Sampling Continuous-Time Signal

[mickonk]

This my homework:
Input signal to system is:

where

H(exp(jw)) is transfer function of ideal low pass filter with cutoff frequency wg=3*pi/4 and zero phase characteristic. Sampling in A/D converter is done with period T=(1/125) seconds.
a) Calculate output signal ya(t)
b) Calculate sampling period T for ya(t)=xa(t)First thing: they said that wa1, wa2 and wa3 have dimension 1/s. Is that mistake? I think that it should be rad/sec.
I recently started studying digital signal processing and I'm not so good yet but here are my thoughts. I know that A/D sampling period tells us that every T seconds A/D converter will take value from input time signal.
Frequency of sampling would be (1/T) [Hz] and it must be at least two times bigger than biggest frequency in input signal. Ideal low pass filter will pass only signals with frequencies lower than cutoff frequency
I know that I should first find amplitude spectrum of input signal using Fourier transform but I don't know how to find x(n). Here is how I would find FT of input signal. We can write last term of xa(t) as $$\sin (wa3t+(\frac{\pi}{2}+\theta))$$ Fourier transform of sine wave $$A\sin (w_0t)$$ is $$Aj\pi[\delta(w+w_0)-\delta(w-w_0)]$$, So FT of first term of $$x_a(t)$$ will be $$1j\pi[\delta(w+w_{a1})-\delta(w-w_{a1})]$$, FT of second term $$(1/2)j\pi[\delta(w+w_{a2})-\delta(w-w_{a2})]$$. What would be FT for third term, since it is time shifted?

 

[mickonk]

This would be (hopefully) amplitude spectrum of first and second term of input signal:

 

[milesyoung]

First thing: they said that wa1, wa2 and wa3 have dimension 1/s. Is that mistake? I think that it should be rad/sec.

An angle in radians is a dimensionless quantity, so, strictly speaking, it's correct to use the unit hertz [1/s] for both frequency and angular frequency. We usually make it explicit, though, by using [rad/s] for angular frequency to avoid any confusion.

What would be FT for third term, since it is time shifted?

Try going through your transform table again (Google one if you need to) - the translation/shifting property of the Fourier transform should be included in just about any of them.

 

[mickonk]

Hi milesyoung.
That's FT pair from my book. Is it wrong?
I found time shifting property in my book: if function is $$f(t\pm t_0)$$, FT is $$e^{\pm jw_0t_0}F(jw)$$. So FT of time shifted term would be $$e^{\pm jw(\frac{\pi}{2}+\theta)} F(jw) $$, right?

 

[milesyoung]

That's FT pair from my book. Is it wrong?

No, sorry, that was my mistake. I'm used to the unitary definition of the Fourier transform, so when I skimmed over it, I just noticed the lack of a scaling factor.

FT is $$e^{\pm jw_0t_0}F(jw)$$

You probably mean $F(\omega)$.

So FT of time shifted term would be $$e^{\pm jw(\frac{\pi}{2}+\theta)} F(jw) $$, right?

Not quite for your function, but it's not something you have to do from scratch. See here:
http://en.wikibooks.org/wiki/Waves/Fourier_Transforms#Fourier_Transform_Pairs

 

[rude man]

If the filter cuts off at w = 3π/4 and the lowest signal frequency is w = 100π, and sampling does not generate frequencies below the signal frequency, then how can anything get past the ideal low-pass filter?