Page 213 - Fundamentals of Radar Signal Processing

(3.44)

Applying a trigonometric identity for sin(θ – ϕ) and equating terms in sinθ and
cosθ on both sides of Eq. (3.44) leads to the following solution for a and a :
21
22

(3.45)

Using Eq. (3.45) in Eq. (3.43) gives the final transformation required

(3.46)

Once the I/Q errors ε, ϕ, γ, and κ are determined, Eq. (3.46) can be used to
compute a new value Q′ for the quadrature channel sample for each measured I-
Q sample pair. The difficulty, of course, is in actually determining the errors;
the correction is then easy. The errors are generally estimated by injecting a
known pilot signal, usually a pure sinusoid, into the receiver and observing the

outputs. Details for one specific technique to estimate gain and phase errors are
given in Churchill et al. (1981); that paper also derives limits to mismatch
correction (and thus to image suppression) caused by noise, which introduces
errors into the estimates of ε and ϕ.
A second method for eliminating I/Q error is based on the idea of
transmitting multiple pulses, stepping the starting phase of each pulse through a
series of evenly spaced values, and then integrating the measured returns. To see

how this technique works, suppose the input signal in Fig. 3.18 is changed to A
sin [Ωt + θ (t) + k(2π/N)] for some fixed integer N and variable integer k; i.e.,
the pulse is one of a series of N pulses, where the initial phase is increased by
2π/N radians on each successive pulse. The extra phase shift propagates to the
output signals

(3.47)

for k = 0, 1, …, N – 1. The development leading to Eq. (3.36) can be repeated
to obtain the complex signal for this case, which is (still suppressing the t