HW3, MAE 171. Spring 2005. UCI
Nasser Abbasi
Give the definition of the starred transform.
is the Laplace transform of
,
where
which is the time domain representation of the input signal
after passing though an ideal sampler.
Mathematically,
where
is the sampling period, and
is the sample number and
is the
transform the sequence
.
Given the standard definition of the
transform, then we can now write
Give the definition of the
transform.
Given a sequence
,
then the one sided
transform of
is defined as
Here we assumed that the number sequence
has its first element at index
.
i.e.
for
For a function
derive a relationship between its starred transform
and its
transform
but
Hence (1) becomes
But given the property that
, then
and (2) can be now be written as
Now, given the one-sided definition of the Z transform of sequence
as
then
by comparing (3) and (4) we see that (3) can be be written as
Which is the required relation to show.
Find the starred transform for
so need to first find the Z transform of
, the discrete time representation of the ideal sampled version of
so we need to first find
Use partial fraction to find the inverse Laplace transform.
Hence
ROC is
and
Hence,
Find the starred transform for
so need to first find the Z transform of
, the discrete time representation of the ideal sampled version of
so we need to first find
roots of denominator are
Let
,
and
,
so now
can be written as
Use partial fraction to find the inverse Laplace transform.
Hence
From Tables,
Hence
Given the signal
List all frequencies less than
rad/sec that are present in
contains a frequency of
rad/sec.
contains a frequency of
rad/sec.
Hence the answer is
To illustrate, this is a small Mathematica script.
is sampled at
rad/sec. List all frequencies present in
less than
rad/sec.
The frequency spectrum of the sampled signal will contain frequencies that are
centered around multiples of
Hence the list of generated frequencies are
substitute for
we get
Sort and consider only frequencies less than
rad/sec, we get
The signal
is applied to a ZOH device. List all frequencies present in
less than
rad/sec.
Since a ZOH is an LTI device and the input is a set of eigenfunctions, then the output will contain the same set of input frequencies but possibly scaled and phase shifted. Hence this is the same answer as part (b) above.
is applied to a samples/ZOH device,
rad/sec.
what is the frequency component in the output that has the largest amplitude?
This will be the component with the frequency closest to the DC frequency.
Looking at the sequence of frequencies generated, which are
sort, and consider only positive frequencies, we obtain
Hence the component with the largest amplitude has a frequency of
Find amplitude/phase of that component.
But
To
find
use the relation that
has the Fourier transform
Then,
the Fourier transform of
is (where here
)
substitute
(3) into (2), then (1) can be written as
but
Hence
(4) becomes
Evaluate
at
rad/sec, and given that
rad/sec, we get
Now
to find the phase of this component.
since
then, since
then
this
is because
at frequency
rad/sec which is smaller than
rad/sec, so we are inside the first hub of the sinc function. i.e.
in this case.
Now
but since
when
,
then
(using equation 3-33 page 105 in book).
Hence
Hence, (5) now becomes
sketch the input signal and the component of part(b) vs. time.
Find the ratio of the amplitude in part(b) to that of the frequency component
in the output at
rad/sec.
but
and
Hence
Hence ratio is
This is expected, as at
we are in the second hub of the sinc function, which will have much less
amplitude.