Bandwidth of PN sequence

I have a simulator that determines the power consumed of a ZigBee
CC2420 based on different parameters.

What I am currently trying to do is by inputting predetermined PN
sequences into the simulator, to generate a trend of the bandwidth of
the particular PN sequences vs. the power consumed of the circuit. The
problem resides in calculating the frequency response and in turn the
bandwidth of the PN sequences.

First, I modulated a half-sine wave carrier with different PN sequence
such that:
if bit = 1 --> positive half cycle
if bit = 0 --> negative half cycle

and calculated the fft in matlab of the resulting waveform and found
that the result only depended on the carrier frequency and not the PN
sequences. Then, I just took the FFT (again in matlab) of just the PN
sequences and got no frequency (DC). Maybe I am approaching this
problem incorrectly. Is there a way of getting the bandwidth (0 -->
pi/2) of a particular bit pattern, such that I can say that 010100100
has a particular bandwidth? I have run out of ideas and would greatly
appreciate any help. 

Thank You
Mohammed Billoo

billoo wrote:

   ...

> if bit = 1 --> positive half cycle
> if bit = 0 --> negative half cycle

This is an extreme form of phase modulation and creates an RF signal of 
very great bandwidth. The power should be independent of the bit 
sequence. During a long series of ones or zeros, the lowest broadcast 
frequency will be twice that of the sinusoidal carrier, and there will 
be many harmonics. (It is the waveform of a full-wave rectifier: for RF, 
very ugly.)

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

"billoo" <mohammed.billoo@gmail.com> wrote in message 
news:1153948710.244919.15290@h48g2000cwc.googlegroups.com...
>I have a simulator that determines the power consumed of a ZigBee
> CC2420 based on different parameters.
>
> What I am currently trying to do is by inputting predetermined PN
> sequences into the simulator, to generate a trend of the bandwidth of
> the particular PN sequences vs. the power consumed of the circuit. The
> problem resides in calculating the frequency response and in turn the
> bandwidth of the PN sequences.
>
> First, I modulated a half-sine wave carrier with different PN sequence
> such that:
> if bit = 1 --> positive half cycle
> if bit = 0 --> negative half cycle
>
> and calculated the fft in matlab of the resulting waveform and found
> that the result only depended on the carrier frequency and not the PN
> sequences. Then, I just took the FFT (again in matlab) of just the PN
> sequences and got no frequency (DC). Maybe I am approaching this
> problem incorrectly. Is there a way of getting the bandwidth (0 -->
> pi/2) of a particular bit pattern, such that I can say that 010100100
> has a particular bandwidth? I have run out of ideas and would greatly
> appreciate any help.
>
> Thank You
> Mohammed Billoo
>

I'm not sure if I understand the question exactly, but here it goes anyways. 
If you are talking about BPSK with a constant amplitude envelope (in the 
time domain), then varying the bit rate (bandwidth) should have little 
effect on the current consumption of the device. It simply spreads the 
available power over a greater spectrum (in the frequency domain). That is 
until the bit rate becomes so great that the phase transitions are a 
significant portion of the modulation waveform. This will load the output 
transistors and the output power will begin to roll off. What the current 
does at this point is really device dependant.

Thomas 

billoo wrote:
> I have a simulator that determines the power consumed of a ZigBee
> CC2420 based on different parameters.
>
> What I am currently trying to do is by inputting predetermined PN
> sequences into the simulator, to generate a trend of the bandwidth of
> the particular PN sequences vs. the power consumed of the circuit. The
> problem resides in calculating the frequency response and in turn the
> bandwidth of the PN sequences.
>
> First, I modulated a half-sine wave carrier with different PN sequence
> such that:
> if bit = 1 --> positive half cycle
> if bit = 0 --> negative half cycle
>
> and calculated the fft in matlab of the resulting waveform and found
> that the result only depended on the carrier frequency and not the PN
> sequences. Then, I just took the FFT (again in matlab) of just the PN
> sequences and got no frequency (DC). Maybe I am approaching this
> problem incorrectly. Is there a way of getting the bandwidth (0 -->
> pi/2) of a particular bit pattern, such that I can say that 010100100
> has a particular bandwidth? I have run out of ideas and would greatly
> appreciate any help.

Hello Mohammed,

A PN sequence can be viewed as simple binary PAM (pulse amplitude
modulation) modulation. The power spectral density of binary PAM is
derived in Proakis [proakiscomm] as

  Syy(f) = (1/T) * |G(f)|^2 * Sii(f),

where Sii(f) is the PSD (power spectral density) of the information
sequence, T is the symbol period, and G(f) is the spectrum of the
pulse shape g(t). In your case g(t) is the half-sinusoid.

