As propagation terminals have evolved, their design has trended more toward a software-based approach that facilitates convenient adjustment and customization of the receiver algorithms. One potential improvement is the implementation of a frequency estimation algorithm, through which the primary frequency component of the received signal can be estimated with a much greater resolution than with a simple peak search of the FFT spectrum. To select an estimator for usage in a Q/V-band beacon receiver, analysis of six frequency estimators was conducted to characterize their effectiveness as they relate to beacon receiver design

Morse, Jacquelynne R.

Nessel, James A.

Zemba, Michael J.

NASA Technical Reports Server

 
Fig. 1. The scalloping of power observed in the peak bin of the FFT 
and the reduction in scalloping when summing neighboring bins. 
455.0 455.1 455.2 455.3 455.4 455.5
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Input Signal Frequency (kHz)
Po
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 (d
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Power in FFT Bins - Positive Frequencies Only, -3dBW Input
(f
s
 = 3276.8 kHz, N = 32768, f
s
/N = 100)
 
 
Peak Bin
Peak ±1
Peak ±2
Peak ±5
All Positive Frequencies
Frequency Estimator Performance for a  
Software-Based Beacon Receiver 
 
Michael J. Zemba, Jacquelynne R. Morse, and James A. Nessel 
Antenna and Optical Systems Branch 
NASA Glenn Research Center 
Cleveland, OH 
 
 
Abstract—As propagation terminals have evolved, their design 
has trended more toward a software-based approach that 
facilitates convenient adjustment and customization of the 
receiver algorithms. One potential improvement is the 
implementation of a frequency estimation algorithm, through 
which the primary frequency component of the received signal 
can be estimated with a much greater resolution than with a 
simple peak search of the FFT spectrum. To select an estimator 
for usage in a Q/V-band beacon receiver, analysis of six 
frequency estimators was conducted to characterize their 
effectiveness as they relate to beacon receiver design. 
I. INTRODUCTION 
As radio communication links are driven to higher 
frequencies (e.g. Ka-band and above) by both spectrum 
congestion and the appeal of higher data rates, the propagation 
statistics at these frequencies must be known in order to design 
efficient communication links. Propagation studies are often 
conducted using beacon receivers, in which a ground terminal 
monitors the power received from a continuous-wave (CW) 
beacon on a geostationary satellite. However, due to system 
electronics and Doppler effects, the received signal will possess 
some nominal drift in frequency which needs to be accurately 
determined before a power measurement can be made.  Several 
techniques are available to system designers to track frequency 
changes in the measured signal, e.g., phase locked loops and 
multi-sampled FFTs [1].  Herein, we focus on frequency 
estimation approaches as applied to the GRC-developed beacon 
receiver and compare the relative performance of each. 
II.  FREQUENCY ESTIMATORS  
A. Background 
The Fast Fourier Transform (FFT) may be easily applied to 
measure the received frequency of a beacon signal by finding 
the peak of the frequency spectrum. However, the resolution of 
the FFT is defined by fs/N, where fs is the sampling frequency 
of the signal and N is the number of acquired points. Thus, 
while the actual frequency of the signal may vary continuously, 
the discrete points of the FFT frequency spectrum are limited to 
integer multiples of fs/N (bins). The resolution of the frequency 
measurement is therefore restricted by the extent to which fs/N 
can be minimized. This requires either increasing the number 
of points collected and/or decreasing the sampling frequency, 
both of which increase the acquisition time and reduce the rate 
at which the propagation measurements can be recorded.  
Furthermore, the peak search alone overlooks additional 
information that is present in the FFT; it is only when the 
frequency of a signal falls exactly into a bin frequency that the 
bin will contain the total power of the signal. In all other cases, 
the power will leak into nearby bins, to an extent governed by 
the distance of the signal frequency from the nearest bin 
frequency [2]. When calculating the power of the signal from 
the peak bin, this introduces a scalloping effect, as shown in 
Fig. 1, where the power is minimized when the frequency is 
between bins and maximized when it is near a bin frequency. 
This effect may be mitigated by summing the peak with several 
nearby bins, but this does not fully eliminate the scalloping and 
will effectively decrease the dynamic range of the beacon 
receiver. 
B. Algorithms 
A variety of algorithms exist that take advantage of the 
power in nearby bins to interpolate the frequency when it falls 
between the bins of the FFT. To characterize the performance 
of these frequency estimators as pertaining to beacon receiver 
design, a selection of algorithms were applied to a simulated 
beacon signal. The frequency estimators considered alongside 
the FFT peak search were the Buneman [3], Quinn-Fernandes 
[4], Quinn’s First Method [5], Jacobsen’s Algorithm [6], and 
MacLeod’s Algorithm [7]. Also considered was a modified 
version of the Quinn-Fernandes method (coined Quinn-
Fernandes-Nessel) that was primed with a priori information – 
namely, a window in which the frequency is known or 
expected to appear, giving the algorithm a search space smaller 
https://ntrs.nasa.gov/search.jsp?R=20140017299 2019-08-31T15:05:05+00:00Z
 
