# Sampling Continuous-Time Signal

Discussion in 'Homework Help' started by mickonk, Apr 9, 2015.

1. ### mickonk Thread Starter New Member

Apr 6, 2015
13
0
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

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 $\sin (w_a_3t+(\frac{\pi}{2}+\theta))$. 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 xa(t) will be $1j\pi[\delta(w+w_a_1)-\delta(w-w_a_1)]$, FT of second term $(1/2)j\pi[\delta(w+w_a_2)-\delta(w-w_a_2)]$ What would be FT for third term, since it is time shifted?

Last edited: Apr 9, 2015
2. ### mickonk Thread Starter New Member

Apr 6, 2015
13
0
This would be (hopefully) amplitude spectrum of first and second term of input signal:

3. ### WBahn Moderator

Mar 31, 2012
18,087
4,917
radian/second and 1/second are the same thing because a radian is dimensionless (its length/length, namely circumference/diameter).