Waveform Modulations and Techniques
2.1 INTRODUCTION
2.1.3 Notation
2.1.3.1 Common Variables
Variables used throughout the chapter are as follows:
t time
j √
−1
π 3.14159265. . .
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
Melvin-5220033 book ISBN : 9781891121531 September 14, 2012 17:23 22
22 C H A P T E R 2 Advanced Pulse Compression Waveform Modulations 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,t x second oscillator signal used to synthesize transmit signal
tr cv time delay on receive, referenced to the center of the range window L O1,r cv first oscillator signal applied on receive
L O2,r cv 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 δf 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
Melvin-5220033 book ISBN : 9781891121531 September 14, 2012 17:23 23
2.1 Introduction 23
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 δfD 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
Melvin-5220033 book ISBN : 9781891121531 September 14, 2012 17:23 24
24 C H A P T E R 2 Advanced Pulse Compression Waveform Modulations 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
Melvin-5220033 book ISBN : 9781891121531 September 14, 2012 17:23 25
2.1 Introduction 25
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