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Hi,

I'm learning digital communication now. I use Proakis' "Digital
Communications" as my text book. Now several terms confused me, which
are projection, correlation and covariance. These terms appeared in
the book many times, but with different formula. Can any body help me
to understand them? thanks very much.

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On Apr 7, 9:16 am, chenyong20...@gmail.com wrote:
> Hi,
>
> I'm learning digital communication now. I use Proakis' "Digital
> Communications" as my text book. Now several terms confused me, which
> are projection, correlation and covariance. These terms appeared in
> the book many times, but with different formula. Can any body help me
> to understand them? thanks very much.


Chenyong, your question is rather generic. Can you point out the
specific
formulae that are confusing you? It will be easier to clarify specific
instances.

Thanks,
Dilip.

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On 7 Apr, 15:16, chenyong20...@gmail.com wrote:
> Hi,
>
> I'm learning digital communication now. I use Proakis' "Digital
> Communications" as my text book. Now several terms confused me, which
> are projection, correlation and covariance. These terms appeared in
> the book many times, but with different formula.

As others already said, it's hard to answer generic questions.

Having said that, it might be helpful to you that the terms you
mention are generic concepts, which can be used in different
contexts. For instance, correlation and covariance are statistical
terms that are all but omnipresent. Chances are they are used
in almost every context where statistics is used. Different formulas
could also be explained by that one uses different estimators
for the same statistic.

You also already know the concept of projections from
basic vector calculus, where projections are computed
in terms of inner products between vectors. It's the same
general idea in complex-valued N-D space, but the
expressions become a bit more involved, particularly
if you project onto planes instead of vectors.

Rune

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On 4=D4=C27=C8=D5, =CF=C2=CE=E79=CA=B142=B7=D6, Rune Allnor <all...@tele.nt=
nu.no> wrote:
> On 7 Apr, 15:16, chenyong20...@gmail.com wrote:
>
> > Hi,
>
> > I'm learning digital communication now. I use Proakis' "Digital
> > Communications" as my text book. Now several terms confused me, which
> > are projection, correlation and covariance. These terms appeared in
> > the book many times, but with different formula.
>
> As others already said, it's hard to answer generic questions.
>
> Having said that, it might be helpful to you that the terms you
> mention are generic concepts, which can be used in different
> contexts. For instance, correlation and covariance are statistical
> terms that are all but omnipresent. Chances are they are used
> in almost every context where statistics is used. Different formulas
> could also be explained by that one uses different estimators
> for the same statistic.
>
> You also already know the concept of projections from
> basic vector calculus, where projections are computed
> in terms of inner products between vectors. It's the same
> general idea in complex-valued N-D space, but the
> expressions become a bit more involved, particularly
> if you project onto planes instead of vectors.
>
> Rune

Hi Dilip & Rune,

thanks for your reply. Well, I'm learning Proakis' book "Digital
Communications". This book is full of mathematics. There are problems
when I'm reading chapter 5 "Optimum Receivers for the additive white
gaussian noise channel".

1. Projection.
"Suppose the received signal r(t) is passed through a parallel bank of
N cross correlators which basically compute the projection of r(t)
onto the N basis functions {fn(t)}, ..."
Question: the book gives the formula as integral[r(t)*fk(t)] from 0 to
T. Does projection f1 onto f2 means integral[f1*f2]? I can't find this
definition in book and wiki.

2. Covariance
Question: The book gives covariance in one part as E(n1*n2), but I
found another formula in the book as E[(x-mx)*(y-my)]. This confused
me. Can you help me to understand their difference? Thanks very much.

