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ACTA IMEKO 
February 2015, Volume 4, Number 1, 111 – 120 
www.imeko.org 

 

ACTA IMEKO | www.imeko.org  February 2015 | Volume 4 | Number 1 | 111 

Frequency‐domain  characterization  of  random 
demodulation  analog‐to‐information  converters 

Doris Bao, Pasquale Daponte, Luca De Vito, Sergio Rapuano 

Department of Engineering, University of Sannio, Piazza Roma, 21, 82100 Benevento, Italy 

 

Section: RESEARCH PAPER  

Keywords: Analog‐to‐Information Converter, Compressive sampling, Testing, Frequency domain. 

Citation: Doris Bao, Pasquale Daponte, Luca De Vito, Sergio Rapuano, Frequency‐domain characterization of random demodulation analog‐to‐information 
converters, Acta IMEKO, vol. 4, no. 1, article 17, February 2015, identifier: IMEKO‐ACTA‐04 (2015)‐01‐17 

Editor: Paolo Carbone, University of Perugia  

Received January 14
th
, 2014; In final form April 4

th
, 2014; Published February 2015 

Copyright: © 2014 IMEKO. This is an open‐access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits 
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited 

Funding: (none reported) 

Corresponding author: Luca De Vito, e‐mail: devito@unisannio.it 

1. INTRODUCTION 

High-speed data acquisition is becoming a relevant topic 
in advanced applications, such as high-speed radar and 
communications, signal analysis, high-speed video 
acquisition, and so on. Moreover, it is a relevant challenge 
in wideband spectrum sensing for software defined radio 
and cognitive radio applications [1], [2]. 

Such demand often is not met by traditional Analog-to-
Digital Converters (ADCs), due to technological limits in 
fast sampling rates [3]. 

The recent studies about compressive sampling (CS) 
drew a possible solution for signals, that can be represented 
by a finite number of non-zero elements in a specific 
domain. 

They demonstrated that, for such class of signals, it is 
possible to reconstruct the original waveform, from a set of 
samples of a lower dimension than that required by the 
Shannon theorem. 

The idea, underlying the AIC, is to spread the frequency 
content of the input signal. In this way, the high frequency 
components, folded back to low frequencies, can be 
acquired by an ADC with a lower sampling frequency than 
that required by the Shannon's theorem for the original 
signal. 

Basing on this concept, different architectures have been 
proposed, implementing the frequency spreading by 
exploiting: (i) non-uniform [4] or random sampling [5], (ii) 
random filters [6], and (iii) random demodulation [3].  

The aim of the paper is to define performance 
parameters and test methods for AICs, starting from the 
state of art of research and the scientific knowledge about 
ADC testing, well summarized in [7] and [8]. To this aim, 
the first step is the application to AICs of standard 
parameters and test methods, actually defined for ADCs in 
order to study how they are influenced by (i) the AIC 
architecture type, (ii) the AIC design parameters, and (iii) 
the circuit non-idealities. 

ABSTRACT 
The paper aims at proposing test methods for Analog‐to‐Information Converters (AICs).  
In particular, the objective of this work is to verify if figures of merit and test methods, currently defined in standards for traditional 
Analog‐to‐Digital Converters, can be applied to AICs based on the random demodulation architecture. 
For this purpose, an AIC prototype has been designed, starting from commercially available integrated circuits. A simulation analysis 
and an experimental  investigation have been carried out to study the additional  influencing factors such as the parameters of the 
reconstruction algorithm. Results show that standard figures of merit are in general capable of describing the performance of AICs, 
provided  that  they  are  slightly  modified  according  to  the  proposals  reported  in  the  paper.  In  addition,  test  methods  have  to  be 
modified in order to take into account the statistical behavior of AICs. 



 

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In the scientific literature few papers can be found, 
facing the AIC testing and most of them take into account 
only a reduced set of Figures of Merits (FoMs) and 
influencing parameters [9]. 

In [10], the authors presented a preliminary investigation 
carried out in simulation and on a first AIC prototype, 
based on a digital oscilloscope, about the application of 
standard ADC FoMs on the AIC. This paper is an extended 
version of such work, in which new results are presented 
and a new AIC prototype is used, based on commercial 
integrated circuits. 

