- 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
- Suppose that a given filter has the frequency response
- 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.
- Consider the sequence yt, and suppose that it satisfies the homogeneous di↵erence equation
for each t.
- Show that yt has solutions of the form
where the Ams are arbitrary constants and the zms 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 nth order ordinary di↵erential equation with constant coe cients has solutions of the form eakt, and the numbers ak 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 yt if yt 5yt 1 + 6yt 2 = 0
and y0 = 0 and y1 = 1.
- 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.
- (a) Plot the sum of the first 39 harmonics for the square wave
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’.