Waveform Modulations and Techniques
2.4 NONLINEAR FREQUENCY MODULATED WAVEFORMS
2.4.3 Design Example
A process for creating an NLFM waveform is demonstrated using the inverse relationship in equation (2.86) and the parametric relationship in equation (2.88). Consider a waveform with a constant time-domain envelope and a −40 dB peak sidelobe requirement. The waveform is expected to sweep its instantaneous frequency from−β/2 toβ/2 during the time interval−τ/2≤t ≤τ/2 andβτ 1. The specific bandwidth and pulse length are undefined at the moment and are left as variables.
A Taylor weighting is chosen to meet the sidelobe requirement and is defined as
WT aylor()=G
1+2
¯n−1
m=1
Fmcos
m2π 2πβ
−πβ ≤≤πβ (2.90)
The coefficients Fmare
Fm =
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎩
(−1)m+1n=1¯n−1
1− m2
SD2+(n−0.5)2
2¯n−1
n=1n=m
1− m2
n2
, m =1,2, . . . , (¯n−1)
0, m ≥ ¯n
(2.91)
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2.4 Nonlinear Frequency Modulated Waveforms 53
where
D= 1
π cosh−110−P S R/20 (2.92)
S= ¯n2
D2+(¯n−0.5)2 (2.93)
and
G= 1
1+2
¯n−1
m=1Fm
(2.94)
The exact shape of the Taylor weighting is a function of the desired peak sidelobe ratio (PSR) and a parameter ¯n. The Taylor weighting in equation (2.90) is centered at baseband and extends over the frequency range−πβ ≤ ≤ πβ. The square of the spectrum magnitude in equation (2.88) is set equal to equation (2.90) or
|X()|2=WT aylor() (2.95)
The requirement for a constant time-domain envelope is satisfied by setting the envelope equal to 1 or
a2(t)=1 (2.96)
Integrating equation (2.88) with respect toyields θ()= k1
2πG
+2 ¯n−1 m=1
βFm m sin
m β
+k2 −πβ≤≤πβ (2.97) The waveform’s group delay is
tgd = −θ()= −k1 2πG
+2
¯n−1 m=1
βFm m sin
m β
−k2 −πβ≤≤πβ (2.98) The constants k1 and k2 are obtained by evaluating the group delay at the boundary conditions. Evaluating equation (2.98) at tgd = −τ
2when= −πβ, and at tgd = τ 2when =πβ, yields
k2=0 (2.99)
and
k1 = − τ
Gβ (2.100)
Making the substitutions for k1and k2into equation (2.98), tgd = 1
2π τ β
+2
¯n−1 m=1
βFm
m sin m
β
−πβ≤≤πβ (2.101)
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54 C H A P T E R 2 Advanced Pulse Compression Waveform Modulations FIGURE 2-17 The
waveform’s nonlinear frequency response will generate a Taylor weighted spectrum.
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4 0.5
time (normalized by τ)
frequency(normalized by
β)
At this point, we want to implement the graphical technique proposed by Cook [26].
First, the group delay in equation (2.101) is evaluated at equally spaced points in . Next, a numerical interpolation is performed on the samples to obtain equally spaced points in time (i.e., group delay) that map to specific values of the independent frequency variable,. The time samples are then multiplied by –1 (flipping them about the group delay axis) and circularly rotated left to right (e.g., a clockwise rotation of a row vector containing the samples) creating new function with time as the new independent variable and instantaneous frequency as the new dependent variable. Figure 2-17 contains a plot of the instantaneous frequency obtained from equation (2.101) after applying a cubic interpolation and the inversion process.
A model for the instantaneous frequency [3] is proposed:
φ(t)=2πβ t
τ +2 M m=1
dksin
2πmt τ
(2.102)
Equation (2.102) asserts that the instantaneous frequency may be modeled as a linear term plus a sum of harmonically related and weighted sine functions. To obtain the coefficients dk, the linear component is first subtracted from both sides of the equation
φ(t) 2πβ − t
τ =2 M m=1
dksin
2πmt τ
(2.103) The right side of equation (2.103) represents the first M terms of the Fourier series of an odd signal. Figure 2-18 contains a plot of the instantaneous frequency with the linear component removed. The response is odd and may be interpreted as one period of a periodic function. Harmonic analysis is applied to the signal in Figure 2-18 to obtain the coefficients dk.