As in [10], the AIC architecture considered in this paper 
is based on the random demodulation, as it does not require 
a high sampling frequency ADC. However, test methods 
and considerations can be easily extended to the other types 
of AIC architectures. 

In particular, in the paper, a characterization of a 
random demodulation AIC has been carried out by 
applying a reconstruction algorithm to the AIC output and 
evaluating the dynamic parameters in the frequency 
domain, defined for ADC testing. To this aim, in a former 
phase, a behavioural model of the random demodulation 
AIC has been simulated, by considering the non-idealities 
introduced by its main building blocks. In a latter phase,  
an AIC prototype has been designed,  by following the 
theoretical descriptions found in literature, and an 
experimental analysis has been conducted on it. 

The paper is organized as follows: In Section 2, an 
introductory description about compressive sampling 
theory is given; in Section 3, the random demodulation 
architecture, which was used both in the simulation and the 
experimental analyses of this work, is described; in Section 
4, the approach followed for AIC testing is explained; then, 
in Section 5, a simulation phase is reported, in which the 
influence of several factors, such as circuit non-idealities, 
AIC design parameters and reconstruction algorithm 
parameters, have been investigated. In Section 6, the FoMs 
defined for traditional ADCs are revised. Finally, in Section 
7, the experimental analysis is presented and results are 
discussed. 

2.  THEORETICAL  BACKGROUND 

The idea underlying CS approach is that many natural 
signals have concise representations when expressed in a 
convenient basis [11]. As an example, audio signals have 
sparse representations in the Short-Time Fourier Transform 
domain, or in the Modified Discrete Cosine Transform 
domain [12]. Another example is given by radar echo 
signals, that, depending on the radar signal type, can have 
sparse representations in the time, frequency, wavelet, or 
time-frequency domains [13]. 

Sparse representations of natural signals, audio, images 
and videos are currently exploited by transform coding 
schemes, such as those used by the JPEG, JPEG2000, 
MPEG, and MP3 standards. 

However, in signal compression, signals are acquired 
using Nyquist rate converters, then they are transformed in 

a proper domain, where less significant coefficients are 
discarded. 

CS, instead, aims to acquire directly the compressed 
version of the signal, without wasting acquisition or 
memory resources, by taking a vector y of observations of 
the signal to be acquired, where the size of y is lower than 
that required by the Shannon theorem to digitize the signal. 

In the past, some other techniques have been proposed 
to overcome the Shannon theorem constraints in some 
specific conditions. The equivalent time sampling of time 
domain signals is an example of such techniques. However, 
equivalent time sampling requires the observed portion of 
the signal to be repetitive. CS, instead is applicable even to a 
non-repetitive signal, providing that a domain can be found, 
where the representation of such signal is sparse. 

For a compressible signal x(t), if x is a the vector of N 
samples of it acquired according to the Shannon theorem, 
its compressed counterpart is represented by a vector y of 
size M < N, such that: 

,= Φxy  (1) 

where,  is a matrix modelling the compression process. 
It can demonstrated that an estimate of x can be 

reconstructed from y according to (1) if a matrix 
transformation  exists, such that: 

,= Ψcx  (2) 

where c has only K < M non-zero elements. 
The above defined condition is not rare in reality, since 

many natural signals are sparse or compressible in the sense 
that they have concise representations when expressed in 
the proper basis. 

By combining (1) and (2), the following expression is 
obtained: 

,= Acy  (3) 

where A =  is an M × N matrix and, therefore, (3) is an 
under-determined linear system in c. 

The system can be solved by finding the solution of (3) 
that minimizes the ℓ0 norm, that is having the highest 
number of non-zero elements in c: 

  .subject toˆ
0

Acycc =argmin=  (4) 

The minimization of the ℓ0 norm is both numerically 
unstable and NP-complete, requiring an exhaustive 

enumeration of all 







K

N
possible locations of the non-zero 

entries in c [14]. Therefore, the solution is approximated 
with that obtained by the minimization of the ℓ1 norm: 

  .subject toˆ
1

Acycc =argmin=  (5) 

In presence of additive noise e, equation (3) becomes: 
eAcy +=  (6) 

and the minimization problem is modified as: 

  .subject toˆ
21

τ<argmin= Acycc   (7) 

or as: 

    .subject toˆ
1

ε<argmin=


 AcyAcc T  (8) 



 

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where and are small positive constants. 
 
