# How to draw a spectrum of Pulse Amplitude Modulated signal?

Started by May 7, 2007
Hello all!

I have a school project to do about PAM (Pulse Amplitude Modulation),
but these stuff aren't explained at all in our school class. I tried
anything. Even in Wikipedia there was almost nothing useful about PAM.
I'm given a triangle signal with period=T,amplitude=A and a pulse
(square) carrier signal at 1MHz.
I can write this signal in a mathematical way, but I don't know how to
find its spectrum.
I have 2 things to do:
1) Write down an algoritm to calculate the spectrum of the modulated
signal.
2) Draw the spectrum of the modulated signal up to 6MHz.

If you help me with 1) I'll be able to do 2)

Thanks!


On May 7, 8:35 am, ivanat...@gmail.com wrote:
> Hello all!
>
> I have a school project to do about PAM (Pulse Amplitude Modulation),
> but these stuff aren't explained at all in our school class. I tried
> anything. Even in Wikipedia there was almost nothing useful about PAM.
> I'm given a triangle signal with period=T,amplitude=A and a pulse
> (square) carrier signal at 1MHz.
> I can write this signal in a mathematical way, but I don't know how to
> find its spectrum.
> I have 2 things to do:
> 1) Write down an algoritm to calculate the spectrum of the modulated
> signal.
> 2) Draw the spectrum of the modulated signal up to 6MHz.
>
> If you help me with 1) I'll be able to do 2)
>
> Thanks!

Thanks for being honest that this is a school project.

To go from the time domain to the frequency domain, you will need to
do a Fourier transform. Hopefully your instructor has covered Fourier
transforms, Discrete Fourier Transforms, and Fast Fourier Transforms.

In the field of Fourier transformation, there are some useful theorems
for shifting and modulation. You should be able to use these to
accomplish what you need. There is even a theorem for the spectrum of
periodic signals.

I hope this points you in the right direction.  Has your instructor
worked a similar problem during lecture? If so try to follow his
steps. But also look at what tools (theorems) he has given you. I
don't think he expects you to invent something new.

Clay


The truth is that our educational system sux - there are too few
lessons to cover the whole material, and even less hours of exercises.
When we are on exercies the only thing we do is clicking in Mathlab
and entering data, and then Mathlab script shows some specter or
graph... And our teacher thinks it is worthless to explain anything at
all 'becouse if you really need it you will find it somewhere'... Most
of my classmates pay to someone to make their school projects but I'm
trying to actually learn something.

On the subject: I know Fourier Transform, and can do FT on my signal
to get it as a sum of coss and sins. We have some theorems that I'm
sure won't go here: about how the FT changes when the signal is
shifted left or right, or if its amplitude is being multiplicated by a
fixed number...
I've seen FFT and DFT only on Mathcad examples and I don't really know
how to use them.

So as far as I undestood by now - I will need a Fourier Transform of
my signal, take frequencies of the gradients from there and here is a
part of my spectrum?

Thank you for your quick response.


On May 7, 10:00 am, ivanat...@gmail.com wrote:
> The truth is that our educational system sux - there are too few
> lessons to cover the whole material, and even less hours of exercises.
> When we are on exercies the only thing we do is clicking in Mathlab
> and entering data, and then Mathlab script shows some specter or
> graph... And our teacher thinks it is worthless to explain anything at
> all 'becouse if you really need it you will find it somewhere'... Most
> of my classmates pay to someone to make their school projects but I'm
> trying to actually learn something.
>
> On the subject: I know Fourier Transform, and can do FT on my signal
> to get it as a sum of coss and sins. We have some theorems that I'm
> sure won't go here: about how the FT changes when the signal is
> shifted left or right, or if its amplitude is being multiplicated by a
> fixed number...
> I've seen FFT and DFT only on Mathcad examples and I don't really know
> how to use them.
>
> So as far as I undestood by now - I will need a Fourier Transform of
> my signal, take frequencies of the gradients from there and here is a
> part of my spectrum?
>
> Thank you for your quick response.

A book that will likely help you a lot is Paul Nahin's "Dr. Euler's
Fabulous Formula." It covers all of the essential elements you need
and then some. It also shows quite a few tricks.

Clay


I've read some reviews on that book and I want to find it. I checked
in some libraries but no one have it. I wish I had a CC to order it
from Amazon...
Do you have this book in some sort of e-book variant (PDF, html or
etc)?


