  We investigate the theory of affine groups in the context of designing radar
waveforms that obey the desired wideband ambiguity function (WAF). The WAF is
obtained by correlating the signal with its time-dilated, Doppler-shifted, and
delayed replicas. We consider the WAF definition as a coefficient function of
the unitary representation of the group $a\cdot x + b$. This is essentially an
algebraic problem applied to the radar waveform design. Prior works on this
subject largely analyzed narrow-band ambiguity functions. Here, we show that
when the underlying wideband signal of interest is a pulse or pulse train, a
tight frame can be built to design that waveform. Specifically, we design the
radar signals by minimizing the ratio of bounding constants of the frame in
order to obtain lower sidelobes in the WAF. This minimization is performed by
building a codebook based on difference sets in order to achieve the Welch
bound. We show that the tight frame so obtained is connected with the wavelet
transform that defines the WAF.
