Event Detection: Qrs-Complexes in Ecg Signals - microLab

BioMedSigProcAnawhich delays each sample of the input sequences by n 0 samples.The transfer function of this system is H(z) = z −n0 , and itsfrequency response becomes H(ˆω) = e −jˆωn0 .Next consider an M-point moving-average filter. Its transferfunction isH(z) = 1 M(1 + z −1 + . . . + z −(M−1)) = 1 1 − z −MM 1 − z −1 .Its frequency response can be expressed asH(ˆω) ˆ= H(z = e jˆω ) = 1 M1 − e −jˆωM1 − e −jˆω= 1 e ( −jˆωM/2 e jˆωM/2 − e −jˆωM/2)M e ( −jˆω/2 e jˆω/2 − e −jˆω/2)( ) 1 sin(ˆωM/2)=e −jˆω(M−1)/2 ,M sin(ˆω/2)} {{ }Dirichlet functionwhere we have used Euler’s formula, and where the indicatedDirichlet function is purely real. We see then that H(ˆω) is theproduct of two functions, the Dirichlet function and e −jˆω(M−1)/2 ,with the latter indicating a shift of (M −1)/2 samples. Clearly,if the filter length M is an odd integer, (M − 1)/2 becomes aninteger, and the meaning of delaying by an integer number ofsamples is clear. In M is, however, an even integer, (M − 1)/2is not an integer, and the delay needs a special interpretation;[MSY98, pp. 192–194] gives such an interpretation by applyinga discrete-time moving average filter to process continuous-timesignals.6 EventDet 35 2014