The ℓ1 norm minimization is a convex optimization 
problem that conveniently can be reduced to a linear 
program known as basis pursuit, whose computational 
complexity is about O(N 3) [14]. 

The minimum number of observations M required to 
successfully reconstruct x is given by the following 
expression [11]: 

( ) ( )log
2M Cμ , K N , ³ ×  (9) 

where, C is a positive constant and ( )μ ,  is a quantity, 
called coherence between  and , having the following 
expression: 

( ) max i j
1 i, j N

μ , N φ ,ψ , 
£ £

=  (10) 

with i and j the i-th row of and the j-th column of 
respectively, and , the dot product. 

The coherence measures the largest correlation between 
any two elements of  and ; If  and contain 
correlated elements, the coherence is large; otherwise, it is 
small [11]. 

By looking at (9), it can be easily seen that the lower is 
the coherence between  and , the fewer observations are 
needed to reconstruct the original signal. It can be 
demonstrated that incoherence can be achieved with high 
probability simply by selecting as a random matrix [14], 
whose elements are drawn from a suitable distribution, 
such as Gaussian, Bernoulli or Rademacher. 

3.  THE  RANDOM  DEMODULATION  AIC  ARCHITECTURE 

The block scheme of the random demodulation AIC is 
shown in Figure 1. Considering an ADC with sampling 
frequency fs, and a signal x(t), whose frequency content 

exceeds the first Nyquist region 







2
0, s

f
. An example of a 

three-tone signal having such characteristics is shown in 
Figure 2a. According to the Shannon theorem, it would not 
be possible to sample x(t), since the components located at 
frequencies greater than fs /2 would produce aliasing. 

It can be observed that for such signal the condition (2) 
is verified as soon as x has a Fourier representation with a 
limited number of non-zero coefficients. In that case, in 
fact, (2) is simply the Fourier synthesis formula. 

 

Therefore, first, a compressed version of x is obtained, 
and, then, the signal could be reconstructed by solving one 
of the problems (5), (7) or (8). 

In order to obtain the compressed version of x(t), it is  
mixed with a Pseudo-Random Binary Sequence (PRBS),  
having a bit rate equal to fp =  fs , with  positive integer 
greater than 1. The bit rate of the PRBS determines the 
maximum frequency component of the original signal that 
the AIC is able to acquire, as it is required to be at least 
twice the maximum frequency component of the signal. 

The spectrum of the PRBS, shown in Figure 2b, has a 
frequency content spread over the whole region  ,f p0,
therefore, the output of the mixer contains the information 
of the original signal, spread over the whole first Nyquist 
band, even if it was originally located outside such region 
(Figure 2c). This allows that after low-pass filtering 
(Figure 2d) and A/D conversion (Figure 2e), the 
information contained in the original signal has not been 
lost. However, it needs to be recovered from the filter 
output. 

A reconstruction algorithm is, then, in charge of 
extracting this information, by solving one of the problems 
in (5), (7) or (8), as previously quoted, giving thus a digital 
representation of the original signal (Figure 2f). 

 

Figure 1: Block scheme of the random demodulation AIC. 

 

Figure 2. Spectra of the signals processed inside the AIC structure. The AIC is 
able of reconstructing a signal even if its spectral content exceeds the first 
Nyquist region. 



 

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The capability of the reconstruction algorithm of 
actually recovering the original signal with high probability 
depends on the undersampling factor , and on the 
sparsity rate K / M. A performance analysis for a random 
demodulator using an integrate and dump filter has been 
carried out in [15], and the results showed that a high 
probability of reconstruction could be achieved if: 









 1log1.7 +
K

N
KM  (11) 

Therefore, the AIC is not able to preserve the information 
contained in the signal if undersampling is below a certain 
value, depending on the number of non-zero element in the 
sparse representation of the signal to be acquired. 

The architecture can be extended by using several 
parallel channels, each driven by a different PRBS. In this 
case, the AIC is capable of acquiring signals even in the case 
undersampling causes the overlapping of several signal 
components, since in each channel, the overlapping appears 
in different form [16]. For sake of simplicity, in this paper 
the analysis has been conducted on a single channel 
architecture. However, results can be easily extended to a 
multi-channel architecture. 

