Practice Problems on Signals and Systems

(ECE301)

## Contents

## Related Subtopics

- Review of complex numbers
- Signal Power and Energy in CT
- Compute the power and energy of a (CT) exponential
- Compute the power and energy of a (CT) sine
- Compute the power and energy of a complex (CT) exponential
- Energy of x(t)= cos(t)+j sin(t)
- Compute the power and energy of t times a step function
- Compute the power and energy of 2 times t squared
- Compute the power and energy of a square root
- Compute the power and energy of a square root times a step function
- Compute the power and energy of 5 j times sin(t)
- Compute the power of 2j

- Signal Power and Energy in DT
- Transformation of the independent variable
- Basic System Properties
- Is the following system invertible?
- Is the following system invertible? New! Waiting for input!
- Is the following system invertible? New! Waiting for input!
- Is the following system invertible? New! Waiting for input!
- Is the following system invertible? New! Waiting for input!
- Is the following system invertible? New! Waiting for input!
- Is the following system stable?
- Is the following system memoryless?
- Is the following system causal?

- Linearity and time invariance of a system
- Linearity and time invariance
- Computing the output of a DT LTI system by convolution
- Compute the output of the following DT LTI system
- Compute the output of the following DT LTI system
- Compute the output of the following DT LTI system
- Compute the output of the following DT LTI system
- Compute the output of the following DT LTI system
- Compute the output of the following DT LTI system
- Compute the output of the following DT LTI system
- Compute the output of the following DT LTI system

- Computing the output of a CT LTI system by convolution
- Computing the Fourier series coefficients of a CT signal
- See subtopic page for a list of all problems on Fourier series of a CT signal

- Computing the Fourier series coefficients of a DT signal
- Fourier transform of a continuous-time signal:
- See subtopic page for a list of all problems on Fourier transform of a CT signal

- Computing the Fourier transform of a discrete-time signal:
- Causal LTI systems defined by linear, constant coefficients difference equations:
- Nyquist theorem
- What is the Nyquist rate for this signal?
- What is the Nyquist rate for this other signal?
- Samping and reconstruction of sinc function.
- Samping and reconstruction of sinc function multiplied by exponential.
- Samping and reconstruction of sinc function multiplied by exponential (another one).
- Samping and reconstruction of sinc function multiplied by exponential (another one).
- Samping and reconstruction of sinc function multiplied by acosine.
- What is the Nyquist rate of a sinc function multiplied by itself?
- What is the Nyquist rate of the multiplication of two sinc functions?

- Modulation
- Z-transform
- Prove the modulation property of the z-transform
- Computation of the z-transform
- Another computation of the z-transform
- Computation of the inverse z-transform
- Another computation of the inverse z-transform
- Prove the scaling property of the z-transform
- compute the z-transform of this function
- Practice Question on z-transform computation
- Practice Question on inverse z-transform computation
- Compute this z-transform
- Compute this z-transform
- Compute this z-transform and obtain Fourier transform
- Obtain the inverse z-transform
- Obtain the inverse z-transform
- Obtain the inverse z-transform
- Obtain the inverse z-transform
- Obtain the inverse z-transform
- Obtain the inverse z-transform
- Obtain the inverse z-transform

# Problems from the official textbook (Oppenheim WIllsky)

- 4.1_ECE301Fall2008mboutin
- 4.2_ECE301Fall2008mboutin
- 4.3_ECE301Fall2008mboutin
- 4.4_ECE301Fall2008mboutin
- 4.5_ECE301Fall2008mboutin
- Discussion about problem 5.31

## Quizzes with solution

- From instructor Jang, Summer 2015
- Quiz 1. Linear and Time Invariant (pdf)
- Quiz 2. Convolution of Two Rectangles (pdf)
- Quiz 3. Continuous-Time Fourier Transform Exercise (pdf)
- Quiz 4. Continuous-Time Fourier Transform Equation (pdf)
- Quiz 5. Nyquist Sampling Theorem (pdf)
- Quiz 6. Discrete-Time Fourier Transform Exercise (pdf)
- Quiz 7. Sampling Theory (pdf)

## More Practice Problems

- On Fourier transform
- Computing the energy and power of a CT signal: two examples
- Laplace transform example
- Frequency and impulse response from diff. eq.
- Example of CT convolution
- Is this system time-invariant?
- Inverse z-transform: summary of theory and practice examples with solutions
- practice problems (mostly on Fourier transform)
- Finale exam practice (written by a student)