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On 8 Apr, 13:55, chenyong20...@gmail.com wrote:
> On 4=D4=C27=C8=D5, =CF=C2=CE=E79=CA=B142=B7=D6, Rune Allnor <all...@tele.=
ntnu.no> wrote:
>
>
>
>
>
> > On 7 Apr, 15:16, chenyong20...@gmail.com wrote:
>
> > > Hi,
>
> > > I'm learning digital communication now. I use Proakis' "Digital
> > > Communications" as my text book. Now several terms confused me, which
> > > are projection, correlation and covariance. These terms appeared in
> > > the book many times, but with different formula.
>
> > As others already said, it's hard to answer generic questions.
>
> > Having said that, it might be helpful to you that the terms you
> > mention are generic concepts, which can be used in different
> > contexts. For instance, correlation and covariance are statistical
> > terms that are all but omnipresent. Chances are they are used
> > in almost every context where statistics is used. Different formulas
> > could also be explained by that one uses different estimators
> > for the same statistic.
>
> > You also already know the concept of projections from
> > basic vector calculus, where projections are computed
> > in terms of inner products between vectors. It's the same
> > general idea in complex-valued N-D space, but the
> > expressions become a bit more involved, particularly
> > if you project onto planes instead of vectors.
>
> > Rune
>
> Hi Dilip & Rune,
>
> thanks for your reply. Well, I'm learning Proakis' book "Digital
> Communications". This book is full of mathematics. There are problems
> when I'm reading chapter 5 "Optimum Receivers for the additive white
> gaussian noise channel".
>
> 1. Projection.
> "Suppose the received signal r(t) is passed through a parallel bank of
> N cross correlators which basically compute the projection of r(t)
> onto the N basis functions {fn(t)}, ..."
> Question: the book gives the formula as integral[r(t)*fk(t)] from 0 to
> T. Does projection f1 onto f2 means integral[f1*f2]?

Yes, although 'inner product' is a more common term for the
integral. But as you remember for vector calculus, an inner
product between vectors a and b is the projection of a onto b.
Or vice versa.

> I can't find this
> definition in book and wiki.
>
> 2. Covariance
> Question: The book gives covariance in one part as E(n1*n2), but I
> found another formula in the book as E[(x-mx)*(y-my)]. This confused
> me. Can you help me to understand their difference?

Covariance is defined as E[(x-mx)*(y-my)] while correlation is
defined as  E[x*y], the difference being whether the mean is
subtracted or not. When it comes to communications, radar
and sonar, one deals with signals that have zero mean.
Authors often forget about the subtract-the-mean terms,
since these are zero anyway.

It is very common for authors to forget about the formal
difference between correlation and covariance, as well as
the special case of zero-mean signals. Lots of people
use these terms and definitions interchangeably. It is
just human nature, so you will have to keep your eyes
open and look for these kinds of things, and maybe
make the effort to be more precise in your own writings.

Rune

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On 2009-04-08 09:49:46 -0300, Rune Allnor <a...@tele.ntnu.no> said:

