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Homework Help: BPSK singlepath Rayleigh channel

  1. Sep 2, 2011 #1
    1. The problem statement, all variables and given/known data
    The aim of this exercise is to derive and simulate the error rate performance
    of binary PSK when the signal is transmitted over a single-path (or frequency
    non-selective) slowly fading channel. The frequency-nonselective channel results
    in multiplicative distortion of the transmitted signal s(t). In addition, the con-
    dition that the channel fades slowly implies that the multiplicative process may
    be regarded as a constant during at least one signaling interval. Therefore, the
    received equivalent lowpass signal in one signaling interval is

    r(t) = [itex]\alpha[/itex] * exp(j * [itex]\phi[/itex]) * s(t) + z(t) 0 [itex]\leq[/itex] t [itex]\leq[/itex] ts

    where z(t) represents the complex-valued white Gaussian noise process with
    power spectral density [itex]N_0[/itex]/2

    Assume that the channel fading is sufficiently slow that the phase shift, [itex]\phi[/itex] can
    be estimated from the received signal without error. Then, we can achieve ideal
    coherent detection of the received signal. The received signal can be processed
    by passing it through a matched filter. Show that for a fixed (time-invariant)
    channel, i.e., for a fixed attenuation [itex]\alpha[/itex], the error rate of binary PSK as a function
    of the received SNR, [itex]\gamma[/itex] is
    [itex]P_b[/itex]([itex]\gamma[/itex]) = Q([itex]\sqrt{}2\gamma[/itex])
    where [itex]\gamma[/itex] = [itex]\alpha^2 E_b / N_0[/itex] and [itex]E_b[/itex] is the energy of the transmitted signal per bit.

    2. Relevant equations

    3. The attempt at a solution
    Can someone give me an idea on how to start..
    Last edited: Sep 2, 2011
  2. jcsd
  3. Sep 3, 2011 #2
    How do I go about it
    i) Take the convolution of r(t) with the impulse response of a filter matched to s(t). Consider output signal and output noise separately.
    ii) Remember that the modulation method is binary PSK, which means the phase shift is either 0 or pi. What is the possible output signal values?
    iii) Before you compute error probability, you must obtain the pdf of the output noise.
    Last edited: Sep 3, 2011
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