<ivanatora@gmail.com> wrote in message
> Hello all!
>
> I have a school project to do about PAM (Pulse Amplitude Modulation),
> but these stuff aren't explained at all in our school class. I tried
> anything. Even in Wikipedia there was almost nothing useful about PAM.
> I'm given a triangle signal with period=T,amplitude=A and a pulse
> (square) carrier signal at 1MHz.
> I can write this signal in a mathematical way, but I don't know how to
> find its spectrum.
> I have 2 things to do:
> 1) Write down an algoritm to calculate the spectrum of the modulated
> signal.
> 2) Draw the spectrum of the modulated signal up to 6MHz.
>
> If you help me with 1) I'll be able to do 2)
>
> Thanks!
>

Well one way to do this is if you can write this signal in a mathematical
way you can generate samples of that signal, that you can later subject to a
frequency transform. If your interested in frequencies up to 6 MHz then
Nyquist says you need a sample rate of at least 12MHz (I personally would go
beyond just Nyquist).  Then generate an array of samples with respect to
time. i.e. t=0 t=8.3E-8 t=1.66E-7 t=2.5E-7 are the first four samples at
Nyquist. Then you can do a DFT or FFT on the samples to get the spectrum.
Whatever the required frequency resolution will determine how many samples
are generated.

Thomas Magma


On May 7, 11:53 am, ivanat...@gmail.com wrote:
> I've read some reviews on that book and I want to find it. I checked
> in some libraries but no one have it. I wish I had a CC to order it
> from Amazon...
> Do you have this book in some sort of e-book variant (PDF, html or
> etc)?

I have the book in a quaint form - hardcover. I bought it in a
bookstore (Barnes & Noble) last summer. I have other books by Nahin,
so I was sure this would be a keeper, and it is.

Clay

On your original problem, find the FT of a solitary pulse. Then think
about how you would find the FT of two pulses side by side using the
Heaviside shifting theorem. Then extend the idea to an infinity of
pulses.


Clay wrote:
> On May 7, 11:53 am, ivanat...@gmail.com wrote:
>> I've read some reviews on that book and I want to find it. I checked
>> in some libraries but no one have it. I wish I had a CC to order it
>> from Amazon...
>> Do you have this book in some sort of e-book variant (PDF, html or
>> etc)?
>
> I have the book in a quaint form - hardcover. I bought it in a
> bookstore (Barnes & Noble) last summer. I have other books by Nahin,
> so I was sure this would be a keeper, and it is.
>
> Clay
>
> On your original problem, find the FT of a solitary pulse. Then think
> about how you would find the FT of two pulses side by side using the
> Heaviside shifting theorem. Then extend the idea to an infinity of
> pulses.

I don't think a PAM spectrum can be deduced in the abstract. One needs
the modulating signal, just as for AM or FM. Adjacent pulses will have
different amplitudes. (The duty cycle needs to be specified too, and in
practice, the splatter filter will round the corners of the pulses.

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

On May 7, 9:02 pm, Jerry Avins <j...@ieee.org> wrote:
> Clay wrote:
> > On May 7, 11:53 am, ivanat...@gmail.com wrote:
> >> I've read some reviews on that book and I want to find it. I checked
> >> in some libraries but no one have it. I wish I had a CC to order it
> >> from Amazon...
> >> Do you have this book in some sort of e-book variant (PDF, html or
> >> etc)?
>
> > I have the book in a quaint form - hardcover. I bought it in a
> > bookstore (Barnes & Noble) last summer. I have other books by Nahin,
> > so I was sure this would be a keeper, and it is.
>
> > Clay
>
> > On your original problem, find the FT of a solitary pulse. Then think
> > about how you would find the FT of two pulses side by side using the
> > Heaviside shifting theorem. Then extend the idea to an infinity of
> > pulses.
>
> I don't think a PAM spectrum can be deduced in the abstract. One needs
> the modulating signal, just as for AM or FM. Adjacent pulses will have
> different amplitudes. (The duty cycle needs to be specified too, and in
> practice, the splatter filter will round the corners of the pulses.
>
> Jerry
> --

He just ends up with a summation where each term is weighted by the
corresponding PAM value. For special sets of PAM values, the summation
can be reduced.

Clay


Clay wrote:

...

