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:

    t1.png

    where

    t2.png

    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


    t3.png
    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:

    t4.png
     
  3. WBahn

    Moderator

    Mar 31, 2012
    17,731
    4,789
    radian/second and 1/second are the same thing because a radian is dimensionless (its length/length, namely circumference/diameter).
     
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