In your case the information sequence I[n] is deterministic and
periodic, so the autocorrelation sequence is just the symbol sequence
and the power spectrum is the Fourier series of this periodic
sequence.  Thus Syy(f) will be a line spectrum with magnitudes that
depend on the Fourier series Sii(f) (or Sii(k/T) and the pulse shape
spectrum G(f).

--Randy

Randy Yates wrote:
> [...]
> In your case the information sequence I[n] is deterministic and
> periodic, so the autocorrelation sequence is just the symbol sequence

I'm sorry Mohammed, but I made an error on this part. First of
all, the autocorrelation sequence is not the symbol sequence.
In fact, since this isn't a stationary signal, the autocorrelation
sequence is a function of two variables and does not have
a power spectrum. It is, however, cyclo-stationary, and I
think there is a way to convert such a process into a
stationary process so that you can obtain the PSD. If
interested, reply for details.

--Randy

billoo wrote:

> I have a simulator that determines the power consumed of a ZigBee
> CC2420 based on different parameters.
> 
> What I am currently trying to do is by inputting predetermined PN
> sequences into the simulator, to generate a trend of the bandwidth of
> the particular PN sequences vs. the power consumed of the circuit. The
> problem resides in calculating the frequency response and in turn the
> bandwidth of the PN sequences.
> 
> First, I modulated a half-sine wave carrier with different PN sequence
> such that:
> if bit = 1 --> positive half cycle
> if bit = 0 --> negative half cycle
> 
> and calculated the fft in matlab of the resulting waveform and found
> that the result only depended on the carrier frequency and not the PN
> sequences. Then, I just took the FFT (again in matlab) of just the PN
> sequences and got no frequency (DC). Maybe I am approaching this
> problem incorrectly. Is there a way of getting the bandwidth (0 -->
> pi/2) of a particular bit pattern, such that I can say that 010100100
> has a particular bandwidth? I have run out of ideas and would greatly
> appreciate any help. 
> 
> Thank You
> Mohammed Billoo
> 
If you parse out the name 'PN' you'll see that it stands for 'Pseudo 
Noise', i.e. fake noise.  It's fake because it's deterministic, but it's 
noise because it has statistics that approach 'perfect' noise.

If you had a sequence of binary numbers that were completely random, 
they'd always take on values of 0 or 1 but each sample would be 
completely independent of all the other samples in the sequence.  If you 
took the Fourier transform of such a sequence it'd have an impulse at f 
= 0 (because a sequence that takes on values of 0 and 1 must have some 
average value), but outside of that it'd have a constant power spectral 
density from -pi to pi.

If your sequence of random binary numbers were completely random, if you 
changed it to take on values of -1 and 1, and if the probability of 
seeing -1 and 1 were each exactly 1/2, then the power spectral density 
would be constant across the board.  This is sort of what you're doing 
when you modulate your carrier, except you're also stretching your 
numbers (which only exist at discrete points) into shaped pulses.

If you use a pseudo random sequence instead of an actual random 
sequence, and if it's a _good_ pseudo random sequence, then its 
statistics will be close to a real random sequence.  That means that its 
Fourier transform will also be flat (possibly with an impulse at DC 
because of the 0,1 thing).

If you then use your PN sequence for modulation you'll end up 
multiplying the spectrum of your sequence (which is flat) by the 
spectral shaping of your pulse shape, which is just the Fourier 
transform of a half-sine.

The upshot of all of this is that for a _true_ pseudo random sequence 
you're not going to do much frequency shaping!  There are things that 
you _could_ do to shape the noise -- you can come up with sequences of 
1's and 0's that have a lot of low frequency content, or that have _no_ 
low frequency content, but the result won't really be 'PN'.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

Thank you for your perspectives, but I feel I have phrased the question
improperly.

By power consumption, I mean regarding the power consumption of the
individual components (DAC, LPF, PLL) as I change the PN sequence, not
a PSD of the PN sequence. I know for sure that the power consumption of
the DAC will change based on the # of 1s and 0s in the PN sequence,
since I have modelled the DAC as an R-2R ladder.