A mathematical model of the AIC can be obtained as it 
follows. Indicating with x the vector of the signal samples, 
if it was acquired using a sampling frequency equal to the 
bit rate of the PRBS fp, and with  11,0,  Np,pp=p
the samples of the PRBS, the output of an ideal digital 
mixer m can be obtained by multiplying x by a diagonal 
matrix, having the PRBS samples on the diagonal: 

Dxm =  (12) 
where: 























1

1

0

00

00

00

Np

p

p=







D  (13) 

Instead, the filter can be modelled by the following 
Toeplitz matrix: 





























12

121

121

00

00

00

LL

LL

L

hh

hhh

hhh=







H  (14) 

where,  TLhhh= 110 h is the truncated impulse 
response of the antialiasing filter of length L. L should be 
chosen large enough that the truncated response guarantees 
a good approximation of the actual one. 

The random demodulation AIC can be obtained by 
choosing a  matrix that decimates the matrix product 
H D, by taking only 1 every  rows. 

4.  APPROACH  TO  AIC  TESTING 

AIC testing is a harder task than traditional ADC 
testing, due to the following reasons: (i) the AIC output 
signal has a strong stochastic behaviour, due to the PRBS 
(compare Figs.2a and 2e); (ii) the AIC output signal is noise-
like for every input, thus, it is hard to analyse either in the 
time or in the frequency domain; and (iii) the parameters of 
the reconstruction algorithm can affect the AIC results. 

The current research has been focused on frequency-
domain  testing, as it is mostly used in telecommunications, 
which is the main application field of AICs. The approach 
followed for AIC testing consists of using a sinewave signal 
as input to the AIC, applying a reconstruction algorithm, 
and characterizing the spectrum of the reconstructed signal, 
by means of the traditional ADC FoMs. Test signal 
frequencies have been chosen such that the coherent 
sampling condition is respected for the reconstructed signal, 
that is they should verify the following condition: 

π,
N

J
=ω 20  (15) 

where, J is an integer number, relatively prime to N. 
The ADC testing methods in the frequency domain can 

be clearly applied to AIC as the involved signals are sparse 
in such domain. As sparsity in the frequency domain has 
been considered, the matrix in (2) must represent a 
transformation between time domain and frequency 
domain. However, since only real-valued test signals have 
been considered, the Discrete Hartley Transform (DHT) 
basis has been used, which is equivalent to the Discrete 
Fourier Transform, but is purely real. 

The generic element i,j of the DHT has the following 
form: 

.ij
N

π
+ij

N

π

N
=ψ ji, 






















 2

sin
2

cos
1

 (16) 

The Dantzig selector [17] has been used as 
reconstruction algorithm, solving the optimization 
problem in (8), by means of the implementation given by 
the ℓ1 -magic MATLAB toolbox [18]. 

Differently from problem (7), which looks for the 
solution in the least square sense, the Dantzig selector 
computes the residual r = y – A c, and, then, evaluates its 
inner products with the columns of A. These inner 
products form various weighted combinations of the entries 
in r and all those are expected to be small [19]. The 
constraint, thus, proceeds by requiring that all these are 
below the threshold . 

Therefore, the parameter represents the maximum 
allowable value for the correlation of the residual r with 
any  column of A. 

In a preliminary phase, the AIC has been modelled and 
some simulations have been carried out in order to 
understand how the FoMs, evaluated on the reconstructed 
signal, can be affected by: (i) circuit non-idealities, (ii) test 
signal parameters, and (iii) reconstruction algorithm 
parameters. 



 

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According to the results of the simulation analysis, the 
following FoMs, have been extended to AICs: Spurious 
Free Dynamic Range (SFDR), Signal to Noise and 
Distortion Ratio (SINAD), and Total Harmonic Distortion 
(THD). 

For such FoMs, the definitions in [20] have been used. 
However, the same approach can be used even considering 
definitions coming from different standards, such as those 
considered in [21]. 

Finally, an experimental phase has been conducted by 
evaluating the revised FoMs on a specifically designed AIC 
prototype. 

