adequate accuracy. The first term in (1) is the wanted linear gain component, which preserves the voltage ratio of interest. The second
and third order terms are nonlinear products, whose coefficients
can be established from the component characteristics in the form
of their intercept points [3]. From a signals point of view then, the
component is defined by its gain and intercept points. Note each
component may be a lumped circuit of individual components that
all have available specification data, but provided the cascaded figures for intercept points and gain are available the lumping of these
sub-components still fits the model.
As the nonlinear coefficients in (1) are relatively small, typically the output is only affected by the higher order products associated with the large input signals. When dealing with the noise
signals (internal and external), we are typically assessing powers
much weaker than can generate observable nonlinear products in

Figure 1.

A sketch of the system model. Internal noise, external noise, and
external signal voltage time series are independently measured and
propagated through RF components in the receive chain, then compared
at processing output.

DECEMBER 2017

a practical system. In propagating the noise voltages through the
system the effect of the gain and noise figure of each component is
of more pertinence. The noise figure is a measure of the internally
generated noise that a device adds to the internal noise level that
presents to the device input. Depending on the level of external
noise impinging on the device, this internally generated noise may
or may not be consequential in affecting the final vr/(ve + vi + vs)
ratio. By separating internal and external noise, the noise figure of
each device can then be accounted for by adding it to the internal
noise as it propagates through each device, while treating external
noise independently. The noise figure is a typically quoted property of components by electronic manufacturers. If we use the linear
equivalent of this, the noise factor F, then we must first transform
the internal noise voltage to noise spectral density (NSD) and then
by definition of the noise factor, F, the output NSD of the component can be written in terms of the input NSD to the component as
N o ( f ) = g ( N i ( f ) + β ( F − 1))

(2)

where g is the linear gain of the component, β = kTo where k =
Boltzmann's constant, and To is the component temperature.
ADC nonlinear behavior is different to RF analogue component
and is not well modelled by intercept points. In modelling the ADC
behavior, we wish to approximate the device transfer function vi
⇒ vo and apply this to our input signals determined from previous
stages. By this method, we can quantify any spectral impacts from
the digitizer. For an ideal ADC, the voltage levels of all quantization levels can be derived from the specification of the number of
bits and the peak input voltage. Assuming ideal sampling we can
then apply the digitizer's transform to our input voltage time series.
As a test of this methodology, a digital broadband noise signal was computer generated with NSD of -130 dBW/RBW (where
RBW is the final spectral processing resolution bandwidth, 1.51
kHz in this case) across 0-50MHz. A digital filter was applied to
the signal so that the NSD between 30 and 50MHz was dropped to
a relatively low value of -190 dBW/RBW. The resultant spectrum
was inverse Fourier transformed to a voltage time series to form
the input to the ideal digitizer. The result of an incoherent average
over 100 runs, with the input (red) and output (blue), is shown
in Figure 2. The NSD in the 30-50MHz band has risen to -167.5
dBW/RBW or about -199 dBW/Hz.
This is the quantization noise generated in the digitization
process. Theoretically the RMS value of this quantization noise is

IEEE A&E SYSTEMS MAGAZINE

Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine December 2017