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

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.

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)