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I thought I knew partial fractions... Help welcome

Started by Jean Castonguay May 25, 2005
Here is a very simple Z-transform:

			z**-1    		     z**-1
	X(z) = ---------------------------- = -------------------------------

	       1 - 0.25 z**-1 - 0.375 z**-2   (1 - 0.75 z**-1)(1 + 0.5 z**-1)

Since I have not found a way to use the circumflex to denote �raised 
to power�, I used �**� instead, and  I used no symbol to denote 
multiplication.

Whether I use long division or residues, I get:
	x[n] = 0,8 [0.75**n - (-0.5)**n]

Moreover, I used two variants of the decomposition  by partial 
fractions unsuccessfully.
Here is what I did.


	1 - first variant:

			 z**-1
	X(z) = -------------------------------
	       (1 - 0.75 z**-1)(1 + 0.5 z**-1)

			        1                        A                B 
	z X(z) = ------------------------------- = --------------- + 
-------------
	         (1 - 0.75 z**-1)(1 + 0.5 z**-1)   1 - 0.75 z**-1)   1 + 0.5 
z**-1

	          1
	A = ------------- = 0.6 since z = 0.75
	    1 + 0.5 z**-1

	          1
	B = -------------- = 0.4 since z = -0.5
	    1 - 0.75 z**-1


		       0.6 z**-1        0.4 z**-1
	Thus X(z) = -------------- + -------------
	            1 - 0.75 z**-1   1 + 0.5 z**-1


	2 - second variant:

			 z**-1                            z 

	X(z) = ------------------------------- = -------------------
	       (1 - 0.75 z**-1)(1 + 0.5 z**-1)   (z - 0.75)(z + 0.5)

		   A          B 
	     = --------- + -------
	       z - 0.75)   z + 0.5

	        1
	A = ------- = 0.6 since z = 0.75
	    z + 0.5

	        1
	B = -------- = 0.4 since z = -0.5
	    z - 0.75


		      0.6 z     0.4 z
	Thus X(z) = -------- + -------
	            z - 0.75   z + 0.5
	

I think I have tried to solve this problem for too long.  I cannot see
my error.  Can you?

Thank you very much in advance.	
-- 
Jean Castonguay
�lectrocommande Pascal

Jean Castonguay wrote:

> Since I have not found a way to use the circumflex to denote �raised > to power�,
On my keyboard, <shift>6 Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Jerry Avins wrote:

> Jean Castonguay wrote:
>> Since I have not found a way to use the circumflex to denote &#2013266091;raised >> to power&#2013266107;,
> On my keyboard, <shift>6
You mean the exclusive OR operator? I always use ** in posts, independent of newsgroup, especially as it could be cross posted. -- glen
"Jean Castonguay" <jcastong@riq.qc.ca> writes:

> Here is a very simple Z-transform: > > z**-1 z**-1 > X(z) = ---------------------------- = ------------------------------- > > 1 - 0.25 z**-1 - 0.375 z**-2 (1 - 0.75 z**-1)(1 + 0.5 z**-1) > > Since I have not found a way to use the circumflex to denote &#2013266091;raised > to power&#2013266107;, I used &#2013266091;**&#2013266107; instead, and I used no symbol to denote > multiplication. > > Whether I use long division or residues, I get: > x[n] = 0,8 [0.75**n - (-0.5)**n] > > Moreover, I used two variants of the decomposition by partial > fractions unsuccessfully. > Here is what I did. > > > 1 - first variant: > > z**-1 > X(z) = ------------------------------- > (1 - 0.75 z**-1)(1 + 0.5 z**-1) > > 1 A B > z X(z) = ------------------------------- = --------------- + > ------------- > (1 - 0.75 z**-1)(1 + 0.5 z**-1) 1 - 0.75 z**-1) 1 + 0.5 > z**-1 > > 1 > A = ------------- = 0.6 since z = 0.75 > 1 + 0.5 z**-1 > > 1 > B = -------------- = 0.4 since z = -0.5 > 1 - 0.75 z**-1 > > > 0.6 z**-1 0.4 z**-1 > Thus X(z) = -------------- + ------------- > 1 - 0.75 z**-1 1 + 0.5 z**-1
This is correct.
> 2 - second variant: > > z**-1 z > > X(z) = ------------------------------- = ------------------- > (1 - 0.75 z**-1)(1 + 0.5 z**-1) (z - 0.75)(z + 0.5) > > A B > = --------- + ------- > z - 0.75) z + 0.5
(1)
> 1 > A = ------- = 0.6 since z = 0.75 > z + 0.5
(2)
> > 1 > B = -------- = 0.4 since z = -0.5 > z - 0.75
(3)
> > > 0.6 z 0.4 z > Thus X(z) = -------- + ------- > z - 0.75 z + 0.5
You mean: 0.6 0.4 X(z) = -------- + ------- z - 0.75 z + 0.5 This follows from (1), (2) and (3) above. Scott -- Scott Hemphill hemphill@alumni.caltech.edu "This isn't flying. This is falling, with style." -- Buzz Lightyear
On Wed, 25 May 2005 21:38:49 UTC, Scott Hemphill 
<hemphill@hemphills.net> wrote:

