Waveform Modulations and Techniques
2.1.2 Key Points
Important concepts developed throughout the chapter are summarized as follows:
• Stretch processing is a technique applied to a wideband LFM waveform to reduce the required processing bandwidth while maintaining the range resolution afforded by the transmit bandwidth.
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• Stretch processing is applied in many high-resolution systems including synthetic aper-ture radars (SARs).
• A stepped chirp waveform consists of several LFM pulses that are shifted in frequency by a constant offset. The waveform provides a reduction in instantaneous bandwidth on both transmit and receive.
• The stepped chirp waveform is sampled at the single pulse bandwidth and is recon-structed in the signal processor to achieve a range resolution commensurate with the composite waveform bandwidth.
• NLFM waveforms employ frequency modulation to shape the spectrum and thereby reduce the range sidelobes. These waveforms do not require an amplitude taper as commonly used with an LFM waveform and thus avoid the associated SNR loss.
• NLFM waveforms are less Doppler tolerant than their LFM counterparts.
• Stepped frequency waveforms achieve a low instantaneous bandwidth on both transmit and receive. These narrowband signals are used to lessen hardware and processor requirements. Fine range resolution is achieved by stepping the frequency of each pulse to create a synthetic wideband waveform.
• An SF waveform is transmitted over a relatively long time interval, thus limiting its application; however, SF waveforms are used in some wideband instrumentation radar systems.
• Quadriphase codes exhibit low spectral sidelobes and thus help to mitigate EMI.
• Quadriphase codes are created by transforming a biphase code into a four-phase code and applying a subpulse with a half-cosine shape. The subpulses overlap creating a nearly constant envelope.
• Mismatched filters may be designed and applied to phase codes to reduce the integrated and peak sidelobe responses with minimal loss in SNR. Mismatched filters may also be used to intentionally shape the sidelobe response.
• The performance of mismatched filters is sensitive to uncompensated Doppler shifts.
2.1.3 Notation
2.1.3.1 Common Variables
Variables used throughout the chapter are as follows:
t time
j √
π 3.14159265 . . .−1
f0 transmit center frequency
frequency in radians per second c speed of light
τ pulse length
td time delay associated with a point target Fs analog-to-digital converter sampling rate ω frequency in radians per sample
Ts analog-to-digital converter sampling period fd Doppler shift
f frequency step size
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T pulse repetition interval tgd group delay
δR range resolution
R range window extent
2.1.3.2 Stretch Processing
Variables associated with stretch processing are as follows:
x(t) transmit waveform
f1 center frequency of first oscillator β LFM waveform’s swept bandwidth θ1 phase of the first local oscillator
f2 center frequency of the second oscillator θ2 phase of the second local oscillator
L O2,tx second oscillator signal used to synthesize transmit signal
tr cv time delay on receive, referenced to the center of the range window L O1,rcv first oscillator signal applied on receive
L O2,rcv second oscillator signal applied on receive xr(t) received signal
y(t) received signal after mixer stages
td time delay relative to the center of the range window ϕ residual video phase
θ composite phase after deramp operation fb beat frequency
Y() spectrum of the deramped signal YM() spectrum magnitude
d d = −τ (1 − |td|/τ)
peak location of sinc’s main lobe peak
null location of sinc’s first null
δ difference betweenpeakandnullin radians per second δ f difference betweenpeakandnullin hertz
δtd time-delay resolution SNRloss signal-to-noise ratio loss
trw time duration associated with a range window BF low-pass filter bandwidth
Y(ω) spectrum of the sampled signal y(n) sampled received signal
n sample index
N number of samples collected from a single point scatterer YM(ω) magnitude of the spectrum
r range delay relative to the center of the range window δω Rayleigh resolution in radians per sample
k discrete Fourier transform (DFT) bin index Y(k) discrete Fourier transform
M discrete Fourier transform size
Nnoi se number of samples containing thermal noise collected over the receive window
SNRβ signal-to-noise ratio using a filter with bandwidthβ
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SNRBF signal-to-noise ratio at the output of a filter with bandwidth BF SNRD F T signal-to-noise ratio at the output of the DFT
f frequency in hertz
fb beat frequency including Doppler shift
td time delay offset associated with range-Doppler coupling
2.1.3.3 Stepped Chirp Waveforms
Variables associated with stepped chirp waveforms are as follows:
β LFM intrapulse swept bandwidth
Nsc number of pulses comprising a stepped chirp waveform
n pulse index
xt x(t) transmit waveform β single-pulse bandwidth
βsc stepped chirp waveform’s composite bandwidth xr cv(t) received waveform
L Or cv(t) local oscillator signal applied to received waveform xB B(t) received waveform mixed to baseband
