Page 218 - Fundamentals of Radar Signal Processing

(3.53)

This asymmetric filter frequency response corresponds to a complex-valued

impulse response, giving rise to the complex output from the single real input.
While Eq. (3.53) states that the value of H(ω) around DC is unconstrained,
in fact it should be close to zero. The filter will then also suppress any DC
component in the signal (not sketched in Fig. 3.21) that may have been
introduced by nonideal mixing in the first analog frequency translation. Thus,
this digital I/Q architecture also makes it easier to suppress mixer bias terms.

This would not be possible if the spectrum had been translated to the lowest
possible IF frequency, namely β Hz, since there would then be no region of the
spectrum around DC that did not contain signal components of interest.
A particularly efficient design for realizing the filter H(z) as a pair of low-
order recursive filters is based on the mathematics of phase-splitting networks;
details are given in Rader (1984). However, the particular design of the filters
is not central to the architecture of the approach.

The final step is to translate the remaining spectral sideband, centered at
ω = π/2, to baseband and to reduce the sampling rate from 4β to the final
0
Nyquist rate β. This can be accomplished by multiplying the complex filter
output by the sequence exp(–jπn/2) = (–j)n and then simply discarding three of
every four samples. Because of the special form of the multipliers, the complex
multiplications could be implemented simply with sign changes and

interchanges of real and imaginary parts, rather than with actual complex
multiplications. This is a consequence of having selected the original sampling
rate to be 4β instead of 3β.
However, this multiplication is not shown in Fig. 3.20 because, in fact, it is
not necessary at all. The spectrum of the decimator output y[n] is related to the
spectrum of according to (Oppenheim and Schafer, 2010)

(3.54)

Equation (3.54) states that the decimation process causes the spectrum to
replicate at intervals of π/2 radians. Since the nonzero portion of the spectrum is
bandlimited to π/2 radians, these replications abut but do not alias; furthermore,
since the spectrum prior to decimation is centered at ω = π/2, one of the