My main concern is to determine the bandwidth of  PN sequences that are
predetermined (i.e. they're stored in a lookup table in ROM and I just
map symbol to a PN sequence). I have tried tp just brute force it:

x = [1 0 1 0 1 0 1 0];
fftx = abs(fft(x));
.
.
.

in Matlab and gotten a frequency response of 0 (i.e. DC). Then I tried
using the PN sequence to modulate a half-sine wave and gotten that the
frequency response is just based on the carrier frequency and not the
PN sequence at all:

t = linspace(0,1,3000);
s = sin(pi*30*t); % only want 1/2 cycle;
pulse = ones(1,300);
x = [1*pulse -1*pulse 1*pulse -1*pulse 1*pulse -1*pulse 1*pulse
-1*pulse 1*pulse -1*pulse];
input = x.*s;
fftin = abs(fft(input));
.
.
.
So my question is there a way to find the bandwidth or a frequency
response of a PN sequence or a digital signal (bit pattern)? By the
way, all of this is just QPSK.

Thank You
Mohammed Billoo

Randy Yates wrote:
> Randy Yates wrote:
> > [...]
> > In your case the information sequence I[n] is deterministic and
> > periodic, so the autocorrelation sequence is just the symbol sequence
>
> I'm sorry Mohammed, but I made an error on this part. First of
> all, the autocorrelation sequence is not the symbol sequence.
> In fact, since this isn't a stationary signal, the autocorrelation
> sequence is a function of two variables and does not have
> a power spectrum. It is, however, cyclo-stationary, and I
> think there is a way to convert such a process into a
> stationary process so that you can obtain the PSD. If
> interested, reply for details.
> 
> --Randy

Okay, Tim you mentioned that I want a PN sequence with low frequency
content or no frequency content (I dont mind the randomness at the
moment). That's the thing that I am exactly looking for. Let's say I
have a PN sequence, how do I know whether it has low frequency, high
frequency, or no frequency content at all. I apologize because I am
about to take DSP next semester and have very little knowledge
regarding digital signals.

Thank You
Mohammed Billoo

Tim Wescott wrote:
> billoo wrote:
>
> > I have a simulator that determines the power consumed of a ZigBee
> > CC2420 based on different parameters.
> >
> > What I am currently trying to do is by inputting predetermined PN
> > sequences into the simulator, to generate a trend of the bandwidth of
> > the particular PN sequences vs. the power consumed of the circuit. The
> > problem resides in calculating the frequency response and in turn the
> > bandwidth of the PN sequences.
> >
> > First, I modulated a half-sine wave carrier with different PN sequence
> > such that:
> > if bit = 1 --> positive half cycle
> > if bit = 0 --> negative half cycle
> >
> > and calculated the fft in matlab of the resulting waveform and found
> > that the result only depended on the carrier frequency and not the PN
> > sequences. Then, I just took the FFT (again in matlab) of just the PN
> > sequences and got no frequency (DC). Maybe I am approaching this
> > problem incorrectly. Is there a way of getting the bandwidth (0 -->
> > pi/2) of a particular bit pattern, such that I can say that 010100100
> > has a particular bandwidth? I have run out of ideas and would greatly
> > appreciate any help.
> >
> > Thank You
> > Mohammed Billoo
> >
> If you parse out the name 'PN' you'll see that it stands for 'Pseudo
> Noise', i.e. fake noise.  It's fake because it's deterministic, but it's
> noise because it has statistics that approach 'perfect' noise.
>
> If you had a sequence of binary numbers that were completely random,
> they'd always take on values of 0 or 1 but each sample would be
> completely independent of all the other samples in the sequence.  If you
> took the Fourier transform of such a sequence it'd have an impulse at f
> = 0 (because a sequence that takes on values of 0 and 1 must have some
> average value), but outside of that it'd have a constant power spectral
> density from -pi to pi.
>
> If your sequence of random binary numbers were completely random, if you
> changed it to take on values of -1 and 1, and if the probability of
> seeing -1 and 1 were each exactly 1/2, then the power spectral density
> would be constant across the board.  This is sort of what you're doing
> when you modulate your carrier, except you're also stretching your
> numbers (which only exist at discrete points) into shaped pulses.
>
> If you use a pseudo random sequence instead of an actual random
> sequence, and if it's a _good_ pseudo random sequence, then its
> statistics will be close to a real random sequence.  That means that its
> Fourier transform will also be flat (possibly with an impulse at DC
> because of the 0,1 thing).
>
> If you then use your PN sequence for modulation you'll end up
> multiplying the spectrum of your sequence (which is flat) by the
> spectral shaping of your pulse shape, which is just the Fourier
> transform of a half-sine.
>
> The upshot of all of this is that for a _true_ pseudo random sequence
> you're not going to do much frequency shaping!  There are things that
> you _could_ do to shape the noise -- you can come up with sequences of
> 1's and 0's that have a lot of low frequency content, or that have _no_
> low frequency content, but the result won't really be 'PN'.
>
> --
>
> Tim Wescott
> Wescott Design Services
> http://www.wescottdesign.com
>
> Posting from Google?  See http://cfaj.freeshell.org/google/
>
> "Applied Control Theory for Embedded Systems" came out in April.
> See details at http://www.wescottdesign.com/actfes/actfes.html

