IEEE - Aerospace and Electronic Systems - January 2020 - 42

Radar Doppler Frequency Measurements-Accuracy Versus SNR in Practical Processors
Table 1.

Normalized Frequency error STD versus SNR
SNR [dB]

-7

-5





sf 

5

M Tr s f; ðM ¼ 32Þ

1/20 ¼ 0.05

4

y1 . . . ym . . . yM T :

(12)

Finally, we define the (M x 2) matrix H that will allow
us to describe the M equations in (8) as the M-dimensional
measurements vector in (11)
H¼

1
0

1
Tr

1
2Tr

...
...

1
mTr

...
...

1
ðM À 1ÞTr

!T
: (13)

The resulted matrix equation is
f ¼ Hx þ y :

(14)

The well-known [9] LS solution of the parameters
vector is
À
ÁÀ1
^LS ¼ HT H HT f:
x

(15)

Well known also is the covariance of the errors of the
estimated parameters
Â
Ã
À
ÁÀ1
^TLS ¼ s 2f HT H :
^LS x
P¼E x

(16)

Inserting (13) in (16) yields
À
ÁÀ1
12
¼ s 2f 4
P ¼ s 2f HT H
À M2
M
"
# (17)
1
À1
M
ð
M
À
1
Þ
ð
2M
À
1
Þ
6
2 M ðM À 1Þ=Tr
:
À1
M=Tr2
2 M ðM À 1Þ=Tr
The error variance of the angular Doppler frequency v
is P2,2, hence
s 2v ¼

12
2

ðMT r Þ ðM À 1=M Þ

s 2f :

(18)

We will convert (18) to Doppler frequency
f ¼ v=ð2pÞ, perform square root and normalize the frequency to create a dimensionless variable by multiplying
the frequency by the CPI, i.e., by MT r ,
sf
MT r s f ¼
p

42

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
3
:
M À 1=M

2



5


1.2

1/25 ¼ 0.04 1/51 ¼ 0.0196 1/85 ¼ 0.0118

We will also arrange the phase measurements errors in
an errors vector
y ¼ ½ y1

0

(19)

The important expression in (19) says that the SNR
dependence of the normalized frequency error MT r s f is
shaped like the SNR dependence of the phase error, multiplied by an expression that is a function of M. Using the
data from Figure 14, we can construct a short table
expressing that dependence.
The two top rows of Table 1 were taken from
Figure 14. From the phase error STD (middle row), the
dimensionless normalized frequency error STD (bottom
row) is calculated using (19). Recall that the normalized
frequency resolution of the conventional FFT processor,
at all SNR values, is MT r Df ¼ 1. Therefore, the values
at the bottom row indicate the ratio s f =Df by which the
frequency error, obtained through phase measurement of
each pulse in the CPI, is smaller than the frequency resolution obtain through the FFT processing of a pulse train.
The bottom row therefore says that over the span
À7 SNR 5 dB, the above approach of measuring
Doppler frequency shift is better than the "ambiguity
function" approach by factors of 20 to 85, respectively.

FIELD TRIAL IN A HIGH-SNR SCENE
In order to create a high-SNR scene, signal transmitted by
a stationary radar was recorded by a receiver on board an
airborne target. The flight direction was slightly off cross
range and the radial distance was about 6 km. The transmitted signal was a coherent train of binary-coded pulses
using a 48 element code, with PRF ¼ 1300 s-1 and tp ¼ 96
ms. The receiver's oscillator was, of course, different from
the transmitter's oscillator and a small constant frequency
difference was added to the Doppler frequency. Phase
unwrapping and outliers' removal were involved in the
processing of the phase measurement. Figure 15 displays
the recorded unwrapped phase of the airborne received
pulses along a flight section of 15 s. Four linear sections
are marked, from which Doppler (þ oscillator bias) frequencies could be calculated accurately. The accumulated
phase of 5000 radians corresponds to a Doppler cycle
count of 800 cycles. At the signal's wavelength of
0.5769 m, this accumulation indicates a range change of
461 m. Therefore, during those 15 s, the average range
rate was about 30 m/s.
A brief historical note: The concept of Doppler count is
associated with the TRANSIT navigation satellite system

IEEE A&E SYSTEMS MAGAZINE

JANUARY 2020
IEEE - Aerospace and Electronic Systems - January 2020

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