A correction to impulse invariance

The classic equations for discrete time filter design by impulse invariance are not correct, in general. In particular, if the impulse response of the causal continuous-time filter is discontinuous at t=0, then the initial sample h[0] of the discrete-time impulse response should equal one-half the value of the discontinuity, not the full value as classically given. Otherwise, the resulting frequency response does not equal the sum of shifted versions of the original response as expected. This effect will typically be seen only in low order filters such as first order lowpass filters or second order bandpass filters.