ECEA108-4 Final Exam PS

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Questions and Answers

If x(n) = {-1, 7, 9, 2, -8, -1, -8}, determine: x(n+2) + x(n-3)

{-1, 7, 9, 1, -1, 7, 1, 1, 1, -16, -1, -8}
{-1, 7, 8, 9, 1, 0, -9, 8, -6, -8, -1, -8} (correct)
{1, -7, -8, -9, -1, 0, 9, -8, 6, 8, 1, 8}
{1, -7, -9, -1, 1, 7, -1, -1, -1, 16, 1, 8}

If x(n) = {-4, 5, -7} and h(n) = {9, 2, 4, -2, 8}, determine y(n) = x(n) × h(n)

{-8, 20, 14} (correct)
{8, -20, -14}
{-8, 20, -14}
{9, -8, 20, 14, 8}

If x(n) = {-3, -4, 7} and h(n) = {3, 5, 8, -8, 7}, determine the convolution of x(n) and h(n)

{-9, -27, -23, 27, 67, -84, 49} (correct)
{21, 4, -57, 103, -27, -23, -21}
{-21, -4, 57, -103, 27, 23, 21}
{9, 27, 23, -27, -67, 84, -49}

If x(n) = {2, 1, 3} and h(n) = {-2, 5, -8, -9, -5}, determine the correlation (rxh(l)) of x(n) and h(n)

{-10, -23, -40, -25, -23, 13, -6} (A) Signup and view all the answers

If x(n) = {9, 5, 8, -2} and h(n) = {3, -1, 4, 6, -8, -7}, solve for x(n) − h(n)

{-3, 1, -4, 3, 13, 15, -2} (B) Signup and view all the answers

Determine the z-transform of the signal: x(n) = 4n u(n)

X(z) = z−4 / z (C) Signup and view all the answers

Determine the z-transform of the signal: x(n) = −5n u(−n − 1)

X(z) = z−5 / (5z −1) (B) Signup and view all the answers

Determine the z-transform of the signal: x(n) = 6n u(−n − 1)

X(z) = 6−z / (6z −1) (D) Signup and view all the answers

Determine the z-transform of the signal: x(n) = nu(n)

X(z) = (1−z−1 )2 / 1 (B) Signup and view all the answers

Determine the z-transform of the signal: x(n) = 3n nu(−n − 1)

X(z) = (1−3z−1 )2 / 3z −1 (D) Signup and view all the answers

Find the inverse z-transform of the signal: X(z) = 1−4z−1 ; |z| > |4|

x(n) = 4n u(n) (D) Signup and view all the answers

Find the inverse z-transform of the signal: X(z) = 1−5z−1 ; |z| < |5|

x(n) = −5n u(−n − 1) (C) Signup and view all the answers

Find the inverse z-transform of the signal: X(z) = 3−z ; |z| < |3|

x(n) = 3n u(−n − 1) (B) Signup and view all the answers

Find the inverse z-transform of the signal: X(z) = z+4 ; |z| > |4|

x(n) = (−4)n u(n) (B) Signup and view all the answers

Given the FIR Filter: y(n) = 0.25x(n) + 0.2x(n − 1) + 0.1x(n − 2) + 0.3x(n − 3), what is the filter length?

4 (A), 3 (B) Signup and view all the answers

An FIR band stop filter with a lower cut-off frequency of 1800 Hz, an upper cut-off frequency of 2200 Hz, and a sampling rate of 8000 Hz, determine the normalized upper cutoff frequency (Ωc).

0.55π radians (C) Signup and view all the answers

An FIR band stop filter with a lower cut-off frequency of 1800 Hz, an upper cut-off frequency of 2200 Hz, and a sampling rate of 8000 Hz, determine the normalized upper cutoff frequency (Ωc).

0.45π radians (B) Signup and view all the answers

Consider the analog signal 3cos100πt, determine the minimum sampling rate required to avoid aliasing.

100 Hz (C) Signup and view all the answers

Given the analog signal, 3cos50πt + 10sin300πt – cos100πt, what is the Nyquist frequency?

300 Hz (A) Signup and view all the answers

What is the sampling period given the sampling frequency of 8000 Hz?

125 µs (C) Signup and view all the answers

Study Notes

Signal Processing Calculations

Sequence Manipulation: Evaluating x(n+2) + x(n-3) gives an output based on shifts of the original sequence.
Convolution Operations: The convolution of two sequences involves combination, effectively merging their effects over time to yield y(n) = x(n) × h(n).

Z-Transforms

Basic Z-Transform: X(z) representations are derived from time-domain signals; the given equations illustrate the necessary transformations based on unit step functions.
Z-Transform of Signals: Signals like x(n) = nu(n) translate to X(z) forms that encapsulate growth behavior influenced by n and unit step u(n).
Inverse Z-Transforms: Important for recovering the time domain from its z-domain counterparts, identifying conditions such as |z| > |a| or |z| < |a| to determine signal behavior.

FIR Filters

Filter Length Determination: For FIR filters like y(n) = 0.25x(n) + 0.2x(n-1) + ... the filter length is based on the number of coefficients involved.
Band Stop Filters: Critical frequencies define how filters alter signal processing; the normalized upper cutoff frequency is calculated from original frequency ranges relative to sampling rates.

Sampling Theorem

Nyquist Frequency: Identifies the minimum rate to avoid aliasing between frequencies in a sampling system. It's calculated as twice the highest frequency present in a signal.
Sampling Period Calculation: Derived directly from the sampling frequency; for instance, at 8000 Hz, the period becomes crucial for timing and synchronization in digital applications.

Frequency Components

Analog Signal Analysis: Analyzing components like 3cos100πt involves breaking down frequency contributions to ensure proper representation in the digital domain for efficient processing.

Feel free to use or combine these study notes for efficient studying and understanding of the key concepts within signal processing!

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Description

Test your knowledge on concepts of signals and systems with this quiz. It includes problems related to signal manipulation and convolution operations. Determine outputs based on given input signals and systems.