EE 448 Final Review

EE 448 Final Review - a) Analog signal gn g a t| tnT w/...

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a) Analog signal g n g a t | t nT w/ highest freq m . Freq-domain exp DTFT G e jw CTFT G a j . G e jw 1 T k G a j w T j 2 k T . 2 T 2 m or T m to avoid aliasing. b) Bilinear transform and an expression (w/ analog freq to digital freq w ). s 2 T 1 z 1 1 z 1 ,  2 T tan w 2 . T 2 -   tan w 2 c) H z is transfer function, stable causal system. All poles of H z must be inside the unit circle is true b/c causality implies ROC extends from outermost pole. Stability implies ROC include unit circle. Therefore, outermost pole should be inside unit circle. Hence, all poles should be inside unit circle. d) Consider LTI system, H z 1 1 3 z 1 1 6 z 2 z 3 . Block diagrams of direct and transposed direct forms. Direct form: Transposed direct form: 2) IIR filter H z 1 z 1 4 2 z 1 . H e jw | w 0 1 and H e jw | w 0 . Squared mag response | H e jw | 2 | H z | z e jw | 2 | 1 e jw 4 2 e jw | 2 | 1 cos w j sin w 4 2cos w 2 j sin | 2 1 cos w 3 j sin w 10 8cos w 2 1 cos w 10 8cos w . 3-dB cutoff freq w c  | H e jw c | 2 1 2 1 cos w 10 8cos w  cos w c 4 5  w c cos 1 4 5 0.6435 . This is a Lowpass filter b/c take the derivative of | H e jw | 2 w/ respect to w  d dw | H e jw | 2 d dw 1 cos w 10 8cos w 18sin w 10 8cos w 2 , range of 0 w , d dw | H e jw | 2 is non-positive, meaning | H e jw | 2 monotonically decreasing function of w . 3) Transfer function stable system is H z z 2/3  z 2 2 z j /2  z j /2 . Zero-pole | z | 0.5 .P lo t  System is not minimum-phase, since there is a zero outside unit circle . Minimum-phase can be obtained by multiplying H z by an allpass filter that reflects the zero inside the unit circle. H min z 2/3  z 2 2 z j /2  z j /2 2 z 1 z 2 z 2/3  2 z 1 2 z j /2  z j /2 z 2/3  z 1/2 z j /2  z j /2 . System is not maximum-phase, since
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EE 448 Final Review - a) Analog signal gn g a t| tnT w/...

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