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Spectral density of a digital signal

Discussion in 'Electronic Basics' started by pozz, Aug 31, 2003.

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  1. pozz

    pozz Guest

    Suppose I want to transmit a bit sequence and I decided to transmit
    x(t) for bit 1 and -x(t) for bit 0 every Ts seconds. X(f) is Fourier
    transform of x(t) and Sx(f) is spectral density of x(t).

    Signal that I transmit by antenna or wire is:
    x_t(t) = SUM_{n=-infty}^{+infty} a_n*x(t-n*Ts)
    where
    a_n = 1 for bit 1 and -1 for bit 0
    Obviously, a_n is a stochastic process related to the sequence of bits.

    Now I want to know the spectral density of my signal x_t(t) on the air
    (or on the wire), i.e. I want to know Sx_t(f).
    Is there a relation between the power spectral density Sx(f) of the
    symbol x(t) and the power spectral of the signal transmitted Sx_t(f)???

    Normally, one calculate Sx(f) and associate it to the power spectral
    density of the signal transmitted... Why?
     
  2. Roy McCammon

    Roy McCammon Guest

    first a little artiface. Lets assume that we have a linear circuit
    with an impluse response of x(t). Then assume that our data signal
    is a sequence of impluses ( either positive or negative ). So then
    the output signal is just the convolution of x(t) with the a_n data
    stream. Thus the Fourier transform of the output signal is product
    of X(f) with the Fourier transform of the input signal. That means
    the PSD (power spectral density) of the output is just the product
    of the PSD of x(t) with the PSD of the data stream. In the special
    case that the data stream is a random stream of impulses ( positive
    and negative ), the PSD of the data is a flat constant independent
    of frequency. Voila, the PSD of the output signal is the PSD of x(t)
     
  3. cirip

    cirip Guest

    Hi Roy,
    Please see in line.


    I am not sure this is right. First of all, the Fourier transform is defined
    for periodic signals only. Your data stream is random, I suspect, so you
    would rather want to talk about power spectral density (PSD).

    According to the Wiener- Khintchine Theorem, the PSD of a random
    process is the Fourier transform of the autocorrelation function of the
    random signal.


    I guess it is. x_t(t) seems to me a magnified translated replica of x(t);
    magnified because the amplitude of x(t) is 1, whereas the amplitude of
    x_t(t) is 2. Translated because the DC component of x(t) is non zero,
    whereas the DC component of x_t(t) is close to 0 assuming that the number of
    ones equals the number of zeros in your message. Based on the properties of
    the Fourier transform, my guess is that the PSD of x_t(t) will be similar to
    the PSD of x(t) less the low frequency components which should be lower
    level on x_t(t).

    The PSD of both x(t) and x_t(t) should be fairly wide. x(t) has a sinc
    (sinx/x) shaped PSD with its first null at f=1/Tbit. I am not sure about
    x_t(t) but I can dig up the derivation for it. I don't think you can apply
    this signal directly to an antenna, though.

    That's valid for linear modulation only. Sx(f) is the PSD of the base band,
    or the raw data. By linearly modulating ( a sort of amplitude modulation) a
    carrier with your x(t) you are actually translating the base band or the PSD
    of x(t) around the carrier frequency.

    For instance if you have a carrier signal c(t)=cos(omega_c*t) and the
    modulating signal x(t)=cos(omega_m*t), by linear modulation you are actually
    multiplying the signals. The result is:
    c(t)*x(t)=cos(omega_c*t) *cos(omega_m*t)
    =1/2[(cos(omega_c-omega_m)*t)+(cos(omega_c+omega_m)*t)

    As you can see there are 2 components, one above and the other below the
    original carrier.
    If your baseband were a sum of cosines (a spectra) they would have been
    translated below and above the carrier. So the base band spectra has been
    translated. around the carrier. That's why analyzing the baseband PSD one
    can easily predict the PSD of the transmitted signal.

    [...]
    In the special
    I thought the PSD of the random rectangular signal is the sinc function.


    Hope it helps,
    Cirip


    Voila, the PSD of the output signal is the PSD of x(t)
     
  4. cirip

    cirip Guest

    Hi Roy,
    OOoooops! I should have said hello pozz
     
  5. Roy McCammon

    Roy McCammon Guest

    the data stream I am using is a stream of impules
    of random polarity. The auto correlation funtion
    of that is a single impulse at Delta_T = 0
     
  6. Hello Roy,

    That would have been correct if the signal were an ideal random signal.
    There's no such signal in the real world. The autocorrelation function of
    the rectangular stream of impulses has a triangular shape centered on t=0.
    As the data rate goes higher and higher, the triangle narrows its base
    becoming a Delta Dirac pulse when the data rate approaches infinity
    (Tbit->0), but that's impractical to achieve.

