Expressible Filtering
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For all subparts requiring symbolic expressions, you should enter your answers using Python syntax. Your answers may contain any of the following:
omega
represents \omegaOMEGA
represents \Omegasin(...)
andcos(...)
represent \sin(\cdot) and \cos(\cdot), respectivelysqrt(...)
represents the square root functione
,1j
, andpi
represent e, j, and \pi, respectivelyRe(...)
andIm(...)
represent {\rm Re}(\cdot) and {\rm Im}(\cdot), respectivelydelta(...)
represents either \delta(\cdot) or \delta[\cdot], depending on the contextu(...)
represents either u(\cdot) or u[\cdot], depending on the context
Consider an LTI system with a unit sample response h[\cdot] given by:
For each of the input signals x_i[\cdot] below, assume that the response of
the system to that input is given by y_i[\cdot], and determine if it is possible to
represent y_i[n] as a single pure sinusoid of the form A_i\cos(\Omega_i n +
\phi_i), where A_i and \Omega_i are positive and real, and \phi_i is
real (but not necessarily positive)?
If so, specify the appropriate values of A_i, \Omega_i, and \phi_i by
entering a single number in each box (square roots, \pi, and fractions are
all OK). If not, write none
in all three boxes.
If any value of a parameter will work, enter any
in that box.