## Description

- Let
*F*be the filter defined by

*y**t *= *x**t x**t *1*,*

and let *G *be the filter defined by

*y**t *= *x**t *+ *y**t *1*.*

Consider the two filters *H*_{1}, which takes the output of *F *and uses it as the input of *G*, and *H*_{2}, 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
*H*_{1 }and*H*_{2 }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

*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.

- Consider the sequence
*y*, and suppose that it satisfies the homogeneous di↵erence equation_{t}

for each *t*.

- Show that
*y*has solutions of the form_{t }

where the *A _{m}*s are arbitrary constants and the

*z*s are the

_{m}*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*

^{a}*, and the numbers*

^{kt}*a*are roots of the auxiliary polynomial.

_{k }- 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
*z*_{1}*,z*_{2 }and try something like*z*_{1 }= 2 and*z*_{2 }= 2 +*✏*and letting*✏*! 0 or solving a di↵erence equation that has auxiliary roots*z*_{1 }=*z*_{2 }= 2 directly. - Find
*y*if_{t }*y**t*5*y**t*1 + 6*y**t*2 = 0

and *y*_{0 }= 0 and *y*_{1 }= 1.

- The theory of gun barrels requires the Fourier series of the function

*f*(*x*) = sin(sin*x*)*.*

- 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

2**Z***.*

**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’.