    Please take a look at the following link.

    http://pdfserv.maxim-ic.com/arpdf/AppNotes/1hfan901.pdf

    Regards,
    Cirip
     
  7. Hey pozz, is this discussion answering your questions or we're just barking
    in the dark? :))

    Cirip
     
  8. pozz

    pozz Guest

    Thanks to all.

    Sure Roy centered the question. I want to know PSD of signal x_t(t),
    ipotetically
    signal transmitted on a channel (air or wire, it's not important).

    Signal x_t(t) is the modulated **digital** signal. So I have a random
    bit-stream
    that I code by a symbol signal x(t).
    For example, x(t) is a rect(t) or a sinc(t) or another signal. For bit 1
    I
    add x(t), for bit 0 I add -x(t), every Ts seconds.
    So, x_t(t) is:
    x_t(t) = SUM_{n=-infty}^{infty} a_n*x(t-n*Ts)
    where a_n=1 for bit 1, -1 for bit 0.

    What is PSD of x_t(t)? What is the relationship with x(t)?

    Roy supposed that x_t(t) is the output of a linear filter with impulse
    response x(t).

    +--------+
    u(t) ---->| X(f) |------> x_t(t)
    +--------+

    u(t) is a random continuous-time stochastic process with impulses at
    t=n*Ts. The amplituted of that impulses is 1 for bit 1 and -1 for bit 0.

    So PSD of x_t(t) is Sxt(f) = Su(f)*|X(f)|^2

    Now what is Su(f)? Obviously, is Fourier transform of autocorrelation
    of stochastic process u(t). But the autocorrelation is a function of
    a single variable only when process is stationary in time. I'm not sure,
    but u(t) isn't stationary... it's periodically stationary, don't it?
    Consider R(t,tau)=E[u(t)*u(t+tau)]. If tau=0, R(t,0) depends from t.
    If t=n*Ts I have identical impulses, if I have t<>n*Ts I have
    u(t)=0 with probability 1.
    How can I consider a PSD for u(t)?
     
  9. Roy McCammon

    Roy McCammon Guest

    pozz,
    as I understood your question, you want to know how
    to get the text book answer. You get that by assuming,
    that the data stream is random, white, stationary
    with auto correlation function an impulse at t=0
    and zero everywhere else.

    Of course that is a convenient approximation. It really
    only needs to be short term stationary to build useful
    systems. After all, the data stream has a finite beginning
    and usually it gets turned off sooner or later, so it is
    not strictly stationary. Many communication circuits
    pre whiten the data stream with simple scramblers
    so even a steam of continuos zeros looks white in the
    short term. These scramblers may be finite state machines
    of they may be software like Zip or PGP or huffman coding.

    -Roy
     
  10. cirip

    cirip Guest

    Hello Roy and pozz,

    Looks like Roy was right. I missed the question, but even now, after reading
    your comments, I am lost.

    Roy, my initial understanding was that pozz was asking something like what
    would be the base band spectrum of a bipolar signal (i.e. 1->a, 0->-a) and
    then what happens when one wants to transmit that data, i.e. modulate a
    carrier with that spectrum. This is usually covered in any digital data
    transmission text book.

    Now, I am curious to understand what is all about and what am I missing.
    Roy, you say: "you want to know how to get the text book answer". Could you
    please point me to a link or something that would clarify what is the text
    book answer you are talking about?

    Thank you,
    Cirip


     
  11. Roy McCammon

    Roy McCammon Guest

    sorry to take so long to respond.

    Here is the original question

    "Suppose I want to transmit a bit sequence and I decided to transmit
    x(t) for bit 1 and -x(t) for bit 0 every Ts seconds. X(f) is Fourier
    transform of x(t) and Sx(f) is spectral density of x(t).

    Signal that I transmit by antenna or wire is:
    x_t(t) = SUM_{n=-infty}^{+infty} a_n*x(t-n*Ts)
    where
    a_n = 1 for bit 1 and -1 for bit 0
    Obviously, a_n is a stochastic process related to the sequence of bits.

    Now I want to know the spectral density of my signal x_t(t) on the air
    (or on the wire), i.e. I want to know Sx_t(f).
    Is there a relation between the power spectral density Sx(f) of the
    symbol x(t) and the power spectral of the signal transmitted Sx_t(f)???

    Normally, one calculate Sx(f) and associate it to the power spectral
    density of the signal transmitted... Why?"

    I boil all that down to what I think is the essential
    question which is this: "why is the power spectral
    density of the transmitted data stream the same as the
    power spectral density of a single symbol under the encoding
    scheme described? The missing information is the assumption
    that the input data stream is assumed to be random and therefore
    "white" (flat with frequency).








     
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