Bilinear Transformation

Gentle comp.dsp Readers,

We just had some homework on bilinear transformations in my Digital
Signal Processing class and I must admit I am confused about
something.  I seek the expert council of the people attending this
group.

Some authors (e.g., [1, 2] define the bilinear transform with a 2/T
factor,

                 2  z - 1
            s =  - ------ .                                   (1)
                 T  z + 1

However, others (e.g., [3]) omit the 2/T factor

                  z - 1
            s =   ----- .                                     (2)
                  z + 1

Proakis and Manolakis explain nicely how the transform (including the
2/T) comes from numerical integration.  Mitra states that, since we
usually begin with a set of requirements in the discrete domain,
inverse map those back to the equivalent analog filter requirements,
then forward transform the result back to digital, the factor of T is
irrelevent.  However, nowhere have I seen it stated explicitly that
"T" is "sampling period," even though that is what that notation
typically means in these contexts.

So basically I have two questions: 1) Is T the sample period, or is it
a parameter that is independent of sample period?  2) When is T
required and when isn't it?

--Randy



[1] John G.  Proakis and Dimitris G. Manolakis.  Digital Signal
Processing: Principles, Algorithms, and Applications.  Prentice Hall,
third edition, 1996.

[2] Sanjit K.  Mitra. Digital Signal Processing: A Computer-Based
Approach.  McGraw- Hill, second edition, 2001.

[3] David J. DeFatta, Joseph G. Lucas, William S. Hodgkiss.  Digital
Signal Processing: A System Design Approach.  Wiley, 1988.
-- 
%  Randy Yates                  % "Maybe one day I'll feel her cold embrace,
%% Fuquay-Varina, NC            %                    and kiss her interface, 
%%% 919-577-9882                %            til then, I'll leave her alone."
%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO   
http://home.earthlink.net/~yatescr

Hello Randy,

I experienced what you have discovered, and basically I handle converting
analog filter designs based on mapping a frequency point in the analog
domain to a point in the discrete domain.


So starting with the bilinear transform

           z-1
s =  c ----
          z+1

And realizing that in terms of LaPlace transforms, real frequencies are on
the imaginary axis and with z transforms they are on the unit circle. And I
wish to match them up. One point in analog -> one point in discrete.


So s becomes j*OMEGA

and

z^-1 becomes exp(-j*omega)

After substituting both of these into the bilinear transform we get


j*OMEGA = j*c*tan(omega/2)

If we now define a digital frequency f relative to the sampling rate

omega=2*pi*f

Then the "c" (your 1/T) may be found to be

c = OMEGA*cot(pi*f)

Often one is converting an "s" equation that has its cutoff normalized to 1
radian/sec, so in this case OMEGA=1.

For your case T=tan(pi*f)

So you can sort of relate the T to the samping period.

IHTH,
Clay S. Turner






"Randy Yates" <yates@ieee.org> wrote in message
news:1xg0hoxa.fsf@ieee.org...
> Gentle comp.dsp Readers,
>
> We just had some homework on bilinear transformations in my Digital
> Signal Processing class and I must admit I am confused about
> something.  I seek the expert council of the people attending this
> group.
>
> Some authors (e.g., [1, 2] define the bilinear transform with a 2/T
> factor,
>
>                  2  z - 1
>             s =  - ------ .                                   (1)
>                  T  z + 1
>
> However, others (e.g., [3]) omit the 2/T factor
>
>                   z - 1
>             s =   ----- .                                     (2)
>                   z + 1
>
> Proakis and Manolakis explain nicely how the transform (including the
> 2/T) comes from numerical integration.  Mitra states that, since we
> usually begin with a set of requirements in the discrete domain,
> inverse map those back to the equivalent analog filter requirements,
> then forward transform the result back to digital, the factor of T is
> irrelevent.  However, nowhere have I seen it stated explicitly that
> "T" is "sampling period," even though that is what that notation
> typically means in these contexts.
>
> So basically I have two questions: 1) Is T the sample period, or is it
> a parameter that is independent of sample period?  2) When is T
> required and when isn't it?
>
> --Randy
>
>
>
> [1] John G.  Proakis and Dimitris G. Manolakis.  Digital Signal
> Processing: Principles, Algorithms, and Applications.  Prentice Hall,
> third edition, 1996.
>
> [2] Sanjit K.  Mitra. Digital Signal Processing: A Computer-Based
> Approach.  McGraw- Hill, second edition, 2001.
>
> [3] David J. DeFatta, Joseph G. Lucas, William S. Hodgkiss.  Digital
> Signal Processing: A System Design Approach.  Wiley, 1988.
> -- 
> %  Randy Yates                  % "Maybe one day I'll feel her cold
embrace,
> %% Fuquay-Varina, NC            %                    and kiss her
interface,
> %%% 919-577-9882                %            til then, I'll leave her
alone."
> %%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO
> http://home.earthlink.net/~yatescr