Fig. 2. The frequency estimate of each algorithm (top) and the 
corresponding power (bottom) for a SNR of -10dB. 
10 20 30 40 50 60 70 80 90 100
454.95
454.96
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Distance Between Bins [%]
Fr
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y 
[H
z
]
Frequency Estimation
 fs = 4550; N = 65536; fs/N = 0.069427; SNR = -10
 f0 = [454.9583 .... 455.0278]
 
 
10 20 30 40 50 60 70 80 90 100
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Distance Between Bins [%]
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IQ Receiver
 
 
FFT Peak
Quinn-Fernandes
Quinn-Fernandes-Nessel
Quinn
Jacobsen
Macleod
Buneman
Actual Frequency
FFT Peak
Quinn-Fernandes
Quinn-Fernandes-Nessel
Quinn
Jacobsen
Macleod
Buneman
 
Fig. 3. The RMS error of all six algorithms as the SNR of the input is 
varied from -30dB to +10dB. 
-30 -25 -20 -15 -10 -5 0 5 10
10-2
10-1
100
SNR [dB]
RM
S 
(D
ev
ia
tio
n
 
fro
m
 
Ac
tu
al
 
Fr
eq
u
en
cy
) [H
z]
RMS (Deviation from Actual Frequency) vs. SNR
 fs = 4550; N = 65536; fs/N = 0.069427
 f0 = [454.9583 .... 455.0278]
 