> On 8 Apr, 13:55, chenyong20...@gmail.com wrote:
>> On 4月7日, 下午9时42分, Rune Allnor <all...@tele.
> ntnu.no> wrote:
>> 
>> 
>> 
>> 
>> 
>>> On 7 Apr, 15:16, chenyong20...@gmail.com wrote:
>> 
>>>> Hi,
>> 
>>>> I'm learning digital communication now. I use Proakis' "Digital
>>>> Communications" as my text book. Now several terms confused me, which
>>>> are projection, correlation and covariance. These terms appeared in
>>>> the book many times, but with different formula.
>> 
>>> As others already said, it's hard to answer generic questions.
>> 
>>> Having said that, it might be helpful to you that the terms you
>>> mention are generic concepts, which can be used in different
>>> contexts. For instance, correlation and covariance are statistical
>>> terms that are all but omnipresent. Chances are they are used
>>> in almost every context where statistics is used. Different formulas
>>> could also be explained by that one uses different estimators
>>> for the same statistic.
>> 
>>> You also already know the concept of projections from
>>> basic vector calculus, where projections are computed
>>> in terms of inner products between vectors. It's the same
>>> general idea in complex-valued N-D space, but the
>>> expressions become a bit more involved, particularly
>>> if you project onto planes instead of vectors.
>> 
>>> Rune
>> 
>> Hi Dilip & Rune,
>> 
>> thanks for your reply. Well, I'm learning Proakis' book "Digital
>> Communications". This book is full of mathematics. There are problems
>> when I'm reading chapter 5 "Optimum Receivers for the additive white
>> gaussian noise channel".
>> 
>> 1. Projection.
>> "Suppose the received signal r(t) is passed through a parallel bank of
>> N cross correlators which basically compute the projection of r(t)
>> onto the N basis functions {fn(t)}, ..."
>> Question: the book gives the formula as integral[r(t)*fk(t)] from 0 to
>> T. Does projection f1 onto f2 means integral[f1*f2]?
> 
> Yes, although 'inner product' is a more common term for the
> integral. But as you remember for vector calculus, an inner
> product between vectors a and b is the projection of a onto b.
> Or vice versa.
> 
>> I can't find this
>> definition in book and wiki.
>> 
>> 2. Covariance
>> Question: The book gives covariance in one part as E(n1*n2), but I
>> found another formula in the book as E[(x-mx)*(y-my)]. This confused
>> me. Can you help me to understand their difference?
> 
> Covariance is defined as E[(x-mx)*(y-my)] while correlation is
> defined as  E[x*y], the difference being whether the mean is
> subtracted or not. When it comes to communications, radar
> and sonar, one deals with signals that have zero mean.
> Authors often forget about the subtract-the-mean terms,
> since these are zero anyway.

Correlation is a rescaled covariance. Correlations are
between -1 and +1. The scaling is by the product of the two
standard deviations. In the case of a time series with
both autocovariance and autocorrelations the scale factor
is just the autocovariance at time zero.

The appearance of formulae withut a mean being subtracted
is a pretty sure sign that an assumption of zero mean was
stated shortly before, or at the beginning of the book or
paper, the formulae.

Stronger coffee in the morning is called for as you got the
zero mean assumption correct but fell on your face over
the scaling of correlations. Maybe too much ice on the roads
even if spring is coming!

> It is very common for authors to forget about the formal
> difference between correlation and covariance, as well as
> the special case of zero-mean signals. Lots of people
> use these terms and definitions interchangeably. It is
> just human nature, so you will have to keep your eyes
> open and look for these kinds of things, and maybe
> make the effort to be more precise in your own writings.

Correlation seems to be much more common in everyday speech
and means appening together. It is a qualitative notion in
speech with the value between +1 and -1 of little import.
Covariance is more of an arithmetical notion.