5.  SIMULATION  ANALYSIS 

During simulation analysis, an explicit behavioural 
model [22] has been considered for each AIC block.  

In Figure 3, the steps carried out for the simulation 
analysis have been drawn. Sinewave signals have been 
generated, with frequency in the range [0.05-0.45] , with a 
record length of N=512 samples. Signal records are then 
given to a mixer model, together with PRBS records of the 

same size. 
As any actual mixer will distort the signals, the ideal 

mixer output yid = x1(t) x2(t) (see Eq. 12) has been replaced 
by a second order polynomial model, as follows: 

             tdx+tcx+txtx+tbx+tax=ty 22212121  (17) 
where, x1(t) and x2(t) are the inputs to the mixer, 
respectively, and a, b, c and d are attenuation coefficients. 

The output of the mixer is given to a lowpass FIR filter, 
with cut-off frequency equal to /4, modelling the 
antialiasing filter. 

The ADC, embedded in the AIC architecture, has been 
modelled as a memoryless nonlinearity followed by a 
downsampling by 4 and an ideal quantizer, where the 
nonlinearity has been obtained, as in [9], using the 
following third order polynomial function: 

     
3

3
3

1 c+

txc+tx
=ty  (18) 

Due to downsampling, the record length is reduced to 
M=128 samples. The output of the ADC model is given to 
the reconstruction algorithm, able of regenerating a record 
of N=512 samples containing the reconstructed signal. 
Finally, the FoMs are evaluated on the averaged spectral 
magnitude of the reconstructed signal. Averaging of spectral 
magnitude is specified by [20] for the evaluation of the 
FoMs in the frequency domain. However, as it is shown in 
Section 5.1, a greater average number is needed than in the 
case of ADCs. 

The above described steps have been executed on several 
signals, when the parameters of the models have been 
varied as shown in the following Subsections. 

Figure 3: Block scheme of the simulation analysis. 

      

  a)  b) 

 

  c)  d) 

Figure 4. Spectral  magnitudes obtained by the  FFT magnitude on a single 
record (a), or by averaging the FFT magnitudes evaluated on 10 (b), 100 (c), 
and 200 (d) records. 

        

  a)  b) 

Figure 5. Averaged spectral magnitudes of the reconstructed signal without 
mixer nonlinearity (a), and with mixer nonlinearity (b). 



 

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 Influence  of  average  number 5.1.

In Figure 4, the spectral magnitude is shown, when it is 
obtained by the FFT (Fast Fourier Transform) magnitude 
of a single record (Figure 4a), or when it is obtained by 
averaging FFT magnitudes evaluated on different records. 
Results are shown for a number of averages equal to 10 
(Figure 4b), 100 (Figure 4c), and 200 (Figure 4d). As it can 
be noted, due to the selection of frequency components 
operated in the reconstruction algorithm, some of them are 
not identified in a single record. Instead, the averaging 
allows obtaining an estimation of the spectral magnitude 
for each frequency component. 

Such behaviour is more evident for higher values of , 
since in this case a higher number of components are 
discarded by the reconstruction algorithm. 

  Influence  of  mixer  nonlinearity 5.2.

The main effect of the mixer nonlinearity is of increasing 
the level of the noise floor, as shown in Figure 5, where the 
averaged spectral magnitudes of the reconstructed signal 
when no mixer nonlinearity has been modelled 
(a=b=c=d=0) (Figure5a), and when a nonlinearity is added, 
with the following values of the attenuation coefficients: 
a=b=0.1, c=d=0.01 (Figure5b). 

Those values have been chosen by looking at the order 
of magnitude of the typical feedthrough and second order 
distortion component attenuation for actual mixers. 

In addition, a particular increase of the level of the 
components, whose frequencies lie at multiples of the 
fundamental, has been observed. 

  Influence  of  ADC  nonlinearity 5.3.

The effect of ADC nonlinearity can be observed in 
Figure 6, where the averaged spectral magnitudes of the 
reconstructed signals are reported in the case the c3 
parameter is set to 0 (Figure 6a), simulating a linear ADC, 
and when the c3 parameter is set to 0.2 (Figure 6b). 
Differently from traditional ADCs, the nonlinearity does 
not act only on a restricted number of components, located 
at multiples of the fundamental, in fact it causes even an 

increase of the noise floor. 

