Signal Analysis Research (SAR) Group - RNet - Ryerson University

Amplitude
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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2007 proceedings.
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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Normalized Frequency
Fig. 5. Third Order DPPT.
This condition is translated into the following requirement on
the range of coefficient aM:
|aM |≤
Π
M!τ M−1 , (21)
∆M when M = 1, we have |a1| ≤ ωs
2 which is the Nyquist
criterion.
The accuracy of the DPPT method depends on many factors
such as the level of noise present, the type of noise, the length
of the chirp signal and the chosen value of τ [10] [11]. The
best signal estimation is achieved when τ = N/M, where
N is the signal length and M is the order of the polynomial
phase of the chirp (M=2 for linear chirp, M=3 for parabolic
chirp).
The SNR (the jamming signal) at the spectral peek ω0
should be at least 14dB for a good detection. Also the number
of points when taking the Fourier transform will affect the
results when estimating the position of the spectral peak.
Increasing the order of the polynomial will also affect the
estimation error. For example, if we consider a third order
polynomial, any error which occurs in the coefficient a3 will
make it impossible to remove this coefficient completely from
the polynomial during phase unwrapping step. Therefore, the
estimation of the next coefficient a2 will suffer and have error
as well. Similarly, the new error in a2 will affect the precision
of a1 and a0.
The computational complexity of the DPPT is determined
based on the number of multiplications needed to synthesize
the chirp having a length of N. The DPPT process
involves the calculation of the ambiguity function and then
taking the Fourier transform of the ambiguity function. The
computational complexity of the ambiguity function calculation
is O(N) and the complexity of the fast Fourier trans-
form is O(N log 2 N). Therefore the total complexity is only
O(N log 2 N).
IV. NUMERICAL AND SIMULATION RESULTS
We used 128 chips per data bit for spreading the message
signal and we assumed a Gaussian channel. We considered
a constant amplitude linear and parabolic chirp as the interference
source. We first evaluated the bit error rate (BER)
resulting from the presence of linear chirp at different jamming
ratios between [0, 60] dB. We assumed SNR (with Gaussian
noise) to be -10 dB for each case. As seen in Figure 6, the bit
error rate increase as the jamming ratio increases. The spread
spectrum system is able to recover the data bits at low jamming
ratio of 10 dB, but as the ratio increases the system fails to
recover the correct data bits.
BER
10 0
10 −1
10 −2
10 −3
10
0 10 20 30 40 50 60
−4
JSR
Fig. 6. BER vs. JSR results for a self-excised SS system.
Next we used the proposed DPPT technique to synthesize
the jamming chirp in the previous spread spectrum system.
The chirp signal used was a linear chirp with normalized
instantaneous frequency varying from 0 to 0.5 Hz. We also
used 128 data chips and 100 data bits for a total of 12800
bits. Table I shows the correlation coefficient between the
reference linear chirp and the synthesized chirp using second
order DPPT. The first simulation was for -0.5 dB signal to
noise ratio (SNR) with signal to jamming ratio (SJR) ranging
from [6,-20] dB, and the second simulation for -5 dB signal
to noise ratio with jamming ratio in the range [6,-20] dB.
The DPPT showed good results since it was able to detect the
chirp under low chirp ratio (0.9879 for -2dB signal to jamming
ratio).
In the next simulation we used a parabolic chirp signal as
the interference source. Figure 8 shows the spectrogram of the
received signal r(t) with interference and noise added.
Table II shows the correlation coefficient between the reference
chirp and the synthesized chirp. A third order DPPT
was applied in the signal. The first simulation was for -0.5
dB signal to noise ratio with signal to jamming ratio changing
in the range [6,-20] dB. And the second simulation for -5 dB
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