> Rune

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On 4=E6=9C=888=E6=97=A5, =E4=B8=8B=E5=8D=889=E6=97=B615=E5=88=86, Gordon Sa=
nde <g.sa...@worldnet.att.net> wrote:
> On 2009-04-08 09:49:46 -0300, Rune Allnor <all...@tele.ntnu.no> said:
>
>
>
> > On 8 Apr, 13:55, chenyong20...@gmail.com wrote:
> >> On 4=E6=9C=887=E6=97=A5, =E4=B8=8B=E5=8D=889=E6=97=B642=E5=88=86, Rune=
 Allnor <all...@tele.
> > ntnu.no> wrote:
>
> >>> On 7 Apr, 15:16, chenyong20...@gmail.com wrote:
>
> >>>> Hi,
>
> >>>> I'm learning digital communication now. I use Proakis' "Digital
> >>>> Communications" as my text book. Now several terms confused me, whic=
h
> >>>> are projection, correlation and covariance. These terms appeared in
> >>>> the book many times, but with different formula.
>
> >>> As others already said, it's hard to answer generic questions.
>
> >>> Having said that, it might be helpful to you that the terms you
> >>> mention are generic concepts, which can be used in different
> >>> contexts. For instance, correlation and covariance are statistical
> >>> terms that are all but omnipresent. Chances are they are used
> >>> in almost every context where statistics is used. Different formulas
> >>> could also be explained by that one uses different estimators
> >>> for the same statistic.
>
> >>> You also already know the concept of projections from
> >>> basic vector calculus, where projections are computed
> >>> in terms of inner products between vectors. It's the same
> >>> general idea in complex-valued N-D space, but the
> >>> expressions become a bit more involved, particularly
> >>> if you project onto planes instead of vectors.
>
> >>> Rune
>
> >> Hi Dilip & Rune,
>
> >> thanks for your reply. Well, I'm learning Proakis' book "Digital
> >> Communications". This book is full of mathematics. There are problems
> >> when I'm reading chapter 5 "Optimum Receivers for the additive white
> >> gaussian noise channel".
>
> >> 1. Projection.
> >> "Suppose the received signal r(t) is passed through a parallel bank of
> >> N cross correlators which basically compute the projection of r(t)
> >> onto the N basis functions {fn(t)}, ..."
> >> Question: the book gives the formula as integral[r(t)*fk(t)] from 0 to
> >> T. Does projection f1 onto f2 means integral[f1*f2]?
>
> > Yes, although 'inner product' is a more common term for the
> > integral. But as you remember for vector calculus, an inner
> > product between vectors a and b is the projection of a onto b.
> > Or vice versa.
>
> >> I can't find this
> >> definition in book and wiki.
>
> >> 2. Covariance
> >> Question: The book gives covariance in one part as E(n1*n2), but I
> >> found another formula in the book as E[(x-mx)*(y-my)]. This confused
> >> me. Can you help me to understand their difference?
>
> > Covariance is defined as E[(x-mx)*(y-my)] while correlation is
> > defined as =C2=A0E[x*y], the difference being whether the mean is
> > subtracted or not. When it comes to communications, radar
> > and sonar, one deals with signals that have zero mean.
> > Authors often forget about the subtract-the-mean terms,
> > since these are zero anyway.
>
> Correlation is a rescaled covariance. Correlations are
> between -1 and +1. The scaling is by the product of the two
> standard deviations. In the case of a time series with
> both autocovariance and autocorrelations the scale factor
> is just the autocovariance at time zero.
>
> The appearance of formulae withut a mean being subtracted
> is a pretty sure sign that an assumption of zero mean was
> stated shortly before, or at the beginning of the book or
> paper, the formulae.
>
> Stronger coffee in the morning is called for as you got the
> zero mean assumption correct but fell on your face over
> the scaling of correlations. Maybe too much ice on the roads
> even if spring is coming!
>
> > It is very common for authors to forget about the formal
> > difference between correlation and covariance, as well as
> > the special case of zero-mean signals. Lots of people
> > use these terms and definitions interchangeably. It is
> > just human nature, so you will have to keep your eyes
> > open and look for these kinds of things, and maybe
> > make the effort to be more precise in your own writings.
>
> Correlation seems to be much more common in everyday speech
> and means appening together. It is a qualitative notion in
> speech with the value between +1 and -1 of little import.
> Covariance is more of an arithmetical notion.
>
> > Rune

Hi Rune and Gordon,

I got your reply and have some clear sight. Thanks for both of you
very much.