> > 2 - second variant: > > > > z**-1 z > > > > X(z) = ------------------------------- = ------------------- > > (1 - 0.75 z**-1)(1 + 0.5 z**-1) (z - 0.75)(z + 0.5) > > > > A B > > = --------- + ------- > > z - 0.75) z + 0.5 > (1) > > 1 > > A = ------- = 0.6 since z = 0.75 > > z + 0.5 > (2) > > > > 1 > > B = -------- = 0.4 since z = -0.5 > > z - 0.75 > (3) > > > > > > > 0.6 z 0.4 z > > Thus X(z) = -------- + ------- > > z - 0.75 z + 0.5 > > You mean: > > 0.6 0.4 > X(z) = -------- + ------- > z - 0.75 z + 0.5 > > This follows from (1), (2) and (3) above. >
You are right: I did not make my &#2013266091;copy and paste&#2013266107; correctly. How dumb of me! I just realized that my error was not in the calculation of the partial fractions but in taking the inverse Z-transform from the partial fractions! I am red-faced. I thank you very much for answering. -- Jean Castonguay &#2013265929;lectrocommande Pascal
glen herrmannsfeldt <gah@ugcs.caltech.edu> writes:

> Jerry Avins wrote: > > > Jean Castonguay wrote: > > >> Since I have not found a way to use the circumflex to denote > >> &#2013266091;raised to power&#2013266107;, > > > > On my keyboard, <shift>6 > > You mean the exclusive OR operator? > > I always use ** in posts, independent of newsgroup, especially as > it could be cross posted.
I concur. Let's keep usenet pure text. The world has already screwed up email by resorting to every variant of HTML/RTF/blah blah blah format known to man. -- Randy Yates Sony Ericsson Mobile Communications Research Triangle Park, NC, USA randy.yates@sonyericsson.com, 919-472-1124
Randy Yates wrote:
> glen herrmannsfeldt <gah@ugcs.caltech.edu> writes: >> Jerry Avins wrote: >> > Jean Castonguay wrote: >> > >> >> Since I have not found a way to use the circumflex to denote >> >> &#2013266091;raised to power&#2013266107;,
If simply typing the circumflex key followed by anything but a vowel doesn't work for you, try putting a space "under" it.
>> > On my keyboard, <shift>6 >> >> You mean the exclusive OR operator? >> >> I always use ** in posts, independent of newsgroup, especially >> as it could be cross posted. > > I concur. Let's keep usenet pure text. The world has already > screwed up email by resorting to every variant of HTML/RTF/blah > blah blah format known to man.
What's non-plain-text about 0x5E? -- Quidquid latine dictum sit, altum viditur.
Martin Eisenberg <martin.eisenberg@udo.edu> writes:

> Randy Yates wrote: > > glen herrmannsfeldt <gah@ugcs.caltech.edu> writes: > >> Jerry Avins wrote: > >> > Jean Castonguay wrote: > >> > > >> >> Since I have not found a way to use the circumflex to denote > >> >> &#2013266091;raised to power&#2013266107;, > > If simply typing the circumflex key followed by anything but a vowel > doesn't work for you, try putting a space "under" it. > > >> > On my keyboard, <shift>6 > >> > >> You mean the exclusive OR operator? > >> > >> I always use ** in posts, independent of newsgroup, especially > >> as it could be cross posted. > > > > I concur. Let's keep usenet pure text. The world has already > > screwed up email by resorting to every variant of HTML/RTF/blah > > blah blah format known to man. > > What's non-plain-text about 0x5E?
Oops. Misidentified the issue here. I was thinking that someone was giving a sequence for typesetting a superscript on certain terminals. -- Randy Yates Sony Ericsson Mobile Communications Research Triangle Park, NC, USA randy.yates@sonyericsson.com, 919-472-1124
Randy Yates wrote:
> glen herrmannsfeldt <gah@ugcs.caltech.edu> writes: > > >>Jerry Avins wrote: >> >> >>>Jean Castonguay wrote: >> >>>>Since I have not found a way to use the circumflex to denote >>>>&#2013266091;raised to power&#2013266107;, >> >> >>>On my keyboard, <shift>6 >> >>You mean the exclusive OR operator? >> >>I always use ** in posts, independent of newsgroup, especially as >>it could be cross posted. > > > I concur. Let's keep usenet pure text. The world has already screwed > up email by resorting to every variant of HTML/RTF/blah blah blah > format known to man.
Randy: I don't know what you mean by "pure text" in this context. '^' is ASCII character 5E, called "circumflex" and often used as a caret. It is the standard exponentiation symbol in some programming languages, as '**' is in others (Fortran et. al.). I have no particular preference for either usage, but '^' is more common in news posts. It seems that Jean would have used it if it had been available to him. '**' is rare enough so he thought it needed explanation. Glen: What language uses '^' to mean XOR? What have I forgotten? Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Randy Yates wrote:

   ...

 > Oops. Misidentified the issue here. I was thinking that someone was
 > giving a sequence for typesetting a superscript on certain terminals.

It's a real issue, and the meaning of "plain text" is no longer clear. 
For instance, some but not all can read x&#2013266098; as x^2 (or x**2). The 
character '&#2013266098;' is $82, but characters above 7F aren't standard and may 
differ from font to font. Is '&#2013266098;' plain text or not?

Jerry
-- 
Should any political party attempt to abolish Social Security,
unemployment insurance, and eliminate labor laws and farm programs,
you would not hear of that party again in our political history.
There is a tiny splinter group, of course, that believes that you can
do these things.  Among them are a few Texas oil millionaires, and an
occasional politician or businessman from other areas.  Their number
is negligible, and they are stupid.
                    - Pres. Dwight D. Eisenhower, November 8, 1954