Question

The pole-zero diagram for a certain digital filter is shown below: 3/4 ** Im(z) -unit-circle Re(z) 1 -1/2 (a) Write the corresponding z-domain transfer function H(z) for this filter (assume that there is no additional gain constant - the poles and zeros depict H(z) completely). (b) This filter is driven by a sum of signals x[n], where x[n] = 3 u[n] + 2 (-1)$^n$ u[n] Write the mathematical form of the filter's steady-state output waveform y[n] in response to x[n] after the filter's transient response has decayed away. [Hint: To what points on the unit circle in the z-plane does each term of x[n] correspond in terms of frequency $\Omega$? This is fundamentally a frequency response calculation - if you're doing elaborate transforms and inverse transforms, you're missing the simplicity of this problem! Use graphical analysis of the pole-zero map to speed the response computation.]

          The pole-zero diagram for a certain digital filter is shown below:
3/4
**
Im(z)
-unit-circle
Re(z)
1
-1/2
(a) Write the corresponding z-domain transfer function H(z) for this filter (assume that there is no
additional gain constant - the poles and zeros depict H(z) completely).
(b) This filter is driven by a sum of signals x[n], where
x[n] = 3 u[n] + 2 (-1)$^n$ u[n]
Write the mathematical form of the filter's steady-state output waveform y[n] in response to x[n] after
the filter's transient response has decayed away.
[Hint: To what points on the unit circle in the z-plane does each term of x[n] correspond in terms of
frequency $\Omega$? This is fundamentally a frequency response calculation - if you're doing elaborate
transforms and inverse transforms, you're missing the simplicity of this problem! Use graphical
analysis of the pole-zero map to speed the response computation.]

The pole-zero diagram for a certain digital filter is shown below:
3/4
**
Im(z)
-unit-circle
Re(z)
1
-1/2
(a) Write the corresponding z-domain transfer function H(z) for this filter (assume that there is no
additional gain constant - the poles and zeros depict H(z) completely).
(b) This filter is driven by a sum of signals x[n], where
x[n] = 3 u[n] + 2 (-1)^n u[n]
Write the mathematical form of the filter's steady-state output waveform y[n] in response to x[n] after
the filter's transient response has decayed away.
[Hint: To what points on the unit circle in the z-plane does each term of x[n] correspond in terms of
frequency Ω? This is fundamentally a frequency response calculation - if you're doing elaborate
transforms and inverse transforms, you're missing the simplicity of this problem! Use graphical
analysis of the pole-zero map to speed the response computation.]

Added by Elena M.

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