Mathematics 433   Assignment #2 Solved

Description

1. Let F be the filter defined by

yt = xt              xt 1,

and let G be the filter defined by

yt = xt + yt 1.

Consider the two filters H₁, which takes the output of F and uses it as the input of G, and H₂, which takes the output of G and uses it as the input of F.

• Explain why F is a discrete di↵erentiator, and G is a discrete integrator.
• Determine the output of H₁ and H₂ if we give the unit impulse t as input.
• Explain why this should be viewed as a discrete version of thefundamental theorem of calculus

2. Suppose that a given filter has the frequency response

H.

• Give a defining equation of the filter.
• Plot the zeroes and poles of the transfer function H (z) in the complex plane.
• Explain why the filter is or is not stable.
• Using a computer, plot the frequency response of the filter. Theabscissa should have units of frequency, in fractions of the sampling rate, and the ordinate should have the units of decibels.

3. Consider the sequence y_t, and suppose that it satisfies the homogeneous di↵erence equation

for each t.

• Show that y_t has solutions of the form

where the A_ms are arbitrary constants and the z_ms are the N roots of the polynomial

.

That is to say,

.

Note that this is the discrete analogue of the well known result from the theory of ordinary di↵erential equations that an n^th order ordinary di↵erential equation with constant coe cients has solutions of the form e^akt, and the numbers a_k are roots of the auxiliary polynomial.

• By once again examining the case of constant coe cient di↵erential equations, deduce what happens to the solution in the case of multiple roots. Likely you will get stuck on doing the general case all at once, so make up an example that you can solve easily to see what is going on. For example, one that has two roots z₁,z₂ and try something like z₁ = 2 and z₂ = 2 + ✏ and letting ✏ ! 0 or solving a di↵erence equation that has auxiliary roots z₁ = z₂ = 2 directly.
• Find y_t if yt 5yt 1 + 6yt 2 = 0

and y₀ = 0 and y₁ = 1.

4. The theory of gun barrels requires the Fourier series of the function

f(x) = sin(sinx).

• Find the first four nonzero Fourier coe cients. If you can not find them analytically, do the integral numerically. If you use a numerical technique, be sure to describe which one you used.
• Plot the function f together with the approximation given by the first four nonzero Fourier coe cients you found.

5. (a) Plot the sum of the first 39 harmonics for the square wave

2Z.

Z.

Let t range over [0,5⇡] for your plot.

• Plot the sum of the first 39 harmonics for the triangle wave defined by

Once again, let t range over [0,5⇡] for your plot.

• Di↵erentiate the series for f(t) term by term, and plot on [0,5⇡] the sum of the first 39 harmonics. Explain why this might have the name ‘buzz’.