EE313 Linear Systems and Signals - Homework #3 Hints

• Applets for convolution

• For each convolution problem, be sure to check the ending point of each interval in the convolution result to make sure that it agrees with the starting point of the next interval.

• Mathematica can be used to convolve two signals expressed as formulas. You do not have to use Mathematica for this problem set.

• Problem 3.1: This is a pointwise system; that is, the current output value only depends on the current input value. The system is nonlinear, time-invariant, causal, memoryless, and stable. In fact, all pointwise systems are time-invariant. Stability in this context means that an input signal that is bounded in amplitude will always give an output signal that is bounded in amplitude, which is also known as bounded-input bounded-output (BIBO) stability.

• Problem 3.2: In this problem, you will compute the convolution manually using the convolution definition, and then plot the convolution result in Matlab.

  This convolution problem involves the convolution two signals of finite duration. You would expect that the duration (extent) of the resulting convolution be the sum of the durations of the two signals. Also, you would normally expect that the convolution result would have three non-zero intervals of interest:

  • amount of overlap between the signals increases
  • signals completely overlap
  • amount of overlap between the signals decreases
  The answer is a trapezoid.

• Problem 3.3: For an LTI system with impulse response h(t), the output y(t) given input x(t) is the convolution of h(t) and x(t). The impulse response uniquely represents an LTI system.

  In the lowpass RC circuit, the initial condition is the initial voltage across the capacitor at t = 0^-. The initial condition is zero. The system passes the all-zero input test. And the system is LTI.

  When a rectangular pulse is the input signal x(t), the convolution can be computed over three intervals of interest: no overlap, partial overlap and complete overlap.

  When viewing the circuit behavior, the capacitor charges when the input pulse is "on" and discharges once the input pulse has turned "off". The system response to a rectangular pulse input is non-zero from t = 0 to t = oo.

bevans@ece.utexas.edu