does anybody know a convinient way of converting between w and f?

In the title, "w" denotes Omega, which is 2*pi*f; "f" is the variable in 
frequency domain.

I am trying to understand Oppenheim's Signal & Systems and Discrete-Time 
Signal processing books. At the same time, I am also using other reference 
books as complementary reading. It is very annoying that some of the books, 
eg. Oppenheim's books use "w" to do everything related to Fourier Transform. 
And other books and our classes use "f" to do everything related to FT. I 
need to convert forth and back which is extremely annoying.

For example, in my problem, I need to invoke some "common" FT pairs and FT 
properties in "f" domain; but the major reference book -- Oppenheim's books 
use "w" everywhere. It is very difficult for my reading.

For instance, I need to use the FT of cos(2*pi*1*t),

In "f" setting, it should be 1/2*(Delta(f-1)+Delta(f+1))
In "w" setting, it should be pi*(Delta(w-2*pi)+Delta(w+2*pi))

I tried to do the conversion from Oppenheim's "w" setting to our "f" 
setting:

Substituting w=2*pi*f,

I got:

pi*(Delta(2*pi*f-2*pi) + Delta(2*pi*f+2*pi))

then I don't know what's wrong...

There are many other such instances that made the reading very difficult.

Esp. they call FT: X(jw), digital signal's FT: X(e^(jw)),

instead of X(f) and X(f),

which are really weird... because only "w"/"f" is the argument, not "jw", or 
"e^(jw)"...

Can anybody throw some lights on how to do conversion back and forth between 
"w" setting and "f"...

I hope there is some clear rules that I can follow...

Thanks a lot,

lucy wrote:

> In the title, "w" denotes Omega, which is 2*pi*f; "f" is the variable in 
> frequency domain.
> 
> I am trying to understand Oppenheim's Signal & Systems and Discrete-Time 
> Signal processing books. At the same time, I am also using other reference 
> books as complementary reading. It is very annoying that some of the books, 
> eg. Oppenheim's books use "w" to do everything related to Fourier Transform. 
> And other books and our classes use "f" to do everything related to FT. I 
> need to convert forth and back which is extremely annoying.
> 
> For example, in my problem, I need to invoke some "common" FT pairs and FT 
> properties in "f" domain; but the major reference book -- Oppenheim's books 
> use "w" everywhere. It is very difficult for my reading.
> 
> For instance, I need to use the FT of cos(2*pi*1*t),
> 
> In "f" setting, it should be 1/2*(Delta(f-1)+Delta(f+1))
> In "w" setting, it should be pi*(Delta(w-2*pi)+Delta(w+2*pi))
> 
> I tried to do the conversion from Oppenheim's "w" setting to our "f" 
> setting:
> 
> Substituting w=2*pi*f,
> 
> I got:
> 
> pi*(Delta(2*pi*f-2*pi) + Delta(2*pi*f+2*pi))
> 
> then I don't know what's wrong...

Actually you got the right answer; if you take the integral of both 
you'll see that Delta(a*x) is equal to 1/a * Delta(x), so 
pi*(Delta(2*pi*f etc.) is equal to 1/2*(Delta(f etc.).
> 
> There are many other such instances that made the reading very difficult.
> 
> Esp. they call FT: X(jw), digital signal's FT: X(e^(jw)),
> 
> instead of X(f) and X(f),
> 
> which are really weird... because only "w"/"f" is the argument, not "jw", or 
> "e^(jw)"...
> 
> Can anybody throw some lights on how to do conversion back and forth between 
> "w" setting and "f"...
> 
> I hope there is some clear rules that I can follow...
> 
> Thanks a lot,
> 
> 
> 

Frankly this kind of notational difference is just something you have to 
get used to.  Every author has their own way of putting things, usually 
in an effort to wake you up to their own way of looking at things.  This 
give you the challenge of figuring out how everything is equivalent.

-- 

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

lucy wrote:

> In the title, "w" denotes Omega, which is 2*pi*f; "f" is the variable in 
> frequency domain.


(snip)

Many problems are much easier to do using w instead of
2*pi*f, and especially Fourier transform problems.

Note also that between the forward and inverse Fourier
transform there is a 1/(2pi) factor that needs to be accounted
for.  Some people, mostly mathematicians, do this with
a 1/sqrt(2*pi) on each transform.   In the w form, more
popular in physics than EE, it comes out nice if you
put  dw/(2*pi) in the w transform, and no ugly sqrt(2*pi)
anywhere.  Measuring frequencies in radian/second makes
sense, measuring time in (2*pi*seconds) doesn't.