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On 8 Apr, 15:15, Gordon Sande <g.sa...@worldnet.att.net> wrote:
> On 2009-04-08 09:49:46 -0300, Rune Allnor <all...@tele.ntnu.no> said:
>
>
>
>
>
> > On 8 Apr, 13:55, chenyong20...@gmail.com wrote:
> >> On 4=E6=9C=887=E6=97=A5, =E4=B8=8B=E5=8D=889=E6=97=B642=E5=88=86, Rune=
 Allnor <all...@tele.
> > ntnu.no> wrote:
>
> >>> On 7 Apr, 15:16, chenyong20...@gmail.com wrote:
>
> >>>> Hi,
>
> >>>> I'm learning digital communication now. I use Proakis' "Digital
> >>>> Communications" as my text book. Now several terms confused me, whic=
h
> >>>> are projection, correlation and covariance. These terms appeared in
> >>>> the book many times, but with different formula.
>
> >>> As others already said, it's hard to answer generic questions.
>
> >>> Having said that, it might be helpful to you that the terms you
> >>> mention are generic concepts, which can be used in different
> >>> contexts. For instance, correlation and covariance are statistical
> >>> terms that are all but omnipresent. Chances are they are used
> >>> in almost every context where statistics is used. Different formulas
> >>> could also be explained by that one uses different estimators
> >>> for the same statistic.
>
> >>> You also already know the concept of projections from
> >>> basic vector calculus, where projections are computed
> >>> in terms of inner products between vectors. It's the same
> >>> general idea in complex-valued N-D space, but the
> >>> expressions become a bit more involved, particularly
> >>> if you project onto planes instead of vectors.
>
> >>> Rune
>
> >> Hi Dilip & Rune,
>
> >> thanks for your reply. Well, I'm learning Proakis' book "Digital
> >> Communications". This book is full of mathematics. There are problems
> >> when I'm reading chapter 5 "Optimum Receivers for the additive white
> >> gaussian noise channel".
>
> >> 1. Projection.
> >> "Suppose the received signal r(t) is passed through a parallel bank of
> >> N cross correlators which basically compute the projection of r(t)
> >> onto the N basis functions {fn(t)}, ..."
> >> Question: the book gives the formula as integral[r(t)*fk(t)] from 0 to
> >> T. Does projection f1 onto f2 means integral[f1*f2]?
>
> > Yes, although 'inner product' is a more common term for the
> > integral. But as you remember for vector calculus, an inner
> > product between vectors a and b is the projection of a onto b.
> > Or vice versa.
>
> >> I can't find this
> >> definition in book and wiki.
>
> >> 2. Covariance
> >> Question: The book gives covariance in one part as E(n1*n2), but I
> >> found another formula in the book as E[(x-mx)*(y-my)]. This confused
> >> me. Can you help me to understand their difference?
>
> > Covariance is defined as E[(x-mx)*(y-my)] while correlation is
> > defined as =C2=A0E[x*y], the difference being whether the mean is
> > subtracted or not. When it comes to communications, radar
> > and sonar, one deals with signals that have zero mean.
> > Authors often forget about the subtract-the-mean terms,
> > since these are zero anyway.
>
> Correlation is a rescaled covariance.

I have some recollection that the 'correlation' is defined
differently, depending on the source. However, my recollections
have been proven wrong in the past, so I have to check
the sources after I get back home.

> Correlations are
> between -1 and +1. The scaling is by the product of the two
> standard deviations. In the case of a time series with
> both autocovariance and autocorrelations the scale factor
> is just the autocovariance at time zero.

This is common in statistics literature, yes. I'm pretty
sure I have seen things be defined differently in DSP
texts - the normalization is not too common there.
But again, I'll have to check my sources.

> The appearance of formulae withut a mean being subtracted
> is a pretty sure sign that an assumption of zero mean was
> stated shortly before, or at the beginning of the book or
> paper, the formulae.

Sure. But you need to keep an eye out for that
assumption.

> Stronger coffee in the morning is called for as you got the
> zero mean assumption correct but fell on your face over
> the scaling of correlations. Maybe too much ice on the roads
> even if spring is coming!

True, driving on icey roads with spike-less winter tyres
is no joke. That 0.5 mm sheet of ice on the asphalt can
really mess up your day. Coffee is not a solution, though,
as you don't make fewer mistakes, just make mistakes
for different reasons.

Rune

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On 8 Apr, 17:07, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 8 Apr, 15:15, Gordon Sande <g.sa...@worldnet.att.net> wrote:

> > On 2009-04-08 09:49:46 -0300, Rune Allnor <all...@tele.ntnu.no> said:

> > > Covariance is defined as E[(x-mx)*(y-my)] while correlation is
> > > defined as =A0E[x*y],

> > Correlation is a rescaled covariance.
>
> I have some recollection that the 'correlation' is defined
> differently, depending on the source. However, my recollections
> have been proven wrong in the past, so I have to check
> the sources after I get back home.
>
> > Correlations are
> > between -1 and +1.