A periodic signal x(t)may be expressed as a sum of weighted of sines and cosines where
x(t)=a0+2 ∞ m=1
[bkcos(m0t)+dksin(m0t)] (2.104)
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2.4 Nonlinear Frequency Modulated Waveforms 55
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
−0.5
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4 0.5
time (normalized by τ) instantaneous frequency (normalized by β)
FIGURE 2-18 The instantaneous frequency after removing the linear component is viewed as one period of a periodic function and is modeled using Fourier series coefficients.
where0 =2π/τ. For odd signals x(t)=2
∞ m=1
dksin(m0t) (2.105)
The Fourier series coefficients dkare obtained by evaluating the integral dk =
τ
x(t)sin(m0t)dt (2.106) whereτ is one period of the waveform or, in this case, the pulse width. The coefficients are obtained numerically by replacing the integral in (2.106) with a summation. Table 2-2
TABLE 2-2 Coefficients Used to Generate an NLFM Waveform
Index Coefficient Value Index Coefficient Value
1 −0.0570894319 16 0.0003386501
2 0.0198005915 17 −0.0002874351
3 −0.0102418371 18 0.0002451937
4 0.0062655130 19 −0.0002100998
5 −0.0042068327 20 0.0001807551
6 0.0029960800 21 −0.0001560755
7 −0.0022222723 22 0.0001352110
8 0.0016980153 23 −0.0001174887
9 −0.0013271886 24 0.0001023710
10 0.0010560642 25 −0.0000894246
11 −0.0008525769 26 0.0000782981
12 0.0006965787 27 −0.0000687043
13 −0.0005748721 28 0.0000604069
14 0.0004785116 29 −0.0000532106
15 −0.0004012574 30 0.0000469533
Melvin-5220033 book ISBN : 9781891121531 September 14, 2012 17:23 56
56 C H A P T E R 2 Advanced Pulse Compression Waveform Modulations FIGURE 2-19 The
NLFM waveform exhibits a shaped spectrum
resembling a Taylor weighting. The amplitude ripple is associated with the finite time and frequency extent constraints placed on the waveform.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
frequency (normalized by β)
amplitude
contains the first 30 coefficients. No attempt is made to determine the number of coefficients or the numerical precision needed to achieve a specified level of performance.
The time-domain phase function is obtained by integrating the expression in equa- tion (2.102)
φ(t)= πβ τ t2−
M m=1
βτ2dk m cos
2πmt τ
(2.107) The resultant baseband, complex NLFM waveform is
x(t)=exp(jφ(t)) (2.108)
The waveform’s spectrum is plotted in Figure 2-19. The envelope of squared spectrum resembles a Taylor weighting. The ripples in the spectrum are a result of the waveform’s finite extent imposed in both domains.
Consider an LFM waveform with a 500 MHz swept bandwidth and a 1 μsec pulse length. Applying the matched filter in either the time or frequency domain generates the compressed response shown in Figure 2-20. The peak sidelobes are approximately FIGURE 2-20 The
NLFM waveform achieves a range compressed response with peak sidelobes slightly above−40dB.
−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02
−60
−50
−40
−30
−20
−10 0
sec)
dB
μ time (
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2.4 Nonlinear Frequency Modulated Waveforms 57
−37 dB; the goal was−40 dB. The peak sidelobe levels are close to the designed-to level. The waveform’s time–bandwidth product does have a direct bearing on sidelobe performance. In general, the higher the time-bandwidth product the closer one gets to the design-to level. Time–bandwidth products greater than 100 are generally required.
For an LFM waveform with 500 MHz swept bandwidth, the time resolution, measured at the –4 dB width, is 0.002μsec. A close examination of the response in Figure 2-20 reveals a−4 dB width of 0.0029μsec. The modulation has degraded the resolution by a factor of 1.45 over that achieved by an LFM swept over the same bandwidth. Of course, a similar degradation in resolution occurs when an amplitude taper is applied to an LFM waveform.