 Influence  of  ADC  quantization 5.4.

The ADC quantization causes an increase of the noise 
floor level. As it can be seen in Figure 7a, the signal is 
perfectly reconstructed (with noise floor level depending 
only on the machine precision) when maximum machine 
resolution is used. Instead, a noise floor arises when 
quantization is used. However, only a small difference in 
the level has been observed for different resolutions. During 
simulation analysis, values of the resolution in the set {8, 
10, 12, 14, and 16} bits have been used. In Figure 7b, the 
averaged spectral magnitudes, related to a resolution of 8 
bits and 16 bits are shown. 

  Influence  of  reconstruction  parameter   5.5.

As the reconstruction parameter  represents a limit to 
the amount of additive noise that can be tolerated on the 
output signal, it controls the selection of the frequency 
components in the reconstructed signal. In fact, for each 
record, a frequency component will be included in the 
reconstructed signal, only if its energy exceeds a certain 
level, depending on the value of . 

On the averaged spectrum, this causes an increase of the 
level of both the noise floor and of the harmonics, for 
decreasing values of . 

Such behaviour can be observed in Figure 8a, where the 
averaged spectral magnitudes of the reconstructed signal are 
shown, where has been set to 0.01, 0.05, and 0.1. 

  Influence  of  the  filter  length 5.6.

The length of the impulse response of the anti-aliasing 
filter determines the capability of the filter of attenuating 
the aliasing replicas. The shorter the filter impulse response 
is, the lower such capability is. Figure 8b highlights this 
behaviour, by showing the averaged spectral magnitudes of 
the reconstructed signals, when filter lengths of 17 and 49 
taps are used. As it can be noted, in the case of a filter 
length of 17 (blue line), the filter is not able to adequately 
attenuate the aliasing replicas, located at the frequencies 

      

  a)  b) 

Figure 6. Averaged spectral magnitudes of the reconstructed signal without 
ADC nonlinearity and with a nonlinearity coefficient c3 equal to 0.1, 0.2, and 
0.3, respectively. 

      

  a)  b) 

Figure  7.  Averaged  spectral  magnitudes  of  the  reconstructed  signal  when 
infinite resolution is used (a), and when a quantization of 8 bit or 16 bit is 
used (b). 

 (22) 



 

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0, 0

2 2π π
ω +ω

 
 

  
. 

 In the case the filter length is of 49 taps (red line), 
instead, the aliasing components are well attenuated under 
the noise floor. 

6.  DEFINITION  OF  THE  FIGURES  OF  MERIT 

As previously mentioned, the FoMs used in this work 
have been defined starting from those of the 
IEEE Std. 1241-2010 [20]. However, the following 
modifications have been introduced: (i) the number of 
averages of the spectral magnitudes has been set to 200, in 
order to take into account the stronger stochastic behaviour 
of the reconstructed signal, and (ii) the frequencies 

0, 0

2 2π π
ω +ω

 
 

  
have been included in the list of the 

harmonics, contributing to the THD. 
The FoMs have been evaluated by averaging the spectral 

magnitude   mX as it follows [20]: 

     1,0,1,2,1
1

 N,=m,mXR=mX
R

=k
kavm  (19) 

over a number of R data records. 

Compared to definitions in [20], no changes have been 
made to the SFDR and SINAD: 

  
  

















mXmax

nX
=SFDR

avm
Sm

iavm

0

1020log  (20) 

   
 








0

2

22

10
3

20log

Sm
avm

iavmiavm

mX

nNX+nX

N

N
=SINAD  (21) 

where, ni is the frequency index corresponding to the 
fundamental component, S0 is the set of frequency indexes 
from 1 to N -1, excluding the two values corresponding to 
the fundamental and its image. 
Instead, THD has been modified as shown in (22), where, 
Nh is the number of the highest harmonics considered,  
mod(∙,∙) is the modulo operator, and nh is the frequency 
index corresponding to a generic harmonic or its aliasing 
component: 

   hih N,±=hN,hnmod=n 2,3,  (23) 
In the considered case Nh has been chosen equal to 10. 