billoo wrote:
(top posting fixed)
> 
> Tim Wescott wrote:
> 
>>billoo wrote:
>>
>>
>>>I have a simulator that determines the power consumed of a ZigBee
>>>CC2420 based on different parameters.
>>>
>>>What I am currently trying to do is by inputting predetermined PN
>>>sequences into the simulator, to generate a trend of the bandwidth of
>>>the particular PN sequences vs. the power consumed of the circuit. The
>>>problem resides in calculating the frequency response and in turn the
>>>bandwidth of the PN sequences.
>>>
>>>First, I modulated a half-sine wave carrier with different PN sequence
>>>such that:
>>>if bit = 1 --> positive half cycle
>>>if bit = 0 --> negative half cycle
>>>
>>>and calculated the fft in matlab of the resulting waveform and found
>>>that the result only depended on the carrier frequency and not the PN
>>>sequences. Then, I just took the FFT (again in matlab) of just the PN
>>>sequences and got no frequency (DC). Maybe I am approaching this
>>>problem incorrectly. Is there a way of getting the bandwidth (0 -->
>>>pi/2) of a particular bit pattern, such that I can say that 010100100
>>>has a particular bandwidth? I have run out of ideas and would greatly
>>>appreciate any help.
>>>
>>>Thank You
>>>Mohammed Billoo
>>>
>>
>>If you parse out the name 'PN' you'll see that it stands for 'Pseudo
>>Noise', i.e. fake noise.  It's fake because it's deterministic, but it's
>>noise because it has statistics that approach 'perfect' noise.
>>
>>If you had a sequence of binary numbers that were completely random,
>>they'd always take on values of 0 or 1 but each sample would be
>>completely independent of all the other samples in the sequence.  If you
>>took the Fourier transform of such a sequence it'd have an impulse at f
>>= 0 (because a sequence that takes on values of 0 and 1 must have some
>>average value), but outside of that it'd have a constant power spectral
>>density from -pi to pi.
>>
>>If your sequence of random binary numbers were completely random, if you
>>changed it to take on values of -1 and 1, and if the probability of
>>seeing -1 and 1 were each exactly 1/2, then the power spectral density
>>would be constant across the board.  This is sort of what you're doing
>>when you modulate your carrier, except you're also stretching your
>>numbers (which only exist at discrete points) into shaped pulses.
>>
>>If you use a pseudo random sequence instead of an actual random
>>sequence, and if it's a _good_ pseudo random sequence, then its
>>statistics will be close to a real random sequence.  That means that its
>>Fourier transform will also be flat (possibly with an impulse at DC
>>because of the 0,1 thing).
>>
>>If you then use your PN sequence for modulation you'll end up
>>multiplying the spectrum of your sequence (which is flat) by the
>>spectral shaping of your pulse shape, which is just the Fourier
>>transform of a half-sine.
>>
>>The upshot of all of this is that for a _true_ pseudo random sequence
>>you're not going to do much frequency shaping!  There are things that
>>you _could_ do to shape the noise -- you can come up with sequences of
>>1's and 0's that have a lot of low frequency content, or that have _no_
>>low frequency content, but the result won't really be 'PN'.
>>
>>--
>>
>>Tim Wescott
>>Wescott Design Services
>>http://www.wescottdesign.com
>>
>>Posting from Google?  See http://cfaj.freeshell.org/google/
>>
>>"Applied Control Theory for Embedded Systems" came out in April.
>>See details at http://www.wescottdesign.com/actfes/actfes.html
> 
> 
 > Okay, Tim you mentioned that I want a PN sequence with low frequency
 > content or no frequency content (I dont mind the randomness at the
 > moment). That's the thing that I am exactly looking for. Let's say I
 > have a PN sequence, how do I know whether it has low frequency, high
 > frequency, or no frequency content at all. I apologize because I am
 > about to take DSP next semester and have very little knowledge
 > regarding digital signals.
 >
Do a Fourier transform on the sequence itself, preferably after 
modifying it so it goes from -1 to 1 instead of 0 to 1 to get rid of the 
DC content.

