Collectively solved problems related to Signals and Systems

• Review of complex numbers
  • Compute the magnitude of these three complex numbers
  • Compute the magnitude of these two CT signals
  • Compute the magnitude of these three DT signals
• 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
  • Compute the power and energy of the following DT exponential signal
  • Compute the power and energy of the following DT signal
• Transformation of the independent variable
  • Cascade the following time delay and time scaling
• 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
  • Is the following system time-invariant?
  • Is the following system linear?
• Linearity and time invariance
  • Is the following system time-invariant?
  • Is the following system linear?
  • Is the following system time-invariant?
• 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
  • Compute the output of the following CT LTI system
  • Compute the output of the following CT LTI system
  • Compute the output of the following CT LTI system
  • Compute the output of the following CT LTI system
• 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
  • Obtain the Fourier series coefficients of this DT sinusoidal
  • Obtain the Fourier series coefficients of this DT pulse-train
  • A page containing several practice problems on computing Fourier series of a CT 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:
  • Compute the Fourier transform of 3^n u[-n]
  • Compute the Fourier transform of cos(pi/6 n).
  • Compute the Fourier transform of u[n+1]-u[n-2]
  • Compute the DT Fourier transform of a sinc
  • Compute the DT Fourier transform of a rect
• Causal LTI systems defined by linear, constant coefficients difference equations:
  • Example of "typical" questions on causal LTI systems defined by difference equations
  • Frequency and impulse response obtained from a difference equation describing an LTI system
  • A tricky example: only attempt if you really understand what is going on
• 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
  • Demodulation of signal modulated with exponential
  • Demodulation of signal modulated with cosine
• 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
  • Fourier transform problem
  • A mistake NOT TO REPEAT when computing Fourier transforms
  • How NOT to take a 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)

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