Randy Yates wrote:

> Gentle comp.dsp Readers,
> 
> We just had some homework on bilinear transformations in my Digital
> Signal Processing class and I must admit I am confused about
> something.  I seek the expert council of the people attending this
> group.
> 
> Some authors (e.g., [1, 2] define the bilinear transform with a 2/T
> factor,
> 
>                  2  z - 1
>             s =  - ------ .                                   (1)
>                  T  z + 1
> 
> However, others (e.g., [3]) omit the 2/T factor
> 
>                   z - 1
>             s =   ----- .                                     (2)
>                   z + 1
> 
> Proakis and Manolakis explain nicely how the transform (including the
> 2/T) comes from numerical integration.  Mitra states that, since we
> usually begin with a set of requirements in the discrete domain,
> inverse map those back to the equivalent analog filter requirements,
> then forward transform the result back to digital, the factor of T is
> irrelevent.  However, nowhere have I seen it stated explicitly that
> "T" is "sampling period," even though that is what that notation
> typically means in these contexts.
> 
> So basically I have two questions: 1) Is T the sample period, or is it
> a parameter that is independent of sample period?  2) When is T
> required and when isn't it?
> 
> --Randy
> 
> 
> 
> [1] John G.  Proakis and Dimitris G. Manolakis.  Digital Signal
> Processing: Principles, Algorithms, and Applications.  Prentice Hall,
> third edition, 1996.
> 
> [2] Sanjit K.  Mitra. Digital Signal Processing: A Computer-Based
> Approach.  McGraw- Hill, second edition, 2001.
> 
> [3] David J. DeFatta, Joseph G. Lucas, William S. Hodgkiss.  Digital
> Signal Processing: A System Design Approach.  Wiley, 1988.

                                       z - 1
I don't know the texts. I do know that ----- is dimensionless, while s
                                       z + 1
has the dimension of 1/t. Therefore, the factor T is necessary for
consistency. Its omission might make no numerical difference in the long
run, but dimensional consistency is important in mathematics if only for
pedagogic reasons. One of my first brouhahas in comp.dsp was in defense
of R.B-J's position on a very similar construction involving impulses.

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

Randy Yates <yates@ieee.org> writes:
> [...]

PS: My homework report is accessible at 

  http://www.uspsdata.org/hw.pdf
-- 
%  Randy Yates                  % "Watching all the days go by...    
%% Fuquay-Varina, NC            %  Who are you and who am I?"
%%% 919-577-9882                % 'Mission (A World Record)', 
%%%% <yates@ieee.org>           % *A New World Record*, ELO
http://home.earthlink.net/~yatescr

Jerry Avins wrote:
(snip)

>                                        z - 1
> I don't know the texts. I do know that ----- is dimensionless, while s
>                                        z + 1
> has the dimension of 1/t. Therefore, the factor T is necessary for
> consistency. Its omission might make no numerical difference in the long
> run, but dimensional consistency is important in mathematics if only for
> pedagogic reasons. One of my first brouhahas in comp.dsp was in defense
> of R.B-J's position on a very similar construction involving impulses.

Well, in physics and engineering, at least.

Not so long ago I was thinking about the different way physics and
engineering look at units in equations.  It seems to me that in
physics variables describe quantities with units, while in engineering
the units are factored out.

Physics will say  F=ma, and F will have the units of m multiplied by
the units of a.  (Though it might have a different name.)

Engineers might say F(in Newtons)=m(in kg)* a (in m/s**2).
One then converts the quantities supplied to the appropriate
units before using the equation.

Certainly there is a lot of overlap between physics and engineering,
and even then it may not be so convincing, but it seems to me to
be at least a little bit true.