7.  EXPERIMENTAL  INVESTIGATION 

For the purpose of the experimental investigation, an 
AIC prototype has been designed, using commercially 
available circuits and bench instrumentation. In the 
following Subsections, the AIC prototype is described, 
then, some details, about the calibration phase needed to 
estimate the  matrix, are given; finally, the experimental 
results are presented. 

  The  AIC  prototype 7.1.

A block scheme of the AIC prototype is shown in 
Figure 9. The mixing process is carried out by means of the 
Analog Devices AD8342 evaluation board, while the 
antialiasing filter has been realized by the first channel of 
the Analog Devices ADRF6510 evaluation board. The 
ADRF6510 is a dual-channel programmable lowpass filter, 
with cut-off frequency tunable in the range [1-30] MHz. 

 

Figure 9. Block scheme of the AIC prototype and the test setup, used during experimental analysis. 

    

  a)  b) 

Figure  8.  Averaged  spectral  magnitudes  of  the  reconstructed  signal  for 

values of  equal to 0.01, 0.05, and 0.1. (a), respectively, and the averaged 
spectral magnitudes of the reconstructed signal, for filter  lengths equal to 
17 and 49 taps (b). 



 

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After mixing and filtering, signal is acquired by means of an 
Analog Devices AD9230 evaluation board, operating at a 
sampling frequency of 50 MS/s. The ADC board is 
connected with a computer through an Analog Devices 
HSC-ADC-EVALC Capture board. 

For the test purpose, sinewave signals have been 
generated by an Agilent E4438C Vector Signal Generator, 
while the PRBS is generated in the computer and 
downloaded, through the GPIB, to the Tektronix AWG420 
Arbitrary Waveform Generator. An Agilent E8663B is used 
to generate the sampling clock for the ADC board. 

In order to synchronize the PRBS generation with the 
acquisition, the HSC-ADC-EVALC Capture board has 
been  configured with a specific firmware allowing external 
triggering. In this case, the acquisition is started on the 
computer, causing  a Ready signal to be set on the HSC-
ADC-EVALC Capture board. Such Ready signal is given 
both to the Tektronix AWG420, to generate the PRBS and 
again to the HSC-ADC-EVAL Capture board to start the 
acquisition. The Agilent E4438C, the Tektronix AWG420 
and the Agilent E8663B have been then synchronized by 
means of a 10 MHz reference, in order to guarantee the 
coherent sampling condition, and controlled by the 
computer by means of a GPIB interface.

   matrix  estimation 7.2.

In order to run the reconstruction algorithm, it is 
necessary to estimate the  matrix in (1). Since it is 
obtained from the PRBS and from the impulse response of 
the anti-aliasing filter through (13) and (14), only the 
estimation of the impulse response of the anti-aliasing filter 
is necessary. 

This is done in a preliminary calibration phase, by 
connecting the Tektronix AWG420, generating a PRBS, 
directly to the input of the filter. Then, the output of the 
filter is acquired. As shown in [23], if the input to a system 
x(t) is a stochastic process, having impulsive autocorrelation 
function, the impulse response of the system can be 
measured as: 

 
 
 

,
r

nr
=nh

xx

xy

0
 (24) 

where, rxy(n) is the cross-correlation function between the 
input and the output and rxx(0) is the autocorrelation of the 
input, evaluated at a lag equal to 0, corresponding to the 
input signal energy. Since in the considered case the 
generated PRBS signal has impulsive autocorrelation, it is 
possible to estimate the impulse response of the filter using 
(24). 

  Experimental  results 7.3.

Using the AIC prototype and the test setup shown in 
Figure 9, a wide experimental investigation has been carried 
out, by evaluating the SFDR, SINAD, and THD, on the 
reconstructed signal. As in simulation tests, sinewave input 
signals have been used. In order to guarantee the coherent 
sampling condition, the test signal frequency has been 
selected, using the following expression: 

pf
N

J
=f 0  (25) 

where, fp is the PRBS bit rate, corresponding to the 
equivalent sampling frequency, and J is an integer coprime 
to N. 

