32 H. Abbas et al.
QREs are defined and what they compute. Readers familiar with QREs will
notice that, when writing the QRE expressions, we occasionally sacrifice strict
syntactic correctness for the sake of presentation clarity.
5 QRE Implementation of Peak Detectors
We now describe the QREs that implement peak detectors WPM and WPB
of Sect.3.2. It is emphasized that even complicated procedures such as these
two algorithms can be described in a declarative fashion using QREs, without
resorting to a programming language or explicitly storing state, etc.
Fig. 3.QREpeakWPM5.1 QRE for WPM
A numerical implementation of a CWT returns a discrete set of coefficients. Let
s 1 <s 2 < ... < snbe the analysis scales and lett 1 ,t 2 ,...be the signal sampling
times. Recall that a QRE views its input as a stream of incoming data items. A
data item for WPM isd=(si,tj,|Wx(si,tj)|)∈D:= (R+)^3 .Weused.sto refer
to the first component ofd,andd.|Wx(s, t)|to refers to its last component. The
input streamw∈D∗is defined by the values from the spectrogram organized
in a column-by-column fashion starting from the highest scale:
w=(sn,t 1 ,|Wx(sn,t 1 )|),...,(s 1 ,t 1 ,|Wx(s 1 ,t 1 )|)
︸ ︷︷ ︸
wt 1...
...(sn,tm,|Wx(sn,tm)|),...,(s 1 ,tm,|Wx(s 1 ,tm)|)
︸ ︷︷ ︸
wtm