Frequency Response of Rational Systems Quote of the Day If knowledge can create problems, it is not through ignorance that we can solve them. Isaac Asimov Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, © Prentice Hall Inc.

Copyright (C) 2005 EEE413 Digital Signal Processing 2 Frequency Response of Rational System Functions DTFT of a stable and LTI rational system function Magnitude Response Magnitude Squared

Copyright (C) 2005 EEE413 Digital Signal Processing 3 Log Magnitude Response Log Magnitude in decibels (dB) Example: –|H(e j )|=0.001 translates into –60dB gain or 60dB attenuation –|H(e j )|=1 translates into 0dB gain –|H(e j )|=0.5 translates into -6dB gain Output of system

Copyright (C) 2005 EEE413 Digital Signal Processing 4 Phase Response Phase response of a rational system function Corresponding group delay –Here arg[.] represents the continuous (unwrapped) phase –Work it out to get

Copyright (C) 2005 EEE413 Digital Signal Processing 5 Unwrapped (Continuous) Phase Phase is ambiguousWhen calculating the arctan(.) function on a computer –Values between - and + –Denoted in the book as ARG(.) –Any multiple of 2 would give the same result –Here r() is an integer for any given value of  Group delay is the derivative of the unwrapped phase

Copyright (C) 2005 EEE413 Digital Signal Processing 6 Frequency Response of a Single Zero or Pole Let’s analyze the effect of a single term If we represent it in dB The phase term is written as And the group delay obtained by differentiating the phase Maximum and minimum value of magnitude