Proakis Derivation

PeteS <peter.smith8380@ntlworld.com> writes:

> In years gone by I taught technically, so I am used to 'hunting down'
> the derivations. In this case, he substituted *twice*, so it's not
> surprising you got caught :)

I don't think it was simply a matter of skipping a step in a
derivation. I think it was a problem of notational inconsistency 
and notational "overloading."

In the same equation he mixes the use of theta(t), which is the phase
component of the polar form of the bandpass signal (a(t)*e(j*(omega*t
+ theta(t)))), with |s_l(t)|, where s_l(t) is the complex lowpass
signal. It would have been more consistent to write a(t) instead of
|s_l(t)|.

If you combine this with the common use of theta as a random phasse
element (e.g., the receiver's carrier phase offset), then you start
to get the dimension of the confusion.
-- 
%  Randy Yates                  % "My Shangri-la has gone away, fading like 
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%%%% <yates@ieee.org>           % 'Shangri-La', *A New World Record*, ELO
http://home.earthlink.net/~yatescr

Randy Yates wrote:
> PeteS <peter.smith8380@ntlworld.com> writes:
> 
>> In years gone by I taught technically, so I am used to 'hunting down'
>> the derivations. In this case, he substituted *twice*, so it's not
>> surprising you got caught :)
> 
> I don't think it was simply a matter of skipping a step in a
> derivation. I think it was a problem of notational inconsistency 
> and notational "overloading."
> 
> In the same equation he mixes the use of theta(t), which is the phase
> component of the polar form of the bandpass signal (a(t)*e(j*(omega*t
> + theta(t)))), with |s_l(t)|, where s_l(t) is the complex lowpass
> signal. It would have been more consistent to write a(t) instead of
> |s_l(t)|.
> 
> If you combine this with the common use of theta as a random phasse
> element (e.g., the receiver's carrier phase offset), then you start
> to get the dimension of the confusion.

I completely agree, as it happens. I am not unfamiliar with the subject 
matter and it took me 10 minutes or so of thinking and looking before I 
saw how he had jumped there.

The equations were perfectly valid, but, like you, I was confused as to 
how they were *derived* (which is a requirement in a text).

Cheers

PeteS

Randy Yates wrote:
> PeteS <peter.smith8380@ntlworld.com> writes:
> 
>> Randy Yates wrote:
>>> Hi Folks,
>>> I'm having an embarrassingly hard time deriving what should be
>>> a simple result in [proakiscomm]. Could someone please show me
>>> how he gets from equation 4.1-22 to 4.1-23? This is the equation
>>> for the energy in a bandpass signal. I don't see where the phi(t)
>>> comes from, for one thing. I have checked his errata at
>>> http://www.mhhe.com/engcs/electrical/proakis/errata.mhtml
>>> and find nothing referring to this equation.
>>> Any help would be appreciated.
>>> --Randy
>>> @BOOK{proakiscomm,
>>>   title = "{Digital Communications}",
>>>   author = "John~G.~Proakis",
>>>   publisher = "McGraw-Hill",
>>>   edition = "fourth",
>>>   year = "2001"}
>>>
>> I pulled out my handy copy of Proakis [really, I have one ;)] and if
>> you jump back a page to 4.1.18 he expands s(t) to be equal to three
>> representations, and I think you will find the answer there :)
> 
> Got it, Peter! Thanks for steering me straight.  Man, that really
> threw me! Kept thinking theta(t) was a random phase and not part of
> the signal. 

Anyway - glad to be of help!

Cheers

Peter