HW3, MAE 171. Spring 2005. UCI

Nasser Abbasi

Problem 3-1

part a

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

Part b

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

Part c

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 asthen by comparing (3) and (4) we see that (3) can be be written as

Which is the required relation to show.

Problem 3-4.

part c

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,

part f

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

Problem 3-15.

Given the signal

part a

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

rad/sec

To illustrate, this is a small Mathematica script.

Part b

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

Part c

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.

Problem 3-16

is applied to a samples/ZOH device, rad/sec.

part a

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

rad/sec

part b

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

part c

sketch the input signal and the component of part(b) vs. time.

part d

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.