Page 212 - Fundamentals of Radar Signal Processing

only, that is, ε = 0 and ϕ 1. In this case Eq. (3.37) reduces to

(3.40)

where the second line is obtained using the small angle approximation cos ϕ ≈ 1

2
– ϕ /2. Note that ϕ is in radians. The relative power of the image component in
decibels is then

(3.41)

As an example, a phase mismatch of 1° gives an image component
approximately 41.2 dB below the desired response.

3.4.2 Correcting I/Q Errors
As shown in Fig. 3.18, the I and Q signals in the presence of mismatch can be
modeled as

(3.42)

where the dependence on time t continues to be suppressed to simplify the
notation. The desired in-phase signal I is Acosθ, and in the quadrature channel is
Asinθ. Is it possible to recover the desired outputs from the available
measurements of Eq. (3.42)?

Consider forming a new I′ and Q′ as a linear combination of the measured I
and Q. Specifically, require that I′ = Acosθ and Q′ = Asinθ. Although it is
straightforward to solve the general problem, it is obvious that the DC offsets
should simply be subtracted, and then a linear combination of the zero-offset
data formed

(3.43)

By inspection, a = 1 and a = 0. The remaining equation is
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