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Biosignal HW

The document outlines a homework assignment for a Biosignal Processing course, consisting of six problems related to system stability, block diagram sketches, frequency resolution, z-transforms, and transfer functions. Students are required to submit handwritten answers on A4 sheets in class. Each problem is assigned a specific point value, totaling 100 points.

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12 views2 pages

Biosignal HW

The document outlines a homework assignment for a Biosignal Processing course, consisting of six problems related to system stability, block diagram sketches, frequency resolution, z-transforms, and transfer functions. Students are required to submit handwritten answers on A4 sheets in class. Each problem is assigned a specific point value, totaling 100 points.

Uploaded by

phamxuanhoc097
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Biosignal Processing

Homework #04
Note:
 The answers must be handwritten on A4 sheets.
 Please submit the answers in class.
Problem 1 (20 points)
Determine the if the system below is stable.
a. 𝑦(𝑛) = 5𝑥 (𝑛 − 10)
b. 𝑦(𝑛) = 𝑥(𝑛) + 0.5𝑥(𝑛 − 1)
c. 𝑦(𝑛) = ∑ 0.75 𝑥 (𝑛 − 𝑘 )
d. 𝑦(𝑛) = ∑ 2 𝑥 (𝑛 − 𝑘 )

Note: A linear system is stable if the sum of its absolute impulse response coefficients is a finite
number.

𝑆= |ℎ(𝑘)| = ⋯ + |ℎ(−1)| + |ℎ(0)| + |ℎ(1)| + ⋯ < ∞

Problem 2 (30 points)


Sketch the block diagram of the following systems:

a. 𝑦(𝑛) = 0.5𝑥 (𝑛) − 0.5𝑥(𝑛 − 2)


b. 𝑦(𝑛) = 0.75𝑦(𝑛 − 1) + 𝑥 (𝑛)
c. 𝑦(𝑛) = −0.8𝑦(𝑛 − 1) + 𝑥(𝑛 − 1)

Problem 3 (10 points)


Consider a digital sequence sampled at the rate of 20,000 Hz. If we use the 8,000-point DFT to
compute the spectrum, determine
a. The frequency resolution
b. The folding frequency in the spectrum.
Problem 4 (10 points)
We use the DFT to compute the amplitude spectrum of a sampled data sequence with a sampling
rate fs = 2,000 Hz. It requires the frequency resolution to be less than 0.5 Hz. Determine the
number of data points used by the FFT algorithm and actual frequency resolution in Hz, assuming
that the data samples are available for selecting the number of data points.
Problem 5 (20 points)
Determine the z-transform of the signal
𝑎. 𝑥(𝑛) = (0.5) sin (0.25𝜋𝑛)𝑢(𝑛)
.
𝑏. 𝑥(𝑛) = 𝑒 cos (0.25𝜋𝑛)𝑢(𝑛)
𝑐. 𝑥(𝑛) = (1 + 𝑛)𝑢(𝑛)
𝑑. 𝑥(𝑛) = (𝑎 + 𝑎 )𝑢(𝑛), 𝑎 ∈ ℝ

Problem 6 (10 points)


( )
Determine the transfer function H(z), given that 𝐻(𝑧) = ( )
and

𝑦(𝑛) = 𝑦(𝑛 − 1) + 2𝑥(𝑛) 10 points

-END-

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