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### Network # Spectral density of a digital signal

P

#### pozz

Jan 1, 1970
0
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?

R

#### Roy McCammon

Jan 1, 1970
0
pozz said:
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?

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)

C

#### cirip

Jan 1, 1970
0
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)

C

#### cirip

Jan 1, 1970
0
Hi Roy,
Please see in line.

OOoooops! I should have said hello pozz

R

#### Roy McCammon

Jan 1, 1970
0
cirip said:
I thought the PSD of the random rectangular signal is the sinc function.

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

C

#### cirip cipcirip

Jan 1, 1970
0
Hello Roy,

Roy McCammon said:
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

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

C

#### cirip cipcirip

Jan 1, 1970
0
Hey pozz, is this discussion answering your questions or we're just barking
in the dark? )

Cirip

P

#### pozz

Jan 1, 1970
0
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)?

R

#### Roy McCammon

Jan 1, 1970
0
pozz said:
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)?

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

C

#### cirip

Jan 1, 1970
0
Hello Roy and pozz,

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

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

Thank you,
Cirip

Roy McCammon said:
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)?

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

R

#### Roy McCammon

Jan 1, 1970
0
cirip said:
Hello Roy and pozz,

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

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

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).

Thank you,
Cirip

pozz wrote:

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)?

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

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