>> I don't think a PAM spectrum can be deduced in the abstract. One needs
>> the modulating signal, just as for AM or FM. Adjacent pulses will have
>> different amplitudes. (The duty cycle needs to be specified too, and in
>> practice, the splatter filter will round the corners of the pulses.
>>
>> Jerry
>> --
>
> He just ends up with a summation where each term is weighted by the
> corresponding PAM value. For special sets of PAM values, the summation
> can be reduced.

Sure. There are even some interesting observations to be made about the
overall statistical properties of the summation. There are some spectral
effects that don't appear when all the pulses have the same amplitude.

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

On May 8, 4:03 pm, ivanat...@gmail.com wrote:
> > Let's start with the basics.  What is the definition of the spectrum
> > of a signal, say x(t)?
>
> Lets say spectrum is when x(t) is discomosed to its simple composits -
> we call them harmonics. So these all harmonics are plotted with their
> amplitudes against frequency.
> So drawing a spectrum is to show these harmonics (Amplitude-Frequency
> Diagram).
> Am I correct? :D

No. Let's be formal here.  I think that's why you are having a hard
time finding the answer to your question.  Although I did cheat a
little
and abused the notation here.  Usually a PAM signal is taken to be
a signal determined by a sequence of random symbols.
In this case you are confusing "the spectrum of a signal" with
"the Fourier transform of a signal".  These are two different things.

Consider a *random* signal x(t) = \sum_n a_n g(t-nT), where a_n
is a sequence of random (PAM) symbols, and g(.) is the modulation
pulse shape.

The spectrum of this signal is the expected value of its square
magnitude in the frequency domain, also called its power spectral
density.

Are you still with me here?

You have the right intuition perhaps, but I think the problem is that
you

probably to lookup a digital communications textbook.  Where I
am trying to steer this discussion is almost always background
material reviewed in a digital comm text.

Julius


ivanatora@gmail.com wrote:
>> Let's start with the basics.  What is the definition of the spectrum
>> of a signal, say x(t)?
>>
> Lets say spectrum is when x(t) is discomosed to its simple composits -
> we call them harmonics. So these all harmonics are plotted with their
> amplitudes against frequency.
> So drawing a spectrum is to show these harmonics (Amplitude-Frequency
> Diagram).
> Am I correct? :D

Yes. That's what spectrum usually means. The amplitudes of the pulses
vary depending on the data transmitted, so the harmonic amplitudes vary
also from pulse to pulse. As a result, harmonics that would cancel when
all pulses are the same will appear in the spectrum.

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

> Let's start with the basics.  What is the definition of the spectrum
> of a signal, say x(t)?
>
Lets say spectrum is when x(t) is discomosed to its simple composits -
we call them harmonics. So these all harmonics are plotted with their
amplitudes against frequency.
So drawing a spectrum is to show these harmonics (Amplitude-Frequency
Diagram).
Am I correct? :D


On May 7, 7:35 am, ivanat...@gmail.com wrote:
> Hello all!
>
> I have a school project to do about PAM (Pulse Amplitude Modulation),
> but these stuff aren't explained at all in our school class. I tried
> anything. Even in Wikipedia there was almost nothing useful about PAM.
> I'm given a triangle signal with period=T,amplitude=A and a pulse
> (square) carrier signal at 1MHz.
> I can write this signal in a mathematical way, but I don't know how to
> find its spectrum.
> I have 2 things to do:
> 1) Write down an algoritm to calculate the spectrum of the modulated
> signal.
> 2) Draw the spectrum of the modulated signal up to 6MHz.
>
> If you help me with 1) I'll be able to do 2)
>
> Thanks!

Let's start with the basics.  What is the definition of the spectrum
of a signal, say x(t)?

Julius


I've know that Mathcad can do some sorts of Fourier Transforms, but I
can't really understand how these functions work. I'll be glad to use
them to verify my calculations...
There is a built-in function fft() (and there are ifft() for inverse
transform and dfft() for discrete transform) but I can't understant
what data should I give to it. In the Mathcad help there is written
fft() takes a vector (a one column matrix?) as input, but what should
this vector contain?


Clay wrote:

...

>> I don't think a PAM spectrum can be deduced in the abstract. One needs
>> the modulating signal, just as for AM or FM. Adjacent pulses will have
>> different amplitudes. (The duty cycle needs to be specified too, and in
>> practice, the splatter filter will round the corners of the pulses.
>>
>> Jerry
>> --
>
> He just ends up with a summation where each term is weighted by the
> corresponding PAM value. For special sets of PAM values, the summation
> can be reduced.

