E12 Assignment 02


1. Convolution is Commutative
Show that convolution is commutative.  In other words show that:

Hint: start with substitution: u=t-λ

2. Convolution is a (Weighted) Average

a) Consider the function


Show that the convolution of h(t) with a function f(t) is simply the average of f(t) over the previous T seconds (i.e., if y(t)=h(t)*f(t) then y(t) is the average of f(t) for the previous T seconds).

b) Now consider the function

 If y(t)=h(t)*f(t) use words to describe the relationship between y(t) and f(t).

Here is a related problem to think about if the answer is not coming to you:  If I give you the high temperature of every day for the last month and ask you to predict the high temperature tomorrow there are several ways you could do it.  One simple way is to average the temperatures over some time period (e.g., one week) and use that as the prediction.  How could you modify the average to get a better prediction (i.e., without looking at trends (slopes) in the data, but only aggregates (integrals/averages...) of the data)?

Note that  for both part a and part b.       

3. Complete Response to Ramp Input, 2nd Order System (1 - the zero state response)

Given a system with impulse response

Use the convolution integral to find the zero state solution if f(t)=tγ(t) (i.e.. f(t)=t for t≥0).  Note: this is a problem you could not have done in E11.   Note also that the integration is somewhat long and tedious but not particularly difficult if you recall (i.e., look it up in an integral table) that

4. Complete Response to Ramp Input, 2nd Order System (2 - complete response)

A system defined by the differential equation

has the impulse response

a) If the input is a ramp (f(t)=2.5·tγ(t)), and the initial conditions are , find the output without solving any additional differential equations by using the results of previous problem.

b) What is the output if the input is f(t)=-2γ(t), and the initial conditions are ?  Do as little additional work as possible.

5. Complete Response, 2nd Order Circuit, Step Input

The circuit shown


with C=0.25F, L=1H, R=5Ω and f(t)=ein(t), and y(t)=eout(t) is defined by the differential equation

and has the impulse response

The input to the system is f(t)=3γ(t), and the initial conditions are

a) Using the impulse response and the convolution integral, find an expression for the zero state response

b) Find an expression for the zero input response

c) Find the complete response

d) Use MATLAB to plot the zero state, zero input and complete response on one set of axes.  Include a legend to differentiate between them.  Turn in the plot and the MATLAB code.

6. Convolution of Two Pulses in Time, General Case

The image below shows two functions, x(t) and yIt), as well as the convolution, z(t)=x(t)*y(t). The function x(t) is a rectangular pulse with height A and width a. The function y(t) is a rectangular pulse with height B and width b. Assume A>B and b>a, as shown.

Determine the values of C, d, e, and f in terms of A, B, a and b.