-- glen

"lucy" <losemind@yahoo.com> wrote in message 
news:clfaoc$19j$1@news.Stanford.EDU...
> In the title, "w" denotes Omega, which is 2*pi*f; "f" is the variable in 
> frequency domain.
>
....................................

You're showing initiative in considering other sources.  The sort of 
notation differences just comes with the territory.  You'll probably want to 
get to be fairly adept at dealing with those differences.  So, hang in 
there.

w and f only differ by a scale factor - which does affect integrals (and 
differentials) but that's about as complicated as it gets.

> There are many other such instances that made the reading very difficult.
>
> Esp. they call FT: X(jw), digital signal's FT: X(e^(jw)),
>
> instead of X(f) and X(f),
>
> which are really weird... because only "w"/"f" is the argument, not "jw", 
> or "e^(jw)"...

Both are valid statements.  X(f) or X(w) refer to the function for all 
(complex) values of f or w.  Often, we are more interested in focusing in on 
the values of X(f) or X(w) only on the jw axis.  Thus: X(jw).  This refers 
to real values of frequency even though there's a "j" there - to separate 
out the dimensions in the complex plane.

I will try to make a leap in 100 words or less:

In digital signal processing the sampling in time causes the transform to 
become periodic.  So, instead of needing to worry about X(w) or X(jw) for 
all values of w from -infinity to +infinity, we remap the jw axis into a 
unit circle so X(e^(jw)).

Fred

[BG] Responses embedded below...

"lucy" <losemind@yahoo.com> wrote in message 
news:clfaoc$19j$1@news.Stanford.EDU...
> In the title, "w" denotes Omega, which is 2*pi*f; "f" is the variable in 
> frequency domain.
>
> I am trying to understand Oppenheim's Signal & Systems and Discrete-Time 
> Signal processing books. At the same time, I am also using other reference 
> books as complementary reading. It is very annoying that some of the 
> books, eg. Oppenheim's books use "w" to do everything related to Fourier 
> Transform. And other books and our classes use "f" to do everything 
> related to FT. I need to convert forth and back which is extremely 
> annoying.

[BG]  It's annoying, but you might as well start getting used to it.  Even 
as you go from one course to another at your university you may find that 
some professors use the "w" notation while others use "f" notation.

> For example, in my problem, I need to invoke some "common" FT pairs and FT 
> properties in "f" domain; but the major reference book -- Oppenheim's 
> books use "w" everywhere. It is very difficult for my reading.
>
> For instance, I need to use the FT of cos(2*pi*1*t),
>
> In "f" setting, it should be 1/2*(Delta(f-1)+Delta(f+1))
> In "w" setting, it should be pi*(Delta(w-2*pi)+Delta(w+2*pi))
>
> I tried to do the conversion from Oppenheim's "w" setting to our "f" 
> setting:
>
> Substituting w=2*pi*f,
>
> I got:
>
> pi*(Delta(2*pi*f-2*pi) + Delta(2*pi*f+2*pi))
>
> then I don't know what's wrong...

[BG] You need to use the property that delta(af) = 1 / |a| * delta(f).

pi*( delta(2pi*(f-1)) + delta(2pi*(f+1)) )
= pi * ( 1/2pi * delta(f-1) + 1/2pi delta(f+1) )
= 1/2 * (delta(f-1) + delta(f+1)

> There are many other such instances that made the reading very difficult.
>
> Esp. they call FT: X(jw), digital signal's FT: X(e^(jw)),
>
> instead of X(f) and X(f),
>
> which are really weird... because only "w"/"f" is the argument, not "jw", 
> or "e^(jw)"...

[BG]  I thought this notation was weird at first too.  However, it makes a 
lot of sense once you know how to think about it.  The reason it's written 
this way is to explicitly connect the various transforms.  For example, in 
analog signal processing you use the Laplace transform X(s) where s is a 
complex variable s=sigma+jw.  The Fourier transform is the slice of the 
complex plane of the Laplace transform where sigma=0.  That is the Fourier 
transform is the Laplace transform evaluated at s=jw.  Hence he writes X(jw) 
when referring to a Fourier transform.  Similarly in discrete time you have 
the Z transform X(z) and the DTFT X(e^(jw)) since the DTFT is equal to the Z 
transform evaluated at z=e^(jw).

> Can anybody throw some lights on how to do conversion back and forth 
> between "w" setting and "f"...
>
> I hope there is some clear rules that I can follow...

[BG]  I think you are on the right track.  All you really need to know is 
that w=2*pi*f and you're set!