Sure. There are even some interesting observations to be made about the
overall statistical properties of the summation. There are some spectral
effects that don't appear when all the pulses have the same amplitude.

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

On May 7, 9:02 pm, Jerry Avins <j...@ieee.org> wrote:
> Clay wrote:
> > On May 7, 11:53 am, ivanat...@gmail.com wrote:
> >> I've read some reviews on that book and I want to find it. I checked
> >> in some libraries but no one have it. I wish I had a CC to order it
> >> from Amazon...
> >> Do you have this book in some sort of e-book variant (PDF, html or
> >> etc)?
>
> > I have the book in a quaint form - hardcover. I bought it in a
> > bookstore (Barnes & Noble) last summer. I have other books by Nahin,
> > so I was sure this would be a keeper, and it is.
>
> > Clay
>
> > On your original problem, find the FT of a solitary pulse. Then think
> > about how you would find the FT of two pulses side by side using the
> > Heaviside shifting theorem. Then extend the idea to an infinity of
> > pulses.
>
> I don't think a PAM spectrum can be deduced in the abstract. One needs
> the modulating signal, just as for AM or FM. Adjacent pulses will have
> different amplitudes. (The duty cycle needs to be specified too, and in
> practice, the splatter filter will round the corners of the pulses.
>
> Jerry
> --

He just ends up with a summation where each term is weighted by the
corresponding PAM value. For special sets of PAM values, the summation
can be reduced.

Clay


Clay wrote:
> On May 7, 11:53 am, ivanat...@gmail.com wrote:
>> I've read some reviews on that book and I want to find it. I checked
>> in some libraries but no one have it. I wish I had a CC to order it
>> from Amazon...
>> Do you have this book in some sort of e-book variant (PDF, html or
>> etc)?
>
> I have the book in a quaint form - hardcover. I bought it in a
> bookstore (Barnes & Noble) last summer. I have other books by Nahin,
> so I was sure this would be a keeper, and it is.
>
> Clay
>
> On your original problem, find the FT of a solitary pulse. Then think
> about how you would find the FT of two pulses side by side using the
> Heaviside shifting theorem. Then extend the idea to an infinity of
> pulses.

I don't think a PAM spectrum can be deduced in the abstract. One needs
the modulating signal, just as for AM or FM. Adjacent pulses will have
different amplitudes. (The duty cycle needs to be specified too, and in
practice, the splatter filter will round the corners of the pulses.

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

On May 7, 11:53 am, ivanat...@gmail.com wrote:
> I've read some reviews on that book and I want to find it. I checked
> in some libraries but no one have it. I wish I had a CC to order it
> from Amazon...
> Do you have this book in some sort of e-book variant (PDF, html or
> etc)?

I have the book in a quaint form - hardcover. I bought it in a
bookstore (Barnes & Noble) last summer. I have other books by Nahin,
so I was sure this would be a keeper, and it is.

Clay

On your original problem, find the FT of a solitary pulse. Then think
about how you would find the FT of two pulses side by side using the
Heaviside shifting theorem. Then extend the idea to an infinity of
pulses.


<ivanatora@gmail.com> wrote in message
> Hello all!
>
> I have a school project to do about PAM (Pulse Amplitude Modulation),
> but these stuff aren't explained at all in our school class. I tried
> anything. Even in Wikipedia there was almost nothing useful about PAM.
> I'm given a triangle signal with period=T,amplitude=A and a pulse
> (square) carrier signal at 1MHz.
> I can write this signal in a mathematical way, but I don't know how to
> find its spectrum.
> I have 2 things to do:
> 1) Write down an algoritm to calculate the spectrum of the modulated
> signal.
> 2) Draw the spectrum of the modulated signal up to 6MHz.
>
> If you help me with 1) I'll be able to do 2)
>
> Thanks!
>

Well one way to do this is if you can write this signal in a mathematical
way you can generate samples of that signal, that you can later subject to a
frequency transform. If your interested in frequencies up to 6 MHz then
Nyquist says you need a sample rate of at least 12MHz (I personally would go
beyond just Nyquist).  Then generate an array of samples with respect to
time. i.e. t=0 t=8.3E-8 t=1.66E-7 t=2.5E-7 are the first four samples at
Nyquist. Then you can do a DFT or FFT on the samples to get the spectrum.
Whatever the required frequency resolution will determine how many samples
are generated.

Thomas Magma