 
FFT Peak
Quinn-Fernandes
Quinn-Fernandes-Nessel
Quinn
Jacobsen
Macleod
Buneman
than the entire spectrum of the FFT. Such a modification may 
be made to any of the algorithms to improve calculation time 
and performance, but may not always be viable if the expected 
frequency is not well known. 
III. ANALYSIS 
To apply the estimators, a noisy sinusoidal beacon signal 
was simulated and swept from one bin frequency to the next. 
At each point, the frequency was estimated with each 
algorithm. The simulation was run with fs = 4550Hz, N=216, 
and f0 = 454.96 to 455.03 Hz, where fs is the sampling 
frequency, N is the number of points, and f0 is the input 
frequency. The signal-to-noise ratio (SNR) was also varied to 
gauge performance in low-power conditions. 
At a high SNR (e.g. +10dB), all six estimation algorithms 
were able to precisely track the frequency of the input signal as 
it swept from one bin frequency to the next with no discernable 
difference between them. The error of the FFT peak search, 
however, was very evident when compared to the estimators. 
Even with a high SNR, the FFT remains limited in resolution 
and can only return either the first bin frequency or the second. 
The scalloping effect is clear in the power measured from the 
FFT results and non-existent in the power from the estimators. 
At a moderate SNR of -10dB, as shown in Fig. 2, the noise 
is beginning to show in all of the estimates but the algorithms 
remain significantly more accurate than the FFT. It also 
becomes apparent that the Buneman algorithm is noisier than 
the other five estimators when nearer to the bin frequencies. 
For a low SNR (e.g. -20dB), the noise is significant, but the 
estimation algorithms continue to offer a much better 
approximation than the FFT alone. The Buneman remains 
noisier than the other algorithms, primarily near to the bin 
frequencies, but is still functional. As the noise grows, the FFT 
begins to oscillate between bins as the two neighboring peaks 
are very similar in magnitude and the noise is significant 
enough to obscure the true maximum. 
To quantify the performance of each algorithm in more 
detail, the root-mean-square (RMS) error of each algorithm, 
with respect to the actual input frequency, was calculated as the 
SNR was varied from -30 to +10 dB. The results are presented 
in Fig. 3. All six estimators exhibited an exponential increase in 
RMS error as the SNR decreased. At approximately -24dB 
SNR, the noise at any point in the spectrum may exceed the 
peak of the FFT, and most of the methods therefore become 
unable to track the frequency. Quinn-Fernandes-Nessel 
manages to survive below this point due to the windowing. 
The calculation time of each estimator was also considered. 
The Buneman was by far the fastest of all algorithms tested, 
taking an average of 6 ms with 216 points. Quinn-Fernandes 
was by far the slowest, taking an average 23 ms, although the 
windowing added to Quinn-Fernandes-Nessel did improve 
calculation time by 1 ms. The rest of the methods were roughly 
equivalent, taking an average of 7.5 ms. These figures should 
be taken in relation to one another, as they will vary with 
hardware and implementation. 
IV. CONCLUSIONS 
Each of the considered estimators were shown to calculate 
frequency to within ±1 Hz given an SNR above -24 dB (with 
fs=4550; N=216), and to run within 23 ms as implemented. 
However, the Buneman was observed to increase in noise near 
the bin frequencies relative to the other algorithms. Although it 
remains better than an FFT peak search, it did introduce 
notable measurement error when implemented in a beacon 
receiver undergoing laboratory test. The current generation 
beacon receivers being developed at NASA’s Glenn Research 
Center have thus implemented a selectively windowed Quinn-
Fernandes-Nessel approach that applies the windowing criteria 
when the power level drops below a set threshold. Similar 
modifications could be made to any of the other methods if 
operation at lower SNR is needed. 
REFERENCES 
[1] W. R. Sylvester, "Theory, Design and Implementation of a Digital 
Receiver for the ACTS Beacons," M.S. Thesis, Dept. of ECE, Virginia 
Polytechnic Institute and State University, Blacksburg, VA, 1992. 
[2] F. J. Harris, "On the Use of Windows for Harmonic Analysis with the 
Discrete Fourier Transform", Proc. IEEE, vol. 66, pp. 51-83, Jan. 1978. 
[3] O. Buneman, “A compact non-iterative poisson-solver,” SUIPR Report 
294, Stanford University l969. 
[4] B.G. Quinn & J.M. Fernandes, “A Fast Technique for the Estimation of 
Frequency”, Biometrika, vol. 78, pp. 489–497, Sept. 1991. 
[5] B. G. Quinn, "Estimating Frequency by Interpolation…," IEEE Trans. 
Sig. Proc., vol. 42, no. 5, pp. 1264-1268, May 1994. 
[6] E. A. Jacobsen, "On Local Interpolation of DFT Outputs," [Online]. 
Available: http://www.ericjacobsen.org/FTinterp.pdf 
[7] M.D. Macleod, "Fast Nearly ML Estimation of the Parameters of Real or 
Complex Single Tones or Resolved Multiple Tones," IEEE Trans. Sig. 
Proc., vol. 46, no. 1, pp. 141-148, Jan. 1998 
 


Frequency Estimator Performance for a Software-Based Beacon Receiver

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