It seems it might be weeks before I get the oportunity
to check my own books, so we might save some time
if somebody could help check what the texts below
say about the difference between correlation and
covariance:

Proakis & Manolakis: "Digital Signal Processing"

(preferably the 2nd edition (1992'ish), since that was
the one I first used), and

Therrien: "Discrete Random Signals and Statistical
   Signal Processing" (1992)

I am 99.9% sure Therrien describes the difference as
I did in the post quoted above.

For completeness, it would be interesting to see
what is said on the matter in

Kay: "Modern Spectral Estimation: Theory and
   Application" (1988)

Marple: "Digital Spectral Analysis: With Applications"
    (1987)

(can't check the Marple book myself, since I only saw
a library copy decades ago)

Bendat & Piersol: "Random Data" (2000)

The one book where I am pretty sure to have seen
the [-1, 1] scaling of the correlation mentioned, is

Shumway: "Applied Statistical Time Series Analysis "
    (1988)

which, as I (possibly wrongly) recall, talks about this
quantity as the correlation *coefficient*, which is
something else than correlation - much the same as
the 'directional cosine' between two vectors is different
from the 'inner product' between the same vectors,
although one might say they express similar relations.

So I'd appreciate if anyone who might have one or more
of these books available could check out what they say
about the difference between correlation and covariance.

Rune

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On Apr 9, 4:21=A0am, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 8 Apr, 17:07, Rune Allnor <all...@tele.ntnu.no> wrote:
>
> > On 8 Apr, 15:15, Gordon Sande <g.sa...@worldnet.att.net> wrote:
> > > On 2009-04-08 09:49:46 -0300, Rune Allnor <all...@tele.ntnu.no> said:
> > > > Covariance is defined as E[(x-mx)*(y-my)] while correlation is
> > > > defined as =A0E[x*y],
> > > Correlation is a rescaled covariance.
>
> > I have some recollection that the 'correlation' is defined
> > differently, depending on the source. However, my recollections
> > have been proven wrong in the past, so I have to check
> > the sources after I get back home.
>
> > > Correlations are
> > > between -1 and +1.
>
> It seems it might be weeks before I get the oportunity
> to check my own books, so we might save some time
> if somebody could help check what the texts below
> say about the difference between correlation and
> covariance:
>
> Proakis & Manolakis: "Digital Signal Processing"
>
> (preferably the 2nd edition (1992'ish), since that was
> the one I first used), and
>
> Therrien: "Discrete Random Signals and Statistical
> =A0 =A0Signal Processing" (1992)
>
> I am 99.9% sure Therrien describes the difference as
> I did in the post quoted above.
>
> For completeness, it would be interesting to see
> what is said on the matter in
>
> Kay: "Modern Spectral Estimation: Theory and
> =A0 =A0Application" (1988)
>
> Marple: "Digital Spectral Analysis: With Applications"
> =A0 =A0 (1987)
>
> (can't check the Marple book myself, since I only saw
> a library copy decades ago)
>
> Bendat & Piersol: "Random Data" (2000)
>
> The one book where I am pretty sure to have seen
> the [-1, 1] scaling of the correlation mentioned, is
>
> Shumway: "Applied Statistical Time Series Analysis "
> =A0 =A0 (1988)
>
> which, as I (possibly wrongly) recall, talks about this
> quantity as the correlation *coefficient*, which is
> something else than correlation - much the same as
> the 'directional cosine' between two vectors is different
> from the 'inner product' between the same vectors,
> although one might say they express similar relations.
>
> So I'd appreciate if anyone who might have one or more
> of these books available could check out what they say
> about the difference between correlation and covariance.
>
> Rune

I'm pretty sure that the scaling that produces the range -1 to +1 is
referred to as the correlation coefficient.

Cheers,
David

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