-- glen

Randy Yates wrote:
> Gentle comp.dsp Readers,
> 
> We just had some homework on bilinear transformations in my Digital
> Signal Processing class and I must admit I am confused about
> something.  I seek the expert council of the people attending this
> group.
> 
> Some authors (e.g., [1, 2] define the bilinear transform with a 2/T
> factor,
> 
>                  2  z - 1
>             s =  - ------ .                                   (1)
>                  T  z + 1
> 
> However, others (e.g., [3]) omit the 2/T factor
> 
>                   z - 1
>             s =   ----- .                                     (2)
>                   z + 1
> 
> Proakis and Manolakis explain nicely how the transform (including the
> 2/T) comes from numerical integration.  Mitra states that, since we
> usually begin with a set of requirements in the discrete domain,
> inverse map those back to the equivalent analog filter requirements,
> then forward transform the result back to digital, the factor of T is
> irrelevent.  However, nowhere have I seen it stated explicitly that
> "T" is "sampling period," even though that is what that notation
> typically means in these contexts.
> 
> So basically I have two questions: 1) Is T the sample period, or is it
> a parameter that is independent of sample period?  2) When is T
> required and when isn't it?
> 
> --Randy
> 
> 
> 
> [1] John G.  Proakis and Dimitris G. Manolakis.  Digital Signal
> Processing: Principles, Algorithms, and Applications.  Prentice Hall,
> third edition, 1996.
> 
> [2] Sanjit K.  Mitra. Digital Signal Processing: A Computer-Based
> Approach.  McGraw- Hill, second edition, 2001.
> 
> [3] David J. DeFatta, Joseph G. Lucas, William S. Hodgkiss.  Digital
> Signal Processing: A System Design Approach.  Wiley, 1988.

If you are seriously interested in approximating a transfer function in 
the s domain with one in the z domain then you want to keep the 2/T. 
The Laplace transform of z is z = e^sT, where T is the sampling time, 
and 2(s-1)/(T(s+1)) is a fairly close approximation when s*T << 1.

Leaving out the 2/T doesn't sit well with me at all, but I don't have 
the context.  I _do_ prefer to construct discrete-time controllers 
without explicitly factoring in the 2/T business -- it makes it harder 
to adjust the sampling rate, but that's usually one of the first things 
to get decided, and by the time you're actually tuning things it's 
usually set in concrete.

-- 

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

"glen herrmannsfeldt" schrieb 
> 
> Physics will say  F=ma, and F will have the units of m
> multiplied by the units of a.  (Though it might have a 
> different name.)
> 
> Engineers might say F(in Newtons)=m(in kg)* a (in m/s**2).
> One then converts the quantities supplied to the appropriate
> units before using the equation.
> 
Well, physics is concerned with how the world works, while
engineering must make a gadget work. 
Ultimately you'll want to build the gadget and then the
dimensions matter a lot.
For thinking about how the world works, you can work in any
unit system you like (as long as the equations are consistent), 
the laws should come out identical.
In Engineering, OTOH, it is important that all players use the
same units (witness the trouble with a space shuttle once 
where one group used inches and the other millimeters). Hence
the obsession with engineers with units. I use to do a unit
consistency check always first, if it fails, I consider the 
equation useless.

Just my 0.02$.
Martin

Hi Randy,

   The "T" is definitely the reciprocal of the 
Fs sampling frequency.

I found that keeping T in the equations allows us 
to mathematically describe the "frequency warping" 
that occurs when we determine what digital-domain 
frequency (that's always in the range of -Fs/2 to +Fs/2) 
that equates to some analog frequency (that can be in 
the range of -infinity to +infinity).

Not that it improves on any of your references, but 
ya' might take a peek at the "Bilinear transform" 
discussion in Section 6.5 of my book.  
(Maybe you've already done that.)  That material 
might contain a little snippet of useful information. 
Who knows. 

Your homework solution(s) required 16 pages huh?
Whew!  That must have taken you a while.