p new pulse index
xB B(m) sampled received waveform
m sample index
Tc time interval supporting the pulse width and receive window yn(m) samples collected from the n-th pulse
yn (m) interpolated signal associated with the n-th pulse
zn(m) output of the digital mixer operation associated with the n-th pulse φn phase correction applied to the n-th pulse
z(m) frequency shifted, phase corrected, and time-aligned stepped chirp waveform Xn(ω) DTFT of the n-th received pulse
X(ω) DTFT of a sampled baseband LFM waveform X∗(ω) spectrum of the matched filter
Yn(ω) spectrum of the n-th pulse having applied a matched filter on receive δ fD F T DFT bin size
k DFT bin index
K length of DFT
P an integer
2.1.3.4 Nonlinear Frequency Modulated Waveforms Variables associated with NFLM waveforms are as follows:
x(t) notional waveform
a(t) waveform’s time-domain amplitude response φ (t) waveform’s time-domain phase response X() waveform’s spectrum
|X ()| magnitude of the spectrum θ () spectrum phase
β bandwidth over which the waveform’s frequency is swept
W() cosine on a pedestal weighting function defined in the frequency domain h parameter associated with cosine on a pedestal tapers
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WTaylor() Taylor weighting function defined in the frequency domain Fm Taylor coefficients
m Taylor coefficient index
¯n n-bar used to define a Taylor weighting function PSR peak sidelobe ratio
a0 average term in Fourier series
bk Fourier series coefficient for even signals dk Fourier series coefficient for odd signals
0 fundamental frequency associated with a periodic signal
2.1.3.5 Stepped Frequency Waveforms Variables associated with SF waveforms as follows:
N number of pulses
R0 range to a stationary point target θ measured phase
n pulse index
θ phase difference between two pulses x(n) sample collected from the n-th pulse X(ω) DTFT of sampled returns
ωR0 frequency in radians per sample; corresponds to the target’s range
R range
δω Rayleigh resolution in radians per sample ωk k-th discrete frequency
Rk k-th discrete range X(k) discrete Fourier transform k DFT bin index
M size of DFT
Rgate location of range gate L physical length of a target RA ambiguous range
ˆx(n) samples containing a Doppler shift
rshift displacement in range due to a Doppler shift
¯rshift normalized range displacement due to a Doppler shift rspread spread in range due to a Doppler shift
¯rspread normalized range spread due to a Doppler shift v radial velocity
vˆ estimate of radial velocity
xcorrect correction factor applied to compensate for a Doppler shift 2.1.3.6 Quadriphase Codes
Variables associated with quadriphase codes are as follows:
cn biphase code indexed by n
qn quadriphase code generated from parent biphase code s a variable having a value of 1 or –1
N length of biphase code p(t) subpulse envelope
τc subpulse width of a biphase code
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z(t) complex signal formed by the quadriphase transformation a(t) envelope or magnitude of z(t)
φ (t) phase of z(t)
y(t) quadriphase code transmitted by a radar and centered at baseband P() spectrum of the half-cosine subpulse
m(t) autocorrelation of half-cosine subpulse QBR ratio of quadriphase to biphase peak sidelobe a peak sidelobe of biphase code
2.1.3.7 Mismatched Filters
Variables associated with mismatched filters are as follows:
ck elements of a biphase or polyphase code indexed by k k phase code element index
K length of the phase code
zm coefficients associated with an M length finite impulse response filter m filter coefficient index
M length of filter
yn output having applied a mismatched filter to a phase code n n-th output sample index
dn desired mismatch filtered response
en error signal or difference between desired response and actual response E sum of the squared error
y column vector containing the filtered response C matrix containing shifted copies of the phase code z column vector containing the mismatched filter H Hermitian operator
d column vector containing the desired response W weighting matrix
w an element of the weighting matrix
2.1.4 Acronyms
Acronyms used in this chapter are as follows:
ADC analog-to-digital converter BTQ biphase-to-quadriphase COHO coherent oscillator CW continuous wave
DFT discrete Fourier transform DTFT discrete time Fourier transform EMI electromagnetic interference ENOB effective number of bits FD frequency domain FFT fast Fourier transform
FMCW frequency modulated continuous wave FSK frequency shift keying
HRR high-resolution range IF intermediate frequency
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ISR integrated sidelobe ratio LFM linear frequency modulated LPG loss in processing gain LS least squares
MF matched filter
MISR minimum integrated sidelobe ratio MMF mismatched filter
MPS minimum peak sidelobe MSK minimum shift keying
NLFM nonlinear frequency modulated PC pulse compression
PRF pulse repetition frequency PRI pulse repetition interval PSP principle of stationary phase PSR peak sidelobe ratio
RF radio frequency SAR synthetic aperture radar SNR signal-to-noise ratio SF stepped frequency TD time domain
WLS weighted least squares