Posted: November 17th, 2015

Mathematics

Mathematics

Problem 20. (Textbook p.642 ) Find several terms in the power series expansion of the
following quotients
!(b)
1
cos(x)
1(c)
1
cos(x) – sin(x)
Problem 21. (Textbook p.632 No. 8) Verify that the inverse hyperbolic sine function
: sinh(x) = e
x-e-x
2
has an inverse (i.e.x = sinh(y)or y = sinh-1
(x)) with a poer series
expansion
sinh-1
(x) = x +
X8
n=1
(-1)n
1 · 3 · · ·(2n – 1)
2 · 4 · · ·(2n)
·
x
2n+1
2n + 1
(Justify your calculation and indicate for which valus of x you are proving the validity of
the expansion, e.g. If the inteval of convergence is a finite interval, what can you say about
the end points?
Problem 22. Obtain the expansion
T =
x
2
+
X8
n=2
(-1)n-1
1 · 3 · · ·(2n – 3)
n!
·
x
n
2
n
for one root of the equation
T
2 + 2T – x = 0,
and show it converges so long as |x| < 1.
Problem 23. Find the radius of convergence for the series
X8
n=1
cnx
n
,
where
cn =
1
v
n2 + 1
+
1
v
n2 + 2
+ · · · +
1
v
n2 + n
.
14.

Signal and systems

Consider the signal
x(t) = ?(t)+ ? m2(t – m)
m?Z
where ? is the rectangular pulse and Z is the set of all integers other than zero. Plot x and show
that it is absolutely integrable and square integrable, but not periodic. Now consider the sequence of
samples cn = x(n) of the signal x. Plot the sequence c and show that it is periodic, but neither absolutely
summable, nor square summable. Hint:
8 1 p2
m2 = 6 .
m=1
2. (Raised cosine) Plot the signal
? 1 1 ??? 4 1 < t =
3
4
3
< t = 1
4
-4
?1 1 p
22 2 + cos 2pt – x(t) = ???1 1 p 1
4 22 2 4 + cos 2pt + – < t = –
?0 otherwise
and find its Fourier transform ˆx = Fx. Plot the Fourier transform. Is the Fourier transform square
integrable? Is it absolutely integrable?
3. (Finite impulse response filter) Design a low pass finite impulse response filter with cuttoff frequency
c = 2400 Hz and sample period P = 1
F where F = 8000 Hz. Ensure the filter satisfies the following
properties:
• has no more that 81 taps,
• affects the amplitude of frequencies in the interval [0, 2300 Hz] by no more than 10%,
• attenuates the amplitude of frequencies in the interval [2500 Hz,F/2] by more than 90%.
Plot the discrete impulse response and magnitude spectrum of this digital filter.

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