I can guarantee you, though, that any PN sequence that's considered 
'good' in general is going to be pretty darn white -- if you want one 
with markedly different spectral characteristics you're going to have to 
make it yourself.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Posting from Google?  See http://cfaj.freeshell.org/google/

"Applied Control Theory for Embedded Systems" came out in April.
See details at http://www.wescottdesign.com/actfes/actfes.html

Random sequences contain all frequencies equally and are considered "white 
noise". This is true to a point when it comes to data modulation. Since most 
modulation schemes have a fixed bit width, it limits the highest fundamental 
frequency to 1/(2*bit width). So if you have a bit rate of 1 MBPS the 
highest fundamental frequency will be 1/(2*1E-6) or 500 KHz. And then 
depending on how fast you transition from bit to bit (phase rise & fall 
times) will determine the amount of harmonic content and spectral spreading 
in the frequency domain.

Thomas

"billoo" <mohammed.billoo@gmail.com> wrote in message 
news:1153973051.598813.204340@b28g2000cwb.googlegroups.com...
> Okay, Tim you mentioned that I want a PN sequence with low frequency
> content or no frequency content (I dont mind the randomness at the
> moment). That's the thing that I am exactly looking for. Let's say I
> have a PN sequence, how do I know whether it has low frequency, high
> frequency, or no frequency content at all. I apologize because I am
> about to take DSP next semester and have very little knowledge
> regarding digital signals.
>
> Thank You
> Mohammed Billoo
>
> Tim Wescott wrote:
>> billoo wrote:
>>
>> > I have a simulator that determines the power consumed of a ZigBee
>> > CC2420 based on different parameters.
>> >
>> > What I am currently trying to do is by inputting predetermined PN
>> > sequences into the simulator, to generate a trend of the bandwidth of
>> > the particular PN sequences vs. the power consumed of the circuit. The
>> > problem resides in calculating the frequency response and in turn the
>> > bandwidth of the PN sequences.
>> >
>> > First, I modulated a half-sine wave carrier with different PN sequence
>> > such that:
>> > if bit = 1 --> positive half cycle
>> > if bit = 0 --> negative half cycle
>> >
>> > and calculated the fft in matlab of the resulting waveform and found
>> > that the result only depended on the carrier frequency and not the PN
>> > sequences. Then, I just took the FFT (again in matlab) of just the PN
>> > sequences and got no frequency (DC). Maybe I am approaching this
>> > problem incorrectly. Is there a way of getting the bandwidth (0 -->
>> > pi/2) of a particular bit pattern, such that I can say that 010100100
>> > has a particular bandwidth? I have run out of ideas and would greatly
>> > appreciate any help.
>> >
>> > Thank You
>> > Mohammed Billoo
>> >
>> If you parse out the name 'PN' you'll see that it stands for 'Pseudo
>> Noise', i.e. fake noise.  It's fake because it's deterministic, but it's
>> noise because it has statistics that approach 'perfect' noise.
>>
>> If you had a sequence of binary numbers that were completely random,
>> they'd always take on values of 0 or 1 but each sample would be
>> completely independent of all the other samples in the sequence.  If you
>> took the Fourier transform of such a sequence it'd have an impulse at f
>> = 0 (because a sequence that takes on values of 0 and 1 must have some
>> average value), but outside of that it'd have a constant power spectral
>> density from -pi to pi.
>>
>> If your sequence of random binary numbers were completely random, if you
>> changed it to take on values of -1 and 1, and if the probability of
>> seeing -1 and 1 were each exactly 1/2, then the power spectral density
>> would be constant across the board.  This is sort of what you're doing
>> when you modulate your carrier, except you're also stretching your
>> numbers (which only exist at discrete points) into shaped pulses.
>>
>> If you use a pseudo random sequence instead of an actual random
>> sequence, and if it's a _good_ pseudo random sequence, then its
>> statistics will be close to a real random sequence.  That means that its
>> Fourier transform will also be flat (possibly with an impulse at DC
>> because of the 0,1 thing).
>>
>> If you then use your PN sequence for modulation you'll end up
>> multiplying the spectrum of your sequence (which is flat) by the
>> spectral shaping of your pulse shape, which is just the Fourier
>> transform of a half-sine.
>>
>> The upshot of all of this is that for a _true_ pseudo random sequence
>> you're not going to do much frequency shaping!  There are things that
>> you _could_ do to shape the noise -- you can come up with sequences of
>> 1's and 0's that have a lot of low frequency content, or that have _no_
>> low frequency content, but the result won't really be 'PN'.
>>
>> --
>>
>> Tim Wescott
>> Wescott Design Services
>> http://www.wescottdesign.com
>>
>> Posting from Google?  See http://cfaj.freeshell.org/google/
>>
>> "Applied Control Theory for Embedded Systems" came out in April.
>> See details at http://www.wescottdesign.com/actfes/actfes.html
>