Brad

"lucy" <losemind@yahoo.com> wrote in message news:<clfaoc$19j$1@news.Stanford.EDU>...
> For example, in my problem, I need to invoke some "common" FT pairs and FT 
> properties in "f" domain; but the major reference book -- Oppenheim's books 
> use "w" everywhere. It is very difficult for my reading.
> 
> For instance, I need to use the FT of cos(2*pi*1*t),

Well, that's just how the world works. Some people define frequency as
f or as w=2pif. Some define the fourier transform as 

   X(w) = integral x(t) exp(iwt) dt

others as 

   X(w) = integral x(t) exp(-jwt) dt

(note the difference between "i" and "-j"). There is a 1/2pi normalization 
factor that needs to go in there somewehere to make the FT and the IFT
consistent. Some put the  factor in the inverse transform, others in the 
forward transform, and others yet as a 1/sqrt(2pi) factor in both 
transforms. 

Once you start working with applications, you'll see that people define 
the most "obvious" details in different ways. I've worked with seismology, 
where the earthquake people, who like to view the earth as a spherical 
globe, prefer to use the "depth" axis as positive upwards, since the 
origo of the spherical coordinate system coincides with the center of the 
earth. Others, like the petroleunm exploration people who work on a local
scale, prefer to use the depth axis as positive downwards, since they 
prefer to think in terms of depth that way.

Nah, I'm sorry, but these kinds of detail are just a fact of life when 
you work in engineering. One might whish things were different, but at 
the end of the day, one just have to deal with the world as it actually is.

Rune

Rune Allnor wrote:
> "lucy" <losemind@yahoo.com> wrote in message news:<clfaoc$19j$1@news.Stanford.EDU>...
> 
>>For example, in my problem, I need to invoke some "common" FT pairs and FT 
>>properties in "f" domain; but the major reference book -- Oppenheim's books 
>>use "w" everywhere. It is very difficult for my reading.
>>
>>For instance, I need to use the FT of cos(2*pi*1*t),
> 
> 
> Well, that's just how the world works. Some people define frequency as
> f or as w=2pif. Some define the fourier transform as 
> 
>    X(w) = integral x(t) exp(iwt) dt
> 
> others as 
> 
>    X(w) = integral x(t) exp(-jwt) dt
> 

And some folks use

     X(w) = c * integral x(t) exp(2 pi i w t) dt

so they do not trip over either radian measure or the "quaint" units
of time that some suggest. In this system the kernel of the integral
has period exactly 1. If you wonder where the 2 pi gets lost to, it
pops up in the derivative of the rescaled "sine" which is no longer
defined as 1 at zero. This is the standard scheme in advanced harmonic
analysis where the "character function" has period 1.

Blackman and Tukey have a nice list of FT pairs and cite much earlier
references for this normalization.

> (note the difference between "i" and "-j"). There is a 1/2pi normalization 
> factor that needs to go in there somewehere to make the FT and the IFT
> consistent. Some put the  factor in the inverse transform, others in the 
> forward transform, and others yet as a 1/sqrt(2pi) factor in both 
> transforms. 
> 
> Once you start working with applications, you'll see that people define 
> the most "obvious" details in different ways. I've worked with seismology, 
> where the earthquake people, who like to view the earth as a spherical 
> globe, prefer to use the "depth" axis as positive upwards, since the 
> origo of the spherical coordinate system coincides with the center of the 
> earth. Others, like the petroleunm exploration people who work on a local
> scale, prefer to use the depth axis as positive downwards, since they 
> prefer to think in terms of depth that way.
> 
> Nah, I'm sorry, but these kinds of detail are just a fact of life when 
> you work in engineering. One might whish things were different, but at 
> the end of the day, one just have to deal with the world as it actually is.
> 
> Rune

On Sat, 23 Oct 2004 21:23:39 -0700, "lucy" <losemind@yahoo.com> wrote:

>In the title, "w" denotes Omega, which is 2*pi*f; "f" is the variable in 
>frequency domain.
>
>I am trying to understand Oppenheim's Signal & Systems and Discrete-Time 
>Signal processing books. At the same time, I am also using other reference 
>books as complementary reading. It is very annoying that some of the books, 
>eg. Oppenheim's books use "w" to do everything related to Fourier Transform. 
>And other books and our classes use "f" to do everything related to FT. I 
>need to convert forth and back which is extremely annoying.
>
>For example, in my problem, I need to invoke some "common" FT pairs and FT 
>properties in "f" domain; but the major reference book -- Oppenheim's books 
>use "w" everywhere. It is very difficult for my reading.
>
>For instance, I need to use the FT of cos(2*pi*1*t),
>
>In "f" setting, it should be 1/2*(Delta(f-1)+Delta(f+1))
>In "w" setting, it should be pi*(Delta(w-2*pi)+Delta(w+2*pi))
>
>I tried to do the conversion from Oppenheim's "w" setting to our "f" 
>setting:
>
>Substituting w=2*pi*f,
>
>I got:
>
>pi*(Delta(2*pi*f-2*pi) + Delta(2*pi*f+2*pi))
>
>then I don't know what's wrong...
>
>There are many other such instances that made the reading very difficult.
>
>Esp. they call FT: X(jw), digital signal's FT: X(e^(jw)),
>
>instead of X(f) and X(f),
>
>which are really weird... because only "w"/"f" is the argument, not "jw", or 
>"e^(jw)"...
>
>Can anybody throw some lights on how to do conversion back and forth between 
>"w" setting and "f"...
>
>I hope there is some clear rules that I can follow...
>
>Thanks a lot,