See Ya',
[-Rick-]

----------------------------------------
On Thu, 14 Oct 2004 23:15:30 GMT, Randy Yates <yates@ieee.org> wrote:

>Gentle comp.dsp Readers,
>
>We just had some homework on bilinear transformations in my Digital
>Signal Processing class and I must admit I am confused about
>something.  I seek the expert council of the people attending this
>group.
>
>Some authors (e.g., [1, 2] define the bilinear transform with a 2/T
>factor,
>
>                 2  z - 1
>            s =  - ------ .                                   (1)
>                 T  z + 1
>
>However, others (e.g., [3]) omit the 2/T factor
>
>                  z - 1
>            s =   ----- .                                     (2)
>                  z + 1
>
>Proakis and Manolakis explain nicely how the transform (including the
>2/T) comes from numerical integration.  Mitra states that, since we
>usually begin with a set of requirements in the discrete domain,
>inverse map those back to the equivalent analog filter requirements,
>then forward transform the result back to digital, the factor of T is
>irrelevent.  However, nowhere have I seen it stated explicitly that
>"T" is "sampling period," even though that is what that notation
>typically means in these contexts.
>
>So basically I have two questions: 1) Is T the sample period, or is it
>a parameter that is independent of sample period?  2) When is T
>required and when isn't it?
>
>--Randy
>
>
>
>[1] John G.  Proakis and Dimitris G. Manolakis.  Digital Signal
>Processing: Principles, Algorithms, and Applications.  Prentice Hall,
>third edition, 1996.
>
>[2] Sanjit K.  Mitra. Digital Signal Processing: A Computer-Based
>Approach.  McGraw- Hill, second edition, 2001.
>
>[3] David J. DeFatta, Joseph G. Lucas, William S. Hodgkiss.  Digital
>Signal Processing: A System Design Approach.  Wiley, 1988.
>-- 
>%  Randy Yates                  % "Maybe one day I'll feel her cold embrace,
>%% Fuquay-Varina, NC            %                    and kiss her interface, 
>%%% 919-577-9882                %            til then, I'll leave her alone."
>%%%% <yates@ieee.org>           %        'Yours Truly, 2095', *Time*, ELO   
>http://home.earthlink.net/~yatescr

Martin Blume wrote:

> "glen herrmannsfeldt" schrieb 
> 
>>Physics will say  F=ma, and F will have the units of m
>>multiplied by the units of a.  (Though it might have a 
>>different name.)
>>
>>Engineers might say F(in Newtons)=m(in kg)* a (in m/s**2).
>>One then converts the quantities supplied to the appropriate
>>units before using the equation.
>>
> 
> Well, physics is concerned with how the world works, while
> engineering must make a gadget work. 
> Ultimately you'll want to build the gadget and then the
> dimensions matter a lot.
> For thinking about how the world works, you can work in any
> unit system you like (as long as the equations are consistent), 
> the laws should come out identical.
> In Engineering, OTOH, it is important that all players use the
> same units (witness the trouble with a space shuttle once 
> where one group used inches and the other millimeters). Hence
> the obsession with engineers with units. I use to do a unit
> consistency check always first, if it fails, I consider the 
> equation useless.
> 
> Just my 0.02$.
> Martin

Even in physics, if the units are inconsistent, you know the equation is
wrong. Consistent units don't guarantee correctness, but inconsistent
units are a sure sign of error. I learned that in physics class.

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

r.lyons@_BOGUS_ieee.org (Rick Lyons) writes:

> Hi Randy,
> 
>    The "T" is definitely the reciprocal of the 
> Fs sampling frequency.
> 
> I found that keeping T in the equations allows us 
> to mathematically describe the "frequency warping" 
> that occurs when we determine what digital-domain 
> frequency (that's always in the range of -Fs/2 to +Fs/2) 
> that equates to some analog frequency (that can be in 
> the range of -infinity to +infinity).
> 
> Not that it improves on any of your references, but 
> ya' might take a peek at the "Bilinear transform" 
> discussion in Section 6.5 of my book.  
> (Maybe you've already done that.)  That material 
> might contain a little snippet of useful information. 
> Who knows. 

Hi Rick,

Of course I would have referred to your book if I had
the opportunity, but your book was at the office and I
did the assignment from home. I will have a look. 

How do other folks solve this problem? It seems that no
matter where I keep my books, they're always in the wrong
place when I need them!

> Your homework solution(s) required 16 pages huh?
> Whew!  That must have taken you a while.

Nah, I'm just a windbag. Besides, once you take out the
title page, TOC, all those figures, and the original problem
text, there isn't much left.

--Randy


-- 
Randy Yates
Sony Ericsson Mobile Communications
Research Triangle Park, NC, USA
randy.yates@sonyericsson.com, 919-472-1124