Sampling Continuous-Time Signal

1. Apr 9, 2015

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?

Last edited: Apr 9, 2015
2. Apr 9, 2015

mickonk

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

Last edited: Apr 9, 2015
3. Apr 10, 2015

milesyoung

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.

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.

Last edited: Apr 10, 2015
4. Apr 10, 2015

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?

Last edited: Apr 10, 2015
5. Apr 10, 2015

milesyoung

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.

You probably mean $F(\omega)$.

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

6. Apr 12, 2015

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?

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