Hi,
  you've encountered the problem we all ran into when 
we first started learning DSP: different authors using 
different notation.  I'm so sympatheic with your 
plight that I attempted to show how to equate three different 
forms of representing "frequency" in Chapter 3 of my 
DSP book.

As Glen said, using "w" makes the algebra of DSP a 
little more simple, and a little more concise.

I always think of the various forms for representing 
"frequency" to be similar to the various forms for 
representing an interval "time".  We can measure an interval 
of time in hours, days, or weeks.  And which "measure" 
do we use?  We use the "measure" that's most convenient 
to the subject we're discussing.  I measure the time 
I sleep in hours.  I measure the time it takes a Birthday 
card to reach my brother in days.  I measure the interval 
of time between receiving paychecks in weeks.

Going back and forth between hertz (f) and 
radians/second (w) is not too hard.  They differ by a 
factor of 2pi.  The notation that puzzled me, at first,
was representing a time-domain sequence's sampling 
frequency (the Fs sample rate in samples per second) 
with the number 2pi as is so common in DSP textbooks.

As far as I can tell the explanation for representing 
the Fs sample rate with the number 2pi goes like this:

Let's say the sample rate is Fs hertz (cycles/second).
I like to call that notation "cyclic frequency."
In terms of "radian frequency", radians/second, the 
sample rate would be represented by 2pi*Fs. 
So it's reasonable to say a sample rate 
is 2pi*Fs radians/second.

Now, if we assume that Fs = 1, we can 
say the sample rate can be represented by 
2pi*1 = 2pi.  So the frequency range of 
discrete sequences can be said to cover the 
range of 0 -to- 2pi.  [Often people find it 
useful to represent the frequency range of 
discrete sequences as -pi -to- pi when they 
working with complex (I/Q) numbers.]

If we use the 0 -to- 2pi notation to 
represent frequency, the units of that frequency 
notation become "radians/sample".  That's a 
weird dimension for frequency but it's correct.
[That "radians/sample" dimension does not 
exist in the analog (continuous) world.]

OK gotta run.  Hope that helped a little.

Good Luck,
[-Rick-]

Rick Lyons wrote:

> Let's say the sample rate is Fs hertz (cycles/second).
> I like to call that notation "cyclic frequency."
> In terms of "radian frequency", radians/second, the 
> sample rate would be represented by 2pi*Fs. 
> So it's reasonable to say a sample rate 
> is 2pi*Fs radians/second.

I believe that "angular frequency" is usually used in physics,
where angles are measured in radians.

Measuring in radians makes the linear and rotational formulae
come out looking similar, without extra 2*pi around.


F=ma                      (torque)= I (theta)
E=(1/2) m v**2            E=(1/2) I w**2
p=mv                      (angular momentum)= I w
d=v t + (1/2) a t**2      (theta)=w t + (1/2) (alpha) t**2


torque is usually tau, but sometimes other letters are used.

theta is angular position, omega is angular velocity,
alpha is angular acceleration.

I is moment of inertia, a rank 2 tensor, so that it should be
E=(1/2) w . I . w     that is, dot product on both sides of I.

-- glen

Rune Allnor wrote:

(snip)

> (note the difference between "i" and "-j"). There is a 1/2pi normalization 
> factor that needs to go in there somewehere to make the FT and the IFT
> consistent. Some put the  factor in the inverse transform, others in the 
> forward transform, and others yet as a 1/sqrt(2pi) factor in both 
> transforms. 

As to the difference between i and j, consider that it is usual
for engineers to look at a system at a given position in space,
and as a function of time.  That is what an oscilloscope does,
for example.

Physics tends to consider at a given time what a system looks
like as a function of position in space.

One result of this difference is that engineers use exp(j w t)
or exp(2 pi j f t)  in the same place that physics uses
exp(-i w t).   A wave travelling in the x direction is described
using exp(i k x - i w t).

As for the 2pi, when w=2 pi f, in the integral then dw= 2 pi df,
a convenient 2 pi to cancel the 2 pi normalization in the
Fourier transform.   (Maybe you said that, already.)

-- glen