Sinewave signals have been generated with frequencies in 
the range [5 - 95] MHz and PRBS bit rate has been fixed to 
200 MHz. Different values of the cut-off frequency of the 
antialiasing filter have been used in the range [10 - 30] MHz, 
while the sampling frequency of the ADC embedded in the 
AIC architecture has been set to 50 MS/s. 

Since the band of the reconstructed signal is in the range 
[0-100 MHz], and the Nyquist frequency of the ADC 
embedded in the AIC is 25 MHz, the AIC allows extending 
4 times the signal bandwidth, compared with the case of the 
ADC embedded in the AIC architecture, running at the 
same sampling frequency. 

Some results of the experimental investigation are shown 
in Figure 10, for a test signal frequency in the range 
[5-95] MHz, with a step of 5 MHz. 

The variations of the values of all three FoMs are in the 
range of some dBs, even beyond the first Nyquist band 
[0-25] MHz, however, performance show a little decrease, 
in SFDR and SINAD, for frequencies approaching 100 
MHz. 

       

       

 

Figure 10. Experimental results for test signal frequency ranging from 5 to 

95 MHz: SFDR (a), SINAD (b), and THD (c), for =0.5. 



 

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It is worth studying the variation of the FoMs versus the 
reconstruction parameter , as well as versus the bandwidth 
of the anti-aliasing filter. 

As shown by the simulation results, the levels of both 
the noise floor and the harmonic components decrease with 
, therefore, an increase in the values of SINAD and SFDR 
is expected, while THD is expected to decrease. This can be 
easily observed in Figure 11, where the trend of the 

considered FoMs versus  is shown. 
In Figure 11, the variation of the FoMs versus the 

bandwidth of the anti-aliasing filter is shown, too. It can be 
observed that the values of SINAD and THD are generally 
better for a filter with a wider band. This is probably due to 
the fact that a greater amount of information passes 
through the filter and can be used for the reconstruction. In 
the case of the SFDR, instead, smaller values have been 
observed with the narrowband filter, and < 1. As it can 
be seen in Figure 12, this due to the aliasing components at 
fs - f0 and fs + f0, that are not attenuated under the noise 
floor level, by the 20 MHz bandwidth filter. In the case of a 
20 MHz bandwidth, in fact, the noise floor level is lower, 
but the aliasing component are higher than the case of a 
14 MHz bandwidth. 

8. CONCLUSIONS  AND  FURTHER  WORK 

In this paper, a frequency-domain characterization of 
AICs has been presented, by concentrating the attention on 
the random demodulation architecture. In a former phase, 
the AIC has been simulated by means of explicit 
behavioural models of its main components, in order to 
evaluate the influence of the circuit non-idealities, the 
design parameters and the reconstruction algorithm to the 
FoMs, currently defined for ADCs. 

The simulation results show that both mixer and ADC 
nonlinearities contribute to increase the noise floor level, as 
well as the level of the harmonic components. This is quite 
different from what happens in traditional ADCs, where 
nonlinearities do not affect the noise floor level. In 
addition, the antialiasing filter can affect the SFDR, as it can 
not completely attenuate the aliasing components. As result 
of the simulation analysis, a modification to the THD 
definition is proposed by including the energy of the 
aliasing components. 

In a latter phase, an experimental analysis has been 
conducted on a specifically designed AIC prototype, using 
sinewaves, by varying the test input frequency, the 
reconstruction parameter , and the bandwidth of the 
antialiasing filter. Such analysis showed that the proposed 
FoMs are in general capable of describing the performance 
of the AIC. However, the FoMs should be observed for 
different values of , since high values of can hide the 
effects of noise and distortion. 

Further work will be directed to analyse the 
Intermodulation Distortion as frequency domain FoMs. 
Then, it can be interesting to observe the FoMs even 
evaluated in the time domain. In addition, the methodology 
presented in this paper can be extended to other AIC 
architectures, such as those based on the random sampling. 
Finally, an alternative approach to AIC testing can be used, 
without relying on the reconstruction algorithm, but 
directly comparing the AIC output, with the expected 
output of the AIC model.  
  

     

      

 

Figure  11. Evaluated FoMs versus  and filter bandwidth, for a test signal 
frequency of 10 MHz. 

 

Figure 12. Averaged spectral magnitude of the reconstructed signal for filter 
bandwidth equal to 14 and 20 MHz. 



 

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