A correction to impulse invariance

• Published in: IEEE signal processing letters Vol. 7; no. 10; pp. 273 - 275
• Main Author: Jackson, L.B.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-10-2000; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Band pass filters; Bandpass filters; Digital signal processing; Discontinuity; Filtering theory; Frequency response; H infinity control; Impulse response; Impulses; Integral equations; Invariance; Laplace equations; Mathematical analysis; Sampling methods; Signal processing; Signal sampling; Time factors
• ISSN: 1070-9908; 1558-2361
